Digital signal processing schemes for efficient ... - IEEE Xplore

Digital signal processing schemes for efficient interpolation and decimation R.A. Valenzuela B.Sc.(Eng.), Ph.D., D.I.C. and A.G. Constantinides B.Sc.(Eng.), Ph.D., C.Eng., M.I.E.E. Indexing terms:

Signal processing, Interpolation, Decimation

Abstract: In this paper a new structure for sampling-rate alteration is presented in which efficiency is achieved by performing all necessary processing at the low sampling rate. Moreover the repeated use of a single processing block makes this structure highly modular and eminently suitable for LSI/VLSI implementation. Particular emphasis is placed on decimating and interpolating by a factor of two. The proposed structures offer very desirable properties in addition to the above and, in particular, in relation to their insensitivity with respect to reduced wordlength performance. Sampling-rate alteration by factors other than two is also examined and design procedures are given. The paper contains extensive tables and graphs to facilitate the design of these structures by estimating the required order and parameters for the given requirements before attempting any optimisation. In the case of interpolating by a factor two an analytic equiripple solution is given. The paper includes some design examples with performance evaluation under different wordlengths.

1

Introduction

There are many areas of application in which samplingrate alteration is necessary or desirable. A typical example is concerned with pulse-code-modulation/frequencydivision-multiplexing (PCM/FDM) conversion and vice versa. The output sampling rate of an N channel PCM/FDM convertor is N times higher or N times lower than the input sampling rate, for the TDM to FDM or the FDM to TDM cases, respectively. Consequently the transmultiplexing function must include interpolation (sampling-rate increase) or decimation (sampling-rate decrease) of some kind. Moreover, owing to the large interpolating factors involved (typically 24 or 64), the complexity of implementation of the technique used to perform this sampling-rate alteration has a considerable effect upon the overall computational burden. In this paper a new structure for sampling-rate alteration is presented in which efficiency is achieved by performing all processing at the low sampling rate. Moreover the repeated use of a single processing block makes this structure highly modular and eminently suitable for LSI/ VLSI implementation. Particular attention will be devoted to decimation and interpolation by a factor of two which is an important processing element in the transmultiplexer designs described in Reference 1. It will be shown that in this case the proposed structure simulates doubly terminated lossless lattice networks thereby preserving their attractive sensitivity properties. A powerful design method based on the work of Reference 1 for the design of electrically symmetrical, reactive (lossless) networks is also presented. Sampling-rate alteration of factors other than a power of two is also considered. A general design method based on nonlinear optimisation techniques which rapidly converge towards the optimal (equiripple) solution is presented. 2

processing is carried out at the low sampling rate. This property can be obtained by specifying a priori the interpolator to be as shown in Fig. 1, as opposed to the procedure involving a rearrangement of a prototype transfer function as suggested in References 2, 3. In the interpolating structure specified (Fig. 1A) the

x( 2 - N ),

Fig. 1A

Interpolating structure

sampling rate is increased N times by interleaving the output from N subfilters H£z~N) operating at the input sampling rate. Similarly, the decimating structure achieves the required iV-fold reduction of the input sampling rate by sequentially feeding one out of every N input samples into each subfilter H{{z~N). One output sample for every N input samples is computed by adding appropriately delayed outputs from all N subfilters (Fig. IB).

Operating principle, transfer function and frequency response

In order to keep the complexity of implementation as low as possible, it is desirable to have a structure in which all Paper 2834G (E10), first received 4th February 1983 and in revised form 29th July 1983 The authors are with the Signal Processing Section, Department of Electrical Engineering, Imperial College of Science and Technology. Exhibition Road, London SW7 2BT, England

1EE PROCEEDINGS,

Vol. 130, Pt. G, No. 6, DECEMBER 1983

Fig. 1 B

Decimating structure

It is required, however, that the output baseband spectrum be identical to the input spectrum (except for the higher sampling rate). This has the consequence that the transfer functions of the individual branches H((z~N) should be of the all-pass type, and hence it is possible to 225

determine the overall transfer function as follows:

Hi (2W),phase response

(1) i=0

As //,(z~N) is to have an all-pass frequency response it can be expressed as a cascade of elementary all-pass sections Hi(z~N)=

(2) gj, k

+

,-N

+aLkz

(3)

-N

The all-pass section of eqn. 3 has N poles and N zeros located at z = ( — aik)~llN and z = ( — «,-,k)1/JV, respectively. Hence, in order to ensure stability, the magnitudes of the coefficients a,- k must be less than unity. Substituting eqns. 3 and 2 into eqn. 1 gives the overall transfer function N-

-1.0

n

(4) t

k

The interpolator frequency response can now be obtained by evaluating the overall transfer function on the unit circle yielding N-l

H(to)=

(5)

where c/>,- t(Afco) is the phase response of the elementary allpass section, and it can be obtained by evaluating eqn. 3 on the unit circle:

But -jNto

_

1 -j tan (Nco/2) 1 + j tan (Nco/2)

(7)

Substituting eqn. 7 into eqn. 6 and rearranging gives tan (Nco/2)

4>itk{Na)) = - 2 tan

(8)

Consequently the interpolating function is achieved if the elementary function i,k( «°) < + Nco,12 -Lk{Na))=

0 ^ co ^s n 71 ^ CO :^ 271

co = mn/N

-mn

-4>iik(2nn - Nto)

(f)Lk{Nco)=

0.15 0.2 0.25 0.3 normalised frequency

0.35 0.4

0.45 0.5

Consequently the structure under consideration has a DC gain which is equal to the sample-rate alteration factor N. Additionally no output energy is found at any multiples of the low sampling frequency. 3

Interpolation/decimation by a factor which is a power of two

The problem of sample-rate alteration by a factor N which is a power of two can be solved by repeated interpolation/ decimation by two and hence attention need only be focused on the case N = 2. Moreover, it will be shown that special attributes of the two-branch structure shown in Fig. 3 make this approach better than the alternative single-step sample-rate alteration. For N = 2 the general structure of Fig. 1 simplifies into two all-pass branches in parallel, as shown in Fig. 3. When used for interpolation, both branches are fed with the lowrate input sequence; the overall output sequence is obtained by interleaving the output from the all-pass branches, as shown in Fig. 3a. For decimation, the input H,(z- 2 ) X(z"2 )

Fig. 2 illustrates the function ,- k(Nco) for at k in the range (0, 1) and N = 2. It is readily seen that this function has the following properties: -NOJ/2^

0.1

Allpass section phase response Fig. 2 4>i t (JVw), N = 2

4-

(6)

0.05

.-2-

(9) (10) (11)

H,(2-2)

(12)

X(z- 2 )

Substituting eqn. 11 into eqn. 5 gives: N-

1

(13)

//(co = mn/N) = X e

2fs

M I H H2(2"2)

i=0

IN /

\

m zj= 0, even 'Cte.

226

m = 0, (co = 0)

m ^ 0, odd

(14)

Fig. 3 Two-branch structure a Decimation b Interpolation IEE PROCEEDINGS,

Vol. 130, Pt. G, No. 6, DECEMBER 1983

sequence is separated into two sequences of half the sampling rate. These are then fed into the all-pass branches. The overall output is then obtained by adding the outputs from the two branches (Fig. 3b). The interpolation function corresponds to a low-pass filter operating at the high sampling rate. Fig. 4a shows

(19)

It is clear that if the phase difference between the all-pass branches is the same as for ideal interpolator then the decimator output spectrum is:

Y(co) = tf

X.(7

(20)

X2(o>) = ftX{co) - X{n - co)]

X2(co]} = X(co) for 0 < co < n/2

(21)

= X(n - co)

Y(a) = ±[*iM X2(z"2)

for n/2 < co < n

(22)

In other words the ideal decimation is obtained as shown in Fig. 4b. The general transfer function given by eqn. 4 evaluated for N = 2 gives Ki

• ~2

•)=n

2fe

%! + !

_,

•n

i

- ~ 2

-2

(23)

i . *•

which represents a stable system provided that the magnitudes of all coefficients at k are less than unity. As discussed before, and also according to the transfer function given above, the structure under consideration consists of two all-pass branches in parallel. Appendix 9 shows that such structures simulate doubly terminated lossless lattice network, preserving their well known sensitivity properties. Without loss of generality eqn. 23 can be rewritten as + z

(24)

where K = Ko + /Cx is the total number of all-pass sections and there are no restrictions on the value of the coefficients ak. The original, stable transfer function is easily found by dividing eqn. 24 by the product of all unstable all-pass sections. It is also seen from eqn. 24 that n, the order of the transfer function, is equal to twice the total number of all-pass sections plus one, i.e. n = 2K + 1

(24a)

The frequency response can be obtained by evaluating eqn. 23 on the unit circle, i.e. Fig. 4

//(co) =

Intermediate signals

J0{2a>) e

+

(25)

where that this effect can be achieved, despite the fact that all processing is performed at the low sampling rate. Moreover, an ideal interpolator is obtained when the phase difference between the all-pass branches is equal to co. In this case the overall phase difference, including the phase contribution due to the unit delay, is such that the output sequences are exactly in phase for 0 < co < n/2 and exactly out of phase for n/2 < co < n for the decimator; it is useful to note that the two low-sampling-rate sequences Xj(n), x2(n) obtained at the input switch by feeding alternate samples to each path (Fig. 3b) can be described as follows: x(n) xAn) = 0

for n even for n odd

(15)

0 [x(n)

for n even for n odd

(16)

x2{n) =

(27)

Eqn. 25 can be rewritten as

x cos [{4>o(2co) -

(17) (18)

Vol. 130, Pt. G, No. 6, DECEMBER 1983

+ co}/2]

(28)

Having in mind the properties of the phase response of the all-pass section, c/>,-fc(Nco),described by eqns. 9-12 it is seen that (29)

has the following properties: l\H(co)\

or in the frequency domain:

(26)

I//(co) |/2 = cos [{0o(2co) — cb{{2co co}/2]

Alternatively:

IEE PROCEEDINGS,

0 o (2co)=

=1

II = 0

for co = 2kn

(30)

for co = (2fe + l)7i

(3D

for co = (2/c + 1)TT/2

(32)

= 1 for all co

(33) 227

The specifications for an interpolator/decimator by two of transition bandwidth a>t are ideally: \H{ a) > n/2 + OJJ2

(35)

Moreover, the two-branch structure of Fig. 3 will always satisfy the required specifications exactly at co = 0 and co = n due to the properties described by eqns. 30 and 31. Furthermore, the symmetry property described by eqn. 33 implies that if co1 is the passband frequency at which the magnitude response is a minimum then co2 = n — cjy

(36)

is a stopband frequency at which the magnitude response is a maximum. For an optimal equiripple response exhibiting Chebyshev type performance in both passband and stopband (Fig. 5), the magnitude response must be equal to the pass-

Newton method has been used to minimise the sum of the squares of the frequency response evaluated on a dense grid of points in the interval of interest [4]. It is observed that the algorithm stops a local minimum at where the number of maxima in the stopband response is equal to the number of coefficients a, k plus one. Since the function 0, fc(2co) which controls the frequency response is monotonic and related only to the coefficient ais) since this is the only maximum that will always occur at a prespecified frequency (cos) (Fig. 6). The design method can now be summarised as follows: (i) Define the stopband edge cos and the number of coefficients Nc, which is in direct relation to the required stopband attenuation as shown in the next Section. Choose M, the number of points into which the interval (cos, n) is to be discretised; 200 points have been found to be sufficient for interpolators/decimators of up to five coefficients. (ii) Minimise (41) where cu,- belongs to the dense grid of frequencies covering the interval (cos, n), by using a nonlinear optimisation technique (based on a corrected Gauss-Newton method for instance [4]). At the starting points the coefficients a{ k are set to be evenly distributed in the range (0, 1). Check that the number of stopband maxima is equal to the number of coefficients Nc. If not, start the minimisation from a different set of coefficients. It has been found that, for most cases, any starting point yields satisfactory results, the only difference being the computation time. (iii) By choosing the coefficient values from the last iteration of the previous step as the new starting point minimise: C2=

\H(ajs)-H(com)\:

(42)

where com is a stopband frequency at which the frequency response is a maximum. These frequencies can be obtained from the evaluation of the frequency response on a dense grid of frequencies. IEE PROCEEDINGS,

Vol. 130, Pt. G, No. 6, DECEMBER 1983

More than seventy interpolators/decimators of up to five coefficients and with normalised transition bandwidths in the range (0.02, 0.3) have been designed using this procedure. In all cases the appropriate number of maxima was obtained at the first attempt (step (ii)). An optimal solution in the equiripple sense was always obtained in less than 10 iterations (step (iii)). Fig. 7 illustrates the relation between transition bandwidth, number of coefficients and stopband attenuation.

Finally, and as an example, the design on interpolators required for the transmultiplexer design described in Reference 10 are presented. The specifications are:

recursive interpolators

From Fig. 7 it is seen that five coefficents are required and by using the appropriate polynomial fit from Table 1, or alternatively from Fig. 12, the attainable stopband attenuation is estimated to be 77.6 dB (using the reviewer's expression one obtains 75.6 dB). The design method previously described requires approximately 37 s of central processor time to finish this design, including compilation time, using a CDC 6600 computer. The final coefficients values are:

140 120

I

80

i

60

Normalised transition bandwidth Stopband attenuation Passband ripple

TO

I 40 1/5

Fig. 7

= 0.02 dB

ax = 0.065227

a2 = 0.223697

a3 = 0.448685

a 4 = 0.667661

0.04 0.08

0.12 0.16 0.20 0.24 0.28 transition bandwidth

0.32 0.36

Stopband attenuation as a function of the transition frequency

The stopband attenuation achieved is 77.4 dB and the result is optimal in the equiripple sense as can be seen from Fig. 13. The passband is practically flat with a ripple of 0.08 x 10~ 6 dB. The coefficient word length is 12 bits for the above

This diagram is particularly useful for determining the minimum number of coefficients necessary to satisfy a given set of specifications. In order to have an easily computable estimation for the attainable stopband attenuation as a function of the transition bandwidth and the number of coefficients, a third-order polynomial fit to the data was made and Table 1 gives the resulting coefficients. In all cases the stopband

•5

Table 1 : Values for the parameters of eqn. 43

S 20

recursive interpolator (N = 1)

35 30

Error %

Nc 1 2 3 4 5

= 70dB

a5 =0.884131

20

u

= 0.075

7.1784 19.62 23.307 32.179 38.482

150.66 269.35 378.13 468.07 679.18

35.83 693.28 976.37 1007.7 2601.0

776.76 1471.4 2068.4 2008.3 6840.3

25

15

0.012 0.013 0.012 0.015 0.005

10 0.04 0.08

0.12 0.16 0.20 0.24 0.28 transition bandwidth

0.28

attenuation is estimated to better than ±0.5 dB, using the estimation formula A = C o + C, dF - C2 dF2 + C 3 dF3

(43)

recursive interpolator N = 1)

1.05

where

0.9

A = stopband attenuation j j 0.7 5

dF — normalised transition bandwidth A simpler formula accurate to within a few dB is given by

•g

0.5

;o

A +6 dF + 0A3

12(N + 0.53) for 0.04 ^ dF ^ 0.3

(43a)

This formula follows from the curves of Fig. 7 and was proposed by one of the reviewers of this paper. The behaviour of the coefficient values as well as the stopband attenuation as a function of the transition bandwidth is illustrated in Figs. 8-12 for 1-5 coefficients, respectively. These data could be of use in reducing the number of iterations required for the design method by improving the choice of the initial set of coefficients values. IEE PROCEEDINGS,

Vol. 130, Pt. G, No. 6, DECEMBER 1983

8

0.45 0.3 015 0

Fig. 8

0

0.04 0.08

0.12 0.16 0.20 0.24 transition bandwidth

0.28 0.32

Coefficient values for optimum designs

AL = I

229

given specifications and only 8 bits for a relaxed 60 dB stopband attenuation.

due to Reference 6, provides a very powerful method for the design of digital elliptic filters. In this method any four

recursive interpolator (N=2)

recursive interpolator (N= 3 ) 105 m •o

90

I 75

o i 60 5

S

30 15

0.04 0.08 0.12 0.16 0.20 0.24 0.28 transition bandwidth

0.32 0.36

0

recursive interpolator (N=2) 1.05

0.9

0.9

j j 0.75 o

0.75

0.28

0.32 0.36

0.12 0.16 0.20 0.24 0.28 transition bandwidth

0.32 0.36

0.5 0.45

0.45

0.3

0.3

0.15

0.15

0 Fig. 9

0.04 0.08

0.12 0.16 0.20 0.24 0.28 0.32 transition bandwidth

0.36

0

Coefficient values for optimum designs

Fig. 10 Nc = 3

Nc = 2

It should be noted that a seventh-order elliptic filter, operating at the high rate, is required to satisfy the given specification, having 11 coefficients and with a coefficientword-length requirement of 16 bits, even for the relaxed 60 dB of stopband attenuation. The low sensitivity of the structure under consideration results from the fact that it simulates a doubly terminated lossless lattice network, as shown in Appendix 9. This property is fully exploited in the next Section in which an analytic design method is presented. 4.2 Analytic design method

Appendix 9 shows that the two-branch structure of Fig. 3 for sample-rate alteration by a factor N = 2 simulates a doubly terminated lossless lattice network through the socalled wave linear transformation. Moreover, the simulated lattice belongs to a class of networks having magnitude response (44)

The theory and design of electrically symmetrical reactive (lossless) networks with particular attention to filters that exhibit Chebyshev-type performance in the passband and stopband has been comprehensively covered by References 1 and 5. This work, together with the well known bilinear transformation for the translation between the analogue and digital domains and the frequency transformations 230

0.12 0.16 0.20 0.24 transition bandwidth

recursive interpolator (N=3)

1.05

c '0

0.04 0.08

0.04 0.08

Coefficient values for optimum designs

of the five parameters n, the filter order, dp, ds,fp and/ s the passband and stopband ripples and cutoff frequencies respectively, can be arbitrarily set to any prescribed value; the remaining parameter and the filter response is then completely and uniquely specified. In the structure under consideration for sample-rate alteration by a factor N = 2, the passband and stopband ripple ds and dp and cutoff frequencies wp and OJS are not independent; in fact they are related through eqns. 37 and 38 as discussed in Section 3. Imposing these relations is a sufficient condition for the previously mentioned design method to yield a unique solution for a given order and transition bandwidth. Therefore the resulting elliptic transfer function must correspond to a lattice network which belongs to the class of networks being simulated by the structure under consideration. The design method for the interpolators/decimators can now be summarised as follows: Step 1 For the required transition bandwidth compute: k = tan 2 {(7i - cof)/4} k' = ^ ( l - k2)

150