DIGITAL FILTER DESIGN WITH OPTIMAL ANALOG PERFORMANCE
Yutaka Yamamoto and Masaaki Nagaharay,
Department of Applied Analysis and Complex Dynamical Systems
Graduate School of Informatics, Kyoto University
Kyoto 606-8501, JAPANAbstract
This paper proposes a new digital filter design methodology, based on sampled-data control theory. In contrast to
the conventional filter designs where the methods are mostly
based on frequency domain approximation techniques, theproposed method makes use of the sampled-data H 1 control theory which has been quite successful in recent years in
the control literature. The novel feature here is that the proposed method can optimize the analog-domain performanceover all frequency ranges, thereby guaranteeing a desirable
performance without breaking the design problem into several different steps, such as linear phase characteristic, optimal attenuation level design, etc. A design example is presented to show the advantages of the present method.
1. Introduction
Digital filter design is an art of approximation which takes
many different specifications into account: linear phaseshift property, smooth pass-band transmission, high attenuation level in the stop band, desirable transition band characteristic, etc. Many guiding quantities are there to help thedesigner [8, 13, 9].
The design is now performed mostly in the discrete-time
domain. To capture the continuous-time performance, thenotion of aliasing is utilized and deviation from the ideal
filter has to be discussed. To bypass the problem of theGibbs phenomenon in the frequency domain windows areoften effective.
One may however note that, in many applications, theperformance we wish to optimize is still in the analog domain: speech/audio is one example; visual images are another. While one may start with the digitized data in whichcase an analog-domain performance cannot be adequately
discussed, there are many other cases where we can discuss
the basic characteristics of the original analog data. For example, in audio recordings, we have a fairly good idea on
how the frequency characteristics are for recorded signals.

 yy@i.kyoto-u.ac.jp
y nagahara@acs.i.kyoto-u.ac.jp

Recovering such signals optimally in the sense of analog
performance is clearly an important issue.
This paper proposes a new digital filter design methodology, based on sampled-data control theory. In contrast to
the conventional filter designs, this design method does notrely on an approximation techniques (e.g., frequency sampling). Instead, it gives rise to an optimal transfer operator,
where the performance is measured by the H 1 norm. In
contrast to the more popular H 2 norm, which measures only
the mean-square performance of the frequency response,the H 1 norm measures the supremum of the gain of thefrequency response. By multiplying a suitable frequency
weighting function, we can control the attenuation level
fairly precisely. The price is that this norm does not makethe underlying signal space a Hilbert space; H 1 is only aBanach space. Hence the standard technique for approximation such as the projection theorem cannot be used, andoptimization in this space is indeed good deal more difficult than that in H 2 which is a Hilbert space. However,
it is more natural and adequate for many applications as aperformance measure and this explains the recent boost ofapplications of H 1 control after this problem was solved ina satisfactory form (see, e.g., [4]).
This development is further generalized to the sampleddata context where measurement and control actions occur
in discrete time. The theory for sampled-data H 1 control
is now fairly complete; the important feature here is thatsampled-data control optimizes continuous-time (analog)
H 1 performance, while maintaining discrete-time control
actions [2]. There are also remarkable differences between
sampled-data and discrete-time designs.
Such a development provides an optimal platform fordesigning digital filters. An attempt is made in [3] for an
multirate filter bank design problem. Other approaches havealso been made, e.g., [7, 10, 11]. However, a low-pass filter
design problem with optimal analog performance has notbeen formulated or solved there.
This paper considers the design of an optimal low-passfilter design when one employs an upsampler. The objective
is to reconstruct the original signal in this situation. Usually this problem is dealt with under the assumption that

-F s -y Swc()-" M x-K z u-H -u P s -z

ydchdA/D()

dch=M()-emhs-F s- S -" M -K z - Hcwc

D/A()

h?
f-e-P s 6+()h=Mc()

Figure 1: Multirate Signal Reconstruction

Figure 2: Signal reconstruction error system

the original signal be band-limited; in practice, however,no signals are entirely band-limited. Instead, we may often know an approximate frequency characteristic that theoriginal signals obey, and it is these original analog signals
we wish to reconstruct optimally.
We thus formulate a digital signal reconstruction problem under the following assumptions:

P (s). An advantage here is that one can use a fast hold device Hh=M thereby making more precise signal restoration
possible. The objective here is to design the digital filter
K (z ) for given F (s), M and P (s).
Figure 2 shows the block diagram for the error system
for the design. The delay in the upper portion of the diagram
corresponds to the fact that we allow a certain amount oftime delay for signal reconstruction. Let Tew denotes theinput/output operator from wc to ec (t) := zc (t) uc (tmh). Our design objective is as follows: the orignal analog signal is subject to a certain frequency characteristic, but not fully band-limited; the digital signal can be upsampled to employ a fasterhold device; the overall analog H 1 performance must be opti-Problem 1 Given stable F (s) and P (s) and an attenuationlevel > 0, find a digital filter K (z ) such that

2. Problem Formulation
Consider the block diagram Figure 1. The incoming signal wc first goes through an anti-aliasing filter F (s) and thefiltered signal yc becomes nearly (but not entirely) bandlimited. F (s) governs the frequency-domain characteristicof the analog signal yc . This signal is then sampled by Sh
to become a discrete-time signal yd with sampling period h.
This signal is usually stored or transmitted with some media(e.g., CD) or a channel.
To restore yc we first upsample the discrete-time signal
yd by factor M :" M : y 7! x : x [k] =ddyd [l]; k = M l; l = 0; 1; : : :0;otherwise.

The signal then becomes another discrete-time signal xd
with sampling period h=M . The discrete-time signal xd
is then processed by a digital filter K (z ), becomes acontinuous-time signal uc by going through the 0-order hold
Hh=M (that works in sampling period h=M ), and then becomes the final signal by passing through an analog filter

ew

supew

This may also be regarded as an optimal D/A converter
design. We will see that performance improvement is possible over a conventional low-pass filter. Even though we do
not explicitly place the constraint on linear-phase property,
it is interesting to note that the obtained filter is very closeto linear phase up to the Nyquist frequency, due to the H 1
performance requirement.

dkT w k2 < ;kw k2
c 2 2 [0 1)kT k1 :=mized.

w

Lc(1)

c;

where kk2 denotes the L2 norm. An advantage of defining
this problem is that we do not need any extra design constraint (such as the linear phase property). Such issues areincorporated into the design diagram Fig. 2.
3. Reduction to A Finite-Dimensional Problem
A difficulty in Problem 1 is that it involves a continuoustime-delay, and hence it is an infinite-dimensional problem.
Another difficulty is that it contains the upsampler " M , sothat it makes the overall system time-varying.
Following the method of [7, 10], however, we can reduce this problem to a finite-dimensional single-rate problem:
Theorem 1 There exist (finite-dimensional) discrete-timesystems Gd11(z ), Gd12(z ) and Gd21(z ) such that (1) isequivalent tokzme (z )Gd21(z )k1 < ;Gd12(z )KGd11(z )(2)e (z ) is the blocking system of K (z ).
where KProof
We first reduce the problem to a single-rate
problem. Define the blocking operator M and its inverse
1byMLLLLM1M:=:=(#M)

1

1z 1zzM 1z M +1

T" M );(-F swc()-e

By this method, our design problem (1) is approximated?
g-e- Ke z - He -P s 6mhsasc-Sh( )kz()

hwhere # M denotes the downsampler# M : x 7! y : y [k] = x [M k]:
Then K (z )(" M ) can be rewritten asdd=5. A Design ExampledL Ke (z)e (z ) := L K (z )LK

K (z )(" M )We present a design example for1MM


1M

10



e (z )GdN 21 (z )k1 < ;GdN 11 (z ) + GdN 12(z )K

where GdN 11 , GdN 12 and GdN 21 are discrete-time systems
obtained by the technique in [2, 10]. Once the problem has
been reduced to such a problem, it can be solved by a control
design toolbox such as those given by MATLAB (see, e.g.,[1]).

Figure 3: Reduced single-rate problemdm0

TF (s) =:1(0:70223s + 1)(7:0223s + 1);

P (s) = 1with h = 1, m = 2 and upsampling factor M = 2. (Incommercial CD players, M is usually 8  32.) An approximate design is executed here for N = M  4 = 8. Fore (z M ):K (z ) = 1 z 1    z M +1 K
comparison, we compare it with the Johnston filter[6] of order 31, which is often used in commercial applications. Fore h defined byUsing the generalized hold H
the CD format, the above frequency response corresponds
to the decay of 20dB/dec beyond 1kHz, and 40dB/dec
Heh : l2 3 v 7! u 2 L2 ; u(kh + ) = ()v[k]beyond 10kHz. In many such applications, we often have 2 [0; h); k = 0; 1; 2; : : :such a frequency characteristic for original analog signals.
8 The computed (via MATLAB [1]) (sub) optimal filter
1 0 0 ::: 0 ; 2 [0; h=M )>>
<0 1 0 ::: 0 ; 2 [h=M; 2h=M ) KSD (z ) is IIR of order 7 with the filter coefficients ( :( ) :=numerator, : denominator) of the transfer function:>>: 23230 0 : : : 0 1 ;  2 [(M1)h=M; h)
0:053136
1:000000
6670:081530 70we obtain the identity
67676 0:252018 7
6 0:255479 767676 0:809684 7
670
Hh=M M1 = He h :7;67:=6=
6 0:917880 7
670
67676 0:512434 767This yields
067674 0:134899 5450ee
Hh=M K (z )(" M )Sh = Hh K (z )Sh :0:010296
0e (z ) is an LTI, single-input/M -output system that satisfies
KHHbaLHence Figure 2 is equivalent to Figure 3. We can theninvoke the technique of [7] to reduce this to a finitedimensional design problem (2).
2
4. Approximation via Fast Sample/Hold
While the procedure above reduces Problem 1 to a finitedimensional H 1 problem, it is in general not numericallysuitable for actual computation; the formulas are quite involved, and not so numerically tractable. It is often moreconvenient to resort to an approximation method. We employ the fast sample/hold approximation [2, 10]. Thismethod approximates continuouos-time inputs and outputs
via a sampler and hold that operate in the period h=M . The
convergence of such an approximation is guaranteed in [12].

baThe impulse response of this filter becomes almost zero after 12  14 steps, so that it can be well approximated by an
FIR filter of 14 taps.
Figure 4 shows the gain characteristics of these filters.
The Johnston filter shows a very sharp decay beyond thecutoff frequency ( [rad/sec]) and the sampled-data designshows a rather slow decay. If the original signals are ideallyband-limited, then a sharp decay such as that of the Johnston
filter would be desirable. But if the original signal is not
fully band-limited, but obeys the frequency characteristic as
specified by F (s), such a sharp cut in the transiiton bandmay not be optimal.
In fact, the reconstruction error characteristic in Figure 5 shows that the performance of KSD (z ) is comparable to that of the 32 tap Johnston filter. Furthermore, weFrequency Response−330−34−10−35−20−36
Gain [dB]Gain [dB]

Frequency Response
10−30−40−37−38−50−39−60−40−70−41−8001234Frequency [rad/sec]

567−42 −210−110010110Frequency [rad/sec]

Figure 4: Frequency response of filter: KSD (z ) (solid) andKJ (z ) (dash)

Figure 5: Frequency response of error system Tew :sampled-data H 1 synthesis (solid) and Johnston filter(dash)have made sure (frequency response plot omitted) that thesampled-data design gives 6-10 dB improvement over theJohnston filter when we increase the upsampling factor toM = 4.

7. References

Furthermore, in the time-domain, the sampled-data design shows a clear advantage over the Johnston filter. Let us
see the time responses against rectangular waves in Figures
6, 7:
While the Johnston filter exhibits a very typical Gibbs
phenomenon, the one by KSD (z ) has much less peakaround the edge. We also note that KSD (z ) is nearly linear
phase, as shown in Figure 8. The Gibbs phenomenon in theconventional design is simply an outcome of the very sharp
cut beyond the cut-off frequency, since it corresponds to thetruncation of a Fourier series with finitely many terms.
To see the effect, the obtained filter is applied to downsampled (by factor 2) CD signals, and less distortion has
been heard compared to that processed by the Johnston filter.

6. Concluding Remarks
We have presented a new method of designing a digital filter
in multirate signal reconstruction problem. An advantage
here is that an analog optimal performance can be obtained,
and this can be advantageous in audio signal reconstruction.
Another advantage is that the design can be done in essentially one process, using MATLAB routines (e.g., [1]), oncethe problem is formulated.

[1] G. Balas, J. C. Doyle, K. Glover, A. Packard and R.
Smith, -Analysis and Synthesis Toobox, MUSYN
and The MathWorks, Inc. (1994)[2] T. Chen and B. A. Francis, Optimal Sampled-Data
Control Systems, Springer, New York (1995)[3] T. Chen and B. A. Francis, “Design of multirate filterbanks by H1 opimization,” IEEE Trans. Signal Processing, SP-43: 2822–2830 (1995)[4] J. C. Doyle, K. Glover, P. P. Khargonekar, B. A. Francis, “State-space solutios to standard H1 and H2control problems,” IEEE Trans. Autom. Control, AC34:831-847 (1989)[5] N. J. Fliege, Multirate Digital Signal Processing, JohnWiley, New York (1994)[6] J. D. Johnston, “A filter family designed for use inquadrature mirror filter banks,” Proc. of IEEE International Conference on Acoustics, Speech and Signal
Processing, pp. 291–294 (1980)[7] P. P. Khargonekar and Y. Yamamoto, “Delayed signal
reconstruction using sampled-data control,” Proc. of35th Conf. on Decision and Control, pp. 1259–1263(1996)[8] T. W. Parks and C. S. Burrus, Digital Filter Design,John-Wiley, New York (1987)[9] P. P. Vidyanathan, Multirate Systems and Filter Banks,
Prentice Hall, Englewood Cliffs, (1993)

Phase plot

1.5

0−50

1−1000.5
Gain [dB]−1500−0.5

−200
−250
−300
−350

−1−400−1.5051015

1234Frequency [rad/sec]

567

Figure 8: Phase plot of KSD

Figure 6: Time response (sampled-data design)
1.5

Processing: Theory and Applications, Marcel Dekker(1994)

10.50−0.5−1−1.50−450051015Figure 7: Time response (Johnston filter)[10] Y. Yamamoto, H. Fujioka and P. P. Khargonekar, Signal reconstruction via sampled-data control with multirate filter banks, Proc. 36th Conf. on Decision andControl, pp. 3395–3400 (1997)[11] Y. Yamamoto and P. P. Khargonekar, “From sampleddata control to signal processing,” in Learning, Control and Hybrid Systems, Springer Lecture Notes inControl and Information Sciences, vol. 241, pp. 108–126 (1998)[12] Y. Yamamoto, A. G. Madievski and B. D. O. Anderson, “Approximation of frequency response forsampled-data control systems,” Automatica, vol. 35,
No. 4, pp. 729-734, 1999.
[13] G. Zelniker and F. J. Taylor, Advanced Digital Signal