Relay Communication with Delay Diversity for Future Communication Systems S. Ben Slimane and Afif Osseiran Radio Communication Systems Dept. of Communication Systems The Royal Institute of Technology (KTH), Stockholm, SWEDEN Emails:
[email protected],
[email protected]
Abstract— In this paper we consider relay communication as a way of increasing the diversity gain in fading wireless environments. The idea is to combine relay communication with delay diversity where each relay introduces a certain time delay to the signal before forwarding. The result is an increased frequency selectivity in the relay channel which can be exploited at the receiver. The obtained results show that, for single carrier signals with frequency domain equalization, considerable performance gain is obtained.
I. I NTRODUCTION The vision of providing coverage in cellular systems at reasonable cost by using relay stations has recently led to tremendous research efforts and emerges as a valuable option for future generations of wireless networks [1], [2], [3], [4], [5]. Communication relaying tries to exploit two fundamental features of the wireless medium: its broadcast nature, and its ability to achieve macro-diversity through providing independent channels and reducing the end-to-end path loss. By introducing different delays at the different relays further performance improvement can be achieved. In fact, these artificial delays introduce frequency selectivity in the overall channel of the radio link [6]. Taking advantage of this frequency selectivity can be done through the use of coded OFDM signals with a proper guard interval or through the use of single carrier transmission with a proper equalization scheme. In this paper we consider single carrier signaling with frequency domain equalization. Our objective is to investigate the possible gain that can be obtained when using relay delay diversity. II. S YSTEM M ODEL The system under consideration is shown in Figure 1 which consists of a transmitter, a receiver, and a set of M relays. We limit our study to low-complexity non-regenerative (amplify-and forward) relays where the received signal from the transmitter is amplified, delayed, and forwarded without decoding and re-encoding. As a relay may not simultaneously receive and transmit, we assume that the relays operate in a time duplex mode where the first time slot is used to receive information from the transmitter (S) and the second slot is used to forward the information to the final destination (D). The parameters hi ’s and gi ’s in Figure 1 represent the coefficients of the radio channel between the transmitter and
Relay i and between Relay i and the receiver, respectively. In this paper we consider both flat fading and frequency selective fading channels. We further assume that the relays are far apart from each other such that the channels can be considered uncorrelated. h1
R1
g1 g2
h2 h0
R2 D
hM −1
S
gM −1
hM RM −1
gM
RM Fig. 1.
Relay communication system.
Let us assume that each Relay i introduces a certain delay τi to the signal followed by transmission. This time delay can be pre-determined or purely random within a pre-defined time guard interval. Looking at the receiver side and assuming a slowly varying frequency selective fading channel, the equivalent lowpass of the received signal at the mobile unit from the base station (first time slot) can be written as r0 (t) =
L−1
h0,l s(t − υ0,l ) + z0 (t)
(1)
l=0
where s(t) is the equivalent lowpass of the transmitted signal, z0 (t) is the additive thermal noise which is modelled as zeromean complex Gaussian process with a power spectral density of N0 , L is the number of paths of the channel, h0,l is the attenuation coefficient of path l and υ0,l is its corresponding delay. The received signals at the different relays from the base station are given by yi (t) =
L−1 l=0
hi,l s(t − τi,l ) + zi (t),
i = 1, 2, · · · , M. (2)
At the second time slot, when the different relays transmit, the received signal at the mobile unit comes from the different relays and can be written as r1 (t)
=
=
M
βi
L−1
gi,l yi (t i=1 l=0 M L−1 L−1
− τi − τi,l ) + zM +1 (t)
βi gi,l hi,k s(t − τi,l − υi,k − τi )
i=1 l=0 k=0 M L−1
βi gi,l zi (t − τi,l − τi ) + zM +1 (t) (3)
+
We extend cyclically this block by G symbols to get a new transmitted block as s˜ = {sN −G , · · · , sN −1 , s0 , s1 , · · · , sN −1 } {˜ s0 , s˜1 , · · · , s˜N +G−1 } where G is chosen such that GTs > Tm and Tm is the maximum delay experienced by the signal. With the guard interval, the transmitted signal can then be written as follows: +∞ s˜k p(t − nTs ), s(t) = k=−∞
i=1 l=0
where τi is the time delay introduced by Relay i, zi (t) is a zero-mean complex Gaussian noise process with power spectral density of N0 , gi,k and υi,k are the fading channel parameters, βi is the amplification coefficient of Relay i which is assumed in this paper as βi = L−1 l=0
1
,
|hi,l |2 + N0 /Es
M L−1 L−1
r 0 = {r0,0 , r0,1 , · · · , r0,N +G−1 } where r0,n =
i ∈ {1, 2, · · · , M }
where Es is the average energy per transmitted symbol. We notice from (3) that the relays have increased the frequency selectivity of the fading channel. The overall channel impulse response from the relays to the mobile unit is given by ht (τ ; t) =
where p(t) is a rectangular pulse of duration Ts . The received block during the first time slot is given by
βi gi,l hi,k δ(τ − τi,l − υi,k − τi )
and ν0,l = υ0,l /T . Ignoring the first G samples of r 0 we get a new block of length N as y0 =
L−1
h0,l Dν0,l s + z 0
l=0
where z 0 is the noise vector and Di s is the right cyclic shift of s by i positions. Taking the Discrete Fourier transform of y 0 we get Y0
=
DFT {y 0 } = {Y0,0 , Y0,1 , · · · , Y0,N −1 } ,
with Y0,k = H0,k Sk + Z0,k ,
(4)
N −1 ki 1 Sk = √ si e−j2π N , N i=0 N −1 ki 1 Z0,k = √ zd,i e−j2π N , N i=0
Hi,k =
L−1
hi,l e−j2πυi,l /(N Ts ) .
l=0
For the second time slot, the received block from the relays can be written as follows: r 1 = {r1,0 , r1,1 , · · · , r1,N +G−1 } where r1,n
=
M L−1 L−1 i=1 l=0 k=0 M L−1
+
s = {s0 , s1 , · · · , sN −1 }, where N is the length of the block and si is the modulated symbol which can be PSK or QAM type.
h0,l s˜n−ν0,l + z0,n
l=0
i=1 l=0 k=0
where we notice that, when the channel is flat, the relays with their different delays will transform the channel into a frequency selective channel. Hence, the receiver can take advantage of this selectivity and improves the quality of the overall link. In order to take advantage of the introduced selectivity with single carrier signals, time equalization can be employed at the receiver. However, time equalization techniques have a complexity that increases with the number of paths. For the flat fading channels, the number of paths is directly related to the number of relays. To avoid this dependency we consider frequency domain equalization where only the cyclic prefix needs to be specified [7]. Frequency domain equalization requires only one-tap equalizer per sub-carrier and can handle fading channels with longer delays. This is suitable for our relay communication scheme as it does not depend on the number of relays. We only need to ensure a proper selection of the delays introduced by the different relays. Let us consider the transmission of a certain block of modulated symbols denoted as
L−1
βi gi,l hi,k s˜n−νi,l,k
βi gi,l zi,n−νi i=1 l=0 M L−1 L−1
+ zM +1,n
βi gi,l hi,k s˜n−νi,l,k + z˜1,n ,
i=1 l=0 k=0
(5)
with νi,l,k = (τi,l + υi,k + τi )/Ts , z˜1,n is a zero-mean complex Gaussian with variance λN0 , and λ=1+
M L−1
βi2 |gi,l |2 .
(6)
i=1 l=0
Ignoring the first G samples of r 1 we get a new block of length N as y1 =
M L−1 L−1
βi gi,l hi,k Dνi,l,k s + z˜1
(7)
After equalization and IDFT demodulation, the received Signal-to-Noise Ratio (SNR)1 when ZFE is employed can be written as follows: −1 N −1 1 Es /σz2 =N Γ= N −1 Γk 1 1 k=0 N
l=0
|H0,k |2 +
where
i=1 l=0 k=0
Γk =
where z˜1 is the noise vector. Taking the Discrete Fourier transform of y 1 we get Y1
DFT {y 1 } = {Y1,0 , Y1,1 , · · · , Y1,N −1 }
=
(8)
with k = 0, 1, · · · , N − 1
Y1,k = Hk Sk + Z1,k ,
(9)
where Sk is as defined above, 1 Z1,k = √ N
Hk
=
=
N −1
ki
z˜r,i e−j2π N ,
−j2π
βi gi,l hi,m e i=1 l=0 m=0 M βi e−j2πkτi /T Hi,k Gi,k , i=1
+ |H0,k |2 Es
N0 is the received SNR experienced at subcarrier k. For the MMSE scheme, the filter tap is given by |Hk |2 + |H0,k |2 λ Gk = |H |2 k + |H0,k |2 + N0 /Es λ
(11)
and the corresponding received SNR after equalization and IDFT demodulation is, after some manipulations, given by N −1 −1 1 Γ=N −1 Γk + 1
i=0
M L−1 L−1
|Hk |2 λ
|Hk |2 λ
kνi,l,k N
where Hi,k is the channel transfer function of the first link and L−1 g0,l e−j2πkτi,l /(N Ts ) Gi,k =
k=0
with Γk as given in (11). sk
Cyclic prefix
Channel zk
sˆk
IDFT
One-tap equalizers
DFT
Cyclic prefix
l=0
is the channel transfer function of the second link. Fig. 2. Frequency domain equalization of single carrier signals. Combining (4) with (9) using Maximum Ratio Combining (MRC) and normalizing we get III. N UMERICAL R ESULTS |Hk |2 2 + |H0,k | Sk + Zk , l = 0, 1, · · · , N − 1 (10) To illustrate the performance of this relay communication Yk = λ scheme we consider coherent QPSK modulation and a block where Zk is now zero-mean complex Gaussian with variance size of N = 128 symbols. Since our objective behind this σz2 = N0 . work is to assess the performance gain introduced by the relay With the help of the cyclic prefix, the frequency selective delay diversity we present our numerical results assuming the fading channel has been transformed into a set of frequency same path loss between the base station and relays, between nonselective channels in parallel. the relays and the mobile unit, as well as between the base A one-tap equalizer per channel can now be used as shown station and the mobile unit, i.e. (Eb /N0 )SD = (Eb /N0 )SR = in Figure 2. Both Zero Forcing Equalizer (ZFE) and Minimum (Eb /N0 )RD . The time delays are assumed random and each Mean Square Equalizer (MMSE) can be considered. IDFT relay can choose its own delay. We will illustrate the link is then used to extract the transmitted symbols of the block performance for both flat Rayleigh fading and frequency from the equalized signal. Since the fading coefficient of the selective Rayleigh fading channels. For the frequency selective combined signal in (10) is the sum of two positive uncorrelated fading channel we consider a 10 path channel according to the random variables, little difference in performance between the model proposed in [8]. two equalizers is expected. Figure 3 illustrates the cumulative distribution function, For the ZFE, the filter tap for the sample Yk is given by Pr(Γ < γt ), as a function of γt for a frequency selective 1 Rayleigh channel and when the average signal-to-noise ratio Gk = . 2 |Hk | 1 The received SNR is the same for all the symbols within the block. + |H0,k |2 λ
Eb /N0 is set to 20 dB. It is observed that by increasing the number of relays, the received SNR improves. We also notice that the SNR distribution of the detection schemes, ZFE, MMSE, are similar especially around the average Eb /N0 .
order for the system. Hence, with delay diversity better relative performance gain is expected as compared to the frequency selective fading case. This is illustrated in Figure 5 where we notice a performance improvement of about 5 dB at a CDF value of 10−3 .
0
10
0
ZFE
10
MMSE
M =1 −1
Probability distribution function
Probability distribution function
−1
10
M =5
−2
10
−3
10
−4
10
10
−2
10
−3
10
With delay diversity (ZFE)
−4
10
With delay diversity (MMSE) Without delay diversity
5
10
15 20 γt , (in dB)
25
30 5
Fig. 3. Cumulative distribution function of the received SNR of the equalized single carrier signal for different number of relays M . Direct link and the different hops are assumed to have the same average (Eb /N0 ).
Figure 4 shows the cumulative distribution function of the received SNR of the equalized single carrier signal with and without delay diversity over a frequency selective fading channel. It is observed that the introduction of different delays at the relays improves the system performance even when the channel is already selective. One can easily see from Figure 4 that the MMSE scheme is more efficient in capturing diversity and hence better improvement is obtained with this scheme as compared to that obtained with the ZFE scheme especially at low values of the SNR threshold γt . 0
10
With delay diversity Without delay diversity Probability distribution function
−1
10
M =5
−2
10
−3
10
−4
10
15 20 γt , (in dB)
25
30
Fig. 5. Cumulative distribution function of the received SNR of the equalized single carrier signal for a flat Rayleigh fading channel and a total of 5 relays.
Figure 6 shows the average bit error probability of our communication link with QPSK modulation as a function of the signal-to-noise ratio of the direct link, γ0 = (Eb /N0 )SD , and for different number of relays when the channel is flat Rayleigh fading. We notice that relay delay diversity has increased the diversity order of the system and as a result good performance improvement is obtained as compared to the case without delay diversity. This performance improvement is obtained without any increase in system complexity since these delays can be introduced in a distributed manner where each relay can introduce its own delay without any coordination with the other relays. Figure 7 illustrates the system average bit error probability for the case of frequency selective fading channel. We notice that increasing the number of relays increases the diversity order of the system even when the channel is already selective. With 5 relays a gain of about 5 dB can be obtained at a bit error probability of 10−5 . Figure 8 shows that the relative gain introduced by relay delay diversity is similar to that obtained in the flat Rayleigh fading channel.
10
ZFE
5
10
IV. S UMMARY
MMSE
15 20 γt , (in dB)
25
30
Fig. 4. Cumulative distribution function of the received SNR of the equalized single carrier signal for the cases: with and without delay diversity when M = 5 relays.
When the channel is flat fading and without delay diversity increasing the number of relays will not increase the diversity
Substantial diversity gain can be obtained in cooperative communication when inducing controlled time delays to distributed relay stations. This procedure increases the frequency selectivity of the overall channel seen at the final destination (relay channel). We showed in this paper that this frequency selectivity is well exploited when single carrier signal with frequency domain equalization is employed. The obtained results showed that relay delay diversity increases the diversity order of the system even when the
Direct only With delay diversity Without delay diversity
−1
10
Bit error probability
Bit error probability
10
−2
10
−3
10
−4
10
0
5
10 15 γ0 , (in dB)
20
25
−1
−2
10
−3
10
−4
10
5
10 15 γ0 , (in dB)
20
−1
−2
−3
−4
0
5
10 15 γ0 , (in dB)
20
25
Fig. 8. Average bit error probability of the relay system with coherent QPSK modulation and MMSE on frequency selective fading channel and a total of 5 relays.
R EFERENCES
Direct only 1 Relay 5 Relays
10
Bit error probability
10
10
Fig. 6. Average bit error probability of the relay system with QPSK modulation and MMSE on flat Rayleigh fading channels and a total of 5 relays.
0
10
Direct only With delay diversity Without delay diversity
25
Fig. 7. Average bit error probability of the relay system with QPSK modulation on Rayleigh fading channels and for different number of relays.
channel is already selective. Relay delay diversity is a very attractive scheme for future wireless communication as it can be implemented in a distributed manner and hence, it does not increase the system complexity. We focused in this paper on the performance of single link without considering the effects of path loss and shadowing. This is a subject of our future study where we are investigating the performance at the system level. One of the interesting points is to study the effects of co-channel interference on the extra diversity introduced by relay delay diversity and the interaction between relay delay diversity and multiuser diversity in cellular/relay systems.
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