Multirate Filters and Wavelets: From Theory to Implementation

Multirate signal processing is an enabling technology that brings DSP techniques to applications requiring low-cost and high sample rates. Field programmable gate arrays (FPGAs) provide system-level hardware solutions for signal processing architects demanding both high-performance and flexibility. This workshop provides an introduction to the fundamental theory of multirate filter techniques and provides simple descriptions of polyphase decimators and interpolators. Wavelet theory has development roots in both mathematics and signal processing communities. In this workshop we will explain wavelet theory from both a signal expansion and filter bank viewpoint. Classes of signal processing problems that can benefit from wavelet techniques will be presented with simple examples. Implementation of mulitrate filters and wavelet structures with FPGAs will be explained for real time applications. INTRODUCTION Digital signal processing (DSP) systems offer degrees of freedom to the system designer that are unparalleled in the analog signal processing domain. One of the most important of these parameters is the availability of a system sample clock. In the context of digital filters, the sample clock is accessed and exploited using multirate filter techniques. Judicious and creative use of multirate processes in digital signal processing systems allow a designer to realize a hardware efficient solution, that in many instances just would not be feasible using single rate signal processing. This paper provides an introductory level overview of multirate filters. The basic theory of multirate techniques is presented along with an explanation of polyphase interpolators and decimators. The hardware realization of multirate systems using field programmable gate arrays (FPGAs) is also examined. 1 Robert D. Turney ([email protected]) and Chris Dick ([email protected]) are Senior Systems Engineers in the CORE Solutions Group, Xilinx Inc., San Jose CA, USA 2 Ali M. Reza ([email protected]) is an Associate Professor in the Department of Electrical Engineering and Computer Science at the University of Wisconsin – Milwaukee, Milwaukee WI, U.S.A Wavelet theory had been developed independently on several fronts. Different signal processing techniques, developed for signal and image processing applications, had significant contribution in this development [1]. Some of the major contributors to this theory are: multiresolution signal processing [2], used in computer vision; subband coding, developed for speech and image compression; and wavelet series expansion, developed in applied mathematics [3]. The wavelet transform is successfully applied to non-stationary signals and images. Some of the application areas are: nonlinear filtering or denoising, signal and image compression, speech coding, seismic and geological signal processing, medical and biomedical signal and image processing, and communication. MULTIRATE FILTERS AND REALIZATIONS Filters, analog or digital, are one of the most widely used signal processing functions. In the analog domain, a filter transfer function is realized by a suitable arrangement of resistors, capacitors, operational-amplifiers (op-amps) and possibly inductors. As with any analog platform, this implementation is exposed to the potential problems that result from temperature dynamics, and component tolerances. The motivation to migrate this type of processing to an all ( or mostly) digital implementation involves eliminating these issues, coupled with the system economics, manufacturability and repeatability issues. One might be tempted to start with an analog prototype design and digitize this system to produce a DSP implementation. In fact, this is precisely the worst approach to adopt. In making a digital clone of an analog system all of the unique features of the DSP domain are not exploited. These features include adjusting the sample rate at various nodes in a system, together with the creative use of deliberate aliasing. Exploiting these degrees of freedom in a digital design is of paramount importance in order to obtain the maximum performance with a minimum amount of digital hardware. Indeed, it may even be the difference between being able to perform a required function at a specified sample rate at all using a DSP implementation. One of the most important techniques that a DSP designer has at their disposal to exploit the time dimension of a problem is multirate filtering. In their most basic form, multirate filters are used to decrease (decimation) or increase the sample rate (interpolation) of a stream of samples. Multirate filters are employed in many diverse applications and for many different reasons. One broad motivating factor is to match the sample rate to the signal bandwidth at all, or most, of the nodes in a system or to alter the sample rate so that signals with differing sample rates may be combined in some manner. Ultimately, multirate filters minimize the arithmetic workload required to perform a specified calculation. In turn, this minimizes the number of clock cycles consumed in a software based implementation, the number of gates in an ASSP (application specific standard part) realization, or the number of logic elements consumed in an FPGA design. Decimation Decimation can be useful if the sample rate of a signal is considerably greater than twice the signal’s bandwidth. The process of sample rate decimation, or simply decimation, involves two steps as shown in Figure 1. The first stage is a bandwidth limiting operation (anti-aliasing filter) while the second step performs the down-sampling. M H(z) x(n) y(m) ANTI-ALIASING FILTER DOWN-SAMPLER Figure 1: Signal decimation – bandwidth reduction followed by down-sampling. The anti-aliasing filter is a lowpass design. This filter must reduce the signal bandwidth in accordance with the Nyquist sampling theorem. The filtered signal’s bandwidth B must satisfy B f s ≤ ' , f f M s s ' / = , where f s ' is the decimated data stream output sample rate, f s is the input sample rate and M is the decimation factor. The implicit assumption employed here is that the band of frequencies B f f s ≤ ≤ / 2 contains no useful information and may be safely discarded in the application. The bandwidth limiting process is often constrained to preserve phase linearity and so a FIR filter, as shown in Figure 2, is employed. a0 a1 z-k a2 z-k a3 z-k aNM -1 z-k