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Enhanced Multicarrier Techniques for Professional Ad-Hoc and Cell-Based Communications (EMPhAtiC) Document Number D4.1 MIMO transmission and reception design schemes Contractual date of delivery to the CEC:
31/08/2013
Actual date of delivery to the CEC:
09/09/2013
Project Number and Acronym:
318362 EMPhAtiC
Editor:
Didier Le Ruyet (CNAM)
Authors:
Rostom Zakaria (CNAM), Didier Le Ruyet (CNAM), Slobodan Nedic (UNS), Xavier Mestre (CTTC), Leonardo Baltar (TUM)
Participants:
CNAM,CTTC,TUM,TUT,UNS
Workpackage:
WP4
Security:
Public (PU)
Nature:
Report
Version:
1.0
Total Number of Pages:
79
Abstract: In this deliverable we have developped different contributions regarding MIMO transmission and reception design schemes for FBMC. Different strategies of receiver are proposed for spatial multiplexing scheme and when considering space time/space frequency block codes. Then the problem of MIMO transmission and reception scheme under high frequency selective channel is considered. The proposed solution with parallel stages is studied for receive-only, transmit-only and full transmit-receive frequencyselective processing. Finally a feasibility study of the use of compact antenna arrays in the context of PMR is performed by considering the transmit antenna array gain.
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Document Revision History Version
Date
Author
0.1
01.06.2013
Rostom Zakaria (CNAM), Didier Le Ruyet (CNAM), Slobodan Nedic (UNS), Xavier Mestre (CTTC), Leonardo Baltar (TUM)
Initial structure of the document
0.2
01.07.2013
Rostom Zakaria (CNAM), Didier Le Ruyet (CNAM), Slobodan Nedic (UNS), Xavier Mestre (CTTC), Leonardo Baltar (TUM)
Initial draft of the document
1.0
10.06.2013
Rostom Zakaria (CNAM), Didier Le Ruyet (CNAM), Slobodan Nedic (UNS), Xavier Mestre (CTTC), Leonardo Baltar (TUM)
Adding introduction and conclusions; making modifications according to comments in the internal review; text polishing
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Summary of main changes
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Table of Contents 1. INTRODUCTION .............................................................................................................................. 4 2. STUDY OF TRANSMISSION SCHEME AND DETECTION ALGORITHM ................................................ 8 2.1 SYSTEM MODEL .................................................................................................................................. 8 2.1.1 Complex-domain formulation ................................................................................................ 8 2.1.2 Real-domain formulation ..................................................................................................... 10 2.2 SPATIAL MULTIPLEXING FOR FBMC-OQAM .......................................................................................... 12 2.2.1 Partial interference cancellation with viterbi detection ....................................................... 12 2.2.2 Coded recursive maximum likelihood detector .................................................................... 18 2.3 STBC/SFBC SCHEMES FOR FBMC-OQAM .......................................................................................... 25 2.3.1 Interference cancellation based STBC .................................................................................. 25 2.4 SCHEMES FOR FBMC-QAM ............................................................................................................... 33 2.4.1 Spatial multiplexing FBMC-QAM ......................................................................................... 39 2.4.2 STBC FBMC-QAM ................................................................................................................. 43 3. MIMO TRANSMISSION AND RECEPTION UNDER HIGHLY FREQUENCY SELECTIVE CHANNELS ....... 50 3.1 INTRODUCTION................................................................................................................................. 50 1.1.1 The MIMO architecture considered here ............................................................................. 50 3.1.1 Practical MIMO-FBMC under frequency selective channels ................................................ 52 3.2 PROPOSED APPROACH ....................................................................................................................... 53 3.2.1 Taylor description of the frequency-domain problem .......................................................... 54 3.2.2 Proposed solution in its full generality and resulting distortion .......................................... 57 3.2.3 Receive-only frequency-selective processing ....................................................................... 58 3.2.4 Specific study for some specific MIMO architectures........................................................... 60 3.3 PERFORMANCE ANALYSIS IN PMR NETWORKS ........................................................................................ 61 3.3.1 Spatial multiplexing with channel inversion at the receiver ................................................ 62 3.3.2 Maximum capacity transmit precoding and channel inversion at the receiver ................... 65 4. COMPACT ANTENNA ARRAYS FOR PMR ....................................................................................... 69 4.1.1 Transmit Array Gain ............................................................................................................. 69 5. CONCLUSION ................................................................................................................................ 75 6. REFERENCES ................................................................................................................................. 77
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Glossary BER BFDM CP CSI EPA EVA FBMC FFT IFFT IIC IOTA ISI LLR LTE MIMO ML MLD MMSE OFDM OQAM OQPSK PaIC PAM PMR QAM QPSK RISI SISO SM SNDR QPSK RISI SNR SFBC STBC SVD TEDS TETRA ULA V-BLAST ZF
Bit Error Rate Bi-orthogonal Frequency Division Multiplex Cyclic Prefix Channel State Information Extended Pedestrian A Extended Vehicular B Filter Bank Multi Carrier Fast Fourier Transform Inverse Fast Fourier Transform Iterative Interference Cancellation Isotropic Orthogonal Transform Algorithm Inter Symbol Interference Log-Likelihood Ratio Long Term Evolution Multiple Input Multiple Output Maximum Likelihood Maximum Likelihood Detection Minimum Mean Square Error Orthogonal Frequency Division Multiplex Offset-Quadrature Amplitude Modulation Offset-Quadrature Phase Shift Keying Partial Interference Cancellation Pulse Amplitude Modulation Professional Mobile Radio Quadrature Amplitude Modulation Quadrature Phase Shift Keying Remaining Inter Symbol Interference Single Input Single Output Spatial Multiplexing Signal to Noise plus Distortion Ratio Quadrature Phase Shift Keying Remaining Inter Symbol Interference Signal to Noise Ratio Space Frequency Block Code Space Time Block Code Singular Value Decomposition TETRA Enhanced Data System TErrestrial TRunked Radio Uniform Linear Array Vertical Bell Laboratories Layered Space-Time Zero Forcing
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1. Introduction
By using multiple antennas at the transmitter and the receiver, it is possible to exploit the space diversity to improve the quality of service or to use the multiplexing gain to transmit multiple streams over separate propagation paths in order to increase the spectral efficiency. The MIMO techniques have been analysed for FBMC in the PHYDYAS. It was shown that the T/2-spaced complex-samples inter-symbol interference inherent in FBMC modulation constitutes a challenge for the implementation of reception and coding techniques in MIMO. These include adaptive equalization, space-time block and trellis coding and Spatial Multiplexing with Maximum Likelihood Detection (SM-MLD) techniques. Some space diversity schemes have been proposed, such as the block-wise Alamouti scheme for FBMC [20] and MISO space time diversity [21]. The performance of the blockwise Alamouti scheme was found to be quite good in case of stationary channel. However, with increasing mobility, the data block length has to be reduced, leading to increased overheads due to the pilots/guard periods. To exploit MISO space time diversity with single delay, a general strategy using Viterbi decoding and iterative decoding have been considered in [21] with performance close to CP-OFDM. Spatial multiplexing with FBMC was also analysed in Phydyas. Since SM-MLD requires the estimation of the interference term, it was proposed in [22] to perform a first MMSE estimation before ML detection. However, the complexity of the algorithm is still high. An iterative ML decoder based on interference cancellation has been presented obtaining significant gain compared to MMSE algorithm. The V-BLAST iteration per subcarrier has been applied to MIMO FBMC making use of the decisions made for previous symbols. Channel estimation for MIMO was also studied, both preamble- and pilots-based. Emphasis was mainly put on the case of flat subchannels (mildly frequency selective or low mobility channels). Optimal training for SISO channel estimation in FBMC was recently designed, again for flat subchannels [19] . Further research is needed to address MIMO scenarios with subchannels that are of significant frequency selectivity. When compact antenna arrays are employed in MIMO systems, the mutual coupling has to be taken into account to fully utilize their potential. It was shown in [23] that the antenna separation is not a fundamental limitation and compact antennas can deliver excellent diversity performance. The authors of [24] have shown that multistreaming is also possible in compact antenna arrays. Small arrays can have excellent multi-user capability and array size does not limit spatial resolution as reported in [25]. But an optimal design of the matching multiport at the antennas input is essential and the signal processing has to be changed in order to reach the optimal solution with compact arrays. The application of compact arrays in highly frequency selective channels will be also considered in the MIMOFBMC context as an extension of the results obtained for flat fading or low frequency selective channels.
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In this deliverable different contributions regarding MIMO transmission and reception design schemes are provided. In chapter 2 after introducing the system models, different strategies of receiver are proposed for spatial multiplexing scheme and when considering space time/space frequency block codes. In section 2.4, an iterative receiver based on interference estimation and cancellation is proposed for FBMC/QAM system. Chapter 3 deals with the problem of MIMO transmission and reception scheme under high frequency selective channel. The proposed solution with parallel stages is studied for receive-only, transmit-only and full transmit-receive frequency-selective processing. Finally, in chapter 4, a feasibility study of the use of compact antenna arrays in the context of PMR is performed by considering the transmit antenna array gain.
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2. Study of transmission scheme and detection algorithm 2.1 System model At the transmitter side, the discrete time FBMC signal is written as follows [12] M 1
s[m] = ak ,n g[m nM / 2]e
j
2 D k ( m ) M 2
e
jk , n
(1)
k =0 nZ
where M is an even number of subcarriers, g[m] is the prototype filter taking values in real D field, is the delay term which depends on the length ( Lg ) of g[m] . We have D = KM 1 2 and Lg = KM , where K is the overlapping factor. The transmitted symbols ak ,n are realvalued symbols which are the real or the imaginary parts of QAM symbols. The additional phase term k ,n is given by
k ,n =
2
(n k ) nk.
(2)
2.1.1 Complex-domain formulation We can rewrite (1) in a simple manner M 1
s[m] = ak ,n g k ,n [m],
(3)
k =0 nZ
where g k ,n [m] are the shifted versions of g[m] in time and frequency. In the case of no channel, the demodulated symbol over the k th subcarrier and the n th instant is determined using the inner product of s[m] and g k ,n [m] rk ,n = s, g k ,n =
s[m]gk*,n[m] =
m =
M 1
a
m = k =0 nZ
k ,n
g k ,n [m]g k*,n [m].
(4)
The transmultiplexer impulse response can be derived assuming null data except at one time-frequency position (k0 , n0 ) where a unit impulse is applied. Then, the equation above becomes
rk ,n =
g
m =
k0 , n0
[m]g k*,n [m]
=
g[m]g[m nM / 2]e
j
2 D k ( m ) M 2
e
j ( k k0 ) n j 2 ( k n )
e
(5)
,
m =
where n = n n0 and k = k k0 . We notice that the impulse response of the transmultiplexer depends on k0 . Indeed, the sign of some impulse response coefficients depends on the parity of k0 . Several pulse shaping prototype filters g[m] can be used according to their properties. In this document, we consider the pulse shapes referred to as the PHYDYAS and IOTA prototype filters, where the overlapping factor is set to K = 4 . Their transmultiplexer impulse response are given respectively in the tables below.
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k 1 k k 1
n3 0.043 j 0.067 j 0.043 j
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n2 0.125 j 0 0.125 j
n 1 0.206 j 0.564 j 0.206 j
n
0.239 j 1 0.239 j
n 1 0.206 j 0.564 j 0.206 j
n2 0.125 j 0 0.125 j
n3 0.043 j 0.067 j 0.043 j
Table 2-1 Transmultiplexer impulse response of the FBMC/OQAM using PHYDYAS filter
k 2 k 1 k k 1 k 2
n3 0.0016 j 0.0103 j 0.0182 j 0.0103 j 0.0016 j
n2 0 0.0381 j 0 0.0381j 0
n 1 0.0381j 0.228 j 0.4411j 0.228 j 0.0381j
n 1 n2 n 0.0381 j 0 0 0.4411 j 0.228 j 0.0381 j 0.4411j 0 1 0.4411j 0.228 j 0.0381j 0.0381 j 0 0
n3 0.0016 j 0.0103 j 0.0182 j 0.0103 j 0.0016 j
Table 2-2 Transmultiplexer impulse response of the FBMC/OQAM using IOTA filter
All the prototype filters g[m] are designed to satisfy the real orthogonality condition given by [12]
Re{ g k ,n [m]g k*,n [m]} = k ,k n,n
(6)
m =
Let us consider the SISO FBMC transmission. When passing through the radio channel and adding noise contribution b[m] , equation (4) becomes [14]
rk ,n = hk ,n ak ,n
hk ,n ak ,n
( k , n ) ( k , n )
g
m =
* k , n [ m] g k , n [ m]
b[m]g
m =
* k , n
(7)
[m],
k ,n
I k , n
where hk ,n is the channel coefficient at subcarrier k and time index n , and the term I k ,n is defined as an intrinsic interference. The autocorrelation of the noise sample k ,n is given by
E k ,n k*,n 2
g
m =
k , n
[m]g k*,n [m]
(8)
where 2 is the variance of the noise b[m] . The most part of the energy of the impulse response is localized in a restricted set around the considered symbol [13]. Consequently, we assume that the intrinsic interference term depends only on this restricted set (denoted by k ,n ). Moreover, assuming that the channel is constant at least over this summation zone, we can write as in [14]
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rk ,n hk ,n (ak ,n
( k , n )k , n
ak ,n
g
m =
k ,n
[m]gk*,n [m]) k ,n
(9)
Iˆk , n
According to (6) and because ak ,n is real-valued, the intrinsic interference Iˆk ,n = juk ,n is pure imaginary. Thus, we can write the demodulated signal as rk ,n hk ,n (ak ,n juk ,n ) k ,n .
(10)
In case of spatial multiplexing configuration, the received vector in FBMC/QAM can be written as: (1N ) r (1) h(11) hk ,n t sk(1),n I k(1),n bk(1),n k ,n k ,n (11) = , ( Nr ) ( Nr 1) (N ) (N N ) (N ) (N ) hk ,nr t sk ,nt I k ,nt bk ,nr rk ,n hk ,n rk , n
s k , n Ik , n
Hk ,n
bk , n
where the entries of the interference vector I k ,n are complex-valued.
2.1.2 Real-domain formulation A just slightly different representation (starting with continuous time and, for the time being different notations) of the transmit OFDM/OQAM modulated signal is defined by x(t )
k 2 2 M 1
j k m d mk g (t mT / 2)e
j 2 k
t mT / 2 T
(12)
k k1 m 0
where the symbols d nk are real valued (Real or Imaginary parts of M QAM symbols, belonging to a block of transmitted data). The index of the K subcarriers are denoted with numbers between k1 and k 2 . The function g t is a real, even, square root Nyquist function of norm 1, with the low-pass bandwidth Bg (1 ) / T , where is the roll off factor. It has finite time duration LgT , LgT , with T denoting the QAM signaling interval. As a modulation, with real valued symbols, of an orthogonal basis
mk t g t mT / 2 e
2 j k
t mT / 2 T
becomes x(t )
k 2 2 M 1
j
k k1 m 0
k m
d mk mk (t ) .
After transmission over an AWGN channel, the maximum likelihood, ML, receiver is obtained by the real part of the scalar product between the received signal r t x t b t and the basis functions mk t , with rmk Re
Lg T
Lg T
r mT / 2 t mk mT / 2 t dt
(13)
where the over-bar denotes complex conjugation. By using orthonormality of basis functions, the sampled version of (12) becomes rmk dmk bmk . (It should be clear from the ICT-EMPhAtiC Deliverable D4.1
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context, that index m corresponds to T/2 instants, as the index m in (12).) Transmission over a dispersive channel
The time-dispersive channel is defined by a finite (complex) impulse response h t for t LhT , LhT with an AWGN noise at the FIR filter output. The received signal is represented by
r t
LhT
LhT
x t h d b t ,
or r (t )
k 2 2 M 1
k k1 m 0
j k m d mk g k (t mT / 2)e
j 2 k
t mT / 2 T
b(t )
(14)
Here g k (t )
t Lg T mT / 2
t Lg T mT / 2
is
a
function
of
g ( )e
j 2k
t T
h(t )d
(15)
time
with duration over the interval t Lh Lg T , Lh Lg T Lg h T , Lg h T . Projection of the received signal over the
orthogonal basis mk t is defined by (13). With mk mT / 2 t j k m g t e
2 j k
t T
,
(16)
L T k2 2 M 1 j k ' n ' d nk'' g k ' t m n'T / 2 g rmk Re j 2 k 't ( mn ')T / 2 t j 2 k LgT T T k 'k1 n '0 e j k m g (t ) e dt taking for a moment b(t ) 0 , and by using k k k and n' m n' ' 1 m( 2 M 1) k k '' mn '' k '' 1 n ''m
rmk
d
(17)
j k '' n '' LgT g t n' 'T / 2 LgT k k '' Re t n ''T / 2 t j 2 k j 2 ( k k '') T T g (t ) e dt e
(18)
Due to the finite length duration in time and frequency of the impulse functions of the modulator and the channel, the received signal is the output of a linear matrix filter represented by real valued samples plus a white Gaussian noise, not accounted for after equation (4). The 3x1MISO model is described by the following relation (m replaced by n)
r k n
1
2 Lg
d
k k " k ,k " n n" n"
k " 1 n" 2 Lg
nk ,
(19)
with reinserting noise samples as nk , where nk'',k '' are the real-domain three impulse responses of the global filters. nk'', 0 is the impulse response for the sub-channel k , nk'',1 represents the interferences from the upper sub carrier of index k 1 and nk'', 1 interferences from the lower sub-channel of index k 1. Note, once again, that the impulse response samples nk'',k '' , the data d nk , as well as noise samples are all real-valued. Due to the very notation used in (19), same filters are used for even and odd T/2 instants, since their definition does not depend on index n. In difference to the complex-domain formulation of the SISO case, data samples corresponding to the particular time-frequency bin are not singled out as in (10) but their received values are determined by those belonging to the set ICT-EMPhAtiC Deliverable D4.1
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of adjacent/neighboring data samples, even in the frequency-flat fading channel. The overall MIMO channel model then becomes a direct extension of the SISO model, by using the fact that, by adequately estimating the real part of the sum of complex signals is the sum of real parts of the individual signals, as will be made use of later. Namely, in the general space division multiplexing case, the counterpart of the ρ-th (ρ=1,2,…,Nr) antenna received signal in equation (11) is
1 2 Lg k k " r ( ) d nn" ( ) nk",k " ( ) nk ( ) . 1t k "1 n"2 Lg Nt
k n
(20)
While in (11) the convolution sums of the data interfering terms and the outer summation are implied, with the (one-tap) complex channel transfer function coefficients imposed individually, the bracketed part of the outer summation in (20) contains the individual realdomain signal samples received from all the Nr transmit antennas.
2.2 Spatial multiplexing for FBMC-OQAM 2.2.1 Partial interference cancellation with viterbi detection In reference [7], the authors have considered the channel model depicted in Figure 2-1, where f0 (ak , ak 1 ,..., ak 1 ) is a function of data symbols and represents the target response expected by the receiver. f1 (ak , ak 1 ,..., ak 1 ) is a function of data symbols and represents a small channel perturbation.
Figure 2-1 Channel model
It should be noted that, in general, both f 0 and f1 may be nonlinear functions. The samples of the signal at the input of the receiver are:
rk = f0 (ak , ak 1 ,..., ak 1 ) f1 (ak , ak 1 ,..., ak 1 ) wk ,
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where wk is the noise contribution. The receiver is composed by a tentative detector producing tentative decisions, and a main viterbi detector which assumes that the channel is described only by f 0 . Before performing the main Viterbi detector, the tentative decisions are used only to cancel the remaining ISI (RISI) represented by f1 . The receiver scheme is depicted in Figure 2-2.
Figure 2-2 Receiver scheme with ISI cancellation using tentative decisions
Given the correct data sequence ak , and a sequence ak( ) for a given error event , let us define:
(k ) = f0 (ak( ) , ak(1) ,..., ak() 1 ) f0 (ak , ak 1 ,..., ak 1 ),
(22)
( ) = [(0 ) , 1( ) ,..., (K)1 ]T , (23) where K is assumed to be the total number of transmitted symbols. The authors in [7] classified the error events in terms of their distance d0 ( ) in the absence of RISI ( f1 = 0 ), which is given, in the presence of correlated noise, by: ( ) 2 (24) d0 ( ) = , H ( )R( ) where R is the normalized noise autocovariance matrix. The events whose distance d0 ( ) is minimum are called "first-order" error events. Similarly, events whose distance is the second smallest are called "second-order" error events, and so on [7]. The conditions for which RISI cancellation is satisfying are summarized as follows [7]: ICT-EMPhAtiC Deliverable D4.1
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1. Errors affecting the main (Viterbi detector) and the tentative detector must be statically independent. 2. The RISI (described by f1 ) must be small enough to guarantee that the main Viterbi detector can make relatively reliable decisions even when the tentative detector makes a decision error, and such that the tentative detector also makes relatively reliable decisions in spite of the ISI. 3. The distance of second-order and higher-order error events that could cause error propagation must be significantly larger than that of first-order error events. Now that the conditions for effective RISI cancellation are summarized, we will attempt to apply them to FBMC. Hence, the problem is, essentially, how to select the functions f 0 and
f1 such as these conditions are fulfilled. The intrinsic interference in FBMC is seen as a twodimensional intersymbol interference (2D-ISI). An extension of the works of Agazzi and Seshadri [7] to 2D-ISI channels was treated in [1] assuming that the noise is uncorrelated (which is not the case in FBMC). Hence in general, the target response f 0 may also represent a 2D-ISI channel. Then, a 2D-Viterbi detector is required to match with f 0 . Designing a 2D-Viterbi is quite challenging. Therefore, for simplicity reasons, we opted to set the additional constraint that the target response f 0 must be one-dimensional and that f1 covers the rest of 2D-ISI. Obviously the receiver complexity depends essentially on the complexity of the Viterbi detector. Therefore, we have to choose a configuration with the least complex Viterbi detector that meets the conditions for effective RISI cancellation. We will select three configurations with different sizes of the target response f 0 . According to the second condition, f 0 must contain the largest coefficients ( k , n ) in each configuration. Hence, from Table 2-1, the selected target responses are [9]: f 0(1) (ak ,n ) = ak ,n ,
(25)
f0(2) (ak ,n , ak ,n1 ) = ak ,n 0,1 ak ,n 1 ,
(26)
f0(3) (ak ,n1 , ak ,n , ak ,n1 ) = 0,1ak ,n1 ak ,n 0,1ak ,n1.
(27)
and The first configuration ( f 0(1) ) corresponds to the whole ISI cancellation which has been studied in the previous section. Since, in FBMC, we have
p ,q
| p ,q |2 = 2 [8], it is easy to
calculate the power of the RISI (represented by f1 ) for each configuration. Regarding the first condition, it is easily satisfied when the tentative detector is different from the main one (Viterbi) [7]. We recall that we consider the case of spatial multiplexing system. Then, we chose the MMSE equalizer as the tentative detector. The third condition concerns the spectrum distances d0 ( ) defined by (24). Hence, for each configuration ( f 0(i ) , i {1, 2,3}) , we compare the non-minimum distances to the minimum one. Then, according to (24), we have to determine the matrix R . Since we
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consider that the target responses are one-dimensional and Viterbi algorithm is performed on each subcarrier "k" , the matrix R is composed only by the coefficients 0,q , q (see equations (4) and (8)) and is given by: 1 R = 0,1 0,2
0,1 1 0, 1
0,2 0,1 1
K K
(28)
In Table 2-3, we summarize the values of the first, second, and third order distances obtained by using (24), and also the power of the RISI for the three considered configurations. We remark that the difference between the second-order and the first-order distances is almost the same for all the configurations ( 0.8 0.03 ), so we consider (as considered in [7]) that the higher-order distances are sufficiently larger than the minimum distance for each configuration. Hence, condition 3) is fulfilled for the three configurations.
First First-order distance Second-order distance Third-order distance Power of the RISI
Configuration 2
Second (1) 0
(f )
Configuration 1.8857
Third (2) 0
(f )
Configuration 1.9189
2 2
2.6668
2.7137
2 3
3.4596
3.2728
1
0.6819
0.3638
( f 0(3) )
Table 2-3 Spectrum distances and RISI power
Now, we have only to determine the configuration(s) for which the second condition is satisfied. Unfortunately, the determination of the RISI power for which the cancellation starts to be effective (or equivalently, error propagation ceases) is not trivial and depends also on the noise variance 2 [7]. We will show in the following (by simulations) that only the third configuration ( f 0(3) ) allows to obtain effective RISI cancellation. As for the receiver complexity, it strongly depends on that of the Viterbi detector. When we consider a spatial multiplexing system with N t transmit antennas, the Viterbi detector has to compute q
i Nt
branch metrics, where q is the number of all possible
symbols ak ,n (constellation size) and i {1, 2,3} is the number of the taps in f 0(i ) . In order to reduce the receiver complexity, we can replace the Viterbi detection algorithm by the MAlgorithm [2] which keeps only a fixed number ( J ) of inner states instead of all the inner states (q
( i 1) Nt
) . Hence, the M-algorithm has to compute only J q
Nt
branch metrics.
In the following, we provide the simulation results concerning the three configurations treated above. Since the motivation of this work is to address the problem of ICT-EMPhAtiC Deliverable D4.1
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optimum detection in spatial multiplexing with FBMC, we have considered the simple 2 2 spatial multiplexing scheme. We assume perfect channel knowledge at the receiver side, and the four Rayleigh sub-channels are spatially non-correlated. The complex data symbols are QPSK modulated (q = 2) . The system performance is assessed in terms of BER as function of SNR and is compared to that of the conventional OFDM with ML detector.
Figure 2-3 Performance of PaIC/Viterbi receivers for 2 2 spatial multiplexing
We call the proposed receivers "PaIC/Viterbi" (for Partial Interference Cancellation with Viterbi detector) followed by an index indicating the considered configuration. Figure 2-3 depicts the performance of the MMSE equalizer (which is our tentative detector) and of the PaIC/Viterbi for the three considered configurations. We clearly notice that PaIC/Viterbi3 exhibits almost the same performance as OFDM, and that the RISI cancellation is effective. Hence, the value of the RISI power given in Table 2-3 Spectrum distances and RISI powerfor the third configuration is sufficiently small so that condition 2) is satisfied. However, a slight degradation of the PaIC/Viterbi-3 performance compared to OFDM is observed beyond 22 dB . Indeed, as we have mentioned at the end of the previous section, the threshold of the RISI power from which the error propagation begins (ineffectiveness of the RISI cancellation) depends on the noise variance 2 . As shown -for a specific example- in [7], the threshold lowers with the SNR increase. As for the first and second configurations, the performance degradation compared to OFDM begins from about 12 dB . This relatively high degradation is due to the high values of the corresponding RISI powers causing error propagation.
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Figure 2-4 Performance of PaIC/M-Algo receiver for 2 2 spatial multiplexing
Now we consider only the third configuration ( f 0(3) ) since the RISI cancellation is effective, and we assess the BER performance when using M-Algorithm instead of the Viterbi one in order to reduce the complexity. In Figure 2-4 Performance of PaIC/M-Algo receiver for 2 2 spatial multiplexing, we show the obtained performance of the MAlgorithm with two different values of J ( J = 2 , and J = 4 ). We notice that with J = 2 we have an SNR loss of about 2.5 dB compared to PaIC/Viterbi-3. This SNR loss is due to the suboptimality of the M-Algorithm when J is small. Moreover, PaIC/M-Algo with J = 2 provides a performance worse than the one provided by the tentative detector (MMSE) as long as the SNR is less than 12 dB . However, with J = 4 , we can observe that PaIC/M-Algo exhibits almost the same performance as PaIC/Viterbi-3 but with much lower algorithm complexity ( 4 inner states instead of 16 ). To conclude, the intrinsic interference in FBMC is a 2D-ISI in the time-frequency plan. In order to avoid a full 2D-Viterbi detector, a receiver based on ISI cancellation is proposed. However, the ISI cancellation is effective only under some strict conditions. One of these conditions is that the ISI must be sufficiently small. Unfortunately, the intrinsic interference, in FBMC, has the same power as the desired symbol. Hence, we have proposed a trade-off between a whole ISI cancellation (Rec-ML) and a full 2D-Viterbi detection. The proposed receiver is composed by a tentative detector giving decisions which serve to partially cancel the interference, followed by a Viterbi detector matching to the non-canceled ISI. Three configurations were treated. The first one is called PaIC/Viterbi-1 (or Rec-ML) and correspond to the whole ISI cancellation. The second one is PaIC/Viterbi-2, where the ICT-EMPhAtiC Deliverable D4.1
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Viterbi detector matches with the two largest coefficients of the transmultiplexer impulse response. The third one is PaIC/Viterbi-3 and the Viterbi detector matches with the three largest coefficients. We have shown, by simulations, that only the PaIC/Viterbi-3 receiver gives the same performance as OFDM. The two other configurations suffer from error propagation because their RISI are not sufficiently small. We have also proposed to replace the Viterbi detector by another based on M-Algorithm in order to reduce the receiver complexity. Indeed, we have shown that using the M-Algorithm with J = 4 , we obtain exactly the same BER performance as PaIC/Viterbi-3 with much lower complexity. 2.2.2 Coded recursive maximum likelihood detector We have seen in the previous sections that the interference cancellation in the proposed receiver schemes is not very effective due to the presence of erroneous tentative decisions. Indeed, it was shown by simulations that there is a BER-performance gap between the proposed receivers and the Genie-Aided performance obtained by perfect interference removing. Since the performance limitation is due to the tentative decision errors, error correction coding may significantly improve the interference cancellation. In this section, we test the proposed receivers in the context of encoded data. At the transmitter side, the binary information enters a convolutional encoder to produce an encoded binary sequence which is bit-interleaved by a random interleaver. Then, the interleaved bits are mapped by a memoryless modulator into symbols belonging q q to a PAM constellation set with cardinality of 2 2 , where is the bit number in the PAM 2 symbol. After that, the PAM symbols are demultiplexed onto N t branches (corresponding to
N t antennas). Over each branch the data are sent to the FBMC modulator and then transmitted through the radio channel. At the receiver side, the N t transmitted signals are collected by N r receive antennas. The signal in each receive antenna branch is FBMC (N )
(i ) r T demodulated, forming thus demodulated vectors rk ,n = [rk(1) , n ,..., rk , n ] , where rk , n are the
FBMC demodulator outputs at the i th receive antenna. In MMSE-ML receiver which is presented in section 2.1, the MMSE equalizer is used to estimate the interference terms which are obtained by taking the imaginary parts of the MMSE outputs. Once the contribution of these interference terms are removed from the received signal vector rk ,n , the resulting signal vector y k ,n is fed to a soft ML detector giving soft outputs in the form of LLR values. Then, these soft values are multiplexed, deinterleaved, and decoded to provide decision bits. Figure 2-5 shows the basic scheme of the MMSE-ML receiver.
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MMSE
Soft MLD
MUX
Decoder
q Nt 2
Figure 2-5 MMSE-ML scheme for FBMC in
Nt N r Spatial Data Multiplexing
The received vector rk ,n is an observation of N t transmitted PAM symbols ak( ,jn) ,
q N t bits d l involved in the received vector rk ,n . The soft 2 ML detector calculates the LLRs of the a posteriori probability (APP) of the encoded bits d l being 1 or 1 . The LLR values for the ML detector are defined as: P(dl = 1| y k ,n ) q LAPP (dl | y k ,n ) = log , l = 1,..., Nt (29) P(d = 1| y ) 2 l k ,n q where is the number of bits that constitute the real symbol ak ,n . Then, in each subcarrier 2 q and half period T / 2 , we have N t soft bits at the soft MLD output. By employing Bayes’ 2 theorem and assuming statistical independence and equiprobability among the bits d l , the LLR can be written as [3], [4]: p(y k ,n | d) d (30) LAPP (dl | y k ,n ) = log k , p(y k ,n | d) d k where the vector d contains all the bits corresponding to the transmitted symbols ak ,n over
j {1,..., Nt} . Hence, there are
all the antennas, and the set
l
(or
l
) contains the vectors d having dl = 1 (or dl = 1 ).
The likelihood density p(yk ,n | d) is given by:
1 exp 2 rk ,n H k ,n a k ,n (d) 2 p(y k ,n | d) = N (2 2 ) r
2
,
(31)
where a k ,n (d) is the transmitted real vector corresponding to the bit-vector d . Substituting equation (31) in (30) and applying the Max-Log approximation, the LLR calculation is simplified by:
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LAPP (dl | yk ,n ) =
1 2
2
min rk ,n H k ,n a k ,n (d)
2
d k
1
q 2 min rk ,n H k ,n a k ,n (d) , l {1,..., Nt } 2 d k 2
(32)
2
The obtained soft information at the ML output should be multiplexed, deinterleaved and fed into the soft-input decoder to recover the transmitted information source bits. As for the Rec-ML receiver, we use the outputs of the MMSE-ML receiver to perform a second interference estimation. This time, the intrinsic interference is estimated by using the decided data bits (available at the MMSE-ML output) which are involved in the symbols ak ,n within the neighborhood of the considered frequency-time position (k , n) . For this, the decided data bits are encoded with the same convolutional code used at the transmitter, interleaved, mapped and demultiplexed repeating exactly the same transmission operations to provide an estimation of the transmitted symbols aˆ k ,n which will serve to calculate an interference estimate. Thus, the interference estimation is improved since the information bits are encoded and some errors would be corrected. Once this interference is estimated, its contribution is canceled again from the received vector rk ,n , and then, we perform once more the soft ML detection. The complete receiver scheme is depicted in Figure 2-6.
MMSE
Soft MLD Delay
q 2
DEMUX
Soft MLD
MUX
Decoder
Mapping
Encoder
Nt
MUX q 2
Nt
Figure 2-6 Recursive ML scheme for FBMC in
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In the following, we show the simulation results for MMSE-ML and Rec-ML receivers and compare their performance to OFDM. We consider a 2 2 spatial multiplexing scheme. Our objective is to test the proposed receiver schemes over a low and high frequency selective channels. For that purpose, we have chosen the Ped-A and the Veh-A channel models. We should note that for both chosen channels, we have not considered the time selectivity. The simulation parameters are the same as the ones used for MMSE equalizer and are summarized in the table below. Table 2-4 Simulation parameters
Complex modulation FFT size ( M ) CP size for OFDM ( ) Convolutional code Sampling frequency
Pedestrian-A QPSK 1024 8 (171,133) 10 MHz
Vehicular-A QPSK 1024 32 (171,133) 10 MHz
We also assume perfect channel knowledge at the receiver side, and the performance is assessed in terms of BER as function of Eb / N0 which is defined, for OFDM, by: Nr M (33) Eb / N 0 = SNR, M qRs Rc where M is the subcarrier number (FFT size), is the cyclic prefix size, Rc is the channel coding rate, Rs is the space-time coding rate, q is the bit number in a complex QAM symbol ( q = 2 for QPSK), and SNR is the Signal-to-Noise ratio. As for FBMC, the expression of Eb / N0 is obtained by nulling the CP duration , so we can write: Nr Eb / N 0 = SNR. (34) qRs Rc
As in the uncoded case, we define the Genie-Aided performance as the fictional one obtained when the symbols serving to estimate the interference are identical to the transmitted ones (perfect interference estimation). We can show that even by neglecting the efficiency loss due to CP, the Genie-Aided receiver outperforms CP-OFDM by about 1 dB in a 2 2 SM with QPSK modulation. Hence, it is interesting to compare its performance to the OFDM and the proposed receivers when using convolutional coding.
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Figure 2-7 BER performance comparison between CP-OFDM and FBMC receivers in 2 2 MIMO case over Ped.-A channel
Figure 2-7 captures the performance of CP-OFDM with ML and that of FBMC with all the proposed receivers (including MMSE) over the Ped-A channel. The curves show that the MMSE-ML scheme outperforms the MMSE equalizer, but the performance is still far from the CP-OFDM with ML. The gain obtained by MMSE-ML with respect to MMSE equalizer is about 2.5 dB at BER = 104 , whereas OFDM-ML provides a 5 dB SNR gain compared to MMSE. However, Rec-ML receiver exhibits almost the same performance as OFDM-ML. It is worth recalling that CP-OFDM performance is obtained with the smallest possible CP size ( = 8 ). Increasing yields a performance degradation for CP-OFDM, and thus, FBMC with Rec-ML receiver will outperform CP-OFDM. For example, as in IEEE 802.16e standard [6], if T we set = = 128 , we obtain a degradation of about 0.48 dB . 8 Regarding the Veh-A channel, Figure 2-8 shows the performance of the different receivers in this propagation channel. First, as in the Pedestrian-A channel case, we remark that a considerable SNR gain is obtained by MMSE-ML receiver compared to MMSE equalizer, we have a gain of about 2 dB at BER = 104 . Secondly, we can clearly observe that the obtained Rec-ML performance is slightly better than that obtained with CP-OFDM from Eb / N0 = 6 dB , and tends to reach the Genie-Aided performance in high Eb / N0 regime.
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Figure 2-8 BER performance comparison between CP-OFDM and FBMC receivers in 2 2 MIMO case over Veh.-A channel
We notice that the potential SNR gain of FBMC over OFDM is not yet completely exploited, and that the interference is not fully removed even in the encoded case [5]. Therefore, further investigations on interference estimation are needed to improve the performance. It is commonly known that soft decoder outputs are more reliable than the hard ones and offer better performance. Therefore, in order to improve the interference cancellation, we estimate the interference by using soft decided symbols. The soft received symbols are determined by calculating their expectation values from their Log-Likelihood ratios (LLR) as
ak ,n =
where Lk ,n
e
Lk , n
1
Lk , n
e 1 are Log-Likelihood ratios about the encoded received symbols and are calculated
using the BCJR algorithm that provides:
Pr ak ,n 1/ yn Lk ,n = log Pr ak ,n 1/ yn
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(35)
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where yn denotes a vector containing the received signals over all the subcarriers and receive antennas.
Figure 2-9 BER performance of the FBMC receiver using soft interference cancellation in 2 2 MIMO case over Ped.-A channel
We test this "soft" Rec-ML receiver in both Ped-A and Veh-A channels with the same parameters as given previously. Figure 2-9 depicts the obtained BER performance of the "soft" Rec-ML receiver in the Ped-A channel. For the sake of comparison, the BER performance of the previous "hard" Rec-ML receiver is also depicted. One can observe a performance improvement by using soft interference estimation. However, there is still a performance gap of about 1 dB at BER = 104 between the Genie-Aided and the soft RecML receiver with 2 iterations.
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Figure 2-10 BER performance of the FBMC receiver using soft interference cancellation in 2 2 MIMO case over Veh.-A channel
As for the case where the Veh-A channel is considered, we remark in Figure 2-10 that the soft Rec-ML with 2 iterations can almost offer the same BER performance as the GenieAided from an Eb / N 0 of about 8 dB . The difference in performance between the cases where Ped-A and Veh-A channel models are used can be explained by the effect of the bitinterleaving. Indeed, this latter attempt to exploit the frequency channel selectivity to improve the diversity. Since Veh-A channel is more selective than Ped-A, the interference estimation is then more reliable.
2.3 STBC/SFBC schemes for FBMC-OQAM 2.3.1 Interference cancellation based STBC The received data symbols are corrupted by inherent interference terms which complicate the STBC decoding [10]. The interference estimation and cancellation procedure is repeated several times to improve the detection. Unfortunately, detection schemes with ISI estimation and cancellation are not always effective due to the error propagation [9]. Therefore, the challenge in ISI estimation and cancellation is mitigating the error propagation through iterations [7]. In order to counteract the error propagation and make the cancellation scheme effective, the authors in [7] showed that a necessary condition to avoid the error propagation is to hold the interference power under a certain threshold, i.e.
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the interference cancellation technique is effective only when the ISI is small enough compared to the minimal distance d 0 between two different symbols. Let us consider the basic 2 1 Alamouti coding: when the first antenna transmits at the k th subcarrier a block [ak ,n ak ,n1 ] , the second antenna transmits the block
[ak ,n1 ak ,n ] . We note that the conjugate operation is aborted because the data symbols are real valued. Without loss of generality, let us consider a specific block with its immediate neighboring symbols. We can write: a0,1 a0,0 a0,1 a0,2 a a1,0 a1,1 a1,2 1, 1 A1 = a2,1 a2,0 a2,1 a2,2 a a a a 0,2 0,1 0,0 0,3 a a1,1 a1,0 a1,3 1, 2 A2 = a2,2 a2,1 a2,0 a2,3 where A1 and A2 are the data blocks transmitted at the first and the second antennas, respectively. For simplicity reasons, let us only consider the interference from the immediate neighborhood of the symbol of interest. Hence, according to (4) we can write the received symbols r1,0 and r1,1 as 1 1 1 1 1 r1,0 = h1 p ,q a1 p , q h2 p ,0 a1 p ,1 h2 p , 1a1 p,0 p,1a1 p, 2 b1,0 p = 1q = 1 p = 1 p = 1 p = 1 1 1 1 1 1 r1,1 = h1 p ,q a1 p ,1q h2 p ,0 a1 p ,0 h2 p,1a1 p,3 p,1a1 p,1 b1,1 p = 1q = 1 p = 1 p = 1 p = 1
The Alamouti decoding is performed by calculating y1,0 and y1,1 as
y1,0 = h1*r1,0 h2 r1,1* * * y1,1 = h1 r1,1 h2 r1,0 After processing, we obtain
Re y = | h |
a
(36)
* Re y1,0 = | h1 |2 | h2 |2 2Re h1*h20,1 a1,0 Re h1*h2 I1,0 Re h1*b1,0 h2b1,1 2
1,1
1
| h2 |2 2Re h1*h20,1
I1,0
where the interference terms
and
I1,1
2Im a
p = 1
p ,0
1 p ,1
2Im a
p,1a1 p,2 *p,1a1 p,2
1
I1,1 =
p = 1
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* Re h1*h2 I1,1 Re h1*b1,1 h2b1,0
are given by
1
I1,0 =
1,1
p ,0
1 p ,0
p ,1a1 p ,3 *p ,1a1 p ,1
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According to Table 2-1, we can show that E{| I1,0 |2 } = E{| I1,1 |2 } = 8 | 1,0 |2 4 | 1,1 |2 2 | 0,1 |2 (37)
1.2646 This remaining term of interference will cause a BER floor in high SNR regime. If this BER floor is not small enough, then iterative interference cancellation receiver will introduce error propagation and, thus, the interference cancellation cannot be effective. Moreover, one can remark the term 2Re h1*h20,1 weighing the useful symbol. Hence, this term increases the BER when it is negative, and decrease the BER when it is positive. Nevertheless, the BER increase is more important than the BER decrease. In this section, we propose some different Alamouti coding schemes where the overall signal-to-interference ratio (SIR) is improved. Let us consider the space time block coding (STBC) defined as a0,1 a0,0 a0,1 a0,2 a0,3 a a1,0 a1,1 a1,2 a1,3 A1 = 1,1 a2,1 a2,0 a2,1 a2,2 a2,3 a0,3 a0,2 a0,3 a0,0 a0,1 a a1,2 a1,3 a1,0 a1,1 A2 = 1,3 a2,3 a2,2 a2,3 a2,0 a2,1 where A1 and A2 are the data blocks transmitted at the first and the second antennas, respectively. We note that in this scheme there are two interleaved Alamouti blocks (shown in red and blue colors). Furthermore, there is an alternating rule in positions of the minus sign between the blue and the red blocks. Let us focus on the decoding of the blue Alamouti block, the received symbols r1,0 and r1,2 are as follow 1 1 1 1 1 r1,0 = h1 p ,q a1 p , q h2 p ,0 a1 p ,2 h2 p ,1a1 p,3 p,1a1 p, 3 b1,0 p = 1q = 1 p = 1 p = 1 p = 1 1 1 1 1 1 r1,2 = h1 p ,q a1 p ,2q h2 p ,0 a1 p ,0 h2 p ,1a1 p,1 p,1a1 p,3 b1,2 p = 1q = 1 p = 1 p = 1 p = 1
The Alamouti decoding for the blue block is performed by calculating y1,0 and y1,2 as * * y1,0 = h1 r1,0 h2 r1,2 * * y1,2 = h1 r1,2 h2 r1,0
(38)
Hence, after processing we obtain: * Re y1,0 = | h1 |2 | h2 |2 a1,0 Re h1*h2 I1,0 Re h1*b1,0 h2b1,2
* Re y1,2 = | h1 |2 | h2 |2 a1,2 Re h1*h2 I1,2 Re h1*b1,2 h2b1,0
where
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2Im a 1
I1,0 =
p = 1
p ,0
1 p ,2
p,1a1 p, 3 *p ,1a1 p,1
2Im a 1
I1,2 =
p = 1
p ,0
1 p ,0
p,1a1 p,3 *p,1a1 p, 1
As in the first case, we can show that E{| I1,0 |2 } = E{| I1,2 |2 } 1.2646 . However, one can remark that, in this case, the only factors that weigh the useful symbol are | h1 |2 and | h2 |2 which are both positives. Therefore, we expect a BER floor level below the classical Alamouti one presented in the previous section. Based on the same idea, we also propose another Alamouti scheme. This time, the Alamouti scheme is the following space frequency block coding (SFBC) defined by: a0,1 a0,0 a0,1 a 1,1 a1,0 a1,1 A1 = a2,1 a2,0 a2,1 a3,1 a3,0 a3,1 a4,1 a4,0 a4,1 a2,1 a2,0 a2,1 a 3,1 a3,0 a3,1 A2 = a0,1 a0,0 a0,1 a1,1 a1,0 a1,1 a6,1 a6,0 a6,1 The received symbols 1
1
r1,0
and
r3,0
are given by
1
1
1
q = 1
q = 1
q = 1
1
1
1
q = 1
q = 1
q = 1
r1,0 = h1 p ,q a1 p , q h2 (1)q 0,q a3, q h2 (1)q 1,q a2, q h2 (1) q 1,q a0, q b1,0 p = 1q = 1
1
1
r3,0 = h1 p ,q a3 p , q h2 (1)q 0,q a1, q h2 (1) q 1,q a0, q (1) q 1,q a6, q b3,0 p = 1q = 1
The Alamouti decoding for the blue block is performed by calculating y1,0 and y3,0 as * * y1,0 = h1 r1,0 h2 r3,0 * * y3,0 = h1 r3,0 h2 r1,0
We obtain:
(39)
* Re y1,0 = | h1 |2 | h2 |2 a1,0 Re h1*h2 I1,0 Re h1*b1,0 h2b3,0 * Re y3,0 = | h1 |2 | h2 |2 a3,0 Re h1*h2 I3,0 Re h1*b3,0 h2b1,0
where
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I1,0 =
2
q = 1
a
1, q 2, q
1,q a4, q (1) q 1,* q a0, q
1
I 3,0 =
2
q = 1
a
1, q 0, q
*1,q a2, q (1) q 1,* q a6, q
Hence, We obtain the same expressions as the STBC ones. However, we can show, according to Table 2-1 that E{| I1,0 |2 } = E{| I3,0 |2} = 12 | 1,1 |2 6 | 1,0 |2 (40) 0.8518 Clearly, the variance of the interference terms I1,0 and I 3,0 is significantly reduced. This fact will guarantee better performance compared to the previous STBC proposal. In the following we provide simulation results of the straightforward implemented Alamouti coding scheme, and the proposed ones. All the receivers apply an iterative interference cancellation. The performance is assessed in terms of BER as a function of the SNR. The performance of the different schemes is compared to the Genie-Aided one which exploits the perfect knowledge of the interference. We assume full channel state information (CSI) knowledge at the receiver side. Figure 2-11 depicts the BER performance obtained with the classical Alamouti scheme in FBMC. The data symbols are OQPSK modulated. One can observe a high BER floor level and the interference cancellation is not effective.
Figure 2-11 Performance of classical Alamouti scheme in FBMC/OQAM using OQPSK modulation
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As for the proposed STBC scheme, the BER performance with OQPSK modulation is shown in Figure 2-12. Clearly, the BER performance of the first iteration is improved and the BER floor is less than 7 103 . Moreover, we can almost reach the Genie-Aided performance by using 3 iterations.
Figure 2-12 Performance of the proposed STBC scheme in FBMC/OQAM using OQPSK modulation
Now, we show the performance of the proposed SFBC scheme in Figure 2-13 where the data is OQPSK mapped. In this case, we observe that with only the second iteration we can reach the Genie-Aided performance. This is thanks to the small residual interference which does not causes significant error propagation.
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Figure 2-13 Performance of the proposed SFBC scheme in FBMC/OQAM using OQPSK modulation
We have also test both proposed schemes with 16-OQAM modulation. Figure 2-14 and Figure 2-15 depict the performance of the proposed STBC and SFBC schemes, respectively. We remark that we need more iterations than in OQPSK to remove the interference. This is explained by the fact that the minimum distance between two different symbols is smaller in 16-OQAM than in OQPSK. Hence, more errors occur when the receive signal is corrupted by interference.
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Figure 2-14 Performance of the proposed STBC scheme in FBMC/OQAM using 16OQAM modulation
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Figure 2-15 Performance of the proposed SFBC scheme in FBMC/OQAM using 16OQAM modulation
In this section, we have considered the application issue of the Alamouti coding scheme in FBMC/OQAM system. Indeed, the self-interference is not removed automatically during the decoding process. We have proposed to use iterative interference cancellation receiver to remove the remaining interference. However, when the interference is not small enough compared to minimum distance between the different useful symbols, the receiver suffers from error propagation and the cancellation is not effective. Therefore, we have proposed some different arrangement in the Alamouti coding such that the signal to interference ratio (SIR) is improved. Thus, we proposed a STBC and SFBC schemes, and we tested them using OQPSK and 16-QAM modulations. We have shown that in both proposed schemes, the BER performance can almost reach the Genie-Aided one.
2.4 Schemes for FBMC-QAM Aiming to reduce the inherent interference, we propose, in this section, an FBMC configuration where QAM symbols are transmitted at each one period T instead of transmitting OQAM (real-valued) symbols at each half a period T / 2 . We note that in this proposed scheme the orthogonality condition (6) is lost. Therefore, an iterative receiver based on interference estimation and cancellation has to be performed. The expression of the transmitted signal s[m] given in (1) becomes
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s[m] = sk ,n g[m nM ]e
j
2 k D ( m ) M 2
e
jk ,2 n
(41)
,
k =0 nZ
where sk ,n are now complex QAM symbols. We note that the phase term k ,2 n can also be aborted, as we dropped the real orthogonality condition. The transmultiplexer impulse response of the proposed FBMC/QAM system can be derived from the impulse response given in Table 2-1 or Table 2-2 by decimation by a factor of 2 in time axis. That yields the coefficients depicted in Table 2-5 and Table 2-6 for PHYDYAS and IOTA filter respectively.
k 1 k k 1
n 1 0.125 j 0 0.125 j
n 1 0.125 j 0 0.125 j
n
0.239 j 1 0.239 j
Table 2-5 Transmultiplexer impulse response of the FBMC/QAM using PHYDYAS filter
k 1 k k 1
n 1 0.0381 j 0 0.0381j
n 1 0.0381 j 0 0.0381j
n
0.4411 j 1 0.4411j
Table 2-6 Transmultiplexer impulse response of the FBMC/QAM using IOTA filter
The expression (9) of I k , n becomes:
I k ,n =
k , n sk ,n ,
(42)
( k , n )*k , n
where k ,n are the coefficients given in in Table 2-5 or Table 2-6, k = k k , and n = n n . It is shown in [8] that, in FBMC/OQAM, we have
p ,q
| p ,q |2 = 2 where p ,q are the
coefficients given in Table 2-1 (or Table 2-2). Hence, the ISI variance in conventional FBMC (FBMC/OQAM) for both prototype filters is 2 = | p ,q |2 ISI ( p , q ) (0,0)
= | p ,q |2 | 0,0 |2 = 1.
(43)
p ,q
Therefore, in FBMC/OQAM, the power of ISI has the same value as the transmitted data variance. Whereas, the ISI variance in the proposed FBMC/QAM system, using PHYDYAS filter, is given by 2 ISI = | p ,2 q |2 ( p , q ) (0,0)
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(44)
p ,q
Hence, thanks to the proposed scheme we have reduced the ISI power to 17.7% . If the IOTA filter is used, we can show that the inherent interference variance in FBMC/QAM is 2 ISI = 0.3956 . We remark that the variance of the interference when using IOTA filter is more than twice the interference variance when using PHYDYAS filter. Therefore, it is natural to expect smaller values of SNR0 in the case of IOTA compared to the PHYDYAS case. Consequently, the error propagation effect begins to occur in the IOTA case before it appears in the case where PHYDYAS filter is used. Moreover, it is worth noting that when the modulation order is increased, the minimal Euclidean distance d 0 between two different symbols decreases. Thus, this makes the condition of ISI cancellation effectiveness more difficult to satisfy. Interference cancellation approaches generally offer the possibility of removing interference with low complexity increase and without enhancing the level of noise already present in the received signal. ISI cancellation scheme is essentially based on using preliminary decisions to estimate and cancel the interference. In single-input single-output (SISO) scenario, we assume that one tap zero forcing (1tap ZF) equalization is used. Hence, assuming that the decided symbols at the equalizer output are correct, we attempt to estimate the interference terms by means of the interference Table 2-5 and Table 2-6. Then, the estimated interference is removed from the received signal and the 1-tap ZF equalization is again performed. These operations can be repeated iteratively many times until convergence. We refer to this receiver as IIC-ZF (Iterative Interference Cancelation with ZF) receiver. Figure 2-16 depicts the basic operations in IIC-ZF receiver. Iter= 1
Iter= 2
Iter= IterMax-1
ZF Eq.
ZF Eq.
ZF Eq.
Inter. estim.
Inter. estim.
Inter. estim.
Ch. H
Ch. H
Ch. H
Iter= IterMax
ZF Eq. Delay
Delay
Delay
Figure 2-16 Block scheme of the IIC-ZF receiver for SISO
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In the following, we provide simulation results of the proposed FBMC/QAM scheme compared to OFDM and also to FBMC/OQAM. The number of subcarriers is set to M = 512 . M = 64 . According to the IEEE 802.16e standard [6], the CP duration for OFDM is set to = 8 We assume perfect channel knowledge at the receiver side, and we use the Veh-A channel model to generate the channels. The complex data symbols are either QPSK or 16-QAM modulated for OFDM and the proposed FBMC/QAM. However, since the conventional FBMC uses OQAM modulation, each transmitted symbol, on each T / 2 , is either 2-PAM or 4-PAM modulated. The system performance is assessed in terms of BER as function of the SNR. For FBMC/OQAM and FBMC/QAM, we define the SNR as N2 SNR = t 2 s ,
where is the signal variance on each transmit antenna, and 2 is the noise variance on each receive antenna. However, for OFDM, the expression of the SNR is defined so that it takes into account the SNR loss due to the CP: M Nt s2 SNR = . M 2 2 s
Figure 2-17 Performance of IC receiver with FBMC/QAM system using IOTA filter and 4-QAM constellation in SISO system
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We will first test the proposed receiver in the SISO case. The tentative detector in this case is the simple one tap ZF equalizer. In Figure 2-17 we present the BER performance of the IIC-ZF receiver with IOTA filter for different numbers of iterations. For comparison we also depict the performance of ZF and the Genie-Aided one. The "Genie-Aided" performance is defined as the performance obtained by assuming a perfect interference estimation, i.e the exact transmitted symbols are involved to estimate the interference. According to the simulation results presented in the figure, the performance of IIC receivers with 3 , 4 and 5 iterations provide the same BER performance. Hence, the IIC-ZF performance converges from the third iteration, but the interference is not successfully removed. We notice that the IIC-ZF receiver does not reach the Genie-Aided performance; we can observe an SNR gap of about 5 dB at BER = 102 . Therefore, we conclude that the interference cancellation is not effective in SISO-FBMC/QAM using IOTA filter, and the proposed IIC-ZF receiver suffers considerably from the error propagation. This is explained 2 = 0.3956 in IOTA case. by the relatively high interference variance ISI
Figure 2-18 Performance of IC receiver in FBMC/QAM system using PHYDYAS filter and 4-QAM constellation in SISO system
Figure 2-18 concerns the case where PHYDYAS filter is used. In this case, we notice that the performance of IIC-ZF receiver converges at the second iteration since IIC-ZF receivers with 2 and 3 iterations exhibit the same BER performance. Moreover, we remark
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that with one iteration we obtain almost the same performance as obtained with 2 or 3 iterations. That means that we can be satisfied with only one iteration in order to limit the computational complexity. Further, we notice that the SNR gap between the IIC-ZF performance and the Genie-Aided one is small enough. We observe an SNR gap of about 0.5 dB in the whole considered SNR region. It is also worth pointing out the impact of the prototype filter choice on the BER performance of the ZF equalizer in FBMC/QAM; according to Figure 2-17 and Figure 2-18 we remark that the ZF performance with IOTA filter is worse than the one obtained with PHYDYAS. This is due to the difference of interference variance 2 2 values for each case (namely ISI 0.18 for PHYDYAS, ISI 0.40 for IOTA).
Figure 2-19 Performance of IC receiver in FBMC/QAM system using PHYDYAS filter in SISO system with 16-QAM constellation
As we mentioned before, when the modulation order is increased, it becomes more difficult to cope with error propagation because the distances between the symbols decrease, whereas the variance of the interference is kept invariant. Since we have seen that the interference cancellation is not effective with IOTA filter, we only test the FBMC/QAM using 16-QAM with PHYDYAS filter, and the BER performance results are depicted in Figure 2-19. In this figure, we compare the performance of IIC-ZF receiver to the Genie-Aided one. We observe that the performance converges from the 6th iteration. A performance degradation with respect to the Genie-Aided can be noted; we have an SNR loss of about 2.5 dB at BER = 102 .
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2.4.1 Spatial multiplexing FBMC-QAM In case of SM configuration with ML detection, the received vector in FBMC/QAM can be written as: (1N ) r (1) h(11) hk ,n t sk(1),n I k(1),n bk(1),n k ,n k ,n (45) = , ( Nr ) (N ) (N N ) (N ) (N ) ( N r 1) hk ,nr t sk ,nt I k ,nt bk ,nr rk ,n hk ,n rk , n
s k , n Ik , n
Hk ,n
bk , n
where the entries of the interference vector I k ,n are complex-valued. We chose to use, first, an MMSE equalizer as the tentative detector and the main detector is a simple classical ML one. Hence, MMSE equalizer provides tentative estimations of the data vectors s k ,n . Then, basing on these tentative estimates, the interference canceller calculates an estimation of the the whole interference vector that should be removed from the received vector rk ,n . It is worth pointing out that the performance of MMSE equalizer in the case of FBMC/QAM is significantly depending on the interference variance because both useful and interference signals ( sk ,n and I k ,n ) are complex-valued. We name this receiver by IIC-ML (Iterative Interference Cancelation with ML). Figure 2-20 depicts the simplified blocks of the proposed IIC-ML receiver. Iter= 1
Iter= 2
MMSE Eq.
Iter= IterMax-1
ML Det.
ML Det.
Inter. estim.
Inter. estim.
Inter. estim.
Ch. H
Ch. H
Ch. H
Iter= IterMax
ML Det. Delay
Delay
Delay
Figure 2-20 Block scheme of the IIC-ML receiver
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Figure 2-21 Performance of IIC-ML receiver in FBMC/QAM system using PHYDYAS filter and 4-QAM constellation for 2 2 spatial multiplexing
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Figure 2-22 Performance comparison between Rec-ML in FBMC/OQAM and IIC-ML in FBMC/QAM using PHYDYAS filter and 4-(O)QAM consellation in 2 2 SM
Regarding MIMO, we have considered the 2 2 spatial multiplexing scheme. The Rayleigh spatial sub-channels are spatially non-correlated. Figure 2-21 depicts the obtained BER performance of the proposed FBMC/QAM using PHYDYAS filter with MMSE equalizer, which is now our tentative detector. We also show in this figure the performance obtained using IIC-ML for different values of iterations and compare them to the optimum performance obtained with the Genie-Aided. We notice that increasing the number of iterations of IIC-ML improves the BER performance and almost converges to the optimum one after 5 iterations, i.e. there is practically no improvement beyond 5 iterations. Hence, IIC-ML receiver performs correctly with 5 iterations. However, as in the SISO case, we observe a slight SNR loss less than 0.5 dB compared to the Genie-Aided performance.
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Figure 2-23 Performance of IIC-ML receiver with FBMC/QAM system using IOTA filter and 4-QAM constellation for 2 2 spatial multiplexing
In Figure 2-22 we compare the performance of IIC-ML in FBMC/QAM with the RecML (proposed in [11]) one in FBMC/OQAM using -for both of them- the PHYDYAS filter. We also show the BER performance of the OFDM using ML detector. For FBMC/QAM, we present only the performance of the MMSE equalizer (tentative detector) and the performance of the IIC-ML after 5 iterations. As for classical FBMC/OQAM, we show the performance of MMSE and the performance of Rec-ML. First of all, one can notice that MMSE equalizer for FBMC/QAM exhibits worse BER performance compared to FBMC/OQAM. This is explained by the fact that the inherent ISI term in FBMC/QAM is complex as the transmitted data symbols (no orthogonality), whereas in FBMC the interference terms are pure imaginary and the data symbols are real-valued (real orthogonality). However, the situation is different with IIC-ML and Rec-ML receivers, we clearly notice that Rec-ML with conventional FBMC/OQAM suffers from the error propagation effect and the BER performance converges to a suboptimal one, whereas the ISI cancellation is effective with IIC-ML in FBMC/QAM system, where we obtain almost the same performance as OFDM-ML. When IOTA filter is used, we notice in Figure 2-23 that the BER performance of the IIC-ML receiver does not converge to the optimal performance. That is, the interference cancellation is not effective and the receiver suffers from error propagations. This is due to 2 = 0.3956 . We observe that the high value of the interference variance which is equal to ISI the IIC-ML performance converges starting from the fifth iteration. IIC-ML receivers with 5
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to 8 iterations exhibit the same BER performance. We can note an SNR loss for the IIC-ML performance with respect to the optimal one of about 7 dB at BER = 102 . Therefore, the proposed IIC-ML receiver can not be performed in FBMC/QAM using IOTA prototype filter. As in the SISO scenario, we test the IIC-ML receiver with PHYDYAS filter when the 16QAM constellation is considered. BER performance is shown in Figure 2-24. In this figure, we depict the BER curves of the IIC-ML with 10 iterations which can be considered as the limit of convergence. As in SISO, we observe a performance degradation compared to GenieAided. We can note an SNR loss of about 2.5 dB at BER = 102 .
Figure 2-24 Performance of IIC-ML receiver with FBMC/QAM system using PHYDYAS filter and 16-QAM for 2 2 spatial multiplexing and 16-QAM constellation
2.4.2 STBC FBMC-QAM As for the STBC decoding, a first Alamouti decoder is used as tentative detector providing tentative estimations of the data symbols. Basing on these tentative estimates, the interference canceler calculates an estimation of the interference, and then its contribution is removed from the received vector rk ,n . After that, the Alamouti decoding is
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again applied, and the operation can be repeated several times. Figure 2-25 depicts the principal block scheme of the proposed receiver.
Iter=1
Iter=2
STBC dem.
STBC dem.
Iter= IterMax-1
STBC dem.
STBC mod.
STBC mod.
STBC mod.
Inter. estim.
Inter. estim.
Inter. estim.
Ch. H
Ch. H
Ch. H
Iter= IterMax
STBC dem. Delay
Delay
Delay
Figure 2-25 Block scheme of the IIC-Alamouti receiver
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Figure 2-26 Performance of IIC-Alamouti receiver with FBMC/OQAM using PHYDYAS filter and 4-OQAM (2-PAM) constellation
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Figure 2-27 Performance of IIC-Alamouti receiver with FBMC/QAM using PHYDYAS filter and 4-QAM constellation
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Figure 2-28 Performance of IIC-Alamouti receiver with FBMC/QAM using 4-QAM constellation with IOTA filter
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Figure 2-29 Performance of IIC-Alamouti receiver with FBMC/QAM using 16QAM constellation and PHYDYAS filter
As for the STBC scheme, we have considered the simple 2 1 Alamouti coding scheme. Figure 2-26 depicts the BER performance obtained of the IIC-Alamouti in FBMC/OQAM with different values of iterations and compare them to the Genie-Aided Alamouti performance. We clearly notice that IIC-Alamouti with conventional FBMC/OQAM suffers from the error propagation effect and the BER performance converges to a suboptimal one. In Figure 2-27 we provide the performance of IIC-Alamouti in FBMC/QAM, we also show the BER performance of the OFDM-Alamouti. For FBMC/QAM, we notice that the performance converges after only 2 iterations. Furthermore, we observe an SNR loss of only 0.5 dB compared to the Genie-Aided performance. Nevertheless, IIC-Alamouti exhibits almost the same performance that obtained with OFDM; this is due to the SNR loss caused by the CP in OFDM. Now, if IOTA filter is used, we can see in Figure 2-28 that the IIC-Alamouti receiver cannot completely remove the inherent interference. We observe an SNR loss of about 4.5 dB after 4 iterations. Then, let us test the IIC-Alamouti with PHYDYAS filter using 16-QAM constellation. The BER performance is depicted in Figure 2-29 where we can observe a degradation with respect to the Genie-Aided of about 2 dB at BER = 102 .
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SISO 4-QAM 16-QAM
SM-ML 4-QAM 16-QAM
STBC 4-QAM 16-QAM
0.5 dB
2.5 dB
0.5 dB
2.5 dB
0.5 dB
2 dB
5 dB
-
7 dB
-
4.5 dB
-
Table 2-7 SNR losses at
BER = 102
Finally, for the sake of summarizing the obtained results, we give in Table 2-7 the values of the SNR losses (with respect to the Genie-Aided performance) at BER = 102 of IIC receivers for the different considered configurations.
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3. MIMO transmission and reception under highly frequency selective channels 3.1 Introduction Current multicarrier-based PMR systems are designed to transmit in relatively narrow frequency bands, which makes them relatively robust to channel frequency selectivity. For instance, the TETRA Enhanced Data System (TEDS) [15] is designed to operate on total bandwidths ranging from 25 kHz to 150 kHz, with an intercarrier separation of 27 kHz (the total number of subcarriers ranges from 8 to 48 depending on the totally occupied bandwidth). This intercarrier separation is designed so that the typical channel frequency response is approximately flat across the bandwidth occupied by each subcarrier, and it can therefore be inverted with a single tap per-subcarrier equalizer. In LTE systems [16], the transmission bandwidth may take values of up to 20 MHz (as opposed to tenths of kHz in TEDS), which provides some strong difficulties when dealing with frequency selective channels. Indeed, if the 27 kHz intercarrier separation fixed in TEDS is considered, the number of subcarriers that needs to be employed can be up to 8192 , which is in stark contrast with the 48 subcarriers employed in TETRA. Given the fact that filterbank-based multicarrier modulations imply the use of FFT/IFFT operations plus pulse filtering at each subcarrier, one can conclude that the computational complexity associated with such a system will be difficult to afford in practice. Thus, the use of high transmission bandwidths in combination with FBMC modulations can only be realized from the practical point of view if the intercarrier frequency separation is allowed to be higher than 27 kHz. However, this in turn implies that the channel frequency response will no longer be approximately flat at each subcarrier, and more sophisticated equalization architectures will have to be used, see e.g. [17] and references therein. In practical terms, if the receiver keeps using a single-tap per-subcarrier equalizer in the presence of a highly frequency selective channel, its output appears to be contaminated by an additional source of distortion superposed to the background noise. This effect is much more evident in MIMO transmissions, basically due to the fact that the distortion effect generated by the multiple channels superposes at the receiver. In this section, we will provide a detailed analysis of this effect, and we will propose an alternative transceiver structure that significantly mitigates this effect. 1.1.1 The MIMO architecture considered here We consider a MIMO system with NT transmit and N R receive antennas. Let H( ) denote an N R NT matrix containing the frequency response of the MIMO channels, so
that the i j th entry of H( ) contains the frequency response between the j th transmit and the i th receive antennas. We assume that the MIMO system is used for the transmission of N S parallel signal streams, 1 N S min N R NT , which correspond to FBMC modulated signals. More specifically, we will denote by s( ) an N S 1 column vector that contains the frequency response of the signal transmitted at each of the N S parallel streams. Hence, each entry of the vector s( ) is the frequency response of a FMBC modulated symbol stream.
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Now, assume that the transmitter applies a frequency-dependent linear precoder, which will be denoted by the NT N S matrix A( ) . The signal transmitted through the NT transmit antennas can be expressed as x() A()s()
where x( ) is an NT 1 column vector containing the frequency response of the signal transmitted through each of the NT antennas. On the other hand, let y ( ) denote an N R 1 column vector containing the frequency response of the received signals in noise, namely y() H()x() n()
where n( ) is the additive Gaussian white noise. We generally assume that the receiver estimates the transmitted symbols by linearly transforming the received signal vector y ( ) . More specifically, we consider a certain N R N S receive matrix B( ) so that the symbols are estimated by
sˆ( ) B H ( )y( ) The whole ideal frequency-selective transceiver chain is implemented in Figure 2-7 for a FBMC-modulated system in which the number of subcarriers was fixed to M .
...
...
...
...
...
...
...
...
...
...
Figure 3-1: Ideal implementation
of a frequency selective precoder A( ) and a
linear receiver B( ) in a FBMC modulation system with M carriers
For the sake of simplicity, we will generally consider here that the matrices A( ) and B( ) are such that
B H ( )H( )A( ) I NS
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where I N S is the N S N S identity matrix1. Indeed, if this condition holds, the output of the receiver contains a noisy version of the transmitted symbol streams, in the sense that
sˆ( ) s( ) B H ( )n( ) Hence, by implementing both transmit and receive filters A( ) and B( ) such that (46) holds, we have been able to retrieve the transmitted symbol streams up to a background noise contribution.
3.1.1 Practical MIMO-FBMC under frequency selective channels Now, the main problem with the MIMO architecture presented in Figure 2-7 comes from the fact that, in practice, the frequency-dependent matrices A( ) B( ) need to be implemented using real filters. In practice, these filters may not have finite impulse response and, even when this is the case, they might be difficult to implement due to the number of coefficients that may be involved. This can be partly solved in multicarrier modulations, as long as it can be assumed that the degree of frequency selectivity is not severe and therefore the channel is approximately flat at each subcarrier frequency band. When this is the case, one can construct the MIMO precoder/receiver operations by applying the matrices A(k ) B(k ) to each subcarrier stream, where here k denotes the central frequency associated with the k th subcarrier. This is further illustrated in Figure 3-2 and Figure 3-3 for the particular case of NT N R 2 antennas in a FBMC modulation transmission, where M denotes the total number of subcarriers and where p[n] and q[n] represent the transmit and receive prototype pulses respectively.
...
...
... ...
...
Figure 3-2: Traditional implementation of the frequecy-selective precoder in multicarrier modulations, for the specific case of subcarriers.
NT
transmit antennas and
M
1
The transceiver architecture proposed below works all the same if this condition does not hold, but we keep it in order to simplify the exposition. ICT-EMPhAtiC Deliverable D4.1
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...
...
...
...
...
Figure 3-3: Traditional implementation of the frequecy-selective receiver in multicarrier modulations, for the specific case of subcarriers.
N R 2 receive antennas and M
As pointed out above, this solution is only effective when the channel frequency selectivity is mild enough to guarantee that each sub-carrier observes a frequency nonselective channel. However this is never the case of practical PMR systems, which will suffer from a non-negligible distortion at the output of the receiver that will severely impair the performance of the MIMO system. This will be later confirmed, both from the theoretical point of view and in simulations. Next, we propose an alternative solution that tries to minimize this effect.
3.2 Proposed approach In this subsection we propose an alternative solution that, with some additional complexity, significantly mitigates the distortion caused by the channel frequency selectivity. We will consider that each of the N S parallel MIMO signal streams modulated and demodulated according to the general transceiver structure shown in Figure 3-4, where P(exp ) and Q(exp ) contain –up to a constant factor– the frequency response associated with the transmit and receive prototype pulses p[n] and q[n] , respectively. The total number of subcarriers M is assumed to be even, and 1… M denote the subcarrier frequencies. Observe also that the real and imaginary parts of the initial frequency-domain complex symbols are sequentially staggered/unstaggered in the time domain (see staggering/unstaggering boxes in Figure 3-4). If the subcarrier frequencies are chosen as k 2M k 1 and the pulse pair ( p[n] , q[n] ) comply with certain bi-orthogonality conditions, this modulation is typically referred to as bi-orthogonal frequency division OQAM (BFDM/OQAM). When the transmit and receive prototype pulses are equal, the resulting modulation is referred to as OFDM/OQAM.
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M
Staggering
De-staggering
De-staggering
Staggering
Synthesis Filter Bank
...
...
...
...
...
...
...
M
Staggering
De-staggering
Analysis Filter Bank
Staggering
De-staggering
Figure 3-4: Filter-bank multicarrier modulator/demodulator considered here.
3.2.1 Taylor description of the frequency-domain problem Let us consider the combination of the FBMC transmission scheme in Figure 3-4 with the frequency selective MIMO precoder A( ) . More specifically, consider the k th subcarrier associated with the nS th MIMO signal stream that is sent through the nT th transmit antenna. According to the modulator in Figure 3-4, and assuming that the ideal frequency-selective precoder matrix A( ) is implemented, this stream of symbols after the expanders will effectively go through a transmit filtering process with equivalent frequency response: P(exp k )A( )n
T
nS
Now, assume that the precoding matrix A( ) is an analytic function in the frequency domain, so that we can express it as its Taylor series development around k , namely
A( ) 0
1 () A (k ) k
where A( ) (k ) denotes the th derivative of A( ) evaluated at k . Using this Taylor series expression, we can describe the frequency response of the ideal transmit chain as
0
1 P(exp k )A( ) (k ) k nT nS
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The idea behind the classical precoder implementation in Figure 3-2 is to truncate this Taylor series development and to consider only its first term. When this is the case, the transmitter frequency response that is effectively implemented takes the form
P(exp k )A(k )n
T
nS
(47)
which corresponds to a single weight multiplication of each subcarrier signal, as shown in Figure 3-2. Here, we suggest to go a bit further and consider the truncation of the above series representation to include its first KT terms, so that the transmitter filter has an effective frequency response equal to KT 1
0
1 P(exp k ) k A( ) (k ) nT nS
(48)
The main advantage of extending this truncation to the case KT 1 comes from the fact that one can effectively implement the above filter by using KT parallel FBMC modulators, together with KT parallel precoders based on single-tap per-subcarrier implementations. To see this, observe that if the prototype pulse p[n] is a sampled version of an original continuous waveform one can actually see p(t ) P(exp k ) M
k
as the frequency response of the sampled waveform
corresponding to the th derivative of p(t ) . Hence, when is relatively close to k , the frequency response in (48) can be equivalently formulated as KT 1
0
1 ( ) ( ) P (exp k )A (k )nT nS M
where P( ) (exp ) is –up to a constant factor– the frequency response associated with the th pulse derivative. Now, observe that each term of the above sum has exactly the same form as (47), replacing the actual precoder matrix A(k ) and the original prototype pulse P(exp ) by their successive derivatives A( ) (k ) , P( ) (exp ) . From all this, we can conclude that the KT -term truncation of the ideal transmit precoder frequency response can be implemented by combining a set of KT parallel precoders that can be implemented as in Figure 3-2. This is further illustrated in Figure 3-5, where we represent the suggested implentation of the transmit precoder when the number of parallel stages was fixed to KT 2 and the number of transmit antennas to NT 2 . We have represented in red the additional stage that needs to be superposed to the original one (in black), which is the same as in Figure 3-2.
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...
... ...
...
...
Figure 3-5: Proposed implementation of the frequecy-selective precoder for the specific case of
NT 2 transmit antennas and NT 2 parallel stages.
We can follow the same approach in order to approximate the ideal frequency selective linear receiver matrix B( ) in combination with the receive prototype pulse
Q(exp k ) . Figure 3-6 illustrates the proposed architecture for the simple case of
...
...
...
...
...
...
...
...
N R 2 receive antennas and K R 2 parallel stages.
Figure 3-6: Proposed implementation of the frequecy-selective linear receiver for the specific case of
NR 2
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3.2.2 Proposed solution in its full generality and resulting distortion From all the above, we can conclude that we can approximate the ideal frequencyselective precoder/linear receiver as depicted in Figure 2-7 by simply increasing the number of parallel stages ( KT , K R ) that are implemented at the transmitter and at the receiver. In this subsection, we investigate the residual distortion effect of this implementation at the output of the receiver, which will be the determining factor in order to fix these two parameters so as to reach a certain performance quality. To that effect, we will make some additional assumptions that will considerably simplify the analysis: The transmit and receive prototype pulses p[n] , q[n] are of length MK , where K is an integer that is typically referred to as the overlapping factor. Furthermore, these pulses are obtained by discretization of smooth analog waveforms p(t ) , q(t ) , so that
MK 1 Ts p[n] p n 2 M
n 1… MK
and equivalently for q[n] , where Ts is the sampling period. Furthermore, the pulses p(t ) , q(t ) and their successive derivatives are null at the end-points of their support, namely at t Ts K 2 . Furthermore, we will assume that p[n] and q[n] are either symmetric or anti-symmetric in the time domain. The complex symbols are drawn from a bounded constellation, and their real and imaginary parts are independent, identically distributed random variables with zero mean and power Ps 2 .
All frequency-depending quantities ( A() B() H() ) are smooth functions of .
Under these assumptions, it is possible to characterize the behavior of the residual distortion at the output of the receiver, assuming that the number of subcarriers is relatively high. We give more specific details below, differentiating among transmit-only, receive-only and full MIMO processing cases. To describe the residual distortion power that is observed at the output of the receiver, it is convenient to define some pulse-specific quantities that will constantly appear. In order to introduce these quantities, consider two general pulses p[n] q[n] of length MK , and let P and Q denote two M K matrices obtained by arranging the original samples in columns. In other words, the k th row of P (resp. Q ) contains the k th polyphase component of the original pulse p[n] (resp. q[n] ). Next, consider two M 2K 1 matrices R p q and S p q obtained as R p q P J M Q S p q J 2 I M 2 P J M Q
where indicates row-wise convolution between matrices, J M is the anti-identity matrix of size M , and denotes Kronecker product. Next, consider four different pulses p1[n] q1[n] p2 [n] q2 [n] all of length MK . We will see next that the distortion power observed at the output of the receiver will strongly depend on the following pulse-specific quantity: ICT-EMPhAtiC Deliverable D4.1
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p1 q1 p2 q2
1 T T R p1 q1 R p2 q2 U S p1 q1 S p2 q2 U M
where U I 2 I M 2 J M 2 and U I 2 I M 2 J M 2 . The quantity is equivalently defined, but swapping U and U . Let us now study the performance of some specific MIMO transceiver architectures. 3.2.3 Receive-only frequency-selective processing Let us assume that the transmitter does not implement any precoder, so that the number of transmitted streams is equal to the number of transmit antennas ( N S NT ) and2 A( ) I NS . In this situation, we propose to use a set of K R stages at the receiver, implemented as shown in Figure 3-6. Then, up to an error of order O M 2( KR 1) , the resulting distortion power observed at the k th subcarrier of the n th symbol stream is given by the expression
Pe k n 2 Ps
p q ( K ) p q ( K R
K R M 2 K 2
R
R)
N S
nS 1
B
KR
k
H
H
2
k nn
(49) S
where q ( K R ) denotes the K R th derivative of the original receive pulse q and
B
KR
k is the
K R th derivative of the receiver matrix B( ) evaluated at k .
Two observations are in order. First, note that the effect of the prototype pulses on residual distortion power is only through the pulse–specific quantity p q ( KR ) p q ( KR ) , which does not depend on the channel frequency response. This suggests that one could design the prototype pulses p q so as to minimize this pulse-specific quantity, thus making the system robust to the general presence of channel frequency selectivity. This is further studied in WP2 of this project, see further [18]. Secondly, we observe that the total residual distortion power that is observed at the n th receive symbol stream is an additive combination of the distortion associated with each of the transmit symbol streams (note the sum from nS 1 to N S in the above formula). This justifies the idea that general MIMO processing is very vulnerable to the presence of highly frequency selective channels, since the higher the number of parallel streams, the higher the residual distortion power that will be observed at the output of the receiver. Transmit-only frequency-selective processing
Assume next that the receiver performs no frequency-selective processing at all, so that N S N R and B I NS . In this situation, we propose to use at the transmitter a frequency-selective processing architecture such as the one shown in Figure 3-5 employing a number NT of parallel stages. In this situation, up to an error of order O M 2( KT 1) , the
2
More generally, it could also be assumed that A( ) is a general constant matrix
independent of . The results presented here are completely equivalent, replacing H( ) by H( )A . ICT-EMPhAtiC Deliverable D4.1
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resulting distortion power observed at the k th subcarrier of the n th symbol stream is given by the expression Pe k n 2 Ps
p ( K ) q p ( K ) q T
T
KT M 2 K 2
T
NS
nS 1
H k A
KT
2
k nn
S
Observe that the expression is very similar to the one obtained using receive-only processing. Hence, from the point of view of the reduction of the residual distortion power, it is equivalent to apply the proposed architecture at the transmitter or at the receiver. In both cases, the residual distortion power will be reduced by the same orders of magnitude. Full transmit-receive frequency-selective processing
Let us finally consider the more general case where both transmitter and receiver use a frequency-selective processing architecture. This is the case, for example, of the architecture that maximizes the channel capacity, as explained below in Section 3.2.3. In this situation, we propose to use a multi-stage frequency-selective architecture at both transmitter and receiver, using the same number of parallel stages at both sides so that KT K R . Indeed, it can be seen that the magnitude of the residual error is always dictated by the minimum between KT and K R , so clearly no additional advantage is obtained when these two quantities are different. In this situation, up to an error of order O M 2( KR 1) , the residual distortion power observed at the k th subcarrier associated with the n th symbol stream takes the form NS
Pe k n An( KnR ) (k ) pN( K R ) qN pN( K R ) qN 2
S
nS 1
NS
2 An( KnR ) (k ) Bn( KnR ) (k ) pN( K R ) qN pN qN( K R ) S S n 1 S
NS
2 An( KnR ) (k ) Bn( KnR ) (k ) pN( K R ) qN pN qN( K R ) S S n 1 S
NS
Bn( KnR ) (k ) pN qN( K R ) pN qN( K R ) 2
nS 1
S
where · and · represent the real and imaginary part, and where we have introduced the quantities ( KR ) nnS
2 R H KR B k H k A (k ) k nn K R K RM S
( KR ) nnS
2 R K R H ( k ) B k H k A k nn K R K RM S
K
A
K
B
In the next subsection, we illustrate all these results with some examples that are of special relevance in practical MIMO settings.
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3.2.4 Specific study for some specific MIMO architectures In this subsection, we provide a more specific study of the proposed solutions for some relevant MIMO architectures that have some practical interest. Spatial multiplexing with channel inversion at the receiver
Let us consider here an NT N R MIMO system where the number of receive antennas is higher than or equal to the number of transmit ones, namely N R NT . Assume that the transmitter does not have access to the channel state information, and simply transmits an independent parallel symbol stream through each of the NT transmit antennas. We assume that the channel matrix H( ) has full rank over all the range of values of . At the receiver, a linear filter performs zero forcing of the resulting MIMO channel, which would be ideally implemented by a matrix of filters with frequency response given by
H
B H ( ) H# ( )
H
( )H( ) H H ( ) 1
Let us now consider the traditional FBMC processing architecture, which is obtained by considering a single term of the Taylor decomposition of B( ) , that is by letting K R 1 . In this situation, the receiver would implement an architecture such as the one shown in Figure 3-3 with B H (k ) H# (k ) . Using (49), we observe that the residual distortion power of the k th subcarrier associated with the n th stream at the output of the receiver can be approximated by
Pe k n 2 Ps
p q ' p q ' M
2
H# (k )H '(k )H '(k ) H H# (k ) H n n
(50)
First of all, observe that the distortion power is inversely proportional to the square of the number of subcarriers. Hence, when the number of subcarriers increases, the distortion power will tend to zero. This is reasonable, because when this happens the channel becomes flat on the bandwidth associated with each subcarrier, and therefore persubcarrier processing becomes optimal. On the other hand, it is clearly observed that in this situation the residual distortion power depends critically on the degree of variability of the channel H( ) in the frequency domain through its first derivative at each subcarrier. Let us consider now the use of K R 2 parallel stages instead of only one as in the traditional per-subcarrier processing implementation. In this situation, we implement the solution as depicted in Figure 3-6, where the first stage applies the linear receiver matrix B H (k ) H# (k ) , whereas the second stage applies the matrix
B '(k ) H H H (k )H(k ) H '(k ) H I NR H(k )H# (k ) H# (k )H '(k )H# (k ) 1
On the other hand, this second stage applies a FBMC modulator but using the derivative of the original prototype pulse, denoted as q '[n] , instead of the original one (see further Figure 3-6). In this situation, the residual distortion power can be approximated by
Pe k n Ps
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NS
nS 1
B '' k H H
2
k nn
(51) S
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where, using conventional matrix algebra, one can obtain
B ''(k ) H H(k ) H# (k )H ''(k ) 2 H# (k ) H '(k )
2
2 H H (k )H(k ) H '(k ) H I N R H(k )H# (k ) H '(k ) 1
By comparing the residual distortion power expressions in (50) and (51) one can readily see that the magnitude of this quantity has been reduced by a factor in the order of O(M 2 ) . We will see in the next section that this implies a substantial improve in terms of the global performance of the system. Maximum capacity transmit precoding and channel inversion at the receiver
Let us now assume that the transmitter has perfect channel state information and that it transmits using the precoder that maximizes the capacity of the system under additive Gaussian noise and Gaussian signalling. To derive the optimum precoder, we must consider the singular value decomposition of the channel matrix, which can be expressed as
H( ) U( )()V H () where U( ) (respectively V( ) ) is an N R N S (respectively an NT N S ) complex matrix with orthogonal columns, and ( ) is an N S N S diagonal matrix containing the positive square root of the symbol stream powers. We will denote by ( ) the N S N S matrix of channel eigenvalues, namely ( ) 2 ( ) . Let s( ) denote an N S 1 column vector containing the frequency response of the N S parallel transmitted symbol streams. It is well known that, under Gaussian signalling and AGWN channel, the transceiver that maximizes capacity is obtained by using the precoder V( ) at the transmit side, namely A() V()
At the receiver, we consider the use of the receiver that inverts the resulting channel, namely
B() U()1 () H()V()1 () To compute the precoders that need to be used at the transmit and receive sides, we restrict our attention to the situation where the positive channel singular values are all simple (the opposite situation is more involved, but not much more difficult to handle). Among all the possible choices for V( ) , we consider here the unique matrix with orthogonal columns such that it has real-valued diagonal entries. Furthermore, we will assume that the diagonal entries of V( ) are non-zero at the frequencies k . In the next subsection we provide a more detailed study of the performance of this method under severe frequency selectivity.
3.3 Performance analysis in PMR networks In this section, we analyze the performance of the proposed precoding/linear receiver architectures in realistic PMR scenarios. We will consider an LTE-like system with 512 subcarriers and an intercarrier separation of 15 KHz, which amounts to 768 MHz of
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transmission bandwidth. An LTE system would only use 300 subcarriers out of these 512 , thus occupying only 5 MHz of the total 768 MHz bandwidth. Here, we will assume for simplicity that all the 512 subcarriers are used. We will also assume that the channel state information is perfectly known at the receiver, and also at the transmitter whenever the use of frequency selective processing is considered. As for the actual FBMC modulation, we considered a sufficiently smooth prototype pulse with overlapping factor equal to K 2 and perfect reconstruction constraints, see further Figure 3-7. The same pulse was used at the transmitter and at the receiver. The number of antennas was fixed to 2 at both the transmitter and the receiver, namely NT N R 2 . All MIMO channels were simulated as independent, static and frequency selective with a power delay profile given by the ITU Extended Pedestrian A (EPA) and Extended Vehicular B (EVA) models [16]. Two different MIMO transceiver architectures were simulated: a pure spatial multiplexing technique with frequency-selective processing at the receiver only and a full frequency selective processing architecture with maximum capacity transmit precoding and channel inversion at the receiver (Section 3.2.3).
Figure 3-7: Prototype pulse used in the simulations.
3.3.1 Spatial multiplexing with channel inversion at the receiver We first consider the case where the transmitter performs no frequency selective processing, so that A( ) I NS and the receiver performs channel inversion, so that
B( ) H# ( ) , see further Section 3.2.3. In order to validate the expressions for the residual distortion in (50) and (51), we considered a noiseless scenario with two fixed channel impulse responses drawn from the Extended Vehicular A and the Extended Pedestrian A models. Figure 3-8 and Figure 3-9 show the eigenvalues of these two channels in the frequency domain (Note that these would be the effective channel responses if the channel was perfectly diagonalised by the optimum precoder/receiver).
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In this setting, we simulated a set of 1000 multicarrier symbols (drawn from a QPSK modulation) and measured the signal to distortion power ratio at the output of the receiver. The results are represented in Figure 3-10 and Figure 3-11 for the EPA and the EVA channel models respectively. In these two figures, solid lines represent the theoretical performance as described by formulas (50) and (51) whereas cross markers are simulated performance values. Observe that there is a perfect matching between them, and the simulated results are virtually indistinguishable from the theoretical ones. As for the actual performance of the transceiver, it must be first pointed out that the performance that can be achieved with the traditional MIMO architecture (which corresponds to the case K R 1 and which basically consists in a per-subcarrier channel inversion) is quite limited. Indeed, even for mild frequency selective channels (such as the one obtained from the EPA model), the maximum achievable signal to distortion is around 40 dB. This corresponds to the maximum signal to noise plus distortion ratio (SNDR) that can be achieved at the output of the receiver, regardless of how low the background noise is. This effect is even more detrimental in the VPA model, for which the maximum SNDR can fall down to 10 dB for some subcarrier groups. Now, observe that the use of K R 2 parallel stages at the receiver guarantees a significant performance improvement in terms of signal quality at the output of the receiver. Gains that can be of up to 20 30 dB are observed in almost all subcarrier frequencies for the EVA channel model, and even higher in the scenario with the EPA channel. Clearly, the proposed receiver is able to remove a significant part of the MIMO residual distortion power at the output of the receiver.
Figure 3-8: Eigenvalues of the MIMO channel used in the first part of the simulations, drawn from the Extended Pedestrian A model.
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Figure 3-9: Eigenvalues of the MIMO channel used in the first part of the simulations, drawn from the Extended Vehicular A model.
Figure 3-10: Signal to distortion power ratio measured at the output of the receiver when the transmitter uses pure spatial multiplexing and the receiver performs zero forcing channel inversion. The simulated channel was the one represented Figure 3-8.
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Figure 3-11: Signal to distortion power ratio measured at the output of the receiver when the transmitter uses pure spatial multiplexing and the receiver performs zero forcing channel inversion. The simulated channel was the one represented in Figure 3-9.
3.3.2 Maximum capacity transmit precoding and channel inversion at the receiver In this section, we analyze the performance of the transceiver presented in Section 3.2.3, which is designed to maximize the link capacity under Gaussian signalling. Figure 3-12 and Figure 3-13 represent the signal to residual distortion power ratio in the noiseless scenario for the channel responses depicted in Figure 3-8 and Figure 3-9 respectively. Results are given for different configurations in the number of parallel stages at the transmit and receive sides. Once again, we observe a perfect match between the simulated performance (cross markers) and the theoretical one (solid lines), which are almost identical. On the other hand, simulations confirm that performance is roughly dictated by the minimum number of parallel stages used at the transmit and receive sides, that is the minimum between K R and KT . For example, in the upper plot of Figure 3-12 we see that, when the transmitter is using only one stage (traditional implementation, i.e. KT 1 ), the use of K R 2 stages at the receiver brings some benefits at some subcarriers (e.g. those surrounding subcarrier 140). However, this gain is not uniform, and we see for example that the traditional receiver (with only one stage, K R 1), outperforms the more complex one at the subcarriers around 80 90 . In any case, we also observe a uniform advantage of up to 20 30 dB when using two stages at both transmit and receive sides. This shows that, when using frequency-selective processing at both transmitter and receiver, the most important ICT-EMPhAtiC Deliverable D4.1
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gains can be obtained by considering the proposed architecture at both sides of the communications link, but using the same number of parallel stages.
Figure 3-12: Signal to distortion power ratio measured at the output of the receiver when the transmitter uses an SVD precoder and the receiver performs channel inversion. The simulated channel was the one represented in Figure 3-8.
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Figure 3-13: Signal to distortion power ratio measured at the output of the receiver when the transmitter uses SVD-type precoding and the receiver performs channel inversion. The simulated channel was the one represented in Figure 3-9.
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4. Compact Antenna Arrays for PMR In this chapter we have performed a feasibility study for the use of antenna arrays with closely spaced antennas for PMR applications. This is what we call compact antenna arrays. In Europe the carrier frequencies chosen for PMR are between f c = 380-385MHz for the uplink and between f c = 390-395MHz for the downlink. The wavelengths in this case lie between = 76 79 cm. Traditionally antenna arrays are arranged with separation of multiples of /2 . If we assume a linear uniform array (ULA), we can see that for arrays of 4 antennas we already get a total length of at least 1.14m. For a base station this is still possible to implement, but it is still quite large. But for a mobile station, even for an array of 2 elements we already get almost 39cm, making it hard or even impossible to employ antenna arrays there. As a consequence we can conclude that the use of compact antenna arrays can be very beneficial for PMR applications. 4.1.1 Transmit Array Gain It is well known from the literature that compact antenna arrays can provide supergain that can become as large as the square of the number of antennas [28] [27]. This is specially true for transmit array beamforming using a lossless ULA and at the end-fire direction, that means in the direction along the axis of the array. In the literature of super-gain some authors state that if the antennas are lossy, the implementation of the compact arrays becomes impratical. The problem is that, although antennas may not have much loss, to achieve super-gain given an arbitrary small distance between the array elements, the electric currents that have to flow in order to radiate a given power may become very large. In other words, for very small separation, the losses inside the antenna can reduce the antenna gain. However in such works, the distance of the array elements is not optimized. We will see later in this section, that if the distance is properly chosen still a much higher gain than the number of array elements can be achieved. But first we need to show how to calculate the loss at the antenna elements. As a representation of the antenna losses we will show here how to calculate the ratio between the radiation dissipation and the radiation resistance. Later we will see that this ratio is necessary in order to calculate the beamforming vector and the array gain. Furthermore, we assume here the use of dipole elements. In this case the antennas can be approximated as a cylindrical wire from the electrical point of view. Also we assume the use 4 of high frequencies such that f > 2 , where r , and are the wire radius, its r 2 permeability and its conductivity. We can equivalently write r > , where is the f wavelength and c is the speed of light. Because wire antennas usually have small diameter 2
