Reconstruction of Signals after Filtering and Sampling Rate Reduction. Abstract-A wave digital filter (WDF) two-port has two input and two output-terminals.
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IEEE TRANSACTIONSON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 4, AUGUST 1985
Reconstruction of Signals after Filtering and Sampling Rate Reduction
Abstract-A wave digital filter (WDF) two-port has two input and two output-terminals. By using these to appropriately connecta WDF to its z A a - ; T . J - 7 A01 202 transpose, the filtering effectis fully compensated exceptfor an all-pass phase, and this independently of any selectivity requirement. This remains true if at the points of interconnection, every second sample is Fig. 1. Representation of a two-port WDF, N . reduced to zero, provided thata certain condition, easilyto be satisfied exactly, is fulfilled.Very efficient solutions are then possible. The method is of interest, e.g., for subband coding. the resulting partial signals, torepeat this process
I. INTRODUCTION NE of the interesting aspects of wave digital filters (WDF’s) is that they are in fact accessible through ports, i.e., terminal pairsconsisting each of one input and one output terminal [l], [2]. Thus, since a classical filter is a two-port, a usual WDF comprises twoinput terminals and two output terminals. This can be made useof to realize branching (directional) filters having exactly complementarytransfer properties [3]-[6]. In the present paper it will be shown how somefurther interesting properties can beobtained by interconnecting two or more WDF’s, taking proper advantage of the fact that each one of these has available more than just one input and one output terminal. The simplest one of the arrangements to bediscussed is analyzed in Section 11. It is shown therethat by appropriately connecting to a first WDF its transpose, the filtering effect is fully compensated, except for an all-pass transfer function. A modification of this arrangement involving decimation by setting to zero, at the points of interconnection, every second sample is discussed in Section 111. Hereagain, full compensation, except for an all-pass function, is achieved if the WDF obeys a suitable condition. This condition can be satisfied rigorously, in particular, for symmetrical and antimetrical filters whose characteristic function is bireciprocal, in which case drastic savings in hardwarecanbeachieved (especially in the symmetrical case) [5]-[SI. The application of the method can be repeated any number of times. Altogether, the methodthus makes it possible to split a signal, after more or less pronounced filtering, into two individual signals, to halve for each one of these
0
as often as desired, yet to reconstruct after this, rigorously, the amplitude-versus-frequency behavior of the original signal, even if the filters involved at the various steps do not exhibit a pronounced selectivity behavior. Furthermore, any wanted degree of phase equalization is achievable, if desired, by adding a suitable phase equalizer. The method is applicable, in particular, for obtaining an efficient alternative solution to the problem of designing filters for use in subband coding arrangements [9]-[17] (Section IV). Much of the theory will be presented without assuming the circuit to be real; theprinciple of wave digital filtering is indeed of interest also in the complex case [SI, [MI, [ 191. Some of the results and the examples (Section VI), however, are specifically referring to real circuits.
11. ARRANGEMENT WITH UNIFORM SAMPLING RATE Consider a WDF two-port, N (Fig. 1). We may write
B = SA
(1)
where
A =
B = (B132)‘
are the vectors of the incident and reflected wave quantities and where S is the scattering matrix of N , the superscript T denoting transposition. The matrix S is also equal to the voltage-wave scattering matrix of the reference filter NR, from which N is derived. Thus, if S’ is the usual (i .e., power-wave) scattering matrix of N R , we have S = R112 SlR-1/2 , R = diag (R1,R2) (2)
R being the diagonal matrix composed of the port resistances R1 and R2. We may assume that (1) refers tosteadystate conditions, i.e., that A1,A2,B1; and B2 are complex constants. Let us designate by NT a transpose of N , i.e., a WDF Manuscript received June 20, 1984; revised December 17, 1984. A. Fettweis is with Lehrstuhl fur Nachrichtentechnik, Ruhr-Universitat, for which B = S’A. It is known that a structure realizing Bochum, West Germany. NT can bederived from N by simply applyingflow reversal J. A. Nossek is with Siemens AG, Munich, West Germany. K. Meerkotter is with the Universitat Paderborn, Paderborn, West Ger- [20]. In particular, the values of the multiplier coefficients in NT are then the same as in N , i.e., for a given N many. 0096-3518/85/0800-0893$01.00
O 1985 IEEE
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2'
1'
Fig. 2. Arrangement whose scattering matrix corresponds to the left-hand side of (4).
the corresponding NT is obtained this way without any approximation. For S and STwe may write
s = (s11 s 2 1
),
).
S T = ( s11 s 2 1 SI2 s 2 2
Sl2 s22
Fig. 3. Arrangement as in Fig. 2, but involving multipliers u(n) and u(n) for halving the sampling rate.
The functions u(n) and u(n) take, alternately, the valin opposite ues 1 and 0, and this either in the same or phases. We can thus write
It is thus easily verified that we have
ETSTES = 1
det S and correspondingly either
where
u(n> = u(n>or u(n> = 1 - u(n>=
4 (1 - ejnnT'21
(9)
where and where
Q = 21rF, det S = det ST
=
F = 1/T.
SllS22-S12S21
is the determinant of S . The left-hand side of (4)can be interpreted as the scattering matrix of the overall two-port defined by the arrangement of Fig. 2. For this arrangement, the transfer functions from 1 to 2' and from 2 to 1' are thus, clearly, equal to zero, and those from 1 to 1' and from 2 to 2' are both equal to det S . On the other hand, we obtain from (2) (6) det S = det S',
Clearly, we may say that immediately after carrying out the multiplications by u(n) and u(n) the effective sampling rate is F/2, or that the effective sampling is there taking place at the rate F/2.No signal is assumed to be fed into terminal 2, and we carry out a steady-state analysis for an input signal of the form a l ( n T ) = AlePnT,
n =
*
:
-1,0,1,2,
*
(10)
where A I is a complex constant. Due to (3a) and (lo), we obtain from Fig. 3 for the input while it is also known that for NR lossless, det S' is an all- signals of N ~ , pass function [21]. Hence, the net effect of the arrangecl(nT) = - A , s,, ($) u(n)epnT, ment of Fig. 2 is then simply to introduce all-pass phase shifts. sll($1 * u(n)ePnT c , ( n ~ )=
-
-
111. ARRANGEMENT WITH SAMPLING RATEALTERATION and thus, using (8) and (9), A. Basic Relationships cl(nT) = A,S21 ($) [ePnT+efP +jW2)nTI, We now modify the arrangement of Fig. 2 in the way shown in Fig. 3. For simplicity,we may assume that N and NT involve only integer delays, i.e., that all transfer funcc2(nT) = AISll($) [epnT+e(pfjni2)nT 1 tions are rational in z = ePT, Le., in
(z
l)/(z
-4
(11)
4
(12)
+ 1)
(7) where in (12) the upper and lower sign holds for the first and second choice in (9), respectively. For computing the where p is the usual complex frequency, and F = 1/T is output signals bl and b2 in Fig. 3, we thus have to deterthe sampling rate. If we have to be more specific,we will mine transfer functions not only at the complex frequency thus write S ($), Sll (4) etc., instead of S , S I 1 etc. , Furp , thus at $ [cf. (7)], but also at p j Q / 2 , i.e., for z thermore, we may assume the sampling instants to occur replaced by e@+ja'2)T= -z, thus for $ replaced by 1/$. at Taking intoaccount(3b), (11), and (12), and grouping terms appropriately, the following result is obtained t n = n Tn ,= * * * -1,0,1,2, $
=
tanh ( p T / 2 ) =
-
+
a .
FETTWEIS et al.: FILTERING AND SAMPLING RATE REDUCTION
bl(nT) = 1 A , det S($)
-
epnT+ A I
1
895
We now assume throughout that NR is indeed lossless. In this case, the matrix S’ is paraunitary, i.e., such that [211 SSS’ = 1, (21) the subscript asterisk notation designating paraconjugation. Recall that for a rational function F = F($) the paraconjugate F, = F*($) is defined by F, = F( -$) if F is real and, more generally, by F, = E*(-$*) if F is complex, the superscript asterisk designating complex conjugation. For a matrix, paraconjugation implies notonly paraconjugation of each entry, but also transposition. One of the equations following from (21) i s
si,s;,*+ s;,s;2*= 0.
(22)
On the other hand, the characteristic~nction,which is of crucial importance in filter design [21], is defined by
*
=
Sil/Sil.
(23)
Consequently, using (22) and (23), we conclude that (20), thus (15), is equivalent to
*(l/$) =
=I= l/**($).
(24)
For real frequencies, i.e., for $ = j 4 with 4 real, we have *.($) = **(I)). Hence, (24) gives in this case in particular
S’ =
;( );
and taking into account (2), we obtain si1 = 811,
I*(l/j4)l = l/I*(j+)l.
(25)
Since replacing w by w f W2 amounts to replacing j + by l/j+, (25) implies a corresponding relationship between attenuation values. Indeed, the efective loss, CY, and the return loss, aR,are known to be given by a =
1 In (1 + ( * I2) ,
aR =
4 In (1 +
(25)
and we have CY
= -In
c ~ R=
Hence, (15) is equivalent to
-In
lS2,1 = -ln)S12), lSlll =
-In
IS221.
For a symmetric and an antimetric filter, we have E211
9,= -9 and
=
\E,
(270)
respectively. Thus, for a symmetric filter, (24) becomes
If (15) holds, and only if this is the case, the output signal atterminal 1’ thus contains nootherfrequency components than the input signal. The transfer function from 1 to 1‘ is then equal to (1/2)detS($), i.e., in view of (6), equal to (1/2)detS’($). However, if the reference filter NR is lossless, detS’($) is an all-pass function (cf. last paragraph of Section 11). Thus, for NR lossless, the transfer function just mentioned is, except for the factor 1/2 (6 dB loss), equal to a simple all-pass function. This remarkable result is valid irrespective of the performance of N, thus evenif this WDF produces, e.g., only a moderate separation of the originalsignal into frequency bands, and this irrespective of the fact that every second sample is reduced to zeroby means of the multipliers u(n) and v(n).
* lW$)
(28)
\E(l/$) = r l/+($).
(29)
W/$) =
and for an antimetric filter,
Evaluating these expressions for $ = 1 and $ = - 1, we obtain from (28), ‘k2(1) = ,k2( - 1) = f 1, and from (29), q2(1) = ,k2( - 1) = =F 1. Hence, for a symmetric filter, the upper choice of sign in (24) can be permitted only if 1) are real, and the lower choice only if *(l) and these quantities are imaginary; for an antimetricfilter, the situation is precisely reverse. For a real filter, *($) is real (i.e., takes real values for $ real). We then conclude from (25) that for 4 = 1, i.e.,
*(-
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foro=Q/4,wehaveIkl =l;thus,dueto(26),c~=c~~= 3 dB. Furthermore, for a real symmetric filter only the upper sign in (28), thus only the first choice in (9), is permitted. Similarly, for a real antimetric filter, only the lower sign in (29), thus only the second choice in (9), is permitted. In either case, (24) (thus (28) and (29), respectively) may therefore be replaced by *(l/$) = l/k($).
-A~S,,($>~P”~,
r--N--l
(30)
A function satisfying such a requirement iscalled bireciprocal (or selfreciprocal [7], [22]). Observe that by including an appropriateconstant in k , the requirement (30) can easily be satisfied for Butterworth and elliptic low-pass and high-pass filters whose transition range is centered at 4 = 1. An important consequence of (30) (k($) being a real function) is that under very general conditions, in particular, if we are dealing with a proper low-pass/high-pass filter, the zeros of the canonicHurwitz polynomial belonging to the filter can be shown to beall located on the unit circle, i.e., on 1 $1 = 1 (cf. Appendix A). Finally, consider still (11) and (12), taking into account that the effective sampling rate for c1 and cz is only F/2. For those values of n for which the samples are not reduced to zero due to the multiplications by u(n) and u(n) (Fig. 3), we have
cl(nTj =
4, AUGUST 1985
c , ( n ~ )= AISll(IC/)e”nT,
which corresponds to transfer functions equal to -S21($) and S1 (4), respectively.
____
Fig. 4. Realization of a symmetric WDF lattice two-port N in terms of the canonic reflectances SIand S,.
Fig. 5. Complete arrangement corresponding to Fig. 3 in the case of a symmetrical WDF lattice two-port N , as given by Fig. 4.
Fig. 6. Arrangement equivalent to that of Fig. 5 if (35) holds.
SI($) =
$1
(z2>,
SA$)
= Z-’$~Z~),
and in the latter
€3. SymmetricFilters
SI($) = z-lsl(z”>, S2($)
For a symmetric filter with R1 = R,, we have
(36)
=
32(z2) (37)
where sil ( * ) and 3, ( are rational functions. This property, clearly, amounts to an immediate reduction in hardware by a factor exceeding2, and furthermore to an additional reduction by 2 due to the halving of the sampling rate at the centerof the arrangement in Figs. 3 and 5. The second of these conclusions follows indeed from the fact that in N only every second sample must be computed and that in N* only every second input sample is different from zero. Altogether, the arrangement of Fig. 5 may thus be replaced by that of Fig. 6, and the amount of hardware required is reducedby a factor largerthan 4 [SI[7] with respect to what might appearat first to beneeded. In fact, ther.e is no longer any explicit multiplication by u(n) and u(n), Le., for c1 and c2 the actual sampling rate is equal to the effective rate F/2. A further important conclusion which can bedrawn from (36) and (37) is that if the zeros of the Hurwitz polynomial belonging to the filter are located on 141 = 1, as discussed above, the critical values of the rational functions (-) and 9, are all 5 0 (cf. Appendix A). This simplifies the WDFrealization of these functions. More specifically, by selecting suitable adaptors, it becomes eveneasier than otherwise to suppress all parasitic oscillations (limit cycles) [7], [24] and to ensure forced response stability as well as optimal scaling [24] , [25]. e )
Si1
=
Si1 =
Si2 = SI1 = S22,
= S 2 , = S12.
Si2
(31)
The two reflectances, S1 and S,, corresponding to the two canonic lattice impedances of N R are given by [21] S1 =
s11
- S21,
S, =
S11
+ S21;
(32)
these canonic reflectances are all-pass functions. We have
k = (S2
+ S,)/(S,
-
SI).
(33)
Assume first that the filter is real. As we have seen after (29) in Section 111-A, only the first choice in (9) is then permitted. The realization of N in terms of SI and S, is shown in Fig. 4 [23], and the resulting full arrangement corresponding to Fig. 3 is given in Fig. 5. In view of (33), (30) is equivalent to SI($)SZ(W = - S l ( W s, ($).
(34)
sl
Clearly, (34) is satisfied if ~ l ( W =
*
SI($),
S2(W) =
=F
S2($)
(35)
where either both upper signs or both lower signs hold. Due to (7), the upper signs imply that SI is even in z and S2 odd, and the lower signs that SI is odd and S, even. In the former case we may thus write
(e)
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FETTWEIS et al.: FILTERING AND SAMPLING RATE REDUCTION
From (44), we also derive
Note that from (32) and (35) it also follows that S11 ( W =
-f.
S21($).
fo*(l/$) = F
Hence, (17) gives
b2 (nT) = f 1 AI det S ($)
*
e@+jnnJT
(38)
which, when substitutedin (- 1)“O,i.e., to
CY*(-
l/$)%*($)
(43), leads to 1 aI2
= F
I c y 1 = 1, (-1)”O = F 1 . where the upper and lower sign corresponds to the upper and lower sign in (35), respectively. Clearly, (38) is sim- Consequently, the upper sign in (24), i.e., the first choice. ilar to. (16). In view of (31) and (32) we have in (9) is only permitted for no odd, and the second choice, no even. for det S = S,S2, (39) On the other hand, if we want to construct a function i.e., the all-pass function det S is equal to the product of 9($) satisfying (24), we may choose eitherfo($) or ho($) the all-pass functions SI and S2. arbitrarily, but the other polynomial is then given by (43) Let us still examinebriefly the most general solution of or (44), respectively. In using either one of these expres(34). Clearly, (34) is also fulfilled if sions, we are still free to choose the unimodular constant $1 = SlOSO, s 2 = S20SO (40) cy, and there is also some freedom to choose the integer where So is an arbitrary all-pass function and where Slo no. Assume, for instance, that fo($)is given. Since ho($) and S2, are all-pass functions satisfying the requirements derived from (43) must be a polynomial, we must choose no 1 degfi($). There is no further restriction however if SIO(l/$) = f $10 ($), S20(1/$) = =I= SZO(rc.1, (41) h(0)# 0, i.e., if&($) does not contain $ as a factor. On either both upper or both lower signs holding in (41). The the other hand, iffo(0) = 0, we must select no = degfo($) since for no > deg fo($)both fo($) and ho($)would have previous theory thus holds entirely for Sloand S20. $ as a.factor. Similarly, if ho($) is given, we may choose Observe that the presence of the common all-pass factor no any integer 2 deg ho($)if ho(0) # 0, but no = deg for So in (40) is simply equivalent to placing one’all-pass cirho($) if ho(0) = 0. cuit of transfer function So in cascade with the input of If the circuit is real, the polynomials ho and fo in (42) - the arrangement in Figs. 5 and 6 and another one in casmay be assumed to bereal, and a must then be equal either cade with the output. In fact, the solution given by (40) + 1 or to 1. The real Hurwitzpolynomial go satisfying to and (41) constitutes themost general solution of (34). This is proved in Appendix B. gogo* = hobo* +fofo* (45) If the symmetricfilter is complex, both signs in (28) are possible. The upper sign leads to the same generalresults is then also fully defined except for an arbitrary choice of as those discussed for the real case, except that the prop- sign. Finally, the canonic polynomialsf, g, and h [21] must erty for the zeros of the Hurwitz polynomial to be located satisfy on \$I = 1 does no longer have to be correct. On the other \E = h/f, gg* = hh* fl* (46) hand, the lower sign in (28) leads to replacing (34) by with g being Hurwitz. We thus must have S , ( $ > S l ( W = -Sd$)S2W$) f = ab*f, h = ab,ho, g = g@b (47) which is satisfied in particular for where a and b are arbitrary real Hurwitzpolynomials, and the resulting scattering matrix is given by ~ l W $ )= .s2($).
+
*
C. General Filters We may write
S’
=; (; -2)
+
(42) where u may be chosen equal to either 1 or - 1. If the circuit is complex, let go still be a Hurwitz polywhere ho($) and fo($)are relatively prime polynomials. nomial satisfying (45); it is uniquely defined except for an Condition (24) may thus also be put into the form arbitrary unimodular factor. Since (46) must still hold, the most general solutions for the canonic polynomials g , and 6 is still given by (47), with a and b arbitrary Hurwitz polynomials, and the resulting scattering matrix is still thus, where we choose no equal to the degree of 9($). given by (48) [21], [26] where u is now an arbitrary unisince lc.“”fo*(l/$)and $“Oho*(l/$) are also relatively prime modular constant. polynomials, we have Note that whether the circuit is realor complex, the inho ($) = CY $“Ofo*(1/$) (43) clusion of Hurwitz polynomials a and b according to (47) simply amounts to adding all-pass factors to the entries of fo ($1 = ==! aV0ho*(1/$> (44) (48). (This corresponds to what has also been discussed where cy i s a constant. quite towards the end of Section III-B.) Such all-pass fac-
W$) = hO($>/f($>
898
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F12
F/2
FIL FIL
F14
Fig. 8. A multistep arrangement with decrease of the sampling rate from F to an effective rate of F/2" and subsequent increase back to F, shown form = 2.
Fig. 7. Arrangement of Fig. 3 modified by inserting two identical circuits with transfer functions Ho, operating at effective rate Fi2.
tors are usually of no avail, i.e., one will usually simply choose a = b = 1. In the theory of complex wave digital filters the concept of one-realness plays a major role. In [18] a condition has been derived which when verified allows us to conclude for our present problem that by properly choosing the various unimodular constants mentioned before the resulting scattering matrix can be made one-real, i.e., such that it is real for $ = 1. This condition had been found to be 9(1)
\E*(-l)
=
V ( 1 ) . *(-l).
IV. MULTISTEPARRANGEMENTS We modify the arrangement of Fig. 3 in the way shown in Fig. 7, Ho being a transfer function in z2,i.e., in = (22
-
1)/(2
+ 1) = 2$/($2 + 1).
I.
c , ( n ~ )= AISll($) * H~ (+2>[ePnT f
Consequently, if (15), or, equivalently, (24) holds, (16) and (17) are replaced by
4 b2(nT) = -4 AlHo($2)
2-" det Sl($J
epnT
det S2($J
det Sm($")
*
=
$,
=
$u+!
2 $ ~ ( $ :+ 11,
u = 1 to m - 1,
(52) and where S , is the scattering matrix of N u , u = 1 to m. Equivalently, (51) may be written, m
H
=
2-"
IT1
det
u=
=
$,(z,),
(53)
1 to m,
$,(z,>
2" =
z (20- 1) .
(54) In the same way we can compute the transfer function from the input terminal to the 2" center points of the overall arrangement (i.e., the 2" points where the effective sampling rate is F/2"). Taking into account what has been said at the very end of Section 111-A, these transfer functions are of the form = Su($u),
m
(50) H,
=
II u=
HUP($,),
(55)
p = 1 to 2"
1
det S($)
- [s,(1 le'^ +jn'2)nT, the main expression of interest being (50). Clearly, the only modification is the appearance of the additional factor Ho($2). The circuits marked H o in Fig. 7 may, of course, in turn be full arrangements of the type of Fig. 3 operating from terminal 1 to 1' . This leads to an overall arrangement which we may represent as shown in Fig. 8. The respective effective sampling rates are indicatedthere by F, Fl2, and Ff4, and M I , e.g., representstherethear-
(51)
where
where, for u
-4 A ~ s ~ ~ ( Ho(G2,[ePnT +) + e(p+jn'2)nT I,
bl(nT) = AIHo($2) * det S($)
=
(49)
This amounts to replacing (11) and (12) by
c , ( n ~= )
H
$1
It is clearly satisfied if (24) holds.
$2
rangement of Fig. 3 from input terminal 1 to the locations where the signals are those marked c1 and c2, etc. Clearly, this process may be repeated as often as desired. The effective sampling rate is then first consecutively reduced by a factor of 2 at each step and then consecutively increased in the same way. The total effective number of samples per unit time ateach vertical cross section remains then clearly invariant. If there are altogether m steps of decrease and m steps of increase andif the filters used are N1 to N,, thus also N i to NT, the overall transfer function, H , turns out to be
where, except possibly for a change of sign, HUp($,) is or Su,21($u)(these functions corequal to either Su,ll($u) responding to the entries in the first column of the scattering matrix of Nu, u = 1 to m),depending on which one of the 2" distinct values of p is being considered. As an example, for m = 2 (cf. Fig. 8) we obtain consecutively,
H 1
=
&,21($)
. S2,21($2),
H2
= -S1,21($)
*
S2,11($2)7
H3 = -S1,11($) * S2,21($2)> H4 = SI,ll($) * S2,,1($2>. In a repeated application of the arrangement of Fig. 3, the halving of the effective sampling rate does not at each
899
FETTWEIS et al.: FILTERING AND SAMPLING RATE REDUCTION
FIL
Fig. 9. Modification of the arrangement of Fig. 8. The second step of decreasing the effective rate by 2 is skipped, with MA having the same allpass transferfunction as that appearing between P and Q.
step have to be applied to all paths. This is illustrated for m = 2 in Fig. 9. We have totake into account,however,that the result (50) had been obtained from Fig. 7 under the assumption that the same Ho had been inserted in both paths. Consequently, in Fig. 9 the transfer function of MA must be the same all-pass function as that determined between thepoints P and Q . This is alw,ays possible without approximation, but one should not overlook the factor i in (16) (or the corresponding higher power of i if more than one stage is being skipped in the process of decreasing and increasing the sampling rate). In the case of symmetrical filters the situation is particularly simple in view of (39). '
bands, but due to the concurrent sampling ratereductions the consecutive bands are always modulated down to a range extending from0 to half the effective sampling rate at the location considered. In view 'of what we have said above, the various frequency bands do not have to be of equal width. As anextremesituation,the consecutive widths may be decreasing by a factor of 2 from one band to the next except for the two smallest bands., which are then of the same width. Furthermore, there may be some advantage to use the same filters at each step, i.e., tohave all N , be the samefor u = 1 to m except of course for the change in (effective) operating (sampling) rate. It is clear that the approach is of interest for subband coding. It is not only efficient in itself, but offers the possibility of using the known attractive properties of wave digital filters: low accuracy requirements for the multiplier coefficients, good dynamic range, same stability properties (under linear and nonlinear conditions) as for nonrecusive digital filters. Furthermore, if one employs the .solution discussed in Section 111-B, the implementation is possible by means of structures that are particularly modular and suitable for high level of parallelism. A further advantage for the present solution is apparent
e-2~[(62/4) -a] + ,-2~u[(Q/4) + w ] = 1 possible irrespective of the degree of selectivity in the var(56) ious filter stages. The various lowest effective sampling rates at which one arrives at the center of the arrange- .Thus, if (W4) w is a stopband frquency, (W4) - w is ments do not have to be all equal since one has the full a passband frequency and vice versa. More specifically, freedom available by generalizing in an obvious way the for any reasonable stopband. requirement, the passband arrangement of Fig. 9. As an extremesituation, one could, loss is totally insignificant. As an example, if the minie.g., have at the center of the arrangement one path with mum stopband is as low as 30 dB, the maximum passband an effective rate of F/2, one with F/4, etc. up to one with loss is already only 0.004345 dB. F/2"-' and two paths with rate F/2", m being as defined Note also thatthe simplest filter which we may consider in Section IV, altogether thus only m + 1 paths. is the one for which the all-pass functions SIand S2 disThere doesoccur, however, an all-pass phase distortion. cussed in Section 111-3 are given by If this is not acceptable, it can be. corrected to any degree required by an additional phase equalization. This may be SI = 1, s2 = z-l, placed, e.g., either in front of the input terminal (thus at the sending end), after theoutput terminal, or at the cen- Such filters are simultaneously WDF's and nonrecursive; ter of the arrangement (as for the circuits with transfer they correspond to the trivial filters mentioned in [13], function Ho in Fig. 7). In the latter case the number of [ 141. It is surprising that despite their extreme simplicity circuits is increased,but since the operating rates are cor- they fully meet the requirementsfor the results of the presrespondingly reduced the total number of arithmetic op- ent theory to be applicable. Of course, the filter perforerations is not altered, although the number of delay ele- mance then achievable from input to the center of the armentsis substantially larger. One may also divide the rangement is much too poor'for mostapplications. Clearly, the results of Section 111-B hold for an arrangephase equalization among the various locations available. The simplest types of filters are, of course, low-pass/ ment such as that of ~Figs.5 and 6, as soon as S1 and S, high-pass combinations, thus withe.g., S,, corresponding are all-pass fuhctions of which one is even in z and the to the transfer functions of a low-pass and SI1to that of other odd. Hence; it is not strictly required that SI and S, the complementary high-pass filter. If such filters are used be realized asWDF's. However,if a realization by a at each step, onemay say that the original frequency band structure other thanthat of a WDF is employed, some of of width F/2 isconsecutively split into smaller and smaller the good properties otherwiseachievable may be lost.
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1
2T
J-
high-pass
Fig. 10. A 7th-order low-padhigh-pass wave digital lattice filter comprising 3 multipliers, yl, y2, and y3.
w-
R/4
0
Rl2
Fig. 12. Loss performance of the WDF lattice filter of Fig. 10 realized with the simple coefficient values listed on the left (solid line) and the highprecision coefficient values listed on the right (broken line).
1
I
I1
I 1
0 0
1
.2
3
I
.5
wlR:flF-
Fig. 13. WDF lattice filter of Figs. 10 to 12. Group delay r versus frequency, in normalized representation, F being the sampling rate, 0 = 2 TF. Fig. 11. Loss performance of the WDF lattice filter of Fig. 10 (high-precision multipliers) compared to that of a 32-tap FIR filter [12].
VI. NUMERICALEXAMPLES
APPENDIX A The statement mentioned in the second last paragraph of Section 111-A. has been shown in [27] to holdin the case of elliptic filters. In orderto prove it for general proper real symmetric and antimetric filters, letf, g , and h be the canonic polynomials [21], [26] of N R . These are real, and g is a Hurwitz polynomial. We have
As an example, we consider a real filter as discussed in Section 111-B, satisfying requirements at least as severe as those given in [12]. This can be achieved with the WDF lattice filter shown in Fig. 10, comprising 3 two-port adap= h/f, **($) = *(I)*), (Ala,b) tors, thus altogether 3 multipliers. The optimum (elliptic function) Chebyshev response is shown in Fig. 11 (solid and furthermore line), where for comparison the result for the 32 tap FIR filter described in [12] is also given (dashed line). There is no need for showing the passband ripple since in view where the upper sign holds in the symmetric case and the lower one, in the antimetric case. of (56) it is totally insignificant. Consider then values of $ with I$\ = 1, i.e., with $* = The ideal multiplier coefficients are y1 = 0.497 012, y2 = 0.160 798, and y3 = 0.169 354. After discrete op- 114; we obtain, using (Alb) and (30), timization,a very satisfactory loss performanceis ob]*($)I = I*($*)\ = I*(W)l = 1 w w ) l tained for thus y l = ?1 , y z = + +323l 73 = Q + & + (57) I*($) = 1 for I$\ = 1 . (A31
*
(Fig. 12). Thus, a realization of the structure of Fig. 10 is Assume next that *($) is holomorphic for I$[ < 1, but possible requiring no multipliers, but altogether only 14 not constant. In this case,we conclude from the maximum adders [i.e., 11 explicit adders shown individually in Fig. modulus theorem that 10, plus 3 implicit adders needed for implementing (57)l. I * W l < 1 for 1$1 < 1 The resulting (normalized) group delay is shown in Fig. 13. The peak value appearing therein could be substan- and furthermore, using (30), tially reduced by adopting less steep transition ranges than those obtained for the elliptic response in Fig. 11. l*(rc.)I > 1 for I$l > 1.
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Consequently, in view of (A2), g can have zeros only for I 4 = 1. The assumption that !I?($) is holomorphic for I$l C 1 holds in particular if !I?($) corresponds to a (proper) lowpass filter. Indeed, the zerosoffare then all located in the stopband, while the lower edge of the latter isnecessarily located at C$ > 1 (thus at lr1.1 > 1) since C$ = 1 has been seen in Section 111-A to be afrequency wherethe loss just reaches 3 dB. We next prove the statement made in Section 111-B that the property just discussed implies the critical yalues of and .) to be real and in fact 5 0 . Indeed, if g is real and if its zeros are located on I$/ = 1, it canbe written in the form
sl(
a )
s2(
g = Go($
+ 1)
g = Go($
+ 1)
+ 1 + 2ai$) +1+24)
($2 i
($2 1
* g,*/g,,
S2
= 5 g2*/g2
where gl and g2 are Hurwitz polynomials such that g = g1g2. Thus, for the factors of and S2 we have, using (49),
,!?,
q2 + $2
1-2 4 -11+
+ 1 + 2Ui$
$2Uj $2Ui
+1 + Ai
- Ag2 z2
where Ai = (1 - @/(l
Ai 2 0.
(B1) SlO($>S20 (1/$) = -S1o(l/$)S20($), which is of the same type as (34). The all-pass functions Sloand S2, may be written in the form [21] wl*/gl,
$20
SlO(W) = (- l)fl‘&o($), S20(l/$) = (- 1)”’S20($), and (- l)n*+ n 2 = - 1,
034)
respectively, which implies indeed (41). Note that in view of (B4), nl + 122 must be odd; this is in agreement with (36) and (37). ACKNOWLEDGMENT The authors are very grateful to S. Gulliioglu and W. Drews for calculating the results shown in Figs. 12 and 13, and to L. Gazsi for many valuable discussions on the subject described in this‘paper. They are indebted to the reviewers for useful comments. [ l ] A. Fettweis, “Digital filters related to classical filter networks,” Arch. Elektr. Ubertr., vol. 25, pp. 79-89, Feb. 1971; also in Digital Signal Processing, vol. 11, IEEE Press Selected Reprint Series. New York: IEEE Press, 1976, pp. 475-485. [2] -“Introduction to wave digital filters,” in Lecture Notes-Course on Digital Signal Processing, Europ. Con$ Circuit Theory Design, Stuttgart, Germany, Sept. 1983, pp. 13-46. [3] -, “Scatteringproperties ofwave digitalfilters,” in Proc. Flor ence Seminar on Digital Filtering,Florence, Italy, Sept. 1972,pp. 1-8. [4]M.Bellangerand D. Didier,“Un filtre numkriqued’onde complCmentaire pour voie tkl6phonique,” Cdbles et Transmission,vol. 31, pp. 497-506, Oct. 1977. [5] W. Wegener, “Wave digital directional filters with reduced number of multipliers and adders,” Arch. Elektr. Ubertr., vol. 33, pp. 239-243, June 1979. [6] A. Fettweis,“Transmultiplexerswitheitheranalogconversioncircuits, wave digital filters, or SCfilters-A review,” IEEE Trans. Commun., vol. COM-30, pp. 1575-1586, July 1982. [7] J. A. Nossek and H.-D. Schwartz, “Wave digital lattice filters with applications in communication systems,” in Proc. IEEE Int. Symp. Circuits Syst., Newport Beach, CA, May 1983, vol. 2, pp. 845-848. [8] K. Meerkotter, “Antimetric wave digital filters derived from complex reference circuits,” in Proc. 6th Europ. Con$ Circuit Theory Design, Stuttgart, West Germany, Sept. 1983, pp. 217-220. [9] D. Esteban and C. Galand, “Application of quadrature mirror filters to split-band coding,” in Proc. IEEE Int. Con$ Acoust., Speech, Signal Processing, Hartford, CN, May 1977, pp. 191-195. [lo] R. E. Crochiere, “On the design of sub-band coders for low-bit-rate speechcommunication,” BellSyst. Tech. J . , vol. 56, pp.747-771, May-June 1977. [ I l l T. A. Ramstad and 0. Foss, “Subband coder design using recursive quadraturemirrorfilters,” in Proc. Europ. Signal Processing Conf., Lausanne, Switzerland, Sept. 1980, pp. 747-752. [12] R. E. Crochiere, “Subband coding,” Bell Syst. Tech.J., vol. 60, pp. 1633-1653, Sept. 1981. [13] T.P. Barnwell, 111, “Subbandcoderdesignincorporatingrecursive quadrature filters andoptimumADPCM coders,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 751-765, Oct. 1982. [14] C. Galand and H. J. Nussbaumer, “New quadrature mirrorfilter structures,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP32, pp. 522-531, June 1984. [15] H. J. Nussbaumer and M. Vetterli, “Computationally efficient QMF filter banks,” in Proc. 1984 IEEE Int. Con$ Acoust., Speech, Signal Processing, San Diego, CA, Mar. 1984, vol. 1, pp. 11.3.1-11.3.4. [16] P. C. Millar, “Mirror filters with minimum delay responses for use in subband coders,” in Proc.I984 IEEE Int. Con$ Acoust., Speech, pp. 27.5.1Signal Processing, SanDiego, CA, Mar.1984,vol.2, 27.5.4. [17] M. W. Smithand T. P. Barnwell, 111, “A procedurefordesigning exact filter banks for tree-structured subband coders,” in Proc. I984 IEEE Int. Con$ Acoust., Speech, Signal Processing, Mar. 1984, vol. 2, pp. 27.1.1-27.1.4. ’
+ ai),
APPENDIX B We want to show that (40) and (41) constitute the most general solution of (34). For this, observe that substitution of (40) in (34) yields
SI0 =
P 2 g 2 ( W = g2($).
REFERENCES
where the ai are positive constants 5 1 and Go is a real constant. On the other hand it is known [21] that SI and S2 are of the form SI =
P l g 1 ( W ) = g1($), Thus, (B2) and (B3) yield
’=
@2g2*/g2
(B2)
where gl and g2 are realHurwitz polynomials, say of degree n1 and 122, respectively, andwhere = l and 0 2 = f1. We may assume that So is the greatest common all-pass factor of S1 and S,, i.e., that gl and g2 are relatively prime. Using (B2), condition (Bl) can be written
(T,
*
gl*($>s2*(1/~>gl(l/$)g2($) = -Sl*(1/$)c.>sz*($>gl($)g2(l/$). (B3)
Since $fl’gl(l/$) and 4bn2g2(l/$)are also Hurwitz polynomials, with coefficients having the same sign as those of gl($) and g2($), respectively, we conclude from (B3) that
’
902
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ON ACOUSTICS, SPEECH, SIGNAL AND
[18] A. Fettweis, “Principles of complex wave digital filters,” Int. J. Circuit Theory Applic., vol. 9, pp. 119-134, Apr. 1981. [19] K. Meerkotter, “Complex passive networks and wave digital filters,” in Proc. Europ. Conf: Circuit Theory Design, Warsaw, Poland, Sept. 1980, V O ~ . 2, pp. 24-35. [20] A. Fettweis, “Reciprocity, interreciprocity, and transposition in wave digital filters,” Int. J. Circuit Theory Applic., vol. 1, pp. 323-337, Dec. 1973. [21] V. Belevitch, ClassicalNetworkTheory. San Francisco, CA: Holden-Day, 1968. [22] E. Bucherl, “Ein Beitrag zurSynthese von Nyquistfiltern,” Frequenz, vol. 27, pp. 2-6, Jan. 1973. [23] A. Fettweis, A. Sedlmeyer, and H. Levin, “Lattice wave digital filters,” Int. J. Circuit Theory Applic., vol. l , pp. 5-10, 1973. [24] L. Gazsi, “Explicit formulae for lattice wave digital filters,” IEEE Trans. Circuits Syst., vol. CAS-32, pp. 68-88, Jan. 1985. [25] L. Gazsi, private communication. [26] A. Fettweis, “Scattering properties of real and complex lossless 2ports,” Proc. IEE, vol. 128, Pt. G, pp.147-148,Aug.1981. [27] G. Bosse and W. Nonnenmacher, “Einheitliche Formeln fur Filter mit Tschebyscheff-Verhalten der Betriebsdampfung,” Frequenz, vol. 13, pp.123-134, Feb. 1959.
Alfred Fettweis (M’56-SM’62-F’75) was born in Eupen, Belgium, on November 27, 1926. He received the degrees in IngCnieur Civil Electricien and Docteur en Sciences AppliquCes from the University of Louvain, Louvain, Belgium, in 1951 and 1963, respectively. From 1951 to 1963, he was employed at various subsidiaries of the ITT Corporation, in particular from 1951 to 1954 and from 1956 to 1963 at the Transmission Laboratory, Bell Telephone Manufacturing Company, Antwerp, Belgium, and from 1954 to 1956 at the Electronic Switching Laboratory, Federal Telecommunication Laboratories, Nutley, NJ. From 1963 to 1967, he was a Professor of Theoretical Electricity at the Eindhoven University of Technology, Eindhoven, The Netherlands. Since 1967, he has been a Professor of Communication Engineering at the Ruhr-Universitat Bochum, Bochum, West Germany. During the summer of 1974 he worked as a temporary member of the Technical Staff at Bell Laboratories, North Andover, MA. He is a recipient of the Prix “Acta Technica Belgica” 1962-1963, the 1980 Darlington Prize Paper Award, the Prix George Montefiore 1980, the IEEE Centennial Medal 1984, and the VDE Ehrenring 1984 of the Verband Deutscher Elektrotechniker.
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Dr. Fettweis is a member of the Rheinisch-Westfalische Akademie der Wissenschaften, Germany, European Association for Signal Processing, Nachrichtentechnische Gesellschaft, Germany, Belgian Telecommunication Engineers’ Society, Sigma Xi, and Eta Kappa Nu.
