Multirate Signal Processing

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1 Chapter 5 Multirate Signal Processing In a software defined radio, one often has to deal with sampled wideband signals that contain a multitude of different user signals. Part of the receiver s task is thus to extract the desired user signal(s) from such a wideband signal. In such cases, the receiver may consist of multiple stages of down conversion and filtering. After each down conversion and filtering stage, the output signal has a lower Nyquist frequency than the input. Thus, it may be decimated before any further processing. As a result, one has to deal with signal processing scenarios where signals with multiple rates are to be dealt with. This chapter discusses the signal processing issues that are pertinent to such scenarios. The most basic blocks in multirate signal processing are M-fold decimator and L-fold expander. These along with their impact on the resulting signal spectra are introduced in Section 5.1. Section 5.2 is devoted to rate-conversion. Through interconnection of decimators/expanders and filters, we develop multirate signal processing structures that allow digital resampling of sampled signals to virtually any arbitrary rate. The rules governing the interconnection of signal processing blocks in multirate systems are reviewed in Section 5.3. Building on the results of Section 5.3, efficient structures for realization of multirate systems, known as polyphase filters, are introduced in Section 5.4. Multistage implementation, a powerful multirate technique for minimization of the complexity of multirate systems, is reviewed in Section 5.6. Cascaded integrator-comb (CIC) filters, a class of very low-complexity (multiplier-free) filters for interpolation and decimation, are introduced in Section 5.7. Some example applications of multirate signal processing techniques are presented in Section

2 100 Multirate Signal Processing Chap M-fold Decimator and L-fold Expander M-fold decimator The M-fold decimator takes an input sequence and produces the output sequence x d [n] =x[mn] (5.1) where M is an integer. In other words, the M-fold decimator only keeps one out of every M elements of and ignores the rest. Figure 5.1 shows the building block that is commonly used for presentation of the M-fold decimator. A demonstration of decimation for M = 2 is presented in Figure 5.2 M x d [n] Figure 5.1: M-fold decimator n x d [n] n Figure 5.2: A demonstration of decimation for M =2. Transform domain analysis The z-transform of the decimated sequence x d [n] is obtained as X d (z) = = x d [n]z n n= n= x[mn]z n. (5.2) Let {, n = multiples of M x 1 [n] = 0, otherwise (5.3)

3 Sec. 5.1 M-fold Decimator and L-fold Expander 101 and note that since x d [n] =x[mn]=x 1 [Mn], from (5.2) we obtain X d (z) = x 1 [Mn]z n = = n= n= n= x 1 [n]z n/m x 1 [n] (z ) 1/M n = X 1 (z 1/M ) (5.4) where the second equality follows because x 1 [n] is zero unless n is a multiple of M. Next, we find the relationship between X d (z) and X(z). For this, we note that x 1 [n] =C M [n] (5.5) where C M [n] is the comb sequence defined as { 1, n = multiples of M C M [n] = (5.6) 0, otherwise. Also, we note that M 1 C M [n] = 1 e j2πkn/m. (5.7) M k=0 Substituting (5.7) in (5.5) and the result in X 1 (z) = n= x 1[n]z n,we obtain X 1 (z) = 1 M 1 e j2πkn/m z n M = 1 M = 1 M k=0 n= M 1 k=0 n= M 1 k=0 Substituting this result in (5.4), we get X d (z) = 1 M X M 1 k=0 If z is replaced by e j2πf in (5.9), we obtain X d (e j2πf )= 1 M (ze j2πk/m) n ( ze j2πk/m). (5.8) ( X z 1/M e j2πk/m). (5.9) M 1 k=0 X(e j2π(f k)/m ). (5.10)

4 102 Multirate Signal Processing Chap. 5 Figure 5.3(b) gives a graphical presentation of this result, for the case where the decimation factor M = 2 and X(e j2πf ) is as shown in Figure 5.3(a). We note that M-fold decimation in the time domain changes the signal spectrum in two ways. (i) The spectrum gets expanded M times across the frequency axis. (ii) The magnitude of the spectrum decreases M times. X (e j2 πf ) A 2 1 (a) 1 X d (e j2 πf )= 1 2 X(ej2πf/2 ) + 1 }{{} 2 X(ej2π(f 1)/2 ) }{{} A/2 2 f 2 1 (b) 1 2 f Figure 5.3: Graphical presentation of the decimation effect in the frequency domain in a two-fold decimator L-fold expander The L-fold expander takes the input sequence and produces the output sequence { x[n/l], n = multiples of L x e [n] = (5.11) 0, otherwise where L is an integer. In other words, to generate the sequence x e [n], the L-fold expander inserts L 1 zeros after each element of. Figure 5.4 shows the building block that is commonly used for presentation of the L-fold expander. A demonstration of expander for L = 2 is presented in Figure 5.5. L x e [n] Figure 5.4: L-fold expander.

5 Sec. 5.1 M-fold Decimator and L-fold Expander n x e [n] n Figure 5.5: A demonstration of expander for L =2. Transform domain analysis The z-transform of the expanded sequence x e [n] is obtained as X e (z) = x e [n]z n = = n= n= n= Also, substituting z by e j2πf, we obtain z Ln ( z L) n = X ( z L). (5.12) X e (e j2πf )=X(e j2πlf ). (5.13) This shows that an L-fold expansion of a sequence in the time domain results in an L-fold compression of the spectrum along the frequency axis. This is demonstrated graphically in Figure 5.6. Note that the compression along the frequency axis results in L-fold repetition of the spectrum over the range 0 f 1. As we shall see later, selection of one of the repetitions and deletion of the rest using a filter allow signal interpolation and modulation in a very efficient way; see Section An alternative interpretation of the above results may prove instructive. Consider the sampled signal x s (t) = δ(t nt s ) (5.14) and its expanded version x se (t) = n= n= x e [n]δ(t nt s /L). (5.15)

6 104 Multirate Signal Processing Chap. 5 X (e j2 πf ) A 2 1 (a) 1 2 f X e (e j2 πf )=X (e j4 πf ) A (b) 2 f Figure 5.6: Graphical presentation of the expansion effect in the frequency domain for a two-fold expander. Because of (5.11), x s (t) and x se (t) are effectively the same signals and, hence, they have the same spectrum. However, since the sampling rate in x se (t) isl times larger than the sampling rate in x s (t), the absolute frequency corresponding to the normalized frequency f =1inX e (e j2πf ) is L times larger than its counterpart in X(e j2πf ) and, accordingly, within each cycle 0 f 1ofX e (e j2πf ), L repetitions of X(e j2πf ) are observed The nature of decimator and expander blocks At this point, it is worth noting that although decimator and expander are linear systems, they belong to the class of time-variant systems. Hence, they cannot be classified as LTI systems and thus the rules of LTI systems are not applicable to multirate systems that inevitably contain decimators and/or expanders. In particular, the commutative rules that are frequently used in manipulation of LTI systems are often inapplicable to multirate systems. 5.2 Rate Conversion Consider a continuous-time signal x(t) and its sampled sequence = x(nt s ). Given, we wish to obtain the sampled sequence x [n] =x(nt s)

7 Sec. 5.2 Rate Conversion 105 of the same continuous-time signal x(t). A naive approach would be to reconstruct the continuous-time signal x(t) by passing the sampled signal x s (t) = n= δ(t nt s ) through a lowpass filter as discussed in Section 4.1.1, and resample the reconstructed signal at the desired rate f s =1/T s. This solution is not attractive because it involves the additional circuitry for digital-to-analog conversion, filtering, and analog-to-digital conversion. In this section, we discuss how multirate signal processing can be used to perform resampling in the digital domain, i.e., without resorting to the use of digital-to-analog and analog-to-digital converters L-fold interpolation L-fold interpolation refers to the case where one wishes to reconstruct a sampled signal at a rate L-times greater than the current rate. In other words, L 1 new samples have to be found and appended after each sample. This can be easily done by passing through the cascade of an L-fold expander and a lowpass filter, as shown in Figure 5.7. L x e [n] Lowpass filter x [n] Figure 5.7: L-fold interpolator. The reason why the setup of Figure 5.7 works can best be explained in the frequency domain by presenting the spectra of the signal sequences, x e [n] and x [n]. These are presented in Figures 5.8(a), (b) and (c), respectively. To emphasize the sampling rate of each signal, the frequency axes are scaled to the actual frequencies in Hertz (and not the normalized frequencies). The figures are self-explanatory and clearly indicate how the cascade of the L-fold expander and the lowpass filter increases the sampling rate L-fold without changing the spectrum of the desired signal. Clearly, the lowpass filter should have a passband that covers the desired signal bandwidth B, and it should have a stopband that begins from the point where the first replica of the signal spectrum begins. A particular and attractive choice of the lowpass filter here is a Nyquist (M) filter with M = L. This is attractive because it assures that the equality x [Ln] = holds exactly, i.e., the original samples will appear at the interpolator output.

8 106 Multirate Signal Processing Chap. 5 X (e j2 πf /fs ) f s B (a) B f s f X e (e j2 πf /Lfs ) Lf s f s B B (b) f s Lf s f Picked by the lowpass filter X (e j2 πf /Lfs ) Deleted by the lowpass filter Lf s f s B B (c) f s Lf s f Figure 5.8: Spectra of the signal sequences, x e [n] and x [n] of Figure 5.7.

9 Sec. 5.2 Rate Conversion M-fold decimation In the context of this section M-fold decimation means reducing the sampling rate of a signal sequence M times. When is bandlimited and decimation does not result in aliasing, M-fold decimation can be established by choosing one out of every M elements of and ignoring the rest. In other words, the resampled signal sequence x [n] is given by x [n] =x[mn]. (5.16) When the bandwidth of is not small enough and direct downsampling causes aliasing, prior to down-sampling, should be passed through a lowpass filter with sufficiently small bandwidth to avoid aliasing, as in Figure 5.9. Clearly, in such cases, the process of resampling results in a loss of part of the spectrum of and thus cannot be recovered from x [n]. Lowpass filter M x [n] Figure 5.9: M-fold decimator. The lowpass filter is to avoid aliasing after down-sampling L/M-fold rate change With the background developed above, we are now ready to discuss the general case of resampling, when T s T s = M L. This can be readily done by first interpolating L-fold, and then decimating the result M-fold, as shown in Figure Here, the lowpass filter has two roles: (i) to cancel the repetitions of the spectral copies of X(e j2πf )asinthel-fold interpolator; (ii) to reject any portion of the spectrum X(e j2πf ) that may result in aliasing after down-sampling. The latter may be required only when M>L. L Lowpass filter M x [n] Figure 5.10: L/M-fold resampling.

10 108 Multirate Signal Processing Chap. 5 Example 5.1: The samples of x(t) = sin(2πt) + 2 cos(4πt) at the rate of 6 samples per second are available. Develop a MATLAB program that takes these samples as input and generates samples of x(t) at the rate 9 (= 3 6) as 2 output. Solution: We should follow Figure 5.10 with L = 3 and M = 2. With f s = 6 Hz, the first and second components in x(t) has the normalized frequencies ±1/6 and ±2/6 = ±1/3. After the L = 3 fold expander, these frequencies will divide by L and within the range of normalized frequency 0 to 0.5 three replicas of them will appear. These are shown in Figure The lowpass filter should be chosen such that it only allows the frequency components at ±1/18 and ±1/9 pass through and the rest of the components are rejected. A MATLAB script that contains the following lines generates x in from x(t) at the rate of 6 samples per second as input and does the desired 3/2-fold rate change and deliver x out. It also plots the results shown in Figure 5.12 and 5.13 L=3; M=2; fs in=6; Ts in=1/fs in; t in=[0:ts in:5];x in=sin(2*pi*t in)+2*cos(4*pi*t in); x e=expander(x in,l); t cont=[0:0.01:5]; x cont=sin(2*pi*t cont)+2*cos(4*pi*t cont); h=l*firpm(60,[0 2/(fs in*l) 4/(fs in*l) 0.5]*2,[ ],[1 1]); x ef=filter(h,1,x e); x out=x ef(1:2:end); Ts out=ts in*m/l;t out=[0:length(x out)-1]*ts out; delay=(ts in/l)*(length(h)-1)/2; x delayed=sin(2*pi*(t cont-delay))+2*cos(4*pi*(t cont-delay)); figure;stem(t in,x in, k, filled ); hold on; plot(t cont,x cont, k ); xlabel( t ),ylabel( x in ) figure; stem(t out,x out, k, filled ); hold on plot(t cont,x delayed, k ); axis([ ]); xlabel( t ),ylabel( x out ) Most of this script is self-explanatory. The reader should also take note of the following points: The vector t cont defines a dense grid of sampling times that is used to approximate the continuous time. In line 5, h is a lowpass filter with proper band edges to select the first two spectral lines of Figure To recover the signal samples at the filter output at the correct level the passband gain of h is set equal to L = 3. Problem 2 guides you to develop an understanding of this point. The filter h introduces a delay equivalent of N/2 samples, where N is the FIR filter order. In matching the interpolated samples to x(t) this delay has been accounted for.

11 Sec. 5.3 Rate Conversion X e (e j2 f ) f Figure 5.11: Fourier transform of the input signal after expansion x in t Figure 5.12: Input signal samples.

12 110 Multirate Signal Processing Chap x out t Figure 5.13: Output signal samples. 5.3 Commutative Rules In classical signal processing, where linear time-invariant (LTI) building blocks are connected together to make a system, the commutative rules are straightforward. Any set of blocks along each branch can be commuted arbitrarily. In multirate signal processing, the presence of decimators and expanders makes this rule not always applicable and thus multirate systems should be treated more carefully. An important case in multirate systems where commutative rules are not always applicable is when decimators and expanders are cascaded as in Figure Considering the case where L = M, one easily finds that {, n = multiple of M y 1 [n] = 0, otherwise but y 2 [n] =, for all n. This shows that, when L = M, the M-fold decimator and L-fold expander blocks are not commutable. On the other hand, if we consider the case where M = 3 and L = 2, it is not difficult to find that both y 1 [n] and y 2 [n], for n =0,1,2,, contain the elements {x[0], 0,x[3], 0,x[6], } which means the commutative rule is applicable in this case. The general rule here is that the M-fold decimator and L-fold expander can be commutated only when M and L are prime with respect to each other. This can

13 Sec. 5.3 Commutative Rules 111 be proved by noting that and Y 1 (z) = 1 M Y 2 (z) = 1 M M 1 k=0 M 1 k=0 X (z L/M e j2πk/m) ( ) X z L/M e j2πkl/m, and the sets {e j2πk/m, k = 0, 1,,M 1} and {e j2πkl/m, k = 0, 1,,M 1} are the same when and only when L and M are prime with respect to each others. M L y 1 [n] (a) L M y 2 [n] (b) Figure 5.14: Cascading decimator and expander blocks. Here, the commutative rule is not applicable, in general. Figures 5.15(a) and (b) show how a unit delay z 1 has to be modified when it is moved from the output of a decimator to its input, and how it should be modified when it is moved from the input of an expander to its output, respectively. These two rules can be confirmed easily as follows. In Figure 5.15(a), y 1 [n] =p[n 1] and p[n] =x[mn]. Hence, y 1 [n] = x[mn M]. On the other hand, q[n] =x[n M] and y 2 [n] =q[mn]. This implies that y 2 [n] =x[mn M] =y 1 [n]. In Figure 5.15(b), p[n] =x[n 1] and { p[n/l], n = multiples of L y 1 [n] = 0, otherwise { x[n/l 1], n = multiples of L = 0, otherwise { x[(n L)/L], n = multiples of L = 0, otherwise On the other hand, q[n] = { x[n/l], n = multiples of L 0, otherwise

14 112 Multirate Signal Processing Chap. 5 and y 2 [n] =q[n L] = { x[(n L)/L], n = multiples of L 0, otherwise = y 1 [n]. p[n] y 1 [n] q[n] y 2 [n] M z 1 z M M (a) p[n] y 1 [n] q[n] y 2 [n] z 1 L L (b) z L Figure 5.15: Modification of the unit delay as it cascades and commutates with decimator and expander blocks. The multirate identities presented in Figures 5.15(a) and (b) can be generalized as in Figures 5.16(a) and (b), where G(z) can be any rational (i.e., a ratio of polynomials of z or z 1 ) transfer function. These which are known as the noble identities, prove very useful in efficient realization of multirate signal processing systems. In particular, we find these very useful in development of the polyphase structures that are presented in the next section. The noble identities can be easily proved by expressing y 1 [n] and y 2 [n] in the z-domain. In Figure 5.16(a), and, thus, P (z) = 1 M Y 1 (z) =P (z)g(z) = 1 M On the other hand, Y 2 (z) = 1 M = 1 M = 1 M M 1 k=0 M 1 k=0 M 1 k=0 M 1 k=0 M 1 k=0 Q (z 1/M e j2πk/m) ( X z 1/M e j2πk/m), ( X z 1/M e j2πk/m) G(z). (5.17) ( X z 1/M e j2πk/m) ( ( G z 1/M e j2πk/m) ) M ( X z 1/M e j2πk/m) G(z) =Y 1 (z). (5.18)

15 Sec. 5.4 Commutative Rules 113 In Figure 5.16(b), Y 1 (z) =P (z L )=G(z L )X(z L ), Q(z) =X(z L ), and, thus, Y 2 (z) =G(z L )Q(z) =G(z L )X(z L )=Y 1 (z). y 1 [n] y 2 [n] p[n] q[n] M G(z) G(z M ) M (a) p[n] y 1 [n] q[n] y 2 [n] G(z) L L G(z L ) (b) Figure 5.16: The noble identities. Finally, the set of commutative rules that are presented in Figure 5.17 are also found useful in the development of multirate systems. The rules presented in Figure 5.17 are for M-fold decimators. The rules are applicable when M-fold decimators are replaced by L-fold expanders. These rules can be verified trivially. a a M M x 1 [n] M x 1 [n] M x 2 [n] x 2 [n] M x 1 [n] x 2 [n] x 1 [n] M M x 2 [n] M Figure 5.17: Trivial commutative rules.

16 114 Multirate Signal Processing Chap The Polyphase Representations The basic idea of polyphase representation can be best explained through a simple example. Consider a filter with the transfer function H(z) = 1+2z 1 +3z 2 +4z 3. By separating the terms with even and odd powers of z, this can be rearranged as H(z) =E 0 (z 2 )+z 1 E 1 (z 2 ) where E 0 (z)=1+3z 1 and E 1 (z) =2+4z 1. Here, E 0 (z) and E 1 (z) are called the zeroth and first polyphase components of H(z). Extending the above idea to a general transfer function H(z) = n= h[n]z n, for an integer M, we can always decompose H(z) as H(z) = which can be written as where n= +z 1. h[nm]z nm n= +z (M 1) H(z) = E k (z) = h[nm +1]z nm n= M 1 k=0 n= h[nm + M 1]z nm (5.19) z k E k (z M ) (5.20) h[nm + k]z n. (5.21) Equation (5.20) is called Type 1 polyphase representation of H(z) (with respect to M) and E k (z) is the respective kth polyphase component. Note that E k (z) vary with M. So, a notation such as E (M) k (z) would have been more logical. This is usually avoided for brevity. A rearrangement of (5.20) leads to where H(z) = M 1 k=0 z (M 1 k) R k (z M ) (5.22) R k (z) =E M 1 k (z). (5.23) This is called Type 2 polyphase representation of H(z) (with respect to M) and R k (z) is the respective kth polyphase component.

17 Sec. 5.5 Efficient Structures for Decimation and Interpolation 115 It is important to note that the polyphase representations are applicable to both FIR and IIR filters. Polyphase representations of FIR filters follow the above equations. Using partial fraction method, an IIR filter, in general, can be expanded to a summation of the terms of the form H l (z) =1/(1 α l z 1 ) (and possibly a FIR term). The polyphase components of each term H l (z) can then be obtained by recalling the identity whose application leads to H l (z) = from which we obtain 1 1+x + + xm 1 = 1 x 1 x M 1 1 α M l z M + z 1 α l 1 αl M z + + α M 1 M z (M 1) l E l,k (z) = 1 α M l z M (5.24) αl k 1 αl M. (5.25) z Efficient Structures for Decimation and Interpolation Filters The interpolation filter (the lowpass filter) in Figure 5.7 has an input in which only one out of every L samples is non-zero. On the other hand, in Figure 5.9 only one out of every M output samples of the decimation filter need to be calculated. These special features of the interpolation and decimation filters can be best captured in computationally efficient structures by using the polyphase representations that were discussed in the previous section Polyphase structure for decimator filters Consider the M-fold decimator shown in Figure 5.18(a). Substituting H(z) by its Type 1 polyphase representation (5.20), we obtain the alternative realization shown in Figure 5.18(b). By shifting the M-fold decimator to the M parallel branches in Figure 5.18(b) and invoking the first noble identity (i.e., the one in Figure 5.16(a)), we obtain the desired polyphase structure shown in Figure 5.18(c). Compared to the direct structure of Figure 5.18(a), the polyphase structure of Figure 5.18(c) offers a number of advantages. These advantages become very eminent when the decimator is to be implemented in hardware (either on an FPGA or a VLSI ASIC chip). To highlight these advantages, a detailed implementation of Figure 5.18(a) is presented in Figure Here, a total of N multiplications and N 1 additions have to be performed for

18 116 Multirate Signal Processing Chap. 5 N 1 H (z) = h[n]z n n=0 M (a) E 0 (z M ) M E 0 (z) z 1 z 1 E 1 (z M ) M M E 1 (z) z 1 z 1 E M 1 (z M ) M E M 1 (z) (b) (c) Figure 5.18: Polyphase implementation of an M-fold Decimator. (a) Direct implementation. (b) Intermediate structure. (c) The desired polyphase structure.

19 Sec. 5.5 Efficient Structures for Decimation and Interpolation 117 computation of each output sample of H(z). Moreover, even though only one out of every M output samples of the filter is needed (because of the M-fold decimator), the N multiplications and N 1 additions have to be performed before arrival of the next sample of the input, as the content of the delay line (the shift register) holding and its delayed samples changes. In other words, although the output is needed at a rate M times slower than the rate of, the filter H(z) has to catch-up with the rate of input for computation of each sample of, while it remains idle for the next M 1 sample intervals of input before starting computation of the next sample. This clearly is a waste of resources. The polyphase components in Figure 5.18(c), on the other hand, operate at the output rate (i.e., the lowest rate in the system) and, thus, remain operational all the time (no rest periods). In addition, the polyphase structure lends itself to parallel processing in a more natural manner. For example, each polyphase component filter may be handled separately through a separate multiply/accumulate processor. z 1 z 1 z 1 h[0 ] h[1 ] h[n 1 ] M Figure 5.19: A direct implementation of Figure 5.18(a). We may note that the above advantages of the polyphase structure of Figure 5.18(c) do not hold if it is to be implemented in a software based system, such as a digital signal processor. In fact, if only the output samples that are required after decimation are to be calculated (which is the case in practice), one finds that the number of additions and multiplications per unit of time in polyphase and direct implementation are exactly the same. On the other hand, the polyphase implementation may bring some structural software complexity that makes it less attractive than the direct implementation.

20 118 Multirate Signal Processing Chap Polyphase structure for interpolator filters Consider the L-fold interpolator shown in Figure 5.20(a). Using the Type 2 polyphase representation (5.22), we obtain the alternative realization shown in Figure 5.20(b). By invoking the second noble identity (i.e., the one in Figure 5.16(b)), we obtain the desired polyphase structure shown in Figure 5.20(c). Comparing the polyphase and direct implementation, here, we note that while in the direct form H(z) has to run at the output rate (the higher rate in the system), the polyphase filters operate at the input rate. Moreover, because of the expander, L 1 out of every L input samples to H(z) are zero. Hence, either many unnecessary/trivial multiplications by zero have to be performed or special control circuitry should be used to only operate on the non-zero samples for computation of each output sample. L N 1 H (z) = h[n]z n n=0 (a) L R 0 (z L ) R 0 (z) L z 1 z 1 L R 1 (z L ) R 1 (z) L z 1 z 1 L R L 1 (z L ) R L 1 (z) L (b) (c) Figure 5.20: Polyphase implementation of an L-fold Interpolator. (a) Direct implementation. (b) Intermediate structure. (c) The desired polyphase structure. Unlike the case of the decimator, the polyphase interpolator structure may also prove useful if it is to be implemented on a software based system. The polyphase structure dictates how the filter coefficients and samples of input are to be stored and accessed from the memory for generation of the

21 Sec. 5.5 Efficient Structures for Decimation and Interpolation 119 output samples. More specifically, one may store samples of in part of the memory and coefficients of each polyphase filter in a separate part of the memory. Then, following the structure of Figure 5.21 (explained below), the outputs of the polyphase filters R L 1 (z), R L 2 (z),, and R 0 (z) are calculated, respectively, and sent to the output in the same order Commutator models In the polyphase structure of Figure 5.20(c), the expander and delay blocks may be replaced by a commutator block that takes successive samples of from the outputs of the polyphase filters R 0 (z), R 1 (z),, R L 1 (z). This is because careful examination of Figure 5.20(c) reveals that the output sequence takes the form,y 1 [n 1],y 0 [n 1],y L 1 [n],,y 1 [n],y 0 [n],y L 1 [n +1], where y 0 [n], y 1 [n],, y L 1 [n] are the output sequences from the polyphase filters R 0 (z), R 1 (z),, R L 1 (z), respectively. Figure 5.21 presents a block schematic of the L-fold interpolator with the delay/add blocks replaced by a commutator. The commutator runs over the outputs of the polyphase filters clockwise, starting with the output of R L 1 (z), and begins the cycle once it has picked the output of R 0 (z). R 0 (z) y 0 [n] R 1 (z) y 1 [n] R L 1 (z) y L 1 [n] Figure 5.21: Polyphase structure of an L-fold interpolator with output samples generated through a commutator. The same concept may also applied to the decimator polyphase structure of Figure 5.18(c) to remove the delay line and the M-fold decimators,

22 120 Multirate Signal Processing Chap. 5 and pass the input samples through a commutator to the polyphase filters E 0 (z), E 1 (z),, E M 1 (z). This is presented in Figure E 0 (z) x[n 1 ] E 1 (z) x[n M + 1 ] E M 1 (z) Figure 5.22: Polyphase structure of an M-fold decimator with input samples distributed through a commutator L/M-fold resampling Here, we discuss how the L/M-fold resampler of Figure 5.10 can be realized efficiently. The cases of L/M > 1 and L/M < 1 are treated separately. The case of L/M > 1 Combining the L-fold expander and the lowpass filter of Figure 5.10 in a polyphase structure, as in Figure 5.21, we obtain the structure of Figure 5.23(a). To accommodate for the M-fold decimator, one can simply rotate the commutator arm at the steps of M polyphase outputs. This is shown in Figure 5.23(b). For example, if L = 3 and M = 2, the output samples are picked from y 2 [n], y 0 [n], y 1 [n], y 2 [n], y 0 [n], y 1 [n],. The polyphase realization of Figure 5.23 is efficient and optimized for implementation in software on a digital signal processor. The tapped delay lines of the polyphase filters R 0 (z), R 1 (z),, R L 1 (z) are filled in with the samples of input at the rate that they arrive. There is no redundant (zero) samples stored. In addition, the commutator arrangement, as discussed above, assures that only the essential samples of output are calculated. When a hardware implementation with maximum speed of operation is required, the polyphase realization of Figure 5.23 can be further improved using the trick that is discussed below for the case of L/M < 1 (see Problem 14).

23 Sec. 5.5 Efficient Structures for Decimation and Interpolation 121 R 0 (z) y 0 [n] R 1 (z) y 1 [n] M R L 1 (z) y L 1 [n] (a) R 0 (z) y 0 [n] R 1 (z) y 1 [n] R L 1 (z) y L 1 [n] The commutator arm rotates at steps of M polyphase output samples. (b) Figure 5.23: Polyphase structure of an L/M-fold resampler. (a) Intermediate polyphase structure. (b) The desired polyphase structure.

24 122 Multirate Signal Processing Chap. 5 The case of L/M < 1 When L/M < 1, the structure of Figure 5.23 becomes inefficient (especially, for smaller values of L/M), even when a software implementation is desired. This is because the polyphase filters have to still cope with the input rate which in the present case is higher than the output rate. A structure that allows operation of the system components (polyphase filters) at the output rate clearly leads to a more efficient structure. Here, in analogy with Figure 5.23(a), to have polyphase filters that operate at the output rate, we may suggest the alternative structure of Figure Examination of this structure soon reveals that, because of the L-fold expander, L 1 out of every L input samples to the polyphase filters E 0 (z), E 1 (z),, E M 1 (z) are zero and therefore a large number of unnecessary multiplications by zero have to be performed. To discuss how one may avoid zero samples, without any loss of generality, we limit our discussion to the simple case where L = 2 and M =3. E 0 (z) L E 1 (z) E M 1 (z) Figure 5.24: An alternative polyphase structure of L/M-fold resampler. Figure 5.25(a) presents a redrawn of Figure 5.24, for the case where L = 2 and M = 3 and the commutator is replaced by delay units and 3- fold decimators (compare with Figure 5.18(c)). This figure can be further rearranged as Figure 5.25(b). In obtaining this figure, we have noted that since 2 and 3 are prime with respect to each other, the blocks 2 and 3 in the top branch commute. In the lower branch, we have used the first noble identity and moved z 2 before 2 and then commuted 2 and 3. For the middle block, we have noted that z 1 = z 2 z 3, then used the noble identity 1 to move z 2 before 2 and the noble identity 2 to move z 3 after 3 and then commuted the blocks 2 and 3. Finally, we note that the blocks within the dashed box A can be replaced by a clockwise commutator. Moreover, the blocks within each of B boxes can be efficiently implemented using (2-fold) interpolation polyphase structures. This final

25 Sec. 5.6 Efficient Structures for Decimation and Interpolation 123 step gets rid of zero samples in the tapped delay lines of the polyphase filters. 2 3 E 0 (z) 2 z 1 3 E 1 (z) 2 z 2 3 E 2 (z) (a) A 3 2 B E 0 (z) z 3 2 B z 1 E 1 (z) B z E 2 (z) (b) Figure 5.25: A more efficient polyphase structure of L/M-fold resampler. (a) Intermediate structure. (b) The desired structure The polyphase identity Figure 5.26(a) is a rather strange structure that appears in some multirate systems and thus its study will prove useful. This is similar to the L/Mresampling structure of Figure 5.10 with L = M. Recalling the polyphase structure of Figure 5.24, for L = M, one finds that only the samples that enter E 0 (z) are non-zero. The inputs to the rest of polyphase components are zero for all n. This is because the expander put M 1 zeros after every element of to make a group of M elements. The commutator passes the first element of the group (which is ) to E 0 (z) and the rest of the elements of the group (which are zeros) to E 1 (z) through E M 1 (z). With this observation, one obtains Figure 5.26(b) as an equivalent structure of Figure 5.26(a). This result is known as the polyphase identity.

26 124 Multirate Signal Processing Chap. 5 M H (z) M (a) E 0 (z) (b) Figure 5.26: The polyphase identity: (a) A strange structure. (b) Its equivalent structure. 5.6 Multistage Implementation We may recall from the discussion of the previous sections that in all cases of sampling rate conversion (Figures 5.7, 5.9 and 5.10), we need to implement a lowpass filter. Moreover, we noted that for large values of L and/or M the length of the desired lowpass filter may become excessively large. The polyphase structures were introduced as a method of implementing these filters efficiently. In this section, we introduce another method that can be used in conjunction with polyphase structures to further reduce the complexity of implementation of interpolation and decimation filters Interpolated FIR (IFIR) technique Consider the problem of designing a narrowband lowpass filter H(z) with the desired magnitude response as in Figure 5.27(a). Recalling the Kaiser s and Bellanger s formulae of Chapter 4, (equations (4.88) and (4.89)), because of the very narrow transition band, realization of this filter may require a very high order. Very high order naturally leads to a large number of multiplications and additions for computation of each output sample of the filter. The following trick may be used to reduce the computational load. We begin with designing a filter G(z) whose magnitude response is a stretched version of the desired response, e.g., as in Figure 5.27(b). Note that in this example G(z) has band edges that are quadruple of those of H(z). Next, by replacing each delay in G(z) by four delays, we obtain a filter with the transfer function G(z 4 ). Comparing G(e j2πf ) and G ( (e j2πf ) 4) = G(e j8πf ), one will find that the magnitude response of G(z 4 ) is obtained by compressing G(e j2πf ) across the frequency axis four times. This leads to the response shown in Figure 5.27(c). Cascading G(z 4 ) and a filter I(z) with the magnitude response shown in Figure 5.27(c) results in a filter with the same magnitude response as the desired response. This is shown in Figure 5.27(d). The key point that makes the cascade design G(z 4 )I(z) more efficient than the direct design H(z) lies in the fact that G(z) has a transition band

27 Sec. 5.6 Multistage Implementation 125 H (e j2 πf ) (a) 0.5 f G(e j2 πf ) (b) G(e j8 πf ) I (e j2 πf ) 0.5 f 0.25 (c) 0.5 f G(e j8 πf )I (e j2 πf ) (d) 0.5 f Figure 5.27: The response of a narrowband filter and a method of its realization through IFIR technique.

28 126 Multirate Signal Processing Chap. 5 that is four times wider than the transition band of the desired response. It thus can be realized with four times less coefficients. I(z) also can be realized with relatively small number of coefficients, because of its very relaxed transition band. Accordingly, the computational complexity of the cascade of G(z 4 ) and I(z) may be only slightly more than one quarter of the complexity of H(z). The terminologies It is instructive to think of the cascade of G(z 4 ) and I(z) as an interpolation process. Replacement of G(z) byg(z 4 ), in the time domain, is equivalent of inserting 3 zeros after every sample of the sequence g[n]. This, in the frequency domain, as noted above, is equivalent of compressing the frequency response G(e j2πf ) along the frequency axis four times; see also the discussion in Section I(z), then, acts as an interpolation (lowpass) filter and replaces the zeros of G(z 4 ) by the right values such that G(z 4 )I(z) has a response that resembles the desired response H(z). Accordingly, I(z) is called interpolation filter and the terminology IFIR follows for obvious reasons. The filter G(z) is called model filter. Ripple sizes We may note that the size of ripples in the passband of G(z 4 )I(z) is determined by the ripple size in the passbands of both G(z 4 ) (which is the same as that of G(z)) and I(z). In the worse case, it is given by the summation of the ripple sizes in the two filters. So, if a desired passband ripple of δ 1 is desired, we may simply design G(z) and I(z) for the passband ripple sizes of δ 1 /2. The stopband ripple size of G(z 4 )I(z) on the other hand is determined by the minimum of the stopband ripple sizes of G(z) and I(z). Hence, if a desired stopband ripple δ 2 is required, we will design both G(z) and I(z) for the stopband ripple size of δ 2. The example given below clarifies these points further. Interpolation factor The example presented in Figure 5.27 arbitrarily assumed an interpolation factor M = 4. Obviously, one could try other values of M as well. An important question to ask here is thus what is the optimum interpolation factor M that minimizes the complexity of the interpolated filter. To find an answer to this question, we may note that increasing the interpolation factor increases the width of the transition band of G(z), and thus, G(z M ) could be realized by smaller number of coefficients. On the other hand, increasing M results in more densely populated images in the response of G(z M ) and accordingly one will find that an interpolation filter I(z) with narrower transition band has to be designed. This increases the complexity of realization of I(z). Therefore, M should be chosen to strike a balance

29 Sec. 5.6 Multistage Implementation 127 between the complexity (the number of non-zero coefficients) of G(z M ) and I(z). Probably the easiest method of answering this question is to examine various choices of the interpolation factor and choose the one that results in the minimum complexity. Example 5.2: It is desired to design a lowpass filter with band edges f 1 =0.02 and f 2 = and the passband and stopband ripples δ 1 =0.01 and δ 2 = Using the Kaiser s formula, find the order of the desired filter, if a direct design was used. Examine various choices of the interpolation factor M and from them choose the one that leads to a design with the minimum complexity. Solution: For the direct design, substituting the specified parameters in the Kaiser s formula, we obtain N H = 507. Hence, we will have a total number of coefficients N H + 1 = 508. For the interpolation filter, if we choose M = 2, for G(z), we will get the parameters f 1 =0.02M =0.04, f 2 =0.025M =0.05, δ 1 =0.01/2 = and δ 2 = and accordingly the order N G = 274. Also, for I(z), we will get the parameters f 1 =0.02, f 2 = =0.475, δ 1 =0.01/2 =0.005 and δ 2 =0.001 and accordingly the order N I =6. Hence, we will have a total number of coefficients (N G+1)+(N I +1) = 282. If we choose M = 3, for G(z), we will get the parameters f 1 =0.06, f 2 = 0.075, δ 1 = 0.01/2 = and δ 2 = and accordingly the order N G = 183. Also, for I(z), we will get the parameters f 1 =0.02, f 2 = = , δ1 = 0.01/2 = and δ2 = and 3 accordingly the order N I = 9. Hence, we will have a total number of coefficients (N G +1)+(N I + 1) = 194. In a similar way, we obtain for M =4: N G = 137, N I = 13, hence, (N G +1)+(N I + 1) = 152; for M =5: N G = 110, N I = 18, hence, (N G +1)+(N I + 1) = 130; for M =6: N G = 91, N I = 23, hence, (N G +1)+(N I + 1) = 116; for M =7: N G = 78, N I = 28, hence, (N G +1)+(N I + 1) = 108; for M =8: N G = 69, N I = 34, hence, (N G +1)+(N I + 1) = 105; for M =9: N G = 61, N I = 41, hence, (N G +1)+(N I + 1) = 104; for M = 10: N G = 55, N I = 50, hence, (N G +1)+(N I + 1) = 107; for M = 11: N G = 50, N I = 60, hence, (N G +1)+(N I + 1) = 112. We observe that, as one would expect, as M increases, N G decreases and N I increases. There should thus be a choice of M that strikes a balance and results in the minimum value of (N G +1)+(N I + 1). In the present case, this balance happens when M = 9. Compared to the direct design, this design reduces the number of non-zero coefficients from 508 to 104; an almost 5 fold reduction.

30 128 Multirate Signal Processing Chap. 5 Figures 5.28(a)-(d) show the magnitude responses of G(z), I(z), G(z 9 ) and G(z 9 )I(z), respectively. Note that how the images in the response G(z 9 ) appear and how they are suppressed by the interpolator filter I(z) G(z) 20 0 I(z) db 40 db f (a) f (b) 20 0 G(z 9 ) 20 0 G(z 9 )I(z) db 40 db f (c) f (d) Figure 5.28: Magnitude response of the various filter designed in Example 5.2. One may also note that it is possible to apply the concept of IFIR design to any filter, including the model filter G(z). In particular, we may start with a small interpolation factor M, decide on the required specifications of G(z) and proceed with an IFIR design for G(z). This is left as an exercise at the end of the chapter Multistage realization of decimation and interpolation filters Consider the L-fold interpolator shown in Figure 5.29(a). Assume that L could be factored as L = L 1 L 2 and, thus, the expander L can be replaced by the cascade of the expanders L 1 and L 2. Moreover, let us apply IFIR design to replace H(z) by the cascade of the transfer functions G(z L2 ) and I(z). These give Figure 5.29(b) as an alternative realization to Figure 5.29(a). Applying the second noble identity of Figure 5.16 to swap the blocks G(z L2 ) and L 2, we obtain the desired structure as in

31 Sec. 5.6 Multistage Implementation 129 Figure 5.29(c). Here, interpolation is done in two stages: (i) L 1 followed by G(z); and (ii) L 2 followed by I(z). Clearly, each of these stages may be replaced by their respective polyphase structures for further improvement on the implementation, as required. L H (z) (a) L 1 L 2 G(z L 2 ) I (z) (b) L 1 G(z) L 2 I (z) (c) Figure 5.29: Development of a multistage interpolation filter. (a) Direct realization. (b) Intermediate structure. (c) The desired structure. To appreciate the computational advantage of the multistage implementation, we compare the number of operations (multiplications and additions) that is required for computation of each sample of output in Figures 5.29(a) and (c). Assuming that H(z) is a FIR filter of order N H and a polyphase structure is used for its implementation, computation of each sample of output requires N H /L additions and (N H +1)/L multiplications 1. In Figures 5.29(c), on the other hand, computation of each output sample requires N I /L 2 + N G /L additions and (N I +1)/L 2 +(N G +1)/L multiplications. The following example gives a better idea on how much we save on the number of operations. Example 5.3: Consider the design and implementation of a 100-fold interpolator. Also, assume that based on the spectral content of the input and the expected accuracy of the interpolator, the design parameters for H(z) are chosen as f 1 =0.002, f 2 =0.004, δ 1 =0.01 and δ 2 = Using Kaiser s formula, find the order of H(z). 1 Note that when H(z) is a linear phase filter, the symmetry of the filter coefficients may be used to reduce the number of multiplications by up to a factor of one half. Here, for the simplicity of the discussion, we have not considered this point.

32 130 Multirate Signal Processing Chap. 5 For L 1 = 2 and L 2 = 50, find the orders of G(z) and I(z). Compute the number of additions and multiplications for the direct and multistage implementations and compare the results. Solution: Substituting the specified parameters in Kaiser s formula, we obtain N H = Accordingly, the direct implementation of H(z), on average, requires 1267 = additions and = multiplications per output sample. With L 2 = 50, the band edges of G(z) should be set at f 1 =0.002L 2 = 0.1 andf 2 =0.004L 2 =0.2. Also, as in Example 5.2, we should choose δ 1 =0.01/2 =0.005 and δ 2 = Substituting these in Kaiser s formula, we obtain N G = 27. For I(z), we find the parameters f 1 =0.002 and f 2 =1/ = 0.016, δ 1 =0.01/2 =0.005 and δ 2 = Using these, we get N I = 196. Using the above results, the multistage implementation, on average, requires =4.19 additions and =4.22 multiplications per output sample. A three fold reduction in the computational load. The concept of multistage implementation may be extended similarly to the decimation filters. This is demonstrated in Figure H (z) M (a) I (z) G(z M1 ) M 1 M 2 (b) I (z) M 1 G(z) M 2 (c) Figure 5.30: Development of a multistage decimation filter. (a) Direct realization. (b) Intermediate structure. (c) The desired structure.

33 Sec. 5.7 Cascaded Integrator-Comb Filters Cascaded Integrator-Comb Filters Cascaded integrator-comb (CIC) filters are an important class of filters that was first proposed by Hogenauer in 1981 and subsequently have become popular because of their very low complexity. These filters allow realization of interpolators and/or decimators with only add/subtract operations. In the next two subsections, we discuss an L-fold CIC interpolator and an M- fold CIC decimator, separately. The combination of the two which results in an L/M-fold resampler will then be straightforward. This is left as an exercise at the end of the chapter L-fold CIC interpolator Recall from Section that an interpolator consists of an L-fold expander followed by a lowpass filter. As shown in Figure 5.8, the output of the expander is a signal whose spectrum within each interval of Lf s, where f s is the sampling rate before the expander, consists of the spectrum of the input signal plus L 1 images of it. The task of the lowpass filter is to remove these undesired images. The lowpass filter thus should have a flat response over the bandwidth of the desired signal (the baseband signal) and sufficient attenuation over the width of the undesired images. In the CIC method, the interpolator lowpass filter is implemented as a cascade of multistages of comb filters and integrators. Figure 5.31 shows a CIC interpolator with N stages of comb filters and the same number of integrators. Each comb filter generates KL zeros (where K is an integer) at frequencies f = 0, 1/KL, 2/KL,,(KL 1)/KL and each integrator generates a pole at z = 0. The zero of the comb filter at f = 0 is thus canceled by the pole of the integrator and, hence, the cascade of each pair of comb filter and integrator results in the transfer function G(z) = 1 z KL 1 z 1 = 1+z z (KL 1). (5.26) Clearly, G(z) is a FIR filter with zeros at f =1/KL, 2/KL,,(KL 1)/KL. Note that the number of zeros in each comb filter is an integer multiple of the interpolator factor L. This is a very important aspect of the CIC interpolator which is discussed in detail later. Substituting z by e j2πf in the first line of (5.26) and simplifying the result, we obtain G(e j2πf )= sin(πklf) e jπ(kl 1)f. (5.27) sin(πf) Since there are N pairs of comb filters and integrators, the realized lowpass filter has the transfer function H(z) =(G(z)) N. (5.28)

34 132 Multirate Signal Processing Chap. 5 z KL z KL z 1 z 1 L N stages of comb filters N stages of integrators Figure 5.31: A CIC interpolator with N stages of comb filters and the same number of integrators. (Initial structure) Substituting z by e j2πf in (5.28) and using (5.27), we obtain ( ) N sin(πklf) H(e j2πf )= e jπ(kl 1)Nf. (5.29) sin(πf) Figure 5.32 presents the plots of magnitude response of H(z), for K =2, L = 5 and three values of N = 1, 2 and 4. The plots are normalized so that their magnitude response at f = 0 be equal to 0 db (f B)/L s (f +B)/L s (2f B)/L s (2f +B)/L s N = 1 Normalized H(e j f ), db B/L N = 2 N = f Figure 5.32: Magnitude response of the CIC lowpass filter for KL = 10 and three values of N = 1, 2 and 4. In the applications of CIC interpolators, the input signal is usually sampled at a rate large enough such that its normalized bandwidth (with respect to the sampling frequency) is relatively small. The expander divides this bandwidth by L when normalized to the new rate. In Figure 5.32, a typical set of band edges of the input spectrum to the interpolator H(z) are shown. As seen, H(z) has relatively flat response over the band of the desired baseband spectrum and introduces deep nulls over the portions of

35 Sec. 5.7 Cascaded Integrator-Comb Filters 133 the band where images of the expanded signal exist. We may also note that by increasing N, the bandwidths and attenuation level of the filter nulls increase and thus results in better cancelation of the signal images. The fact that the signal images coincide with the nulls of H(z) is a consequence of the use of an integer K. With K = 2 the images happen at the alternate nulls of H(z). Another observation in Figure 5.32 is that when B/L is not sufficiently small, the passband of the CIC filter may suffer from a nonnegligible distortion. Such distortion may be compensated by pre-distorting the signal spectrum prior to interpolation. The reader is encouraged to explore further these properties of the CIC interpolator. The CIC interpolator of Figure 5.31 can be further simplified by using the second noble identity to switch the position of the L-fold expander and the comb filters. This results in the structure shown in Figure The advantages brought by this structure are L-fold improvement both with respect to the delay sizes and the operating frequency of the adders since: (i) the delay size in each comb filter is reduced from KL to K and (ii) the comb filters are run at the input rate which is L times slower than the output rate. z K z K z 1 z 1 L N stages of comb filters N stages of integrators Figure 5.33: A CIC interpolator with N stages of comb filters and the same number of integrators. (Simplified structure) An import practical point that should be noted here is the numerical stability of the CIC structure. At first glance, one may believe that because of the unity feedback gain in integrators, the output of such integrator will sooner or later overflow and as a result the CIC interpolator will fail. In the original paper of Hogenauer it is noted that as long as the CIC filter is implemented in fixed point with two s complement arithmetic, the unity feedback of the integrators does not cause any numerical problem. This is because when a sequence of add/subtract operations are implemented in fixed point and two s complement is used for number representations, as long as the final result can be accommodated within the specified word length, intermediate overflow of the numbers is of no consequence. Thus, if two s complement number representations are used, the only design consideration that requires attention is the range of the system output. In particular, we note that the passband gain of the interpolator filter is given by H(e j0 )=(KL) N and the expander gain at f = 0 is 1/L. Hence, the passband gain of the CIC interpolator is K N L N 1. Accordingly, the re-

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