Design of 10 GHz Sampling Rate Digital FIR Filters with Powers-Of-Two Coefficients Benjamin Parent1, Jonathan Müller2, Andreas Kaiser2 1ISEN, 2

IEMN-ISEN Lille, France firstname.name @ isen.fr Abstract— Deep sub-micron CMOS technologies have enabled the development of highly digital radios for wireless communications at low Gigahertz frequencies [5]. Meanwhile, nanometer scale CMOS allows to push the RF frequencies to the millimeter wave frequency range. Particularly 60GHz radio has emerged as the candidate for high-data-rate (10 Gb/sec), short-distance (1 to 10m) wireless communication systems. Radio channels are 2GHz wide, triggering new challenges for the digital circuit designers with sampling frequencies in the GHz range. This paper presents an algorithm implemented for designing low power and very high-speed finite impulse response (FIR) digital filters. FIR filters are a commonly used block for digital signal processing but also widespread in wireless communication in digital radio transmitter.

Andreia Cathelin STMicroelectronics Crolles, France [email protected]

Fig. 1: Spectrum mask of the IEEE 802.15.3c standard

Keywords— Search heuristic, digital signal processing, digital radio, 60GHz transmitter, digital filter design. INTRODUCTION The system under study is based on the IEEE 802.15.3c standard using OFDM modulation with QPSK encoding targeted at wireless transmission of uncompressed video signals. 4 radio channels of 2GHz bandwidth in the 57GHz – 65GHz band are defined. The digital baseband output signal processor is sampled at 2.5GS/s. Figure 1 shows the targeted 60GHz transmitter architecture. The D-to-A conversion is merged with the mixer into a so-called DRFC (digital-to-RF converter). For the standard targeted here, signal-to-noise ratio and spectrum-mask specifications require 7 bit resolution and a digital interpolation filter in front of the DRFC. Details of the system can be found in [5]. The digital data stream at the baseband output needs to be oversampled from 2.5Gs/s to 10Gs/s. Images of the baseband spectrum appear at multiples of the initial sampling frequency and have to be filtered. After filtering and conversion to the analog domain, the first residual image appears at a 10GHz offset.

Fig. 2: Digital transmitter architecture

The function of the interpolator here is to oversample the I and Q signals of the transmitter from 2.5Gs/s to 10Gs/s and filter the images resulting from oversampling. Due to the high sampling frequency, FIR filters have been preferred over IIR structure. But the performance of the interpolator is not just a question of architecture: although FIR greatly simplifies the design compared to the IIR alternative, it forces the designer to tackle frequencies in the 10GHz range. I.PREVIOUS WORK AND NEW PARADIGM Pursuing a previously developped idea [5], and since filtering requires numerous arithmetic operations (multiplications and additions), only coefficients in the form of powers of two (wired bit shifts) were used in order to avoid complex multiplications that are impractical at these sampling rates. This also greatly reduced the power consumption and area. Simplifications were also directly made at the FIR structure level: a “divide and conquer” approach has been proposed to reduce the design complexity. Instead of designing one block at

10 Gs/s, a 2-step upsampling and partitioning of the filter in several “subfilters” was suggested. Fig. 3 shows the resulting architecture. The interpolator was composed of two identical FIR filters with 6 coefficients each working at 5Gs/s and two identical FIR filters with 3 coefficients each working at 10Gs/s. Input data is on 7 bits in order to meet the SNR requirements. Each subfilter’s coefficients were found by a Genetic Algorithm already presented in [5]. This algorithm is able to generate filters based on a targeted frequency response mask, minimizing the ripple in the band and maximizing outband attenuation.

However, tackling sums of Po2 would heavily increase the combinatorial complexity if applied as such. •

On the other hand, convolution (the angular stone of linear systems to which filters belong) is in turn based on additions and multiplications. Sums of Po2 might therefore be obtained by this means.

Definitely this idea is not restricted to this single specific application but might be adapted for any applications where high speed FIR filters are to be used. The next section presents the methodology followed to efficiently generate powers-of-two subfilters and the heuristic developped to search for optimal cascades. II.ALGORITHM DETAILS

Fig. 3: Internal architecture of the interpolator

Beside the design simplification achieved by cascading filters a deterioration of the flatness in the band of the system occured. This system fullfills the EVM requirements for the QPSK OFDM defined in the standard but is not able to satisfy requirements for the 16 QAM OFDM signal. The reason being quite obvious: Cascading leads to a better out-of-band attenuation but it tends to increase oscillations in the band and degrades the global EVM performance. The algorithm presented here is a posteriori able to fullfill the entire standard with a cascade structure style. The initial idea was to automate the search of all possible cascade of several (upsampled) 5GHz FIR filters with other 10GHz ones. The principle herein explored is that a global maximally flat transfer function might be achieved by cascading FIR subfilters with limited orders whose oscillations in the band might compensate each other and vanish. Fig. 4 illustrates this idea. Two FIR filter responses are shown: FIR1 transfert function tends to attenuate energy out of bandwidth whereas FIR2 response brings high ripple in the neighborhood of cutting frequency. Separately these two filters are clearly unfit, but together, the frequency response fullfills all requirements.

A. Powers-of-two Coefficients The binary representation of data is such that multiplication by factors that are powers of two are obtained as bit shifts and thus are particularly more simple and hence faster to compute than other multiplications. Although powers-of-two (Po2) is a very coarse grained subset of the real line, our first attempt was based on the hope that high order FIR filters could compensate that poor quatization. Moreover, were only allowed: •

negative Po2 (right bit shifts only) to avoid overflows

•

and limited negative Po2 to avoid underflows.

Finally each coefficient had to be choosen among the following – further on referred to as – “dynamic set” Dd:

For example D3 = {-1, -0.5, -0.25, 0, 0.25, 0.5, 1}. If NC coefficients are to be tuned, then the search space a priori includes (2d+1)Nc possible Dd filters (i.e. filters whose coefficients are within Dd). B. Enumeration Strategy Avoiding Redundancy Since FIR filters that have proportional coefficients might be considered as equivalent “up to a constant gain” (as power amplification is), only one representent of each (equivalence-) class was kept in order to minimize the computing time. Classically, we impose the biggest coefficient to be equal to 1. With this optimization, the search space for NC coefficients is now:

Fig. 4: Idea of ripple compensation with the cascade of different FIR frequency responses

Beyond this naive motivation lies another belief: •

On the one hand, powers-of-two (Po2) coefficients represent a very coarse grained subset of the real line.

A generic (whatever NC) brute force procedure has been implemented to enumerate all possible non-proportional filters whose Po2 coefficients are within the previously described dynamic set.

Thanks to the above optimization, all D7-filters with orders between 1 and 10 could be generated in less than 23 seconds on Matlab. C. Fitness evaluation In order to discriminate good filters from unfit ones, two fitness evaluation functions were introduced (see [5] and Fig. 5). Briefly we wanted to minimize both •

F1: oscillations in the pass band: evaluated as the standard deviation of the dB gain,

•

and F2: energy transmitted outside the band estimated on base of the area between the targeted mask and the filter under study.

Since these two parameters have distinct impacts on the performances of the filter it would have been difficult (at least tricky) to balance the one against the other. Therefore these two criteria have been kept to build up a fitness vector (F1, F2) leading to multicriterion optimization [6]. This evaluation stage was already optimized and took Matlab 25 minutes to evaluate the whole set of D7 -filters with orders ranging from 1 to 10. Fig. 5 presents an example of one filter evaluation.

Cascading filters results in convolving the corresponding coefficients and leads to global filters made of sums of Po2. In this simple way, we partially benefit from sums of Po2 in a very comprehensive manner. E. Assembling Powers-of-two Filters A pool of Po2 filters was therefore constituted according to rules described in the next section and an algorithm was implemented in charge of randomly exploring all possible assemblies and returns the 300 best solutions. The number of cascaded filters: Nf (although tunable) was here fixed at 5. F. Constitution of the Pool As previously mentioned, both 10GHz and upsampled 5GHz D7-filters were used, with orders in the range 1...10. These filters were (brute-force-)generated but only a subset of them, fullfilling the following conditions, were kept in order to reduce the search space size: •

standard deviation in the band is less than 1dB,

•

filters are not dominated (in the sense of the partial order defined by F1 and F2) by more than 100 other filters, favoring most promising solutions.

(This selection process took 10 minutes.) III.RESULTS The random search algorithm developed in Matlab ran for 9 hours on a single core 1.66GHz Intel processor. A number of Nf = 5 filters, chosen among the pool, were to be cascaded. Fig. 6 displays the fitness scores (F1, F2) of the cascaded filter found by the algorithm (circles). Small crosses show scores of single Po2 filters from the pool (“+” for 5GHz filters and “×” for 10GHz). It can be seen that the heuristic, although very simple, was able to deeply improve the quality of the filters with regards to those of single Po2 filters.

Fig. 5: Evaluation process showing F1 and F2 D. Sums of powers-of-two In a previous work [5], we investigated high order filters (up to 20) that were supposed to compensate the coarse grained sampling of Po2 coefficients. In this article we explore lower order filters that are to be cascaded (convoluted). If the whole set of sums of Po2 coefficients was to be browsed, each coefficient would have to be choosen in the new dynamic set:

then the search space to investigate would reach the huge size of (2*2^d)NC...

Fig. 6: (F1, F2) view of the pool of Po2 filters (crosses) and of optimal cascaded filters (circles)

Solutions from the Pareto front are detailed in the following table whereas four of them are plotted on Fig. 7. For each identifier, the number of coefficients before (5GHz) and after (10GHz) the upsampler is given and the two fitness scores are indicated.

particular, the global filter can be a combination of several cascaded filters even working at different sampling frequencies. It has been successfully applied to the high sampling rate interpolation filters in a digital transmitter architecture for 60GHz high data rate wireless communications. REFERENCES [1] [2]

[3] [4] [5]

IV.CONCLUSION This paper presented a methodology able to select powersof-two filters from a large set of possible solutions. In

[6]

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Fig. 7: Four examples of cascaded filters picked up from the Pareto front of the solutions found by the algorithm