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4.+Design+of+Discrete-Time+Filters

# 4.+Design+of+Discrete-Time+Filters - 4 Design of...

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4. Design of Discrete-Time Filters 4.1. Introduction (7.0) 4.2. Frame of Design of IIR Filters (7.1) 4.3. Design of IIR Filters by Impulse Invariance (7.1) 4.4. Design of IIR Filters by Bilinear Transformation (7.1) 4.5. Design of FIR Filters by Windowing (7.2)

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4.1. Introduction 4.1.1. Overview A discrete-time filter is a discrete-time system which passes some frequency components and stops others. An ideal discrete-time filter is best in frequency selectivity. It passes the frequency components in the pass band distortionlessly and stops the frequency components in the stop band completely. Unfortunately, it is noncausal and cannot be implemented in real time. This is undesired in many applications. In such a case, we need to design a causal discrete-time filter which approximates the ideal discrete-time filter functionally. The goal of the design is an impulse response, a frequency response, a system function, a linear constant-coefficient difference equation or another. An ideal filter can be approximated by an IIR or FIR filter. An IIR filter usually needs less cost, i.e., less computation and memory, and an FIR filter usually has a better performance, especially in the phase
response. If a generalized linear phase is needed, we usually use an FIR filter. 4.1.2. Analysis of Ideal Filters Consider an ideal lowpass filter. The analysis can be extended to other types of ideal filters. Over period [  , ), the frequency response of an ideal lowpass filter is defined as . otherwise , 0 | | ), j exp( ) ( H c  (4.1) Let x i (n)=A i exp(j i n) be a frequency component of the input signal. Then, the corresponding output signal of this filter is y i (n)=A i exp(j i n)H( i ). (4.2) If | i |  c , then

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  . ) n ( ) n ( sin ) n ( h c (4.5) y i (n)=A i exp[j i (n  )], (4.3) i.e., x i (n) is passed with a constant delay . If | i |> c , then y i (n)=0, (4.4) i.e., x i (n) is stopped completely. Thus, the ideal lowpass filter has the best frequency selectivity. The impulse response of the ideal lowpass filter is Since h(n) 0 for n<0, the ideal lowpass filter is noncausal and cannot be implemented in real time. However, in practical applications, a causal lowpass filter is often required. In such cases, we need to find a causal lowpass filter which approximates the ideal lowpass filter functionally. Depending on a specific application, this causal lowpass filter can be an IIR or FIR filter.
4.2. Frame of Design of IIR Filters Conventionally, the design of a discrete-time IIR filter involves a transform from a continuous-time IIR filter into the discrete-time IIR filter. It consists of three steps: 1. Find the specifications on the continuous-time IIR filter from the specifications on the discrete-time IIR filter. 2. Find the system function of the continuous-time IIR filter from

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4.+Design+of+Discrete-Time+Filters - 4 Design of...

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