- 1151 - Interpolation of Nonuniformly Decimated Signals via Sampled-data H ∞ Optimization M. Nagahara †1 , M. Ogura †2 , and Y. Yamamoto †3 † Graduate School of Informatics, Kyoto University, Kyoto, Japan 1 nagahara@ieee.org, 2 ogura@acs.i.kyoto-u.ac.jp, 3 yy@i.kyoto-u.ac.jp Abstract: In this paper, we consider signal interpolation of discrete-time signals which are decimated nonuniformly. A conventional interpolation method is based on the sampling theorem, and the resulting system consists of an ideal filter with complex coefficients. On the other hand we adopt sampled-data H ∞ optimization, which can take account of intersample behavior, and the optimal filter with real coefficients is obtained. An example shows the effectiveness of our method. By examples, we also show that there is the optimal decimation pattern. Keywords: nonuniform decimation, interpolation, sampled-data control 1. INTRODUCTION Interpolation is a fundamental operation in digital sig- nal processing, and has many applications such as signal reconstruction, signal compression/expansion, and resiz- ing/rotating digital images, see [11], [3]. If digital data to be interpolated are located uniformly on the time axis, the uniform interpolation is executed by an expander and a digital filter (called an interpolation filter) [11], which is conventionally designed via the sampling theorem. Periodic nonuniform interpolation (or decimation) also plays an important role in signal processing, such as signal compression by nonuniform filterbanks [8], super- resolution image processing [9], and time-interleaved AD converters [10]. The design has been studied by many researchers [12], [8], [13], [4], [5], in which the design methods are based on the generalized sampling theorem. The optimal filter (or the perfect reconstruction filter) is an ideal lowpass filter with complex coefficients [12], [11]. Since the ideal filter cannot be realized, approxi- mation methods are also proposed, see in particular [12], [13]. On the other hand, real signals such as audio sig- nal (esp. orchestral music) breaks the band-limiting as- sumption in the sampling theorem, that is, they have some frequency components beyond the Nyquist fre- quency. In view of this, we have to take account of the whole frequency range in designing interpolation sys- tems. Sampled-data H ∞ optimization [1], [7] is very ad- equate for this purpose. In this article, we first define nonuniform decima- tion/interpolation. This definition includes the block dec- imation introduced in [8]. Then we formulate the interpo- lation problem as a sampled-data H ∞ optimization. The optimal filter is given by a periodic system, which can be realized by a multirate filterbank. Design examples show the effectiveness of our method. Moreover, we consider by examples what is the optimal decimation pattern. We show that although the decimation rate is the same, the optimal value can differ if the pattern differs. That is, the performance depends on the decimation pattern. This property can be used in designing signal compression. 2. NONUNIFORM DECIMATION AND INTERPOLATION Consider a discrete-time signal x := {x 0 ,x 1 ,x 2 ,...} shown in Fig. 1. Then nonuniform decimation by M := [1, 1, 0] (we call this a decimation pattern) is defined as follows (see Fig. 2). (↓ M)x := {x 0 ,x 1 ,x 3 ,x 4 ,x 6 ,...}. (1) That is, we first divide the time axis into segments of length three (the number of the elements of M), then, in each segment, retain the samples corresponding to 1 in M and discard the one corresponding to 0. This deci- mation includes so-called block decimation, in which the first R 1 samples of each segment of R 2 samples are re- tained while the rest are discarded [8]. By using our nota- tion, the block decimation R 2 : R 1 is represented as ↓ M with M = [1,..., 1 � �� � R1 , 0,..., 0 � �� � R2−R1 ]. Then we consider interpolation. First, we define the nonuniform expander ↑ M with M = [1, 1, 0] by (↑ M)x := {x 0 ,x 1 , 0,x 2 ,x 3 , 0,x 4 ,...}. That is, we first divide the time axis into segment of length two (the number of the elements 1 of M), then insert 0 into the portion corresponding to 0 in M. Ap- plying this to the decimated sequence (1), we have v := (↑ M)(↓ M)x = {x 0 ,x 1 , 0,x 3 ,x 4 , 0,x 6 ,...}. Then, the interpolation is completed by filtering v by a digital filter K (see Fig. 3 (a)). 3. DESIGN OF INTERPOLATION FILTER 3.1 Design Problem In this section, we consider a general case where the decimation pattern is defined by M := [b 1 ,b 2 ,...,b M ], b i ∈{0, 1}. SICE Annual Conference 2008 August 20-22, 2008, The University Electro-Communications, Japan PR0001/08/0000-1151 ¥400 © 2008 SICE