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7 FILTER DESIGN TECHNIQUES 7.0 INTRODUCTION Filters are a particularly important class of linear time-invariant systems. Strictly speak- ing, the term frequency-selective filter suggests a system that passes certain frequency components and totally rejects all others, but in a broader context any system that modifies certain frequencies relative to others is also called a filter. While the primary emphasis in this chapter is on the design of frequency-selective filters, some of the tech- niques are more broadly applicable. Also, we concentrate on the design of causal filters, although in many contexts filters need not be restricted to causal designs. Very often, noncausal filters are designed and implemented by modifying causal designs. The design of filters involves the following stages: (1) the specification of the desired properties of the system, (2) the approximation of the specifications using a causal discrete-time system, and (3) the realization of the system. Although these three steps are certainly not independent, we focus our attention primarily on the second step, the first being highly dependent on the application and the third dependent on the technology to be used for the implementation. In a practical setting, the desired filter is generally implemented with digital computation and used to filter a signal that is derived from a continuous-time signal by means of periodic sampling followed by analog-to- digital conversion. For this reason, it has become common to refer to discrete-time filters as digital filters, even though the underlying design techniques most often relate only to the discrete-time nature of the signals and systems. When a discrete-time filter is to be used for discrete-time processing of continuous- time signals in the configuration of Figure 7.1, the specifications for both the discrete- time filter and the effective continuous-time filter are typically (but not always) given in the frequency domain. This is especially common for frequency-selective filters such 439 
