Stationarity Condition for Fractional Sampling Filters

INDIAN INSTITUTE OF TECHNOLOGY DELHI Stationarity condition for Fractional sampling filters by Pushpendre Rastogi Report submitted in fulfillment of the requirement of the degree of Masters of Technology under guidance of Dr. Brejesh Lall Department of Electrical Engineering June 2011 Certificate This is to certify that the report titled \Stationarity condition for Fractional sampling filters” being submitted by Pushpendre Rastogi to the Department of Electrical Engineering, Indian Institute of Technol- ogy, Delhi, for the award of the degree of Masters of Technology, is a record of bona-fide research work carried out by him under my guidance and su- pervision. In my opinion, the report has reached the standards fulfilling the requirements of the regulations relating to the degree. The results contained in this report have not been submitted to any other university or institute for the award of any degree or diploma. Dr. Brejesh Lall Assistant Professor Department of Electrical Engg., Indian Institute of technology, Delhi New Delhi - 110016 i Declaration of Authorship I, Pushpendre Rastogi, declare that this report titled, `Stationarity condition for Fractional sampling filters' and the work presented in it are my own. I confirm that: This work was done wholly while in candidature for a masters degree at this University. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Signed: Date: ii Acknowledgements I would like to thank my project supervisor Dr. Brejesh Lall for encouraging me and helping me with my research for this project. I would also like to thank all the SRC members for listening to my presentations and giving their inputs. I believe that both the Praise and criticism i received during my presentations helped me in staying focused on my work. iii INDIAN INSTITUTE OF TECHNOLOGY DELHI Abstract Dr. Brejesh Lall Department of Electrical Engineering Masters of Technology by Pushpendre Rastogi Cyclostationary processes are non-stationary processes whose statistical characteristics repeat in time with a period greater than 1. In general Cyclostationary processes when passed through LTI systems remain Cyclo- stationary with the same period. However there are special cyclostationary signals which when passed through specific LTI filters can become stationary. One such type of cyclostationary signal is the output of an interpolator for a WSS input. Such a cyclostationary signal can be stationarized by an LTI system subject to certain conditions. This work aims to generalize this by characterizing an LTI system which can stationarize the output of a general fractional sampling filter. Contents Certificatei Declaration of Authorship ii Acknowledgements iii Abstract iv List of Figures vii 1 Introduction1 1.1 Literature Survey........................2 1.2 Outline..............................3 1.3 Approach and Contributions..................3 1.4 Notations used.........................4 2 Preliminaries5 2.1 Introduction...........................5 2.2 Wide sense stationary process.................6 2.3 Cyclostationary process.....................6 2.4 Power spectral density.....................7 2.5 General Linear Time Varying systems.............7 2.5.1 LSIV systems......................8 2.5.2 LPTV and LTI systems................8 2.6 Bifrequency maps........................9 2.7 M-fold decimator........................9 2.8 L-fold interpolator....................... 10 2.9 Polyphase decomposition.................... 13 2.10 Blocking............................. 14 2.11 Pseudocirculant matrices.................... 15 2.12 Stationarity condition for Fractional sampling filters..... 16 2.13 Concluding Remarks...................... 17 v Contents vi 3 {LT I(h) condition for Fractional sampling filters 18 3.1 Introduction........................... 19 3.2 Analysis............................. 20 3.3 Solution............................. 22 3.4 Conclusions........................... 23 4 Concluding Remarks 25 4.1 Scope for future research.................... 25 Bibliography 27 List of Figures 1.1 Block Diagrams for Basic Multirate systems.........2 1.2 Fractional sampling filter....................3 2.1 Bispectra of 3-fold Decimator................. 11 2.2 Bispectra of 3-fold Interpolator................ 12 −1 2.3 Multirate implementation of BM and BM .......... 15 3.1 Our aim is to find {LT I condition for the system enclosed by dotted lines......................... 19 3.2 Simplification using well known Nobel identity........ 22 3.3 Final simplified Block diagram................. 22 vii Chapter 1 Introduction Cyclostationary signals are stochastic processes. These processes are not periodic functions of time themselves rather they exhibit periodically varying statistical properties [1]. Such processes occur as output and inter- mediate signals in Multirate digital filtering which is used in a variety of applications such as subband coding, voice privacy systems, transmultiplex- ers, and adaptive filtering [2] to name a few. Generally in multirate digital signal processing, time-varying linear systems such as decimators, interpolators, and modulators are put together in complicated interconnections with linear filters which are time invariant [2]. Some work has been done previously to understand the way in which the statistical behavior of a signal changes as it passes through such systems. However this work addresses the problem that how one may convert a Cyclostationary process to a stationary process by passing it through an Linear Time Invariant system. 1 Chapter 1. Introduction 2 (a) M-fold Decimator (b) L-fold Interpolator Figure 1.1: Block Diagrams for Basic Multirate systems 1.1 Literature Survey We now present a brief overview of the existing literature in the area of Study of Cyclostationary signals and multirate systems and specifically the effect of Multirate systems on the statistical properties of cyclostationary signals. A concise survey of the literature on cyclostationarity is presented in the work of Gardner et. al. [1]. A detailed analysis of pseudocirculants for LPTV system simplification, can be found in the Pioneering work of P. P. Vaidyanathan and Sanjit K. Mitra [3]. Other relevant contributions of Vaidyanathan with his Co-authors are as follows. His book Multirate Systems and Filter Banks [4] provides a good introduction to the field. His work with Vinay P. Sathe studies the effect of multirate systems on the second order statistical properties of stochastic signals [2]. In a later work done in 2000 with Sony he generalizes the above results by using the technique of Bispectra and Bifrequencies [5]. Two sources were referred to understand the concept of Bispectrum and Bifrequency maps. First was the 1984 paper by Loeffler and Burrus [6] and second was the book Multirate Digital Signal Processing by Crochiere [7]. Lastly the chapter Cyclostation- ary Signal analysis [8] in Digital Signal Processing Handbook was found to be very informative and helped clear up confusion in the different forms Chapter 1. Introduction 3 of Cyclostationary signal representation that may arise because the names used to refer to the same representations are often different in different works. Figure 1.2: Fractional sampling filter 1.2 Outline The report is organized as follows. In Chapter 2 we provide a review of the definitions, properties and results that we would use in our work. We cover the definitions of stationary and cyclostationary discrete random processes. We also include definitions of various multirate concepts such as Decima- tors Interpolators and Fractional Sampling filters. After that a review of polyphase decomposition of signals and systems is given. Then the effects of Blocking are studied on both signals and systems. Relevant results and derivations are given alongside the definitions for easy referencing. Lastly we review the Pseudocirculant conditions and the Stationarity condition on a Fractional Sampling filter such that it gives stationary output for a stationary input. 1.3 Approach and Contributions We first define a term to conveniently denote the conditions on an lti system such that when it is cascaded to a system h then a stationary signal applied as input to the cascade produces a stationary output. Then we Chapter 1. Introduction 4 state our problem statement in terms of this term. We address the problem of specifying the above mentioned conditions for a Fractional sampling filter. In order to do that, analysis of the problem was done by looking at the bispectra of a general Fractional sampling filter. Several interesting conclusions were drawn from this study of the bispectra. The final solution was then found out by using the powerful Nobel identities and by using a previous result given by P.P. Vaidyanathan and Vinay Sathe in [2] for the simple Interpolator. 1.4 Notations used Superscripts (∗), (T ) and (y) denote the complex conjugate, matrix (or vector) transpose, and conjugate transpose respectively. Boldface letters are used for matrices and vectors. The (i; j) element of a matrix B is denoted by [B](i;j). Lower-case letters are used for 1-D and 2-D discrete sequences, whereas upper-case letters are used for 1-D and 2-D Fourier transforms. Z and R denote the set of integers and real numbers respectively. The space of all finite norm M-component vector sequences is denoted by l2(M).

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