Input-Output Gains of Linear Periodic Discrete ... - Semantic Scholar

Input-Output Gains of Linear Periodic Discrete-Time Systems with Application to Multirate Signal Processing Shahriar Mirabbasi Bruce Francis Electrical and Computer Engineering University of Toronto Toronto, Ontario M5S 1A4 Canada [email protected] [email protected] Tongwen Chen Electrical and Computer Engineering University of Calgary Calgary, Alberta T2N 1N4 Canada [email protected] Abstract

Several input-output gains of linear periodic systems are de ned, in particular, G(`2; `2 )| the worst-case `2 norm of the output over all inputs of unit `2 norm|and G(RMS; RMS)|the worst-case RMS value of the output over all inputs of unit RMS value. It is proved for linear periodically time-varying (LPTV) systems that these two gains are equal, a known fact for LTI systems. In addition, the relationship between two recently introduced generalized frequency responses for LPTV systems is derived. Finally, M-channel maximally decimated lter banks are considered, where, except in the ideal case of perfect reconstruction, aliasing distortion, magnitude distortion, and phase distortion are present. It is shown how these are kept small if the lter bank is designed by a method that optimizes the gain G(`2 ; `2) of an error system.

1 Introduction The notion of input-output gain of a lter is of intrinsic interest. For example, the maximum magnitude of the frequency response of a lter is one possible notion of gain, and it is useful in quantifying the degree of noise attenuation. The purposes of this paper are to de ne several notions of gain of a general linear periodic system, to compare the results with the usual case of a linear time-invariant (LTI) lter, and to show the relevance of gain to aliasing, magnitude, and phase distortion in multirate lter banks. This paper is a followup to [6] and [3]. In [6] linear periodic systems were studied in the frequency domain by de ning a generalized frequency response. Also, several multirate lter design problems were formulated as model-matching problems, where the generalized frequency response of an error system is optimized. In [3] basically the same technique was used to design multirate lter banks, with, however, a somewhat dierent notion of generalized frequency response. To review the LTI case, consider a discrete-time, possibly complex, signal x(k). We will be interested in three measures of the size of x(k):

1

1. The `1 norm: kxk1 := sup jx(k)j: k

This is the zero-to-peak amplitude of the signal. For example, the `1 norm of the complex exponential x(k) = ej 2fk equals 1. 2. The `2 norm:

kxk2 :=

X

k

jx(k)j

!1=2

2

:

Thus, kxk22 is the energy of x(k). Note that for kxk2 to be nite, x(k) must converge to 0 as k tends to 1 and to ?1; thus x(k) = ej 2fk does not have nite `2 norm. 3. The RMS value: !1=2 N X 1 2 RMS (x) := Nlim : jx(k)j !1 2N + 1 ?N

This is most useful for periodic signals. For example, the RMS value of x(k) = ej 2fk equals 1. (The RMS value is not quite a norm, because a nonzero signal could have zero RMS value|for example, a signal of nite duration.) Now consider an LTI, stable (but not necessarily causal), discrete-time lter with frequencyresponse function H(f), input x(k), and output y(k). We de ne four notions of gain for this lter: 1. Consider applying the exponential input x(k) = ej 2fk , having kxk1 = 1; the output is an exponential too (with a dierent magnitude and phase). Sweeping over all frequencies in the range ?1=2  f  1=2 and taking the maximum of kyk1 de nes one type of lter gain; let us denote it by G(exp; `1 ). The rst argument, exp, signi es that the input is an exponential, the second that the `1 norm of the output is taken. Thus G(exp; `1 ) := sup fkyk1 : x(k) = ej 2fk ; ?1=2  f  1=2g: This notion of gain would be relevant when the input is a pure tone and the amplitude of the output is important. As is well-known, since exponentials are eigenfunctions of LTI systems, when the input is x(k) = ej 2fk , the output is y(k) = H(f)ej 2fk , whose `1 norm equals jH(f)j. So the maximum output amplitude equals max jH(f)j: ?1=2f 1=2

This quantity is denoted kH k1 , the L1-norm (or Chebyshev norm) of the frequencyresponse function H(f). Thus, there is an explicit formula for the gain G(exp; `1 ) in terms of the lter frequency-response function, namely, G(exp; `1 ) = kH k1: 2. A second lter gain can be de ned by taking `2 norms: G(`2 ; `2) := sup fkyk2 : kxk2 = 1g: Here, the rst argument signi es that the input can be any signal of unit `2 norm, the second that the `2 norm of the output is taken. Thus G(`2; `2 ) is the worst-case output `2 norm over all inputs of unit `2 norm. In mathematical terms, it is the induced norm of the lter considered as a bounded operator on the space `2 of all square-summable signals.

2

This notion of gain would be relevant when the energy of signals is important. It is a fact1 that this gain too equals the L1 norm of the frequency-response function: G(`2 ; `2) = kH k1: (1) 3. A third lter gain can be de ned by taking RMS values of the input and output: G(RMS; RMS) := sup fRMS (y) : RMS (x) = 1g: Here, the rst argument signi es that the input can be any signal of unit RMS value, the second that the RMS value of the output is taken. It is a fact2 that this gain too equals the L1 norm of the frequency-response function: G(RMS; RMS) = kH k1 : (2) 4. Finally, we might be interested only in pure tone inputs, in which case we could modify the preceding de nition and restrict the input to be an exponential: G(exp; RMS) := sup fRMS (y) : x(k) = ej 2fk ; ?  2f  g: Note that the inequality G(exp; RMS)  G(RMS; RMS) holds automatically, because the RMS value of ej 2fk equals 1 and therefore the maximization is over a larger class in the de nition of G(RMS; RMS). It is easy to derive that this too equals the L1 norm of the frequency-response function: G(exp; RMS) = kH k1: Which of these four gains is relevant would depend on the application. It is an important observation that for LTI stable lters, all four are equal, and equal to the L1 norm of the frequency-response function. Since the above four notions are de ned only in terms of input and output, they certainly apply to linear systems in general, in particular to linear, periodically time-varying (LPTV) systems, such as multirate systems. In this more general context, G(exp; `1 ) and G(exp; RMS) are felt not to be important, because exponential inputs are not eigenfunctions, and therefore are not particularly distinguished. In this paper we focus on G(`2 ; `2) and G(RMS; RMS) for LPTV systems and prove in Section 3 that G(`2 ; `2) = G(RMS; RMS): Section 2 derives the connection between the generalized frequency responses of [6] and [3]. This is illustrated by some standard examples in Section 4. Finally, Section 5 studies M-channel maximally decimated lter banks. Except in the ideal case of perfect reconstruction, these suer from aliasing distortion, magnitude distortion, and phase distortion [7]. It is shown how these are kept small if the lter bank is designed by the method of [3]. This introduction concludes by extending the above gain concepts to multi-input, multi-output LTI systems. The de nition of L1 norm extends as follows: If H(f) is the frequency-response matrix of a multi-input, multi-output LTI system, its L1 norm is de ned by taking the maximum singular value: kH k1 = max  [H(f)]: f max 1 2

See for example Section 5.2.6 in [4] for a proof in continuous time; the discrete-time case is proved similarly. Section 5.2.6 in [4].

3

The Hilbert space `2 consists of all square-summable discrete-time signals de ned over the time set Zof all integers, and `n2 denotes the direct sum of n copies of `2 ; thus, elements of `n2 are square-summable vector-valued signals. The norm of a signal x(k) in `n2 is de ned to be

kxk2 =

X

k

x(k) x(k)

!1=2

;

where  denotes complex-conjugate transpose. If H(f) is the frequency-response matrix of a stable LTI system with input x(k) of dimension m and output y(k) of dimension p, so that H(f) is p  m, then the induced norm from `m2 to `p2 equals the L1 norm of H(f), that is, (3) sup kyk2 = kH k1 : kxk2 =1

This generalizes equation (1). Finally, the RMS value of a vector-valued signal x(k) is de ned to be !1=2

N 1 X RMS (x) = Nlim x(k) x(k) : !1 2N + 1 ?N Again, if H(f) is the frequency-response matrix of a stable LTI system with input x(k) of dimension m and output y(k) of dimension p, then sup RMS (y) = kH k1 : (4) RMS (x)=1 This generalizes equation (2).

2 Frequency-Response Functions of LPTV Systems There are two natural ways to describe an LPTV system in the frequency domain; Shenoy et al. [6] introduced one of these ways, using invariant subspaces, and they remark that this is equivalent to the other way, via blocking. The purpose of this section is to show this equivalence explicitly.3 We begin with the de nition of frequency response from [6] (the continuous-time analog of this was introduced in [1]). It is convenient to de ne the exponential signal of frequency f: ef (k) := ej 2fk : Consider a periodic signal u(k) of period M. It has a discrete Fourier series: MX ?1 u^(n)ek=M (n) (5) u(k) = M1 n=0 u^(n) =

MX ?1 k=0

u(k)e?n=M (k):

(6)

That is, u^(n) is the DFT of u(k). Now consider modulating u(k) to get a signal of the form x(k) = ef (k)u(k): (7) Substitution of (5) into (7) shows that x(k) is a linear combination of the complex exponentials of frequencies f; f + M1 ; : : :; f + MM? 1 ; It is interesting to note that an entirely analogous development has been going on in control theory, where two equivalent notions of frequency response have been developed for sampled-data systems, which are LPTV in continuous time [2] [1] [8]. 3

4

the coecients being the DFT coecients of u(k), multiplied by 1=M. Indeed, the subspace of all signals of the form (7) as u(k) ranges over all M-periodic signals is precisely the M-dimensional subspace Sf := span fef ; ef +1=M ; : : :; ef +(M ?1)=M g: Turning to systems, let H denote an LPTV system of period M. (Operators in the time domain will be written in boldface.) As shown in [6], Sf is an invariant subspace for H. Thus an input to H of the form x(k) = ef (k)u(k); u(k) M-periodic will produce an output of the form y(k) = ef (k)v(k); v(k) M-periodic. The vectors formed from the DFTs of u(k) and v(k) are related by an M  M matrix, denoted HAC (f): 3 2 3 2 u^(0) v^(0) 7 6 7 6 .. .. 5: 5 = HAC (f) 4 4 . . u^(M ? 1) v^(M ? 1) This matrix, HAC (f), is called the alias component (AC) matrix in [6] (where it is denoted H(f)) in view of its prior occurrence in the literature on multirate lter banks, and is a generalization 1 of frequency-response function. In the de nition of HAC (f), f ranges over the interval ? 2M  1 f  2M . The second way to get a frequency-response function for H is to block its input and output to get an LTI system, and then take the ordinary frequency-response matrix of the blocked system. More speci cally, the blocked system, denoted HB , is the M-input, M-output system de ned in Figure 1. Let the frequency-response matrix of HB be denoted HB (f). This blocking approach -

"M

-

j

-

H

6

z

.. .

-

#M

-

-

#M

-

z ?1

.. .

z ?1

j

?

6

z

-

"M

-

.. .

.. .

6

z

-

z ?1

"M

-

#M

Figure 1: Blocking H to get the LTI system HB . was the one used in [3].

5

-

Now what is the relationship between these two approaches, that is, between HAC (f) and HB (f)? To answer this, de ne the following matrix: 2 3 32 1 1


1 1 0


0 6 0 e?j 2f


76 1 ej 2=M


ej 2(M ?1)=M 77 0 76 V (f) := M1 664 .. 6 7: 7 . . .. .. .. .. 5 4 .. . 2 5 . . . 0 0


e?j 2(M ?1)f 1 ej 2(M ?1)=M


ej 2(M ?1) =M Notice that the DFT matrix appears here. Theorem 1 For an M -periodic LPTV system H, the AC matrix HAC (f) and the frequencyresponse matrix HB (f) of the blocked system are related by the equation V (f)HAC (f) = HB (Mf)V (f): (8) From this theorem and the equation V (f) V (f) = M1 I follows the next result. Corollary 1 For each f in the range ? 2M1  f  2M1 , max [HAC (f)] = max [HB (Mf)]: Thus

max

1 f  1 ? 2M 2M

max [HAC (f)] = 1max 1 max [HB (f)] =: kHB k1 : ? 2 f  2

This corollary in turn is important because, as shown in Proposition 3 of [6], the gain G(`2 ; `2) equals the maximum over frequency of the maximum singular value of the AC matrix. So we can summarize as follows. Corollary 2 G(`2 ; `2) = 1 max 1 max [HAC (f)] = kHB k1 ? 2M f  2M

j

Proof of Theorem 1 Solely to simplify the notation, the proof is given for the case M = 2, in which Figure 1 becomes Figure 2. It is convenient to let D and U denote, respectively, the -

"2

-

-

H

6

z

-

-

#2

-

-

#2

-

z ?1

"2 Figure 2: Special case M = 2.

operations in the time domain of downsampling (by 2) and unit time delay. The adjoint operators D and U then stand for upsampling and unit time advance. The operator representation of the system in Figure 2 is therefore 

 D HB = DU H  D UD  :

6

It can be checked that the two operators 



D DU ;



D U D



are inverses. Fix f, let u(k) be an arbitrary 2-periodic signal, and set x(k) = ef (k)u(k). Then y := Hx has the form y(k) = ef (k)v(k), v(k) 2-periodic. From the equation y = Hx we get in turn 

so that



D DU y =







D DU Hx    D = DU H D U D   D = HB DU x; 





D x DU



D D DU ef v = HB DU ef u:   D Now DU ef v is the 2-dimensional output in the following system: ef (k)v(k)

-

#2

-

-

#2

-

(9)

z ?1

This output can be determined to be   v(0) e?j 2f v(1) e2f (k): Similarly,  D  ef u =  u(0)  e2f : DU e?j 2f u(1) From (9) we therefore have     u(0) v(0) e = H 2 f B e?j 2f u(1) e2f : e?j 2f v(1) Now HB is LTI and its input in (10) is an exponential of frequency 2f; therefore     u(0) v(0) e = H (2f) 2 f B e?j 2f u(1) e2f ; e?j 2f v(1) which can be written as     1 0 u(0) e : v(0) e = H (2f) 1 0 B u(1) 2f v(1) 2f 0 e?j 2f 0 e?j 2f Cancelling e2f gives     1 0 v(0) = H (2f) 1 0 B 0 e?j 2f v(1) 0 e?j 2f

7





u(0) : u(1)

(10)

(11)

Next, substitute the DFT equations     v(0) = 1 1 1 v(1) 2  1 ej     u(0) = 1 1 1 u(1) 2 1 ej

v^(0) v^(1) u^(0) u^(1)

 

into (11) to get      1 1 0 1 1 v^(0) = 1 H (2f) 1 0 1 ?1 v^(1) 0 e?j 2f 2 0 e?j 2f 2 B that is,     u^(0) : V (f) vv^^(0) = H (2f)V (f) B (1) u^(1) Finally, by the equation de ning HAC (f), namely,     v^(0) = H (f) u^(0) ; AC v^(1) u^(1) we get     u ^ (0) u ^ (0) V (f)HAC (f) u^(1) = HB (2f)V (f) u^(1) : Since u(k) was arbitrary, it follows that V (f)HAC (f) = HB (2f)V (f):



1 1 1 ?1





u^(0) ; u^(1)



3 Energy Gain and RMS Gain

Theorem 2 For an M -periodic LPTV system H

G(`2 ; `2) = G(RMS; RMS): For  the proof, we must study the relationships between the `2 norms and RMS values of v = vv0 and x in the system 1

v0-

"2

-

j

x-

6

z

v1

"2



and between y and w = ww0 in the system 1 -

#2

z ?1 y

-

8

#2

w-0 w1

Lemma 1

kxk2 = kvk2 ; kyk2 = kwk2

and

p RMS (x) = p1 RMS (v); RMS (w) = 2 RMS (y) 2 Proof We shall prove the case of x and v; the other is similar. The explicit relationship between v(k) and x(k) is   x(2k) v(k) = x(2k ? 1) : Thus, kvk22 = = and

1 X

k=?1

(jx(2k)j2 + jx(2k ? 1)j2)

1 X

k=?1

jx(k)j2

= kxk22;

"

RMS (v) = "

= "

=

#1=2

N X 1 (jx(2k)j2 + jx(2k ? 1)j2) lim N !1 2N + 1 k=?N

2N X 1 lim jx(m)j2 N !1 2N + 1 m=?2N ?1

#1=2

2N X 4N + 2 1 jx(m)j2 lim N !1 2N + 1 4N + 2 m=?2N ?1

#1=2

#1=2 1=2 " 2N X 4N + 2 1 = Nlim lim jx(m)j2 !1 2N + 1 N !1 4N + 2 m=?2N ?1 p = 2 RMS (x): 

j



Proof of Theorem 2 Again, we take M = 2 to simplify the notation. Form the blocked system HB in Figure 3. From the lemma, the ` induced norm of H (from x to y) equals that of HB v0-

"2

-

2

x

-

H

6

z

v1-

y

-

#2

w0-

-

#2

w1-

z ?1

"2 Figure 3: The blocked system.

(from v to w), and the RMS induced norm of H equals that of HB . But the `2 and RMS induced norms of HB are equal because it's LTI. 

9

4 Examples This section provides illustrations of the two notions of frequency response for some standard examples.

Example 1 Consider the trivial case where M = 2 and H is LTI, with frequency-response

function H(f). As shown in [6], the AC matrix is   0 HAC (f) = H(f) 0 H(f + 1=2) : Equation (8) yields the blocked frequency-response matrix  j 2f [H(f) ? H(f + 1=2)]  1 H(f) + H(f + 1=2) e ; HB (2f) = 2 e?j 2f [H(f) ? H(f + 1=2)] H(f) + H(f + 1=2) 4 or in terms of z-transforms   H(z) + H(?z) z[H(z) ? H(?z)] ; HB (z 2 ) = 21 (1=z)[H(z) ? H(?z)] H(z) + H(?z) or in terms of polyphase components H(z) = E0 (z 2) + (1=z)E1(z 2 )   E E 0 (z) 1 (z) HB (z) = (1=z)E (z) E (z) ; 1 0 the familiar pseudocirculant matrix [7].



Example 2 Consider the downsampler-upsampler combination in Figure 4. As shown in [6] the AC matrix of the input-output system is -

F(z)

-

#2

-

"2

-

G(z)

-

Figure 4: Downsampler-upsampler combination.     1 G(f) 0 1 1 F(f) 0 HAC (f) = 2 0 G(f + 1=2) 1 1 0 F(f + 1=2) ; or equivalently     F(f) F(f + 1=2) ; HAC (f) = 12 G(fG(f) + 1=2) which in terms of z-transforms is     1 G(z) HAC (z) = 2 G(?z) F(z) F(?z) :

Equation (8) yields in terms of polyphase components F(z) = E0(z 2 ) + (1=z)E1(z 2 ) (type-1) and G(z) = R1(z 2 ) + (1=z)R0(z 2 ) (type-2)     R 1 (z) HB (z) = (1=z)R (z) E0(z) E1(z) : 0



Example 3 Figure 5 shows a multirate lter bank. It is well-known that the system H from Note that on the unit circle z = ej2f , so f + 1=2 is equivalent to ej2(f +1=2) = ej ej2f = ?z . Note also that we follow the customary abuse of notation writing H (f ) for frequency-response function and H (z ) for transfer function. 4

10

x(k)

-

F0 (z)

-

#2

-

"2

-

G0(z)

-

F1 (z)

-

#2

-

"2

-

G1(z)

-

Figure 5: Multirate lter bank.

j

x^(k)

?

-

x(k) to y(k) is LPTV, of period 2. To analyze the system in the frequency domain, bring in the 2  2 matrix E(z) whose elements are the type-1 polyphase components of the analysis lters:     F0(z) = E(z 2 ) 1 : F1(z) z ?1 Next, the 2  2 matrix R(z) whose elements are the type-2 polyphase components of the synthesis lters:     G0(z) G1 (z) = z ?1 1 R(z 2 ): Use of these in Figure 5 together with the \noble identities" leads to the input-output equivalent system in Figure 6 (see Figure 5.5-3 in [7]). -

x

#2

-

z ?1

-

"2

-

#2

j

z ?1

R(z)E(z) -

-

"2

-

?

x^

-

Figure 6: Equivalent system in terms of polyphase components. Now block the input x and output y in Figure 6 as shown in Figure 7. This latter system, v0v1-

"2

-

z

h

6

x

-

#2

-

z ?1

"2

-

-

"2 z ?1

R(z)E(z)

#2

-

-

"2

Figure 7: Blocked system.

HB , has 2-dimensional input and output, 







v = vv0 ; w = ww0 ; 1 1

11

-

h

-

#2

w0-

#2

w1-

z ?1

?

x^

-

and the frequency-response matrix from v to w is simply   0 1 R(z)E(z): z ?1 0 Therefore   HB (z) = z ?0 1 10 R(z)E(z) and hence G(`2 ; `2) = kRE k1: We conclude this example with a derivation of the AC matrix for this system. This can be done either by using the above HB (z) together with equation (8) or by considering the fact that the system here is the parallel connection of two downsampler-upsampler combinations of Example 2. Hence we have         1 1 G G 0 (z) 1 (z) HAC (z) = 2 G (?z) F0(z) F0 (?z) + 2 G (?z) F1(z) F1 (?z) 0 1    1 G (z) G (z) F (z) F ( ? z) 0 1 0 0 = 2 G (?z) G (?z) F1(z) F1 (?z) 0 1   1 F F 0 (z)G0 (z) + F1 (z)G1 (z) 0 (?z)G0 (z) + F1 (?z)G1 (z) = 2 F (z)G (?z) + F (z)G (?z) F (?z)G (?z) + F (?z)G (?z) : 0 0 1 1 0 0 1 1



5 Distortions in Maximally Decimated Filter Banks Consider the M-channel maximally decimated lter bank of Figure 8. As discussed in [7, pages x(k)

-

F0 (z)

-

#M

-

"M

-

G0(z)

-

F1 (z)

-

#M

-

"M

-

G1(z)

.. .

.. . -

FM ?1(z)

-

.. . -

#M

-

"M

j j ?

.. . GM ?1(z)

-

-

?

x^(k)

-

Figure 8: M -channel maximally decimated lter bank.

^ in terms of X(z) (ignoring coding and 224{225] the expression for the reconstructed signal X(z) quantization errors) is MX ?1 MX ?1 1 ` ^ X(zW ) Fk (zW ` )Gk (z); X(z) = M `=0 k=0

12

where W := e?j 2=M ; equivalently, ^ = X(z) where

MX ?1 `=0

A` (z)X(zW ` );

MX ?1 A` (z) := M1 Fk (zW k )Gk (z): k=0

It is more convenient to write this in terms of frequency f: ^ = X(f)

MX ?1 `=0

A` (f)X(f ? `=M);

The presence of shifted versions X(f ? `=M); ` > 0, is due to the decimation and interpolation operations. ^ therefore suers from three sources of distortion [7]: The reconstructed signal X(f)

Aliasing Distortion It is clear that aliasing is not present for every possible input x(n) if and only if A` (f) = 0; 1  `  M ? 1: This suggests the following performance indicator as a measure of aliasing distortion: AD := max f

MX ?1 `=1

jA`(f)j

!1=2

2

:

Magnitude and Phase Distortions If the aliasing terms are somehow eliminated by forcing A` (f) = 0 for ` > 0, then the overall system is LTI with frequency-response function A (f). In this case, if A (f) is not allpass (i.e., jA (f)j is not constant), the system is said to have 0

0

0

magnitude distortion, and if A0 (z) has nonlinear phase, it is said to have phase distortion. In many cases, lter banks are required to reconstruct x(k) with at most a constant scaling and a time-delay error. That is, the desired system from x(k) to x^(k) is LTI with frequency-response function Td (f) = ce?j 2fm ; c > 0; m  0: This suggests the following indices for magnitude and phase distortions: MD := max jjTd (f)j ? jA0 (f)jj f PD := max j\Td (f) ? \A0 (f)j = max j ? 2fm ? \A0 (f)j: f f

As claimed in [7], elimination of all three errors at the same time is almost impossible due to practical limitations. However, a good design should have small values for these distortions. The design method of [3] can be summarized as follows. It is assumed that the analysis lters have already been designed for good coding of the input. Then the synthesis lters are designed to minimize the gain of the error system. More precisely, denote by T : x 7?! x^ the time-domain operator of the lter bank, by Td the ideal system with frequency response Td (f), and by J the gain G(`2 ; `2) of the error system Td ? T. The design method of [3] minimizes J. It will now be shown that this design method does indeed produce small distortions by showing that the three distortions are small if J is small. The precise bounds are as follows.

13

Theorem 3 For the M -channel lter bank, AD  J; MD  J; and PD  arcsin(J=c): Proof Following the results of [6] and Example 3, the AC matrix for the M-channel lter bank

is

14

2

MX ?1 6 6 HAC (f) = M1 6 4 k=0

3

Gk (f) Gk (f ? 1=M) 77 7 .. 5 . Gk (f ? (M ? 1)=M) 



Fk (f) Fk (f ? 1=M) : : : Fk (f ? (M ? 1)=M) ;

or in terms of elements HAC (f) = [hn;` (f)]M M ; where MX ?1 hn;`(f) := M1 Gk (f ? n=M)Fk(f ? `=M): k=0 Also let P(f) denote the AC matrix for the ideal system Td , that is, 3 2 Td (f) 0 ::: 0 7 6 0 Td (f ? 1=M) : : : 0 7 P(f) = 664 .. 7: . .. . . . 5 . . . . 0 0 : : : Td (f ? (M ? 1)=M) Since the gain G(`2; `2 ) of an LPTV system equals the L1 norm of its AC matrix and since the AC matrix of Td ? T equals P(f) ? HAC (f), we have that J = kP ? HAC k1 : (12) For the rst row of HAC (f), we have MX ?1 1 h0;`(f) = M Gk (f)Fk (f ? `=M) = A` (f): k=0 Since the L1 norm of each block of a matrix is less than or equal to L1 norm of that matrix, by considering the rst row of P(f) ? HAC (f), we have from (12)

 

Td ? A0 ?A1 : : : ?AM ?1  J; 1 which implies kTd ? A0 k1  J (13) and

 

?A1 ?A2 : : : ?AM ?1  J: (14) 1 The left-hand side of (14) equals sup f

MX ?1 `=1

jA`(f)j

!1=2

2

:

Thus AD  J. Since jjTd (f)j ? jA0(f)jj  jTd (f) ? A0 (f)j; (13) implies MD  J. Finally, from (13) j \Td (f ) ? jA0(f)jej \A0 (f )  J; 8f; ce

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and hence j [\A0 (f )?\Td (f )]  J; 8f: c ? jA0(f)je

#"! p

This says that for every f, the point jA0(f)jej [\A0 (f )?\Td (f )] lies in the disk with center c and radius J: 6

 B  B      

J c

-

arcsin(J=c) Therefore, the absolute value of the angle of this point, namely, j\A0(f) ? \Td (f)j, is less than or equal to arcsin(J=c). Since this is true for every f, PD  arcsin(J=c):



Example To see how tight are the upper bounds for AD, MD, and PD in Theorem 3, consider the two-channel lter bank studied in Example 2 of [3]. The analysis lter F0(z) is the lowpass, FIR, linear-phase lter of order 19 obtained using the Hamming window. The highpass F1(z) is chosen simply by F1(z) = F0(?z). The synthesis lters G0(z) and G1(z) are obtained by minimizing the quantity J with the ideal delay system Td (z) = z ?21 . (Note that c = 1.) The optimal G0 and G1 are IIR with high orders and the optimal performance index Jopt = 0:0298. For this system, the true values for AD, MD, and PD can be computed from their de nitions and are as follows: MD PD AD 0:01489 0:0150 0:855 Since the corresponding J is 0:0298, upper bounds for the three distortions are given by Theorem 3: MD PD AD 0:0298 0:0298 1:7077 It is interesting to note that in this example the three upper bounds dier from the true values roughly by a factor of two.

6 Conclusion In summary, the `2 induced gain and the RMS induced gain are equal for LPTV systems, in particular, for multirate systems. We advocate optimizing such gains for an error system as an eective design approach. For the multirate lter bank example, this design method automatically suboptimizes more traditional measures of distortion.

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References [1] M. Araki and Y. Ito. Frequency-response of sampled-data systems I: open-loop considerations. Technical Report, Division of Applied System Science, Kyoto University, 1992. [2] B. Bamieh and J.B. Pearson. A general framework for linear periodic systems with application to H1 sampled-data control. IEEE Trans. Auto. Control, vol. 37, pp. 418{435, 1992. [3] T. Chen and B.A. Francis. Design of multirate lter banks by H1 optimization. IEEE Trans. Signal Processing, to appear. [4] S.P. Boyd and C.H. Barratt. Linear Controller Design: Limits of Performance, PrenticeHall, Englewood Clis, 1991. [5] R.G. Shenoy. Analysis of multirate components and application to multirate lter design. Proc. ICASSP, pp. III-121{III-124, 1994. [6] R.G. Shenoy, D. Burnside, and T.W. Parks. Linear periodic systems and multirate lter design. IEEE Trans. Signal Processing, vol. 42, pp. 2242-2256, 1994. [7] P.P. Vaidyanathan. Multirate Systems and Filter Banks, Prentice-Hall, Englewood Clis, 1993. [8] Y. Yamamoto. Frequency response and its computation for sampled-data systems. Technical Report, Division of Applied System Science, Kyoto University, 1992.

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