Low complexity multi-rate IF sampling receivers using CIC ... - CiteSeerX

Low complexity multi-rate IF sampling receivers using CIC filters and polynomial interpolation Henk Wymeersch and Marc Moeneclaey Digital Communications Research Group Dept. of Telecommunications and Information Processing Ghent University, Sint-Pietersnieuwstraat 41, 9000 GENT, BELGIUM E-mail: {hwymeers,mm}@telin.ugent.be

Abstract This contribution deals with a fully digital multirate radio receiver with IF sampling. Timing correction and sample rate conversion are performed by a polynomial interpolator. By performing interpolation prior to matched filtering a significant reduction in complexity may be achieved. Low degradations are attainable when the receiver design parameters are carefully chosen. If this is not possible, digital anti-aliasing filters are required. We investigate the computational complexity and BER performance when efficient cascaded integrator-comb (CIC) filters are employed to perform anti-aliasing. We show how some of the side-effects of CIC filters may be overcome and how CIC parameters may be selected to provide acceptable BER degradations for all symbol rates.

1

Introduction

In digital packet-based multi-rate bandpass communication, the symbol rate can vary from one packet to the next. Since it is impractical to let the sampling clock frequency depend on the symbol rate, we assume a fixed-rate sampling clock. Consequently, the signal, sampled at a rate 1 Ts , must be resampled a rate which is a fixed multiple (N) of the variable symbol rate 1 T . This operation is known as sample rate conversion (SRC). As packets may arrive with unknown inter-arrival times, timing synchronization needs to be performed. Both sample rate conversion

and timing correction can be performed by a polynomial interpolator (IP) [1]. To reduce receiver complexity, matched filtering (MF) is performed after the interpolator: this allows us to keep the matched filter taps independent of the symbol rate, thus significantly reducing the overall complexity of the receiver. Furthermore, to eliminate the need for identical analog in-phase and quadrature branches, the IF signal (the signal after RF to IF conversion) is sampled directly. When sampling an IF signal, high-frequency (HF) components in the signal may cause aliasing [2] at the output of the interpolator. Although interpolators have good anti-imaging properties, they are poor anti-aliasing filters [3]. To overcome this, we may either limit the design space so that degradation due to aliasing is minimized [4] or filter the signal by a digital anti-aliasing filter (AAF) prior to interpolation. Here we investigate the second solution by employing an efficient AAF which is independent of the symbol rate.

In this contribution we consider decimating cascaded integrator-comb (CIC) filters [5] to perform anti-aliasing. We evaluate these filters in terms of bit error rate (BER) performance and computational complexity. Several side-effects of CIC filters are discussed and solutions suggested. We show that by careful selection of the CIC filter parameters low BER degradations (below 0
1 dB at a BER of 10  3 ) are achievable for all considered symbol rates.

#  "('8+-:9

#  &"('*),+synchronizer

 

   

analog AAF

.  .,/10325476

ADC

fixed

 

N

MF interpolator

   ! "$# $ %

Figure 1: multi-rate receiver with IF-sampling. ; aˆ n < are the detected soft symbols

2

System description

S:TQSVU WYX1Z\[ MON P

We consider multi-rate burst transmission where the symbol rate during a burst is constant, but can change from one burst to the next. We denote the signal after RF to IF downconversion by r = t >@? ℜ A sT = t B τ > e j2π fIF t e jθ CED n = t >

=

L

I (A)

I

I

MQNQR

L I

C(R)

C(R)

L

=

C(R)

(B)

I

I

C(R)

R

L I

R

C(1)

C(1)

C(1)

(1)

where n = t > is real additive gaussian noise (AWGN) with power spectral density equal to N0 2 and sT = t > is the transmitted linearly modulated signal with corresponding symbol rate 1 T . The delay (τ) can vary from packet to packet but is assumed constant within a packet. The symbol interval, T , can take on values from an interval = Tmin F Tmax > . Consequently, the bandwidth B of the transmit pulse is in a corresponding interval = Bmin F Bmax > . As we are not dealing with estimation matters, the propagation delay (τ), the carrier phase (θ) and the IF ( f IF ) are assumed to be known at the receiver. After filtering r = t > by a fixed analog AAF, this signal is converted to the digital domain by the (ideal) analog-to-digital converter (ADC) as illustrated in the block diagram in Fig. 1. The resulting samples at rate 1 Ts are converted to baseband by a simple complex rotation. The signal after down-conversion consists of a baseband (BB) part and a high-frequency (HF) part. The latter is located in the frequency domain at periodic extensions of f ?GB 2 f IF , with period 1 Ts . Denoting rem = x > as the fractional part of x, i.e., rem = x >H? x B f loor = x > , we define r ? rem = 2 fIF Ts > . Hence, the shifted HF components are centered at f ?I= k B r > Ts , k JLK . The resulting samples are then processed by a CIC filter of order L with decimation factor R, a polynomial interpolator and the matched filter. After decimation the soft symbols (aˆn ) are obtained. Only the interpolator

Figure 2: description of CIC filters changes according to the symbol rate.

2.1 CIC filters and interpolation Polynomial interpolators are time-varying filters which are able to correct arbitrary delays and perform arbitrary sample rate conversion. Moreover, they can be implemented in an efficient Farrow structure [6]. Decimating CIC filters are low-complexity decimation filters with good anti-aliasing properties. They are generally used in form (B) in Fig. 2: a CIC decimation filter of order L with decimation factor R (R F L J^] ) consists of L integrators followed by an order R decimator and L first order differentiators. A mathematically equivalent (but computationally more demanding) form is shown as (A) in Fig. 2. Unfortunately, in combination with an interpolator, the use of CIC filters has several sideeffects: the interpolator has to work at a lower rate (1 = RTs > instead of 1 Ts ), reducing its performance. Secondly, CIC filters suffer from a severe passband ’droop’, especially for high symbol rates. The latter problem will be tackled in the next section. Solutions to the former problem were discussed in [7] in the context of software radio receivers: employing a polyphase filter structure allowed the interpolator to

3

0

linear interpolator CIC(4,2) overall filter

−5

As a reference we consider the optimum receiver configuration, consisting of the following stages: analog IF to baseband conversion, analog matched filtering followed by synchronous sampling at the symbol rate. We define the BER degradation of the proposed receiver (from Fig. 1) with respect to the reference receiver as

−10

magnitude [dB]

Receiver design

−15

−20

−25

BER deg [dB] ? 10 log10 −30 −3

−2

−1

0

f.Ts

1

2

3

Figure 3: magnitude frequency response for CIC = 4 F 2 > filter

operate at the higher rate. This is similar to [8] where a set of R parallel CIC filters were placed before the interpolator. An extra control structure forwarded the correct samples to the interpolator. Here we propose a simpler solution: by dropping the decimator in (A), the resulting filter, which we denote by CIC = R F L > , has equal input and output rates. It now serves solely as an anti-aliasing filter and not as a decimator.

c

Eb N0 = Eb N0 > re f d

(3)

where Eb N0 is the SNR needed to reach the same BER as in the reference receiver which operates at a SNR of = Eb N0 > re f . The BER degradation (3) is computed according to the method outlined in [9]. We further maximize this BER degradation over all considered symbol rates, to obtain the worst case situation. The resulting maximum degradation is the performance measure of the proposed receiver.

3.1 Computational complexity

The design parameters ; rF Ts F R F L F N < each have a different impact on receiver complexity: the length of the matched filter after the interpolator is proportional to N, the operating speed of the ADC and The frequency magnitude response of a CIC = R F L > the CIC filter before the interpolator increase with is given by (up to an irrelevant constant): 1 Ts , while r has essentially no impact on receiver complexity. For practical reasons it is preferred to keep fIF as low as possible. R has no impact on the L sin = π f RTs > computational complexity, though higher R results in _
H = f >@?`_ (2) _ R sin = π f Ts > _ longer delays. Computation time of the CIC filter is _ _ _ _ proportional to L.

As can be seen in Fig. 3, a CIC = R F L > has nulls for f Ts ? kR, for k JaK , k b? 0 mod R and is flat around f Ts ? k. It can easily be shown that while increasing the number of stages in the CIC filter (i.e., the order, L) improves the alias rejection, it also increases the passband droop. When we combine with the frequency response of a polynomial interpolator, we have an interpolating filter that has very attractive anti-aliasing properties, especially if the HF signal components happen to be centered around zeros of the CIC filter: rTs ? kR, k b? 0 mod R.

3.2 Reducing aliasing As fIF has no impact on the complexity of the receiver, we assume it fixed to f IF ?e= 1 D 2k > Ts , k J K , or equivalently r ? 1 2. Hence, HF components are centered at f ? 2 Ts D k Ts . When R is even, zeros of the CIC filter frequency response coincide with the central frequencies of the HF components, resulting in maximum anti-aliasing. If we keep the remaining design parameters fixed, increasing R or L will reduce aliasing of HF components but at the

0

0

10

−1

BER degradation [dB]

BER degradation [dB]

10

10

−2

10

IF − L = 1 IF − L = 2 IF − L = 3 BB − L = 1 BB − L = 2 BB − L = 3

IF BB

−3

10

−1

10

−2

1

2

3

4

R

5

6

7

10

8

0

0.1

0.2

0.3

0.4

0.5

r

0.6

0.7

0.8

0.9

1

Figure 4: BER degradation for BB and IF sampling Figure 5: sensitivity of the BER degradation to the for a CIC = R F L > filter when r ? 0
5 IF for a CIC = 2 F 2 > filter same time will reduce the useful signal component. These counteracting effects will result in an optimum (minimum) of the BER degradation. It is instructive to also examine the effect of the CIC filters on the BB component separately, neglecting the HF components. The corresponding BER degradation will serve as a lower bound.

Fig. 5 shows for that optimal CIC filter the sensitivity to r (and thus to the IF). For a fairly wide range of r around 1 2 the degradation will be low. Notice also the local minima around r ? 0
15 and r ? 0
85. According to [4] this is because those HF components with the most energy happen to fall outside the bandwidth of the matched filter for all T .

4

5

Performance results

We have determined the degradation for a linear interpolator at a reference BER of 10  3 , assuming an uncoded system with BPSK signalling with Tmax Tmin ? 2, Bmax ? 0
75 Tmin and Tmin Ts ? 11
1 for N ? 6 and r ? 1 2. The analog AAF is modeled as a 4-th order analog Butterworth filter with a 3 dB cut-off frequency at f ? 0
86 Tmin . Fig. 4 shows the BER degradation maximized over all considered symbol rates as a function of R for L ? 1 F 2 F 3. Also included is the corresponding BER degradation when only the BB component is considered. From the latter curves we conclude that R must be below 5, 4 and 3 for L ? 1, 2 and 3, respectively, if we wish to keep the degradation below an acceptable level, say 0
1 dB. As expected, for fixed L the IF curves exhibit local minima for even R. For R ? 1 (i.e., no CIC filter present) degradations are unacceptably high. From Fig. 4 we see that an optimal choice of CIC filter is CIC = 2 F 2 > .

Conclusion and remarks

This contribution dealt with the design of a multirate receiver with IF-sampling. A low complexity receiver was proposed whereby sample rate conversion and timing correction is performed by a polynomial interpolator followed by a fixed matched filter (i.e., independent of the symbol rate). This generally leads to a significant BER degradation due to aliasing of high-frequency signal components in the matched filter bandwidth. By employing efficient digital anti-aliasing filters based on CIC filters, acceptable degradations (below 0.1 dB for a BER of 10  3 ) are achievable for all considered symbol rates.

Acknowledgement This work has been supported by the Interuniversity Attraction Poles Program - Belgian State - Federal Office for Scientific, Technical and Cultural Affairs.

References [1] F. Gardner. "Interpolation in digital modems - Part I: Fundamentals". IEEE Trans. Comm., 41(3):501–507, 1993. [2] A. M. Guidi and L. P. Sabel. "Digital demodulator architectures for bandpass sampling receivers". Proceedings of the 7th International Tyrrhenian Workshop on Digital Communications, pages 183–194, 1995. [3] T. Hentschel, M. Henker and G.P. Fetweiss. "The digital front-end of software radio terminals". IEEE Personal Communications, 6(4):40– 46, August 1999. [4] H. Wymeersch and M. Moeneclaey. "BER performance of software radio multirate receivers with nonsynchronized IF sampling and digital timing correction". In Proc. ICASSP’03, Hong Kong, April 2003. [5] E.B. Hogenauer. "An economical class of digital filters for decimation and interpolation". IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-29(2):155–162, April 1981. [6] C.W. Farrow. "A Continuously Variable Digital Delay Element". In Proc. IEEE International Symposium on Circuits and Systems, pages 2641–2645, Espoo, Finland, June 1988. [7] L. Lundheim and T.A. Ramstad. "An efficient and flexible structure for decimation and sample rate adaptation in Software Radio receivers". In Proc. ACTS Mobile Comm. Summit, pages 663– 668, June 1999. [8] D. Babic, J. Vesma and M. Renfors. "Decimation by irrational factor using CIC filters and linear interpolation". In Proc. ICASSP’01, Utah, May 2001. [9] K. Bucket and M. Moeneclaey. "The Effect of Interpolation on the BER Performance of Narrowband BPSK and (O)QPSK on Rician Fading Channels". IEEE Trans. Comm., 42(11):2929– 2933, 1994.