DIGITAL FILTER DESIGN WITH OPTIMAL ANALOG PERFORMANCE Yutaka Yamamoto and Masaaki Nagahara, Department of Applied Analysis and Complex Dynamical Systems Graduate School of Informatics, Kyoto University Kyoto 606-8501, JAPAN Abstract This paper proposes a new digital filter design methodol- ogy, based on sampled-data control theory. In contrast to the conventional filter designs where the methods are mostly based on frequency domain approximation techniques, the proposed method makes use of the sampled-data con- trol theory which has been quite successful in recent years in the control literature. The novel feature here is that the pro- posed method can optimize the analog-domain performance over all frequency ranges, thereby guaranteeing a desirable performance without breaking the design problem into sev- eral different steps, such as linear phase characteristic, op- timal attenuation level design, etc. A design example is pre- sented 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 phase shift property, smooth pass-band transmission, high atten- uation level in the stop band, desirable transition band char- acteristic, etc. Many guiding quantities are there to help the designer [8, 13, 9]. The design is now performed mostly in the discrete-time domain. To capture the continuous-time performance, the notion of aliasing is utilized and deviation from the ideal filter has to be discussed. To bypass the problem of the Gibbs phenomenon in the frequency domain windows are often effective. One may however note that, in many applications, the performance we wish to optimize is still in the analog do- main: speech/audio is one example; visual images are an- other. While one may start with the digitized data in which case 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 ex- ample, in audio recordings, we have a fairly good idea on how the frequency characteristics are for recorded signals. yy@i.kyoto-u.ac.jp 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 method- ology, based on sampled-data control theory. In contrast to the conventional filter designs, this design method does not rely on an approximation techniques (e.g., frequency sam- pling). Instead, it gives rise to an optimal transfer operator, where the performance is measured by the norm. In contrast to the more popular norm, which measures only the mean-square performance of the frequency response, the norm measures the supremum of the gain of the frequency response. By multiplying a suitable frequency weighting function, we can control the attenuation level fairly precisely. The price is that this norm does not make the underlying signal space a Hilbert space; is only a Banach space. Hence the standard technique for approxi- mation such as the projection theorem cannot be used, and optimization in this space is indeed good deal more diffi- cult than that in which is a Hilbert space. However, it is more natural and adequate for many applications as a performance measure and this explains the recent boost of applications of control after this problem was solved in a satisfactory form (see, e.g., [4]). This development is further generalized to the sampled- data context where measurement and control actions occur in discrete time. The theory for sampled-data control is now fairly complete; the important feature here is that sampled-data control optimizes continuous-time (analog) 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 for designing digital filters. An attempt is made in [3] for an multirate filter bank design problem. Other approaches have also been made, e.g., [7, 10, 11]. However, a low-pass filter design problem with optimal analog performance has not been formulated or solved there. This paper considers the design of an optimal low-pass filter design when one employs an upsampler. The objective is to reconstruct the original signal in this situation. Usu- ally this problem is dealt with under the assumption that