Effects of finite word length on the performanceof split-band coding systems using quadrature

mirror filtersS.W. Foo, B.E., M.Sc, C.Eng., M.I.E.E., and Prof. L.F. Turner, B.Sc, Ph.D.

Indexing terms: Codes and decoding, Filters

Abstract: The paper reports on an investigation into some problems relating to the implementation of sub-band coding systems using nonrecursive quadrature mirror filters. Particular attention is given to the noiseresulting from the use of finite-word4ength registers for data storage and arithmetic operations. Formulas forthe evaluation of the bounds and variances of the errors due to quantisation of the filter coefficients, roundoffin arithmetic computations and the quantisation for compression encoding are derived. The derivations arebased on a statistical model of the noise sources. An extensive simulation has been carried out and the resultsof the simulation have been found to be in excellent agreement with the theoretically predicted results.

1 Introduction

The use of quadrature mirror filters in the sub-band coding ofspeech has recently received considerable attention [1—3]. Insub-band coding, down sampling (or decimation) of datasequences results in frequency aliasing. The use of quadraturemirror filters in sub-band coding allows for the cancellation ofthis frequency aliasing, even for filters with a small number oftaps and a wide transition width [ 1 ] .

In a previous paper [41, a method was put forward for thedesign of quadrature mirror filters so that the peak-to-peakripple of the frequency response of a split-band system usingquadrature mirror filters could be minimised. In this paper,the errors resulting from the use of finite-word-length registersare considered and the results of an experimental verificationof the theory derived are presented. One of the aims of thepaper is to provide some useful guidelines for the implemen-tation of sub-band coding systems using quadrature mirrorfilters; and, in particular, to provide information as to theword-length that should be used in storage and arithmeticoperations.

The essential equations relating to the operation of quadra-ture mirror filters in split-band systems are given in Section 2,together with a discussion of the noise model used in thesubsequent analysis. In Section 3, the structure and coefficientscaling in quadrature mirror filters are discussed. The effects ofquantisation at various points of the system are discussed inSection 4 and some experimental results are given in Section 5.

2 Theory of quadrature mirror filters and noise modelused

The analysis of quadrature mirror filters relates to the split-band system shown in Fig. 1. In a simple split-band system,

x(n) decimation irterpotalion

fs V2

Fig. 1 Band splitting with quadrature mirror filters

Paper 2138G, first received 16th April and in revised form 14th July1982The authors are with the Electrical Engineering Department, ImperialCollege of Science and Technology, London SW7 2BT, England.Mr. S.W. Foo is on leave from the Ministry of Defence, Singapore.

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

the input signal x(i) is lowpass fitered to B(=fs/2) andsampled at the Shannon rate of fs. The lowpass filter //j andthe highpass filter Hu are then used to split the spectrum intolower and upper parts of equal bandwidth. The filtered signalsXj(n) and xu(n), which each occupies a bandwidth of approxi-mately half B, are then decimated by retaining only alternatesamples. Compression encoding is then usually carried out onthe decimated signals yi(n) and yu(n) before transmission tothe receiver.

At the receiver, the signalsyi(n) andyu(n) are interpolatedto obtain the signals «j(«) and uu(n). The interpolated signalsare then filtered and combined to give s(n) which is intended tobe a replica of the original signal x(n).

Filters Hl and Hu whose frequency responses are mirrorimages of each other are referred to as quadrature mirrorfilters [1]. Thus, if the z-transform Hi(z) of Hh expressed interms of its impulse response h^k), is

t ht(k)z -ft

fe =

then the z-transform Hu(z) of Hu is given by

N-\

k = 0

0)

(2)

It can be shown [1 ] that the z-transform S(z) of the signal s(n)is related to the z-transform X(z) ofx(n) by

S{z) = R(z)X(z)+A(z)X(-z)where

R(z) = {M2){Hl{z)Pl{z)+Hu{z)Pu{z)}

and

The term>4(z)A'(— z) in eqn. 3 is due to frequencyBy choosing

Pt(z) = gHx{z)

and

Pu(z) = -gHu{z) = -gH^-z)

where g is a constant, then

A(z) = 0

and the aliasing effect is thus eliminated.

0143-7089/82/060257 + 13 $01.50/0

(3)

(4)

(5)

aliasing.

(6)

(7)

(8)

257

Further, if Ht is a symmetrical FIR lowpass filter of evenorder, then on imposing the constraints,

g = 2

and

whereH? (eJ'")= Ht(eiuJ) exp [j(N- l)w/2]

the Fourier transform of Ht; it can be shown that

S(z) =

and

s(n) = x ( w -

(9)

(10)

(11)

(12)

From eqn. 12, it is thus clear that the signal is perfectly recon-structed with a delay of (vV — 1) samples.

The effects on the accuracy of systems using a finite-word-length number representation can be studied if it is assumedthat the error introduced by rounding to M bits is a randomnoise having a uniform probability distribution in the interval(-Q/2, (2/2), in which Q{= 2~{ x)) is the quantisation stepsize. In order to simplify the following analysis, it will beassumed that:

(i) any two different samples from the same noise sourceare uncorrelated

(ii) any two samples from different noise sources areuncorrelated

(iii) each noise source is uncorrelated with the input.

This means that the noise source is modelled as a discretestationary white-noise process with uniform power-densityspectrum of magnitude A2 = Q2 /12.

In the case of split-band systems using quadrature mirrorfilters, as shown in Fig. 1, the following four categories ofnoise source can be identified:

(i) the quantisation of samples of the original analogueinput, {x(t)}, into a set of discrete levels. Since the analysis isstraightforward, it will not be dealt with in this paper

(ii) the representation of the coefficients of the filters by afinite number of binary digits

(iii) the rounding of values resulting from multiplications ofnumbers

(iv) the compression coding of the split-band signals {>"/(«)}and {yu(n)}.

In the analysis which follows, it is assumed that all quantisedvalues are represented by fixed-point binary numbers of Mbits, that negative numbers are represented in 2's complementform, and that all numbers are left justified with a magnitudeless than unity.

3 Structure and coefficient scaling of filters

In order that all the numbers and data shall be less than unity,it is necessary that scaling of the input data and all filter coef-ficients be carried out. If it is assumed that the values of thesequence {x(n)} have magnitude less than unity, then in orderto ensure that the magnitude of the filtered sequence {*,(«)} isless than unity, 'sum scaling' has to be performed on the coef-ficients {hi(k)}. That is, a new set of coefficients {hi(k)} areobtained in which

N-l

ht(k) = h\{k)l X \h\{k)\ for k = 0, 1, . . . ,N- 1fe=0 (13)

Thus

ft=0X \K{k)\l X \K{k)\ = 1

fe=0 fe=0(14)

Since hl(k) = hl(N-l-k) for A:=0, 1 , . . . , (N-l), itfollows that

N/2-1

I 2fe = 0

and that

N/2-1£ |/*,(*)l = 1/2 for N > 2

fc = 0

(15)

(16)

This implies that

\ht(k)\ < 1/2 f o r k = 0 , I , . . . , N - l (17)

so that

h2(k) < (1/2)1/7^)1 for k = 0, 1, . . . ,N- 1(18)

N-l N-l

!>?(*) < X = 1/2 for N > 2

(19)

and

J V / 2 - 1

X h2(k) < (1/2)0/2) = 1/4 (20)

If, in implementing the filters, a direct-form shared-multiplierstructure, as shown in Fig. 2, is used, then the filtered outputis given by

N/2-1xt(n) = X

fe =and hence

N/2-1

X lfy

J V / 2 - 1

X = 1 as \x(n)\ < 1

This means that \xi(n)\ < 1 for all n and the filtered sequence{xt(n)} thus has values less than unity.

Fig. 2 Shared multiplier structure

It should be noted that, although the partial sum [x(n —k) +x(n — N+ 1 + k)] may exceed unity, the producthi(k)[x(n — k) + x(n — N + 1 + k)] is always less than unityand the shared-multiplier structure can thus be adopted whenall the numbers are represented in 2's complement form. It isimportant to note that the number of multipliers required

258 IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

with a structure of this kind is only half the number of stagesin the filter.

The same arguments apply to the filter Hu.At the receiver, the interpolated sequence {ut(n)} has

alternate zero values and, if a shared-multiplier structure isused for the filter, then the filtered output is given by

N/2-1

'/(«) = I Piimutn-V + utn-N+l+k)]fe = O

Since one of the terms of [ut(n — k) + ut(n — N + 1 + k)] iszero,

and thus

N/2-1

I'IOOI < I \Pi(k)\ft = O

J V / 2 - 1

This means that \tt(n)\ < 1 if Z \Pi(k)\ = 1, which can befe = O

achieved by dividing the original set of coefficients {p'i(k)}byN / 2 - 1 JV-l

2 |p;(*)|,orequivalently,by(l/2) 2 \p\(k)\.fe=O fe=O

When the coefficients are so scaled

\Pi(k) < 1

N - l

fe = O

and

for A: = 0,l,...,N-l

= 2

J V - l

fe =pf(k) forJV > 2

(21)

(22)

(23)

Similar arguments apply to the filter Pu.When scaling is carried out as proposed, the output sequence

J V - l N-l

has to be multiplied by [ 2 \h[{k) I] [(1/2) 2 |p,'(ifc)|], soft=O fe=O

that its values are of the same order of magnitude as those ofthe input sequence.

4 Effects of quantisation and finite word length in systemsusing quadrature mirror filters

4.1 Effects of quantisation of filter coefficientsDeviation of the filter coefficients from their 'infinitelyaccurate' values produces a distortion of the overall systemresponse and the question arises as to the level of distortionresulting from the use of finite-word-length coefficients. Theanalysis below provides the designer with a method of esti-mating the expected deviation for a given accuracy of coef-ficient representation.

The effects of coefficient quantisation on (i) the resultantfrequency response R(eiu}), and (ii) the output sequence{s(n)} will now be considered.

Let the 'infinitely accurate' coefficients associated with thefilters /// and Pt be h^k) and pt(k), respectively, and let hiik)and Pi(k), k = 0, I,. . . ,N— 1 denote these coefficients whenthey have been rounded to M bits. The quantisation errors arethen seen to be:

eh(k) =

and

«»(*) = Pi(k)-Pi(k)

(24)

for A: = 0, 1, ...,N-\ (25)

where

\eh(k)\

and

\ep(k)\

(2/2

(2/2

(26)

(27)

Let Eh and Ep represent the random variables consisting of{eh} and {ep}, respectively. From the assumed noise model, itfollows that the variance of both Eh and Ep is equal to A2,where A2 = Q21\2.

If it is assumed that:

Hu(z) = Ht(-z) = Hl(-z) + Eh(-z) (28)

Pt(z) = Pl(z)+Ep(z) = 2Hx{z) (29)

and

Pu{z) = -2Hu{z) = -2Hl(-z) = -Pti-z) (30)

where Eh(z) and Ep(z) are the z-transforms of the respectivesequences {eh} and {ep}, then it can be shown that the z-transform of the noisy output sequence is given by

Sc(z) = (\/2){Hl(z)Pl(z)-Hl(-z)Pl(-z)}X(z)

(-z)] [Pl(-z)+Ep(-z))}X(z)

z)X(z) (31)

where

gcr(z) = (l/2){H1(z)Ep(z) + P^E^z) + Eh(z)Ep(z)

-Hl(-z)Ep(-z)-Pl(-z)Eh(-z)

-Eh{-z)Ep{-z)} (32)

The z-transform of the time-domain output error sequence,denoted ^ ( z ) , is then given by

= Sc(z)-S(z) = &cr(z)X(z) (33)

It should be noted that there is no term representing the z-transform of a frequency folded sequence, and hence cancel-lation of frequency aliasing is retained even when finite-word-length coefficients are used.

4.2 Error bounds and statistics associated with finite-word-length coefficients

The analysis of the error &cr can best be carried out by con-sidering its various component terms separately, and this hasbeen done in Appendix 9.1. If it is assumed that

2N-2= Z ecr(k)z

k

-k (34)

then, using the results derived in Appendix 9.1, it can be seenthat the absolute bound on the sequence { e ^ ^ l i s given by

|ecr(A:)i < (1/2)

((2/2)

for 0 < k < (N/2) - 1

(G) + (Q2/4)(k + 1)}

for 0 < k < 2N - 2

1)}

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 259

= (C/4) [6+ fi(*+1)1

(or(N/2) < k <N-\

< (l/2)(2){(fi/2) + (Q) + (Q2/4)(2N- 1 - * ) }

forN-l < k < (3N/2)-2

< (l/2)(2){(G/4) + ((2/2) + «22/4)(2JV- 1 -A:)}

for (3W/2) - 1 < k < 2N-2 (35)

The largest value of the bound is (3(2/2)[1 + C/V0/6)l, andthus, in general,

(3G/2)[1 forO

(36)

Since & c r is a linear function of the random variables $ci,i = 1,6 (defined in Appendix 9.1), which were assumed to beuncorrelated, superposition applies and hence the mean offee,, is given by

&cr = 0 /2) I (gd) = 0

and its variance is

var(g c r) = (1/2)2 X [var(€d)l1=1

(37)

+ [2N/(2N- 1)] + [N2A2/(2N- 1)] }A2

= [5N/(4(2N- 1))] {1 + (2WA2/5)}(A2) (38)

It should be noted that the contributions to the above variancefrom fbcS and § c 6 are small for all practical cases where A2 aresmall.

If the error of the overall frequency response is defined tobe

2N-2&*cr(eJ") = X ecr(n)e-*"n (39)

n = 0

then since ecr{k) = ecr(2N-2 - k), for k = 0, 1,. . . , 2N-2, eqn. 39 can be expressed as

N-2

In = 0

r(eJoJ) =\ X 2ecr(n)cos[(N-l-n)co]

(40)

and since the factor e~i(jJ(-N l) does not affect the magnitudeof the response, it can be neglected.

Let

(41)

Then

N-2

In = 0

2N-2 2N-2T 6

< X l€cr(«)l < X (1/2) Xn=0 n=0

y=o

7 = 0Y(G/2) Y \h(k)\ + ((2/2) x \P(k)\

(G/2)(G/2) X (1)fe = 0

(G/2)(2) + ((2/2)2 (N)}

= (3NQ/2)[l + (NQ/6)) (42)

In order to find the statistics of the error response, an approachsimilar to that of Chan and Rabiner [51 can be adopted. It canbe shown that the error in the overall frequency response haszero mean and a variance given by

N-2

= X 4e2cr(n)cos2[(N-l-n)u}+e2

cr(N-l)

N-l

= var(6 c r ) 1 + X 4COS2(«CJ)

If the weighting function W(u>) is defined to be

i V"1 I |(l/2)+ 4 2- c o s <*>n |

n=l

then

Expressed in closed form, fV2(co) is given by

(43)

(44)

(45)

It can be shown that W2(0) = W2(n) = 1, and 0 < W2{u) < 1for all A , thus:

a2r(co) < var(g c r)(47V-3)

< [5A^(47V-3)/4(2A^- 1)1 [1 + (2iVA2/5)l A2

(46)

In the limit for large N and 0 < CJ < n,

Urn W2(co) = 1/2

The convergence of W2(co) to 1/2 is not uniform for all u>.However, for co in the range [5, n — 8 ], for any 5 > 0, theconvergence is uniform, as can be seen from Figs. 3a and b.Thus, in the range 0 < GO < n, the bound given by exprs. 46 foralr is overestimated by a factor of approximately two for largeN.

Next, consider the time-domain error of the output

260 IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

sequence. From eqns. 33, Alternatively, as x2(*) < 1,

and so, using expr. 36,

2JV-2

fe =\eCT(k)\\x(n-k)\

2JV-2

(47)

From expr. 47, it can thus be seen that the error e^n)depends on the specific input sequence.

O.2TT 0.4TT 0.6TT 0.8TTangular f requency,uo

1.00

0 0.2 IT 0.4 T 0.6 T 0.8T TTangular frequency ,uu

b

Fig. 3 Variation of weigh ting function with angular frequency

(a)N= 8, (b)N= 32

In order to obtain an expression for the error which isindependent of {*(«)}, it should be noted that, as \x(n)\ < 1,

2JV-2

fe = 0(1)

1) (48)

The random variable g ^ , consisting of {ea(k)}, has a meanequal to zero and variance given by

2JV-2

fe =e2

cr(k)x2(n-k)

2AT-2

= var(gcr) X x2(n-k) (49)

(2WA2/5)| A2(50)

4.5 Effects of arithmetic roundoffErrors result when the product of two M-bit numbers exceedsM bits and rounding has to be performed. For the systemsimplemented in the manner described in Section 3, multipli-cations are only performed within the filters, and noise is thusadded to the output sequences {*,(«)}, (*u(«)}, {?,(«)}, {'«(")}from the four filters. Let Xt(z), Xu(z), T^z), Tu(z) represent,respectively, the z-transform of these sequences, and letExi(z), Exu{z), Etl(z), Etu(z) represent the z-transform of therespective noise sequences. The z-transform of the noisy filteroutputs is thus given by:

Xt(z) = *,(z)

Xu(z) = Xu(z)

f,(z) = r,(z)

and

f (z\ = T (7\

(51)

(52)

(53)

(54)

Using Xt(z), Xu(z), ft(z), fjz) in place of Xx(z), Xu(z),Ti(z)> Tu(z), the z-transform of the noisy output sequence canbe shown to be

Sa(z) = S(z) + (l/2)[Pl(z)Exl(z)+Pl(z)Exl(-z)]

+ Etl{z) + (l/2)[Pu(z)Exu(z)+Pu(z)Exu(-z)}

+ Etu<?) (55)

Thus, the z-transform of the output noise sequence, ( z ) , isgiven by

£«,(*) = Sa(z)-S(z)

= (l/2)[Pl(z)Exl(z) + Pl(z)Exl(-z)) +Etl(z)

+ (l/2)[Pu(z)Exu(z)+Pu(z)Exu(-z)} +Etu(z)

(56)

4.4 Analysis of errors arising from arithmetic roundoffThe magnitudes of the errorsEx l ,Ex u ,E t l and Etu depend onthe structure of the filters and on the positions within thefilters where rounding is carried out. Two possibilities exist.The first possibility is that in which all multiplication productsare represented exactly and rounding is performed only afterthey are summed. In this case, noise is only superimposed onthe filter output. The output roundoff noise {eQ(n)} isuniformly distributed between ( - G/2, 0/2) with zero meanand variance A2. This error is independent of the form andstructure of the filters. With the second possibility, all multi-plication products are rounded before they are summed, withthe result that different structures will give rise to differenterror magnitudes.

The second possibility for rounding will now be consideredin some detail. A shared-multiplier structure, as described inSection 3, is assumed. M

Consider the z-transform Exl(z) = 2 exl(k)z~k, in whichfe = 0

each exl{k) is equal to the sum of errors contributed by therounding of the products, that is,

A T / 2 - 1

= X f o r (57)

and

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 261

N/2-1

< Z \eq(n)\ < (Q/2)(N/2) = NQ/4 (58)

Exl has zero mean and a variance given by

N/2-1

var(^) = Z e\(n) = (N/2)(A2) (59)

Similar results can be obtained for Exu(z), Etl(z) and Etu(z).The analysis of the resultant error can be carried out by

considering separate components. Let

k=o

N-l

n = 0

Thus,

< -'OI <lM=0

and on using eqn. 22 it can be seen that

\eai(k)\ < (NQ/4)(2) = NQ/2

The mean of &a l *s given by

N-l

fbol = Z Pl(n)exl(k~n) = 0n = 0

and its variance is given by

i V - l

(60)

(61)

I \Pt(n)\1=0

(62)

(63)

var(gal) =

(A72)(A2)(2) = AAA2 (64)

Similar results can be obtained for &a2(z) =Pl(z)Exl(—z),€ a 3 00 = PJz)Em{z) and ga 4(z) = PB(z)£•„,(-z).

Thus, if

,-h

fe =

then

lco.(*)l < (l/2){|e«i(*)l + \ea2(k)\ + \ea3(k)\ + \ea4(k)\}

+ \etl(k)\ + \etu(k)\ = (3NQI2) (65)

If it is assumed that 6>a\, &a2, 6 a 3 , Sa4, Etl and Etu areuncorrelated, then the mean of the random variable g a s is zeroand its variance is given by

var ( g a , ) < (1 /4) {NA2 + NA2 + A^A2 + WA2 } + (N/2)A2

+ (N/2)A2 = 2NA2 = NQ2/6 (66)

which is independent of the magnitude of the input signal.

4.5 Effects of combined coefficient quantisation andarithmetic roundoff

In this Section, the errors resulting from both coefficientquantisation and the arithmatic roundoff are analysed. The

262

practically important case in which the same word length isused throughout for data storage and arithmetic operationswill now be considered.

The z-transform of the noisy output sequence is

Sb(z) = (\/2)[Hl(z)Pl(z)-Hl(-z)Pl(-z)\X(z)

+ (l/2)Pl(z)[Exl(z) + Exl(-z)} + Etl(z)

- (l/2)^(-z) [Exu(z) + Exu(-z)] + Etu(z)

= S(z) + ^ ( z ) + ga 8(z) + g,.s(z) (67)

where

&iste) = (l/2)Ep(z)[Exl(z) + Exl(-z)}

- (1/2) Ep(-z) [Exu(z) + Exu(-z)] (68)The z-transform of the output noise sequence, due to com-bined coefficient quantisation and arithmetic roundoff, is thus

Note that this consists of separate contributions from coef-ficient quantisation and arithmetic rounding and from theinteraction term &,-s(z).

4.6 Error analysis associated with coefficient quantisationand arithmetic roundoff

Consider the interaction term & ,-8(z), and let

where

= Z en(k)z-k = Ep(z)Exl(z)fe=0

N-l

- I-n=0

(69)

(70)

Using exprs. 58, and assuming that the random variablesEp andExi are independent, it can be seen that

n=0

< ( Q/2) (NQ/4) (N) = (NQ)2/8 (71)

and similar results can be obtained for the other three terms ofate)Therefore,

< 4(1/2) (NQ)2/S = (NQ)2/4

and

+ | € . (*) | + | eu(k) I

<(Q/4)(l8N+3N2Q-NQ-6)

(72)

(73)

The random variable Stl has zero mean and its variance isgiven by

var(Sn) = (A2)(7VA2/2)AZ1(On=0

= (iV2/2)(A4) (74)

Similar results can also be obtained for the other three productterms of 6 IS(z). The random variable fei8 consists of linearcombination of four random variables which are assumed to beuncorrelated. The mean of fbis is thus zero and its variance is

var(£ i8) = 4(1/4) (W2/2) (A4) = (iV2/2)(A4) (75)

Also, if it is assumed that SC f i ,Sa 8 and &fe are uncorrelated,

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

then the mean of fbbs is

and its variance is

var (gb s) = var (£„.) + Var ($„) + var (g l s)

<(5N/4) [1 + (2WA2/5)] (A2)

N-l

(76)

= (13JV/4) (A2) + (W2A4) (77)

As A2 is usually small, the contribution of the error variancedue to interaction can be ignored. Var (tbbs) can be effectivelyobtained by simple addition of the error variances of the com-ponent sources.

4.7 Effects of quantisation for transmission bit ratereduction

In order to transmit the split-band signal at low bit rates, thedecimated sequences {yi(n)} and {yu(.

n)} a r e compressioncoded using only very few bits. In the following analysis, linearquantisation is assumed for each individual element of thesequence. With this assumption, the quantisation of thedecimated sequences is the same as that obtained through thequantisation and subsequent decimation of the sequences{*/(«)}, {*«(«)}•

Let %i(z) and %u(z) denote the z-transforms of the quan-tised sequences {xj(n)} and {xu(n}}, respectively, and defineEpl(z) and Epu(z), such that

Epl(z) = Xt(z)-Xt(z) (78)

Epu(z) = Xu{z)-Xu{z) (79)

The z -transform of the noisy output sequence is then

Sp{z) =

(80)

and if the z-transform of the noise sequence imposed on theoutput is defined to be fbps(z), then

+ Epl(-z)]

(81)

From eqn. 81, it should be noted that part of the noise is, infact, aliasing noise. Thus frequency aliasing cannot be com-pletely eliminated if the signals {yi(n)} and {yu(n)} arequantised. The frequency spectrum of the output noise in-cluding the aliasing noise depends on the spectra of {yi(n)},{yu(n)} and the frequency responses of the filters Pt andPu-

4.8 Analysis of errors with quantisation for transmissionbit rate reduction

Consider

As

so

N - l

Pi(k)ePi(" - k )

(82)

(83)n = 0

N - l

In=0

<(G/2) I IPi(*)K(G/2)(2) = Q (84)n = 0

The random variable & pi has zero mean and a variance givenby

N - l

var(€Pi)= Y, Pi(k)el(n~k)n=0

= (A2) f1pl2(A:)<(A2)(2)

n=0

Similar results can be obtained for the other terms.Thus, if

= £ eps(k)zfe = 0

-fe

then

< 0/2) (G + G + G + G) = 2(2

(85)

(86)

(87)

and if it is assumed that the four terms of &ps(z) are uncor-related, then the mean of feps is zero and its variance is

var(gp s)<4(l /2)2(2A2) = 2 A2 = Q2/6 (88)

The error variance var (length of the filters.

5 Simulation results

is thus independent of N, the

Computer simulations were carried out using a CDC 6500computer which operates to 28 significant digits when themachine is used in the double-precision mode. The resultsobtained from computations using double-prescision arithmeticwere taken as being 'infinitely accurate'. The system structureused was that described in Section 3. The impulse response andcharacteristics of the filters used are given in Appendix 9.2.

The error variance a2r(co) of the frequency response of the

overall system, using finite-word-length coefficients withfilters of length N = 8, 16 and 32, are shown in Figs. 4a, band c, respectively. The input used in this case was an impulse.

As indicated in Section 4.2, the bound (exprs. 46) is over-estimated by a factor of approximately two for large N. Thus,most of the points lie at a distance below the bound, especiallywhen N is large.

Using the theoretical estimate of variance, lines of deviationof a^ijjS) and 2ocr(cS) from the 'infinitely accurate'maximumpeak-to-peak ripple have been derived, and in Figs. 5a, b and cthese are superimposed on the graphs, showing the maximumpeak-to-peak ripple of the overall frequency response of thesystems when the filter coefficients are quantised with differentnumbers of bits. It can be seen that the actual peak-to-peakripple of the system frequency response for coefficientsquantised with the number of bits shown are within 2acr(co)of the nominal values for the three cases investigated withfilters of length N = 8, 16 and 32. The peak-to-peak ripple insome cases is smaller than the nominal values as the methodadopted for the filter design does not necessarily provide aglobal optimual solution.

The error variance of & „ , defined in Section 4.2, wascomputed using (a) speech, (b) Gaussian distributed randomdata, and (c) uniformly distributed random data. The com-puted variances, together with the theoretical bounds (exprs.50) are shown in Figs. 6a, b and c for systems with filters oflength 8, 16 and 32, respectively. The speech input comprised512 samples obtained by the analogue-to-digital conversion ofa segment of voiced speech. The speech was sampled at 8k-samples/s. The mean square value of these samples (after gainadjustment) was 0.036039. The Gaussian random samples

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 263

were obtained from a normal distribution having zero meanand a standard deviation of 0.25, samples with magnitudegreater than 1 being discarded. The mean square value of the512 samples was 0.0640976. The uniformly distributed dataranged from — 1 to 1 and the mean square value of the 512samples used was 0.344866.

From Figs. 6a, b and c, it can be seen that the error vari-ances computed from simulation results are below the theore-tical bounds. For a given word length, the magnitude of theerror variance for the various inputs is seen to be approxi-mately in proportion to the mean square value of the respec-tive input. This is in agreement with the prediction givenin eqn.49.

Simulation results of errors arising from arithmetic round-off, together with the theoretical values calculated usingexprs. 66 for systems using filters of length 8, 16 and 32, areshown in Figs, la, b and c, respectively. It can be seen thatthe experimental results using the three different inputsequences are in excellent agreement with the theoreticalbounds.

Figs. 8a, b and c show the results of a computer simulationwhen both the coefficients and the multiplication productsare rounded. The results are shown for the three types of inputdata and filter length of 8, 16 and 32. Also shown in theFigures are the theoretical bounds of var (£bg), the error vari-ance due to coefficient quantisation and arithmetic roundoff.

10.-3

10',-5

7

0

10"

10"

10"

- 210

10'

§10

2i_ -8

210

1610

-1210

io2

10*

results

11 13 15 17 19 21 23number of bits

a

bound

results

X X

11 13 15 17 19 21 23number of bits

b

results

7 9 '.1 13 15 17 19 21 23number of bits

cFig. 4 Error variance a\r (us) of frequency response of system dueto coefficient quantisation

(a)N= 8,(b)N= 16, (c)N= 32

264

0.072

0.060

0.048

0.036

0.024

0.012

0.00011 13 15 17

number of bitsa

19 21 23

0.072

13 15 17number of bits

0.175

o, 0.150aa- 0.125

a

& 0.100oA 0.075

0.050

QO25

0.00013 15 17

number of bitsc

19 21 23

Fig. 5 Illustration of deviation of normalised peak-to-peak ripplefrom nominal value due to coefficient quantisation

(a)N= 8,(b)N= 16, (c) N = 32

+ a+ 2a— a

—2a

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

It should be noted that as the values of var ($„) are depen-dent on the values of the input as indicated by eqn. 49, itsbound (exprs. 50), derived on the assumption that all inputdata take on the maximum values, is overestimated for mostpractical cases. As var (ft>a) is one of the constituent terms ofvar (§b , ) , the bound on var (&&,), given by eqn. 77, is alsooverestimated for most practical cases.

For errors arising from quantisation associated with trans-mission bit rate reduction, the theoretical values and resultscalculated from experimental data for various values of M arecompared in Fig. 9. It can be observed that there is excellentagreement between the two sets of values for the range of Mconsidered, especially for M larger than 4.

-810

<- -102 1 0

-1210

-14

bound

10 12 14 16number of bits

a

18 20 22

-610

-810

coo -1 0

-1210

-1410

bound

A •

10 12 14 16number of bits

b

18 20 22

6 Conclusions

Problems relating to the implementation of split-band codingsystems employing quadrature mirror filters have been inves-tigated and the theoretical bounds and statistics have beenderived for errors arising from the various sources of quan-tisation involved. The results derived provide guidelines forthe design and implementation of split-band systems. A simu-lation has been performed and excellent agreement has beenfound between the simulation results and those predictedtheoretically. The closeness of the agreement indicates thatthe assumptions relating to the various noise models holdgood in practice.

7 Acknowledgment

One of the authors, S.W. Foo, is in receipt of a DSO scholar-ship, and he wishes to thank the Ministry of Defence, Singaporefor this financial support.

8 References

1 ESTABAN, D., and GALAND, C : 'Application of quadraturemirror filters to split band voice coding schemes'. Proceedings ofthe 1977 international conference on acoustics, speech and signalprocessing, Hartford, CT., May 1977, pp. 191-195

2 BARABELL, A.J., and CROCHIERE, R.E.: 'Subband coder designincorporating quadrature filters and pitch prediction'. Ibid., Wash-ington DC,1979,pp. 530-533

3 RAMSTAD, T.R., and FOSS, O.: "Subband coder design usingrecursive quadrature mirror filters'. Proceedings of European Asso-ciation for Signal Processing conference, Lausanne, Switzerland,1980, pp. 747-752

4 FOO, S.W., and TURNER, L.F.: 'Design of nonrecursive quadraturemirror filters',IEE Proc. G, 1982,129, (3), pp. 61-68

5 CHAN, D.S.K., and RABINER L.R.: 'Analysis of quantisationerrors in the direct form for finite impulse response digital filters',IEEE Trans., 1973, AU-21, pp. 354-366

6 HOGG, R.V., and CRAIG, A.T.: 'Introduction to mathematicalstatistics' (Macmillan, 1971)

,65

-710

-910

io11

-1310

bound

10 12 14 16 18number of bits

c

20 22

Fig. 6 Error variance of output due to coefficient quantisation

(a)N=2,,(b)N = 16, (c) N = 32Inputo speechA Gaussian distributed+ uniformly distributed

IEE PROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 265

9 Appendixes

9.1 Analysis of the error &i

Let

it follows that

2N-2

fe=Oecl(k)z-k = Hx{z)Ep{z) (89)

Then

Hi

, since

(z)Ep (2) = IN

N-l

« = o-1 N - l

I;=o

106

o ~8§ 1 0

, 6 1 0

-1210

10

-I10 r

-610

> 10

-1010

-1210

12

I I»=o y=

(90)

theoretical values

14 16 18number of bits

a

20 22

theoretical values

10 12 14 16 18number of bits

b

20 22

U(91)

where the summation is over all i, / such that 0 < i < N — 1,0<j<N-l and i + / = it.

In general, for 0 < k < A — 1,k

Cd(*) = Z ^ O N M * " " * ) (92)m =0

2AT-2 -fe

m =0(93)

The following points should be noted:(i)Scil2) contains an odd number (27V — 1) terms(ii) since hl(n) = hl(N- 1 -n), n = 0, 1 , . . . , N- 1 is

symmetrical, it follows that ecl(k) = ecl(2N — 2—k), k = 0,\,...,2N-2

(iii) e^iV— 1) has the maximum number of terms of theform fc,(*)ep (*) of all the ec l (k), k = 0, 1,. . . , 2 /V- 2.

From eqn. 92, it can be seen that

£ \hi{m)\\ep{k-m)\<{QI2) Xm =o m =0

KQ/2) (1/2) = 0/4 forI((2/2) (1) = 0/2 for {Nl2)<k<N-\

(94)

10?

-710

-910

-11in

# \

-

-

-

-

A. theoretical values

o \

\

* \

1 1 1 1

10 12 14 16 18number of bits

c

20 22

Fig. 7 Error variance of output due to arithmetic roundoff

(a)N=8,(b)N= 16, (c) N =32Inputo speech^ Gaussian distributed+ uniformly distributed

266 IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

and similarly, from eqn. 93

7N-2-H

I \hi{N-\-m)\\ep(k-N+\+m)\Im =o

2N-2 -ft

Im =0

\hl(N-l-m)\

f(G/2)(l) = G/2 for N-\<k<(3N/2)-2[ ((2/2) (1/2) = (2/4 for (3N/2) - 1

- 2(95)

-610

S 10

-1010

-1210 10 12 U 16

number of bitsa

18 20 22

ios

10

-1010

-1210

theoretical values

10 12 U 16 1ftnumber of bits

b

20 22

192

10

-6s

-810

" \

\<

-

-

1

* \ theoretical

o \

6 \

i i

values

© \

7 9 11number of bits

13 15

Fig. 9 Error variance of output due to quantisation for transmissionbit rate reduction

Inputo speechA Gaussian distributed+ uniformly distributed

10

S - 7

I

10 12 14 16 18number of bits

c

20 22

Fig. 8 Error variance of output due to coefficient quantisation andarithmetic roundoff

(a)N= 8,(b)N= 16, (c) N = 32Inputo speech^ Gaussian distributed+ uniformly distributed

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 267

The mean of § c l is given by

IN -2 \)(

I ecl(k))/(2N-l)fe=o /N - l N - l

and its

var

= Ii = O

variance i

(So,)

=

is given by

[l/(2tf-l)J

[1/(2AT-1)]

2N-2

7 P2 mfe=O

N - l N - l

1=0 j = 0

N-l

(96)

< [N/2(2N-l)]A2

Similar results can be derived for the term

Sc2(z) = Hl(-z)Ep(-z)

In the case of § c 3 (z) , on using

2N -2ScaGO = I ec3(k)z~k = Px(z)Eh(z)

fe=o

(97)

(98)

(99)

and applying similar arguments to those used in connectionwith £ c l (z), it can be seen that

\ec3(k)\< Z IPiWlkp(A:-m)|<(e/2) £m =0 m =0

|(G/2)0) = Q/2 for 0<A:<(/V/2)-l

l(G/2)(2) = G for (N/2)<k<N-l

(100)and

2N-2 -fe

fn =0

2N-2 -fe

m =0

f(G/2)(2) = G for N-Kk<(3N/2)-2

" [(G/2)(l) = G/2 for (3N/2)-\<k<2N-2

(101)

The mean of the random variable £c3 is zero, its variance isgiven by

N - l N - l

var(gc3) =i=o y=o

X»=o

= [27V/(2/V-l)] (A2) (102)

and similar results can be obtained for the term

Sc4(z) = Pl(-z)Eh(-z)

Next, if

2AT-2

^cs(^) = I ec5(fc)z-fe = ^h(^)^p(^)fe=0

then since

(103)

(104)

it follows that

«=oN - l N-l

Z Z e/=o y=o

ePU)z-j

(105)

Z eh(m)ep(k -m) for 0<k<N-lm =o

2JV-2 -fe

Z eh(N-l-m)ep(k-N+ 1 + m)m =0

for N-Kk<2N-2 (106)

and hence

\eh(m)\\ep(k-m)\m =0

<((22/4) Z (1)m =o

= (G2/4)(A:+1) for 0<k<N-\ (107)

Similarly,

|ec5(A:)|<((22/4)(2/V-l-/t) for

J V - K K 2 J V - 2 (108)

If £"h and Ep are assumed to be independent, rather than justuncorrelated, then the mean of the random variable EhEp isequal to the product of the means of Eh and Ep and is thuszero. With independent Eh and Ep, the variance is then equalto the product of the variances of Eh and Ep and is thusequal to (A2)2 = A 4 .

Hence 6>cs n a s z e r 0 mean, and a variance

N - l N - l

Z Zi=o ; = o

var(gc5) =

= [7V2/(27V-1)](A4)

Similar results can be obtained for

gc6(z) = Eh(-z)Ep(-z)

(109)

(110)

268

9.2 Characteristics of quadrature mirror filters used in thesimulation

The following filters are designed using the method proposedin Reference 4. The grid density used is 12. Nominal passbandresponse is OdB. Only the lowpass responses {h^k), k = 1,/V}are given, as the response of the other filters in the split-band

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982

system can readily be derived from knowledge of the lowpassresponses.

9.2.1 Length of filter (N): 8.Normalised passband cutoff frequency: 14/96 = 0.1458Normalised stopband cutoff frequency: 44/96 = 0.4583h(0) = + .54920980658135847062^-02h(l) = - .64441133083122211809^- 01h (2) = + .61265467158468209490^ - 01/i(3) = + .49450790193418641529£ + 00Stopband attenuation:

.63513318493080046368£ - 02 = - 43.94 dBMaximum normalised peak-to-peak ripple of the systemresponse: 0.0324388426

9.2.2 Length of filter (N): 16Normalised passband cutoff frequency: 38/192 = 0.1979Normalised stopband cutoff frequency: 68/192 = 0.3542h (0) = + .12036986232979960262^ - 02h{\) = - . 13947183910049689982£ - 01h(2) = + .86878452750951197325^ - 03h(3) = + .39541353135492923529^-01/?(4) = - .20123800393657739950^- 01h(5) = - .97166531339110921907E - 01h (6) = +.11024493072165109992^+00h(T) = + .47627724001768050680^ + 00Stopband attenuation:

.62030172343726271870^-02=-44.15 dB

Maximum normalised peak-to-peak ripple of the systemresponse: 0.0342363782

9.2.3 Length of filter (N): 32Normalised passband cutoff frequency: 86/384 = 0.2240Normalised stopband cutoff frequency: 116/384 = 0.3021h(0) = + .5 3 77003606143 793 5040£" - 03fc(l) = -.54374312401735415346£'-02h{2) = - .79028701844009622402£ - 0 3h(3) = + .81981463902728243769^-02h(4) = - .25020466621689314679£ - 0 3fc(5) = - . 14123528885178776032^ - 0 1h(6) = + .28329773005496698388^-02h(l) = + .22591238868226283681£-01h(S) = - .83615826500139088530£ - 02h(9) = - .34912955265598788586£ - 0 1h(!0) = + .19495516603424058872^ - 01h{\ 1) = + .54937320697082723837^ - 01h{\7) = - . 44265408015541034044£- 01h{\3) = - .99626628146938275104£ - 01h(\4) = + .13256233665235574877^ + 00h(l5) = + .46406696724962884217^+00Stopband attenuation:

.50916435318935652383^ - 02 = - 45.86 dBMaximum normalised peak-to-peak ripple of the systemresponse: 0.0310370518

Foo Say Wei received the B.E. degree inelectrical engineering from the Universityof Newcastle, Australia, and the M.Sc.degree in industrial engineering from theUniversity of Singapore.' Since graduatinghe has been working in the Ministry ofDefence, Singapore, first as head of theElectrical Branch and subsequently ashead of a research department. He ispresently on leave and is working for thePh.D. degree in electrical engineering at

the Imperial College of Science and Technology, London.

Laurence Turner served an apprentice-ship with AEI, during which time heobtained ONC and HNC certificates. Hethen studied at the University of Birm-ingham from which he received the B.Sc.and Ph.D. degrees. After a period as aresearch fellow at the University of Birm-ingham, he joined Standard Telecom-munications Laboratories Ltd., where heworked on aspects of pattern recognitionand data transmission. He subsequently

joined the Electrical Engineering Department at ImperialCollege, where he is at present Professor and head of theDigital Communications Section.

IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 269