IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-23, NO. 4, APRIL 1976

The Design of Multidimensional FIR 185

Digital Filters by Phase Correction M. T. MANRY, STUDENT MEMBER, IEEE, AND J. K. AGGARWAL; FELLOW, IEEE

Abstract-The present paper presents a new technique for the +zs@ of multidimensional FIR digital filters. Given the desired amplitude respqi?se, the truncation error, of a low-order approximation to a high-order filiki,‘is iteratively decreased by varying the phase response. Convergence of t6e iterative algorithm is discussed. Examples illustrating the use of the technique are included. The design method is formulated for two- dimensional filters, however filters of any dimension may be designed by this technique.

I. INTRODUCTION

T HE LAST SEVERAL YEARS have seen an increas- ing uve of two-dimensional digital filters for the

processing of aerial and satellite photos, the enhance- ment of two-dimensional geological data and medical X-rays, and in the pre-processing of digital images in pattern recognition applications. The present paper devel- ops and illustrates a new technique for the design of multidimensional filters. However, in the following, the design technique is formulated for the two-dimensional case.

A two-dimensional FIR digital filter is described by the Z-transform transfer function

M-l N-l ~(Z,,Z*)= 2 2 knnz;“z; (1.1)

m=O n=O

or by the difference equation

M-l N-l 57 mn= 2 2 hkldtn-k,n-I

k=O I=0 (‘4

where {h,,,,} is the filter coefficient spatial sequence, {dmn} is the’ input spatial sequence, and { gm} is the output spatial sequence.

Several techniques [1] exist for designing one- dimensional FIR digital filters and a few [2] of these techniques have been extended to two dimensions. A fast technique, for the design of a special class of two- dimensional FIR digital filters, is by McClellan [3]. The simplest design method, and the one most easily extended to two dimensions, is the Fourier series coefficient tech-

Manuscript received March 3, 1975; revised November 19, 1975. This work was supported in part by NSF Grant GK42790 and in part by AFOSR Grant 72-237 1.

The authors are with the Electronics Research Center and the Depart- ment of Electrical Engineering, The University of Texas at Austin, Austin. TX. 78712.

nique. As shown by Rabiner [4] for the one-dimensional case, this method can be approximated by using the @crefe Fourier transform (DFT).

For every spatial sequence {h,,} of length (M, N), we can find a corresponding frequency domain sequence { Hk,}, which is the DFT of {h,,,,}. The DFT relationship, denoted as h,,,eHkl, is described as

M-l N-l Hkr = 2 2 h,, Uk”V’”

m=O n=O (1.3)

and M-l N-I

h,,,,=(l/MN) c 2 Hk,U-k”V-‘n (1.4) k=O I=0

where U = exp ( - j2rr/ M) and V = exp ( - j27r/ N). Thus a filter described by (1.1) is specified by the MN points of the spatial sequence, {h,,}, or by the MN points of the frequency domain sequence, { Hkl}. The frequency domain sequence is expressed by an amplitude sequence, { 1 Hk,l}, and a phase sequence, {ok,}, where

In general, for a given amplitude sequence, distinct choices of the phase, sequence { okkl}, lead to distinct im- pulse response sequences, {h,,,,}.

Let T,, be the set of nonnegative integer pairs (m,n), where O<m<M-1 and O<n<N-1, and let T be a subset of To. Given a spatial sequence { h,,},(m,n) E T,,, we wish to truncate {h,,} to form a filter

M-l N-l iqZ,,Z,)= 2 2 imz;lz; (1.6)

m=O n=O

such that &,,,, = h,, if (m, n) E T and &,, = 0 if (m, n) SE T. In general T may be any set of nonnegative integer pairs satisfying the condition above, however in the present paper the choice of T is restricted to

T={(m,n)EZ2~0~m&M,,0<n<N,} (1.7)

where M, < M- 1, N, < N- 1, acd Z* denotes the set of all integer pairs. In this case F(Z,,Z,) is a low-order approximation of F(Z,,Z,). In the following sections a method is presented for picking spatial sequences, {h,,,,,}, such that the truncation errQr between F(Z,,Z,) and

186 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, APRIL 1976

j(Z,, Z,) is minimized. Further, the objective is to design and a low-order filter with truncation error small enough so that the operation of windowing is unnecessary. Zii, = IGL,l exp (j&L, ). (2.8)

As seen in (2.7) the amplitude sequences {IHl,I} are II. THE PHASE CORRECTION ALGORITHM restricted by the equation I HL,I = ( Hk,l for every i, which

Given the desired amplitude sequence { IHk,l} and the implies that the amplitude sequence is held to be the same for every iteration. Further, a simple application of

s:t T, the truncation error between F(Z,,Z,) and parseval’s equation leads to F(Z,,Z,) may be written as

M-l N-l

mzo n~o(h,fm)2=W~N)M~’ N%l IfLl*=E P-9) k=O I=0 M-l N-l

z({ek,})= x x hm!* m=o n=O (m,n) ET

and

lh,fJ < El’* (2.10) M-l N-l =(‘/W2N2)) c I2

mro n=O (mn) bT

‘Hkh?xp(jek,)U-kmv -In 2

P-1)

a function of the phase sequence {8,,}. The expression (2.1) is arrived at by the use of the earlier equations (l.l), (1.4)-(1.6). Z({O,,}) is a measure_ of the goodness of the approximation of F(Z,,Z,) by F(Z,,Z,). In this section we derive an iterative method for picking phase sequences, {Ok,}, so that the error function Z({O,,}) is decreased with respect to {Ok,}.

For the ith iteration, where i is a nonnegative integer, the high-order filter and its truncated version are de- scribed, respectively, as

M-l N-l

F’ (Z,,Z,) = 2 x h;,,Z;tz; m&O n=O

(2.2)

and

M-I N-l 3(Z,,Z,)= % 2 i;,z;“z; (2.3)

m=O n=O

where

$;,, = h,&, (m,n)E T =o, (m,n)E T.

(2.4)

It may be emphasized that i is a superscript and not an exponent. The frequency domain sequences, {HL,} and {HL,}, are obtained from spatial sequences as

for every i, m, and n. Four equivalent equations for the error between

F’(Z,,Z,) and &‘(Z1,Z2), where i andp are nonnegative integers, are:

i) Z@=(l/4~*)f//F~(exp(-iw,),expo) -97

- Fp (exp (- jtii), exp ( - jw,))l*do,dw, (2.11)

M-l N-l

ii) Z@= 2 x (hL,-- &,)* m=O n=O

(2.12)

M-l N-l

iii) Z*=(l/MN) x x IH&--&,I* k=O I=0

(2.13)

and

M-l N-l

iv) Z@‘=(l/MN) 2 2 IHk,j2+jZ?[,12 k.=O I=0

-2~Hk,IIEi,qlcos(e&-&,). (2.14)

Equations (2.12) and (2.14) follow directly from the evaluation of (2.11) and (2.13) respectively. Equation (2.13) follows when

M-l N-l

hA,,=(l/MN) 2 2 H,j,U-kmV-‘n (2.15) k=O I=0

and M-l N-l

&,=(l/MN) x x fi[,U-k”V-,n (2.16) k=O I=0

and

ha;,++&.

The amplitude and phase relationships are I

Hti, = l&l exp (jell )

are substituted into (2.12). Using (2.4) in (2.12) and letting (2.6) p = i, we find that

M-l N-l

(2.17) I”= x x (h,&)*

(2.7) m=O n=O (m,n)ET

MANRY AND AGGARWAL: MULTIDIMBNSIONAL FIR FILTERS 187

and using (2. I), (2.5), (2.7), and (2.17), that

z({el, })=z”. (2.18)

From (2.18) it is obvious that an algorithm which itera- tively produces a decreasing error sequence

p={z~,zll,..., 1” ,... }

also decreases the function Z ({ ZZ,,}) iteratively. Given the desired amplitude sequence {]ZZk,]} and the

set T, the first step of the phase correction algorithm is to choose an initial phase!, sequence { I$$}. In order for the sequences { Zz;,} and {A,&,} to be real, all phase sequences, including the initial phase sequence, must satisfy the sym- metry condition

e~1=27Tn:I-eiM-k~nr,[N-l~hl (2.19)

where {n:,} is an integer sequence, and [M- klM,[ N - IIN, are integers modulo A4 and N, respectively.

The numbers ZE and i,, are specified, in the algo- rithm’s first step, to provide an exit from the algorithm. The integer i,,, is the maximum value of i allowed. In other words, only (i,, + 1) iterations are allowed. ZE is a nonnegative real number such that if I’-‘*‘-’ - I” < ZE for some i, the algorithm is halted. The steps of the algorithm

Fig. 1. Block diagram for original algorithm.

are given Step 0:

Step 1: Step 2: Step 3: Step 4:

below. Given { ]Hkrl}, { 0,“l}, T, ZE, and i,,,, compute (HE) from (2.7). Set-i-equal to 0. Calculfte {Zz;,,} using the relationship (2.5). Get {h,&} from (2.4). Calculate I” using (2.17). It may be noted, from (2.12) and (2.17) that

Step 5: Step 6: Step 7:

Iii < zi,i- 1.

Exit if I’-‘si-l-Zii<ZE or i=i,,,. Compute {Z&} from (2.6) and (2.8). Increase the index i by 1. Using the equation

(2.20)

Step 8:

q, = 6;; 1

calculate {ZZL,} from (2.7). Go to Step 2.

(2.21)

Whereas I’- ‘*i- ’ can be expressed by (2.14), I’,‘-’ can be expressed, using (2.14) and (2.21), by

M-l N-l

Z iJ-‘=(l/MN) z x (]Z+]&$ (2.22) k=O I=0

This means that

zi,i- 1 < Ii- l,i- 1. (2.23)

The implementation of the phase correction algorithm is shown in Fig. 1. In the figure we have implemented

Step 7 with

~~,=IH,,l(~~~‘/l~~~‘l) (2.24)

which follows from (2.8) since

exp(jf$’ )=Z?L,-‘/Ifi;,-‘I. (2.25)

By alternately decreasing the truncation error in the spatial and frequency domains, the algorithm generates new spatial and frequency domain sequences for each iteration. Inequalities (2.20) and (2.23) imply that the sequence P is nonincreasing, so we have shown that the phase correction algorithm decreases I({ e,,}) or leaves it the same.

If tn,, = cc and Is = 0 the infinite sequence

w=(z~,zlO,z~~,. . . ,zii,zi+l.i,. . .} (2.26)

is generated.

Theorem I

If zii=zi+I,i or zi+l,i=zi+l,i+I for SOme finite non- negative integer i, the sequence W converges in a finite number of terms.

Proof: case 1, zii=zi+Li: By using (2.13) and (2.22) one

obtains M-l N-l M-l N-l kIxo ,To If&- &12= kxo [X0 (If&l - IP~,l)2. (2.27) = 5

188 ’ IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, APRIL 1976

This implies that {@,} = {@,} and that {e$‘} = { $,}. However, there is often room for improvement. Inlthis Since the same operations are performed in each iteration, section a modified algorithm, which increases the rate { ep} = {e;,} for every nonnegative integer p. of convergence of the error sequence, is presented. In

case 2, Ii+ 1,i = Ii+ l,i+ 1.. By using (2.22) one obtains particular, an algorithm that approximates the high-order filter

M-l N-l

I’ J+1J=(l/d4iv> 2 2 pk,Iexp(je;:l ) k=O /=o

M-l N-l

?(Z,,Z,)= 2 x l&J;“z; (3.1) m=O n=O

- lz$$ exp (je;: 1 )I2

M-l N-l by the truncated filter = x 2 (h;‘-&,)’ M-l N-l

m=O n=O FqZ1,Z2)= 2 2 g,z;“z; (3.2) M-l N-l

= mso nzo(h,$-&,)2+ M$ Ni’(hG1)2.(2.28) m=O n=O

m=O n=O where the conditions (mn)E T (m,n)eT

~~~ Ii+ l,i= Ii+ l,i+ 1, (2.17) and (2.28) imply that the c;, = ii;,, (m,n)E T

right-hand side of (2.28) equals = 0, (m,n)E T (3.3)

M-l N-l

,s, nzo (h2 l)’

(m,n)l& T

and that

M-l N-l

mIxo nlzo (h2’-KL)2=o. (2.29)

(m,n)E T

Equ@on (2.29) implies that {k: ‘} = {LA,} and {e:;’ ‘} = {f$!!}, which leads to {BL:2} = { eL:i}. The same opera- tions are carried out for each iteration and therefore {ep+q={epj f or every nonnegative integer p. Q.E.D.

Corollary

The sequence W is always convergent. Proof: This is proved for a special case in theorem 1.

From [5] all bounded, nonincreasing sequences of real numbers converge. The members of W are bounded by 0 and Zoo, and W is nonincreasing as seen in (2.20) and (2.23). The corollary follows. Q.E.D.

The convergence of the error sequence W is proved above, but this does not imply the convergence of the sequences,

and

Y={{h~~}r{h~,},~~~,{h~,},~~~},. The convergence properties of X and Y will be dis-

cussed in a separate paper. and

III. INCREASING THE RATE OF CONVERGENCE

Normally the phase correction algorithm reduces the error function I({ e,,}) significantly in the first few itera-

and

are satisfied, is described in the following. The phase sequences of the high-order filters are denoted by {&} as in (3.4). As with the original algorithm every high-order filter F’(Z,, Z,) has { ]Hk,]} as its amplitude sequence. An expression for the error between i’(Z,,Z,) and Fp(Z,,Z2) is

M-lN-1 pP= 2 x (p//yg2. (3.5)

m=O n=O .

The error sequence of the modified phase correction algo- rithm is given by

*I={foO,flO,. . .,I;‘i,I;‘+l,i,. . . 1. (3.6)

In a manner analogous to the original algorithm, the first spatial sequence is generated from { ]Hk,]} and {q,}, using the equation

{ k’k} = ZDFT { V4Aew(jd,)}. (3.7) Further, the first two iterations of the modified algorithm are identical to the first two iterations of the algorithm of Section II, meaning that

l;,, = h,&

for i = 0,l. For i = 2 the equations above also hold, but for i > 2 we form a second truncated sequence { yk,} in each iteration by using

MANRY AND AGGARWAL: MULTIDIMENSIONAL FIR FILTERS 189

where b’ is a real number chosen to accelerate the conver- gence of the algorithm. It may be assumed that in the first steps of the present algorithm, b” and 6’ are zero. The number b’ is chosen so that the sum of squared errors

M-l N-l

Ki= 2 x (/?;,-(f&‘+bi(~;,-1-h;;2)))2 (3.9) m=O n=O

between { LA,} and { Z& ’ + b i (6;; I - F& 2)} is minimized. This is achieved by differentiating (3.9) with respect to 6’ and setting the result equal to zero. This gives

M-l N-l x 2, (~~,-~~~‘)(~~~1-~~~2)

b’= m=O n=O

M-l N-l . (3.10)

x 2 (p;l-p3 m=O n=O

Given the sequence {y;,}, the operations

{H~,}=DFT{Y~~}

i=i+l

~~,;,=lH,,l(I?k,‘/l~~~‘l)

(3.11)

(3.12)

(3.13)

and

{~~~}=zDFT{z$,} (3.14)

are carried out for the implementation of the algorithm. This leaves the algorithm at the truncation step. Because of the way the sequences {y:,} are generated one cannot guarantee that

p<fi-l,i-1. (3.15)

Therefore, if (3.15) does not hold, let

(3.16)

b’=O (3.17)

and continue the iteration. In other words, if an iteration of the modified algorithm fails to produce a smaller error number than the previous iteration, we repeat the iteration with b i = 0, which is equivalent to performing an iteration in the original algorithm. The final error sequence I@ has the property that

fii<fi-l,i-1 (3.18)

for all i greater than 0. If the iteration fails in the above fashion the value of the error function for two successive iterations is the same, meaning

-.. zrt=I;‘-l,i-I (3.19)

However, this does not imply that the modified algorithm has converged. Whenever (3.16) is implemented, the equa- tion

p&l&-2 mn

Fig. 2. Block diagram for modified algorithm.

holds in the following iteration, giving a zero denominator in (3.10). This is easily remedied by replacing IL;2 by K&3 in (3.10) in an iteration following the implementation of (3.16).

The block diagram for the modified phase correction algorithm is shown in Fig. 2. As with the original algo- rithm, the numbers Z, and i,,, provide exits. The steps of the algorithm are given below in detail. As seen in (3.10), b’ can’t be calculated if i < 2. Therefore, (3.17) is used for i=O and 1.

Step 0:

Step I: Step 2:

Step 3: Step 4: Step 5:

Step 6: Step 7: Step 8: Step 9: Step IO: Step 11: It may

algorithm

Given {If&l>, { ~~~}, T, Z,, and i,,,, set i equal to zero and compute { Z?,$} using

fi”l = IfhI exp (A$). (3.20)

Calculate {LA,} using (3.14). If (3.15) holds, continue to Step 3; otherwise skip to Step 6. Perform the truncation in (3.3). Exit if I;‘- 1*i- ’ - I;‘i < Z, or i > i,,,.

If i < 2 set b’ = 0; otherwise calculate 6’ from (3.10); go to Step 7. Perform (3.16) and (3.17). Compute { y;,} using (3.8). Find { Z?L,} from (3.11). Increase i by, 1. Calculate {Hi,} using (3.13). Go to Step 1.

be added that the extra steps of the modified do not add significantly to the computation

time of an iteration. Although faster converging error sequences have been observed when using the modified

190 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, APRIL 1976

algorithm, an analytical proof of the fast convergence is iv) It is specifically assumed in this section that T is not available at this time. However, this approach is described by (1.7), meaning that (2.3) may be rewritten as useful in many optimization problems where the dimen- sionality is too high for the Fletcher-Powell algorithm [6] to be practical.

P’(Z,,Z,)= z 2 2z;“z;. (4.6) m-0 n=O

IV. PHASE SEQUENCES AND PHASE ERROR If P’(Z,,Z,) is a linear phase filter with positive symmetry and if LAN,, iio, &, and ih,, are nonzero for:ome values

In this section a technique for choosing the initial phase elf. m, n, k, and 1, the phase response G’(w1,w2) ‘of sequence {f?,“,} is formulated and a method of comparing F’(Z~~Z2) can be expressed as the phase sequence {f?;,} to a set of desired phase sequences in developed. The following notation and ob- sji(w,,w2)= -(M1/2)U1-(N1/2)W2+~i(Wl,W2)7r (4.7)

servations are needed in this section. The frequency response of the filter F’(Z,,Z,) is ex-

where 7 i (w,, wz) is a function taking on values 0 or 1.

pressed in amplitude-phase form as F) The DFT_. that is used to get {Hi,} from {h;,,} and

{ Hl,} from {h;,,} is of order (M,N). This means that w, and o, are restricted by

F’ (q ( -jq>, exp ( -h2)) ‘,

= IF’(exp( -jw,),exp( -jw,))lexp(jfi’(w,,w,)) (4.1) w, = (Za/M)k, O<k<M-1 (4.8) w2= (2a/N)I, O<l<N-1 (4.9)

where I F’(exp ( - j,,), exp ( - jo2))l is the amplitude re- and that sponse and Gi(wl,w2) is the phase response. Salient prop- erties of the frequency response are as follows.

H;,= Fi((2n/M)k,(2v/N)I)

i) The symmetry conditions for the amplitude and Z&f,= Pi ((2a/M)k,(2a/N)Z). phase responses are

IF’ (exp ( -k), exp ( -iw2))I = IF’ (exp (jwd9 exp (h2))I Thus from (4.5), (4.8), and (4.9) a positively symmetric 1’ mear phase filter, Fi(Z1, Z,), has a phase sequence ex-

(4.2) pressed as

and fl&=Q’((2+V)k,(2?r/N)i)

~2’(wl,w2)=~~(w~,02)2~-~(-w~, -02) (4.3) where pi(wl, w2) takes on integer values. Conditions (4.2)

= -(K/2)(2a/M)k-(L/2)(2m/N)l

+ yi((2n/M)k,(2n/N)I)a. (4.10)

and (4.3) are necessary and sufficient for {h&} to be a real sequence. Let S’ be a subset of To such that (k,l)E S’ if y’((2a/M)

ii) Linear phase responses for digital filters may be k,(2n/N)l) =O, $? S’ otherwise. Equation (4.10) can now expressed as be expressed as

ei = kl -(K/2)(2m/M)k-(L/2)(2n/N)l (4.11) cqw,,o,)= -~~,/2-po2/2+~1T/2+y~(wI,w2)~ (4.4)

for (k, I) E S’ and where (Y, p, and .$ are nonnegative integer constants, with ,$ equal to 0 or 1, and y i (o,, w2) is a function taking on values 0 or 1. If [= 1 in (4.4), the filter is said to have negative symmetry and if [=O, the filter has positive symmetry.

iii) The results of this section are derived for filters with positive symmetry but they are easily derived for nega- tively symmetric filters as well.

If F’(Z,,Z,) has linear phase and positive symmetry, @(w,, w2) may be expressed as

f-li(o,,w2) = -(K/2)w, - (L/2)w,+ yi(wl,w2)a (4.5)

8’ k,= -(K/2)(2n/M)k-(L/2)(2w/N)l+n (4.12)

if (k,l)g S’. It should be emphasized that a linear phase response implies a linear phase sequence; however a linear phase sequence such as (4.10) does not necessarily imply a linear phase response. For a filter F’(Z,,Za to have the linear phase response of (4.5), the necessary and sufficient coefficient constraint is

h A,, = h:-m,L-n, m<Kandn<L, = 0, m>Korn>L.

(4.13)

Theorem 2 where K and L are nonnegative integers. If h&-,, hLo, hik, and h&-,,, are nonzero for some values of m, n, k, If the initial phase sequence { 6$} is described by (4. lo), and I, the only possible values ‘of K and L in (4.5) are with (K, L) = (M,, N,), every phase sequence {e;,} can also A4 - 1 and N - 1, respectively. be described by (4.10) with (K, L) = (M,, N,).

MANRY AND AGGARWAL: MULTIDIMENSIONAL FIR FILTERS

Proof: Let (K,L)=(M,,N,) and let U and V be as defined in Section I. Note that

exp (je;, ) = UkM1/2V’NI/2

if (~,Z)E S’ and

(4.14)

exp (je;, ) = - UkMg/2v’NI/2 (4.15)

if (k,l)eS’. For m < M, and n ( N,, (2.4), (4.14), and (4.15) imply

M-l N-l $,=(l/~jq x x IHk,Iu-k(“-M,/2)v-I(n--N,/2)

k=O I=0 (kl)es’

M-l N-l -(~/MN) x 2 IHk,lu-k(m-M,/2)v-l(n-N,/2).

k=O I=0 (k.l) cts’

(4.16)

The fact that M, - m - M,/2= -(m - M,/2) and N, - n - NJ2 = -(n - N,/2) leads to

M-l N-l

‘&m.N,-n +/MN) x x IHk,IUk(“-M,/2)l/l(n-N,/2)

k=O I=0 (k0e.s’

M-l N-l

- (l/MN) 2 x IH,,IUk(-W) v+N,/2)

k=O I=0 (kl) gs’

= & = &, (4.17)

for m G M, and n G N, . Using (4. lo), (4.13), and (4.17)

@,= -(M,/2)(2+M)k-(N,/2)(2vr/N)I

+ ?((2v/M)k,(2n/N)+ (4.18)

Equations (2.2 1) and (4.18) imply

8 i+l= kl -W,/W+W- (N,PWlNY

+ yi((2a/M)k,(2n/N)l)Ir. (4.19)

Replacing fi((2n/M)k,(2n/N)I) by yi”((2n/M) k, (2a/N)Z) in (4.19), { $,?‘} can be expressed by (4.10) if {fZ&} can be expressed by (4.10), when (K,L)=(M,, NJ. Since {f?,“,} is described by (4.10) with (K,L)= (M,, NJ, the theorem holds by induction. Q.E.D.

Corollary

If { 0,“,} is described by (4.10) error sequence W converges in tions.

and (K,L)=(M,,N,), the a finite number of itera-

Proof: Assuming (K,L) = (M,, N,) and (4.10) holds for i=O, theorem 2 implies (4.10) holds for every i. The function yi((2n/M)k,(2r/N)Z) takes on one of two val- ues for every one of the MN ordered pairs (k,l) in To. Therefore there are only 2MN possible candidates for the function yi((2n/M)k,(2r/N)l) and fewer than 2MN

191

when (2.19) holds. Since W is strictly decreasing before convergence and since different error values Z” corre- spond to different phase sequences {BL,}, there are fewer than 2MN iterations. Q.E.D.

The corollary to theorem 2 implies that if {ZZ:} is properly chosen the phase correction algorithm may be used to design linear phase response filters. However, in designing such filters, numerical problems in the FFT implementation of the DFT cause the symmetry condition of (4.13) to be violated. In other words

$, = i&-,,,+ + CA,,, m < M, and n < N,

where {e&} is an error sequence with nonzero members. The result is that the phase sequences {e;,} may not be described by (4.10) and W does not converge in a finite number of iterations. This can be remedied by implement- ing the arithmethic replacement statement

In general the limit Z”O”O of the sequence W depends upon the initial phase sequence {B,$], as illustrated in the next section on numerical results. In other words, the algorithm converges to a local minimum rather than to a global minimum. It may be possible to guarantee conver- gence to a global minimum in the 1 - D case. by making use of minimum phase versions of the filter coefficient polynomials. However, polynomials of two or more di- mensions generally do not have minimum phase versions and such a procedure cannot be extended to two or more dimensions. At present, the only way to insure that this error Z oOm is small is to pick {e,“,} such that Zoo is small. If Zoo is small, the fact that W is nonincreasing precludes the possibility that Zm” is large. Let

fl,o,= - (2+V)(K/2)k- (2m/N)(L/2)1 (4.20)

where K and L are nonnegative integers which are to be chosen such that Zoo is minimized. Typically, Zoo is minim- ized for some (K,L) close to (M,,N,). However, if (K,L) =(M,, N,), there are a finite number of iterations, using the corollary to theorem 2, and the limit of W is not much smaller than Zoo. For this reason one should not use (K,L)=(M,,N,) in (4.20) unless a linear phase response filter is mandatory.

Theorem 3

If {e,“,} is specified by (4.20), then

zoo= E- 16 Ix 2 qtm-K12M,[2n-L12N (4.2 1) m=O n=O

where

%n”lHkll (4.22)

where I Hk,l = 0 for k > M or I > N, and where the. DFT is for sequences of length (2M,2N).

192 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, APRIL 1976

Proof: From (2.9) and (2.17),

Zm=E- $ 2 (h;,,)‘. m=O n=O

(4.23)

Equation (4._20) leads to exp (jf$) = Uk_K/2V,L/2 and there- fore, using U=exp(-j2~/2M) and V=exp(-j2m/2N),

M-l N-l )&+/MN) 2 x IHk,lU-k(m-K/2)v-‘(n-L~2)

k=O I=0

ZM-1 ZN-1 =(4/(2~2~)) x 2 IHk,I~-k(2m-K)~;-‘(2n-L’

k=O I=0

=4q[2m-K12~,12n-L12~. (4.24)

On substituting 4q[2m-&,[zn-L],, for hz, in (4.23) one obtains the result. Q.E.D.

After using the ZDFT to find {qmn} from {I Hk,l}, one can easily find from (4.21) the values of K and L for which Zoo is minimized.

With the exception of the special case of theorem 2 and its corollary the phase sequences {e&} produced by the phase correction algorithm are in general nonlinear. Lin- ear phase sequences are usually desirable and it is there- fore profitable to compare the actual phase sequences {/$,} to linear phase sequences. Let the linear phase sequences, which we shall compare the actual phase sequences to, be described by

/3,“,=&(2vr/M)(D/2)k-(27r/N)(Q/2)1 (4.25)

where D, Q, and 6 are nonnegative integer variables and S is 0 or 1. A convenient measure for the error between { Z$,} and the linear phase sequences { 0ikdl) is

I&I’bP (jog ) - exP(j% )I”

(4.26)

Two advantages in using Ji as a measure of phase error are:

and’ the sequences {R (D, Q)}, { IHk,l}, and { HL,} are of length (2M,2N), with 1 Hk,l = H,f,=O for k > M or I > N.

Proof: From (4.25), (4.26), and (2.9),

i) phase differences of 2~ are equivalent to zero phase differences and

M-IN-1 M- lN- 1

Ji=2 x x IH,,12-2max k=OI=O

ii) phase errors in pass regions are weighted more heavily than phase errors in transition and reject regions.

=2( MNE-ma;((- ~)~~~~~~~lHk,IH;;UkD/2V,Q/2)). Suppose that for some positive A smaller than VT,

10,$-O;,/=A,(k,Z)ETO. (4.27)

Equation (4.27) implies that the error between {@,} and a linear phase sequence equals A at each of the MN DFT points in the frequency domain, and that

c9;,=0,d,+A,(k,l)ET,, (4.28)

If (4.27) holds and (4.28) is substituted into (4.26), the fact that lexp (je,“,) - exp (j(0,$ + A))12 = 2( 1 - cos (A)) implies

that

k-1 N-l

Ji= c 2 IHk,122(1-cos(A)) k=O I=0

M-l N-l

=2(1-‘-(A)) x c lHkr12 k=O I=0

=2MNE(l -cos(A)). (4.29)

For a given value of A, if

Ji < 2A4NE (1 - cos (A)) (4.30)

then

Ie,d,-e;j<~ (4.3 1)

will probably not be true for all (k,l) in To, but it will tend to be true in the pass regions, since the error in the pass regions contributes heavily to the value of Ji.

Given that (4.31) is desired for some positive A < 7, Ji can be calculated every few iterations and the algorithm can be halted when (4.30) is not satisfied. If the initial phase sequence is a linear one such as given in (4.20), (4.30) will hold at least for i=O.

It is assumed that the specific values of 6, D, and Q in (4.25) are not of critical importance. If the values are needed, they are easily determined by using the following theorem.

Thewgm 4 ,.:.. ”

For given values of E and { Hl,},

Ji=2MN(E- yy(lR(D,Q)I))

where

R (D, C? )‘dHk,IH~,

Replacing U iI2 by fi and V iI2 by ? and taking the conjugate above leads to

/(4/(2M2N))

ZM-1 ZN-1 . kxo x IH~,[H$?-~~~-,~ .

P I-0 1)

MANRY AND AGGARWAL: MULTIDIMENSIONAL FIR FILTERS

...... 111Brn...........@1#11..... m ...

... ..mlrmym...........qammm ....

4 ..mlrd~............~~~mmm .... ... ..mm119.............#*mma .... .. ..mmmrm9.............99mm9m ... .mmmmmmmm...............9m99mmm m mmmm9mmmm...............9mm9mmm fi mmmmmmmm.................mmmmmm m rmrmmmm...................mmmm~ m mmmmm.......................mmm m mm ........................... ..m ................................ .................... ..? ......... ................................ ................................. ................................ . . . . . . . . . . . . ..a................. ................................ ................................ . ..? ............................ ,.... . . . . . mm... mama1 maam 11181 aman .mmm . ..* 1 . . ..I . . ..I . . ..a

.........................

.........................

........................ .

................... ..amm m i...................mmmam m im..............~..mmmmmm a i9m.....~.........9mmmmmm m .mm...............9mmmmmm m ~rm9.............99mm9m ... 19m9.............99m9m .... 1m9).............m9mmm .... ~mmmm...........9mmmmm ....

Fig. 3. (lHk,j} for Filters 1.1, 1.2, and 1.3.

Using ZM-I ZN-1

R(D,Q)=(4/(2M2N)) 2 2 ~Hk,~H,j,~-kD~-‘Q k=O /=O

the result follows.

V. EXAMPLES AND DISCUSSION

The numerical implementation of algorithms described in previous sections, together with the design of filters and a comparison of their frequency responses is presented in this section., The computation was performed on the CDC-6600 at The University of Texas at Austin. The responses are printed in 16 grey levels with darker levels representing more positive quantities. For every grey level picture the range between the maximum and minimum intensities, of the function being displayed, was linearly divided into the 16 levels. Except for the linear phase response filters, every filter was designed in 100 iterations, with i,,, =99 and ZE=O.

Example 1

The de&red amplitude response is a ring described by

A(q,+)= 1, 1 <(w:+w:)“2< 2

= 0.02 3 otherwise (5.1)

(w21 < r. The values of @, N,M,, N, are respectively. The amplitude sequence Fig. 3. Using the initial phase sequence

where lwij < r and 32, 32, 7, and 7, { ( Hk,(} is shown in of (4.20) with K and L determined from theorem 3 to be 8, Filter 1.1 was designed by the algorithm of Section II and Filter 1.2 was designed by the modified algorithm of Section III. The amplitude and phase responses of Filter

Fig. 4. Amplitude response of Filter 1.2.

Fig. 5. Phase response of Filter 1.2.

1.2 are shown in Figs. 4 and 5, respectively. The responses of Filter 1.1 are almost identical to those of Filter 1.2, and are not given. The coefficients of Filters 1.1 and 1.2 are presented in Tables I and II. The initial and final error numbers for Filters 1.1 and 1.2 are normalized using the number E and given by

(Z”O/E)=(I”c@/E)=0.13185 (5.2)

(Z99Y99/E)z0.102859 P-3)

194 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. APRIL 1976

TABLE I COEFFICIENTS OF FILTER 1.1

TABLE II COEFFICIENTS OF FILTER 1.2

l’s represent log((Iii/E)-. ,028)

2’S re,xese”t log((?/E)-.1028)

and

(1”99,99/E)x0.10368. (5.4)

One gets an idea of the rate of convergence of I” and pi as i+99 from the plot of log((Z”/E)-0.1028) and log((I”“/E)-0.1028) versus i, given in Fig. 6. As th_e figure shows, the modified algorithm’s error sequence W is nearer convergence after 40 iterations than the original algoirthm’s error sequence W after 100 iterations. The coefficients of Filters 1.1 and 1.2, in Tables I and II, are significantly different.

Filter 1.3, also with the desired amplitude sequence 01 Fig. 3, had the linear initial phase sequence of (4.20) with (K, L) = (M,, N,) = (7,7). The filter was designed using the original algorithm and its amplitude and phase responses

Fig. 7. Amplitude response of Filter 1.3.

Fig. 8. Phase response of Filter 1.3.

are given in Figs. 7 and 8. The coefficients are given in Table III. The error sequence W, for this filter, converged at the fourth iteration, which is in agreement with the corollary to theorem 2. The normalized error numbers for Filter 1.3 are

(5.5) and

(Z3*3/E)~0.1302. (5.6)

MANRY AND AGGARWAL: MULTIDIMENSIONAL FIR FILTERS

TABLk III COEFFICIENTS OF FILTER 1.3

As predicted by theorem 2, the phase response of Fig. 8 is linear. The error numbers of (5.5) and (5.6) show that P3(2,,Zz) is not significantly better than P’(Z,,Z& The final filter is usually a small improvement over the origi- nal filter when (4.20) holds and (K,L)=(M,,N,).

Comparing Figs. 4 and 7, one can see that Filter 1.2, with the nonlinear phase, has a significantly smaller mag- nitude of ripple in the reject regions. Whereas Filter 1.3 has four distinct ripples in its passband, the passband ripple of Filter 1.2 is not detectable in the figure. The passband rejectband boundaries are distinctly diamond shaped for Filter 1.3 but are more nearly circular for Filter 1.2. Comparing Figs. 5 and 8, one can see that the phase response of Filter 1.2 is not wild in the passband, but fairly linear.

Filter 1.4 was obtained by windowing Filter 1.3. Each coefficient Q, of Filter 1.3 was replaced by hmw,,,,, in Filter 1.4, where

W mn = 1 - w,,,,/ R

rmn= [(m-M,/2)2+(n-N,/2)2]“2

R- [(M,/2)2+(N1/2)2]“2

and

a = 0.6.

The window in this example is a generalization of a window found in [8]. The amplitude response of Filter 1.4 is shown in Fig. 9. Comparing Figs. 7 and 9, it is apparent that although windowing reduces amplitude response rip- ples it also decreases the steepness of the response in transition regions.

Example 2

For this example the desired amplitude response is Fig. 10. (I&(} for Filters 2.1 and 2.2.

A(q,aJ= 4 l4+l%l~2.1 =o, otherwise (5.7)

where ]wr] < 7r and ]w2] < s. This corresponds to a low-pass filter with a diamond-shaped pass region. Both filters of this example were designed by the modified algorithm

with M, N, M,, and N, equal to 64, 64, 7, and 7, respectively. The desired amplitude sequence for the filters is shown in Fig. 10.

The initial phase sequence of Filter 2.1 was (4.20) with (K,L)= (8,6), using theorem 3. The amplitude and phase

Fig. 9. Amplitude response of Filter 1.4.

............... ................................................. ............... ................................................. ............... ................................................. ............... ................................................. ................................................................ ............... ................................................. ............... ................................................. ................................................................ ............... ................................................. ................................................................ ................................................................ ................................................................ ................................................................ ................................................................ ................................................................