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U. Glaeser

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where A(ω)<br />

, B(ω)<br />

, X(ω)<br />

, and Y(ω)<br />

are the DTFT of a[<br />

n],<br />

b[<br />

n],<br />

between FIR and IIR filters can be summarized as follows:<br />

© 2001 by CRC Press LLC<br />

x[<br />

n],<br />

and y[<br />

n],<br />

respectively. The difference<br />

FIR filter: An FIR filter has A(ω)<br />

= 1, so its frequency response is formed as a linear combination of<br />

complex exponential functions that is equivalent to a polynomial. Hence, the design problem can<br />

be formulated on a linear vector space and very efficient mathematical optimization methods are<br />

available for approximating the desired frequency response. The design methods are simple, and<br />

often guarantee convergence to an optimal solution. Finally, since FIR filters do not have feedback<br />

they do not suffer stability and sensitivity problems.<br />

IIR filter: In contrast to the FIR case, IIR filters are rational functions, so the design problem is<br />

inherently nonlinear. No elegant mathematical method can guarantee convergence to the global<br />

optimum. In addition to the difficulty of numerical design, IIR filters might exhibit instabilities<br />

where a finite input can generate infinite output and high sensitivity, and where roundoff noise<br />

can be amplified; however, IIR filter design has more design freedom, so IIR filters can have the<br />

same performance as FIR filters but with many fewer filter coefficients.<br />

FFT Implementation<br />

It is possible to implement a digital filter in the frequency domain with the fast Fourier transform (FFT)<br />

algorithm [7]. The implementation requires one FFT of the input signal, one multiplication of vectors,<br />

and one inverse FFT. The length of the FFT determines a block length so the signal must be segmented<br />

into sections for both the input and output. The frequency domain implementation actually uses circular<br />

convolution, so some care is needed to get the correct outputs. The FFT-based method of convolution<br />

is used in special circumstances because it is only practical for real-time systems when the FIR filter<br />

length is rather long—the major drawback is that it requires a large amount of buffer memory for the<br />

block processing.<br />

Adaptive and Time-Varying Filters<br />

Another important class of FIR filters is the class of adaptive filters [8], which find widespread application<br />

in areas such as equalizers for communication channels. The filter coefficients in an adaptive filter are<br />

continually changing as the input changes, so the filter design problem is quite different for these filters.<br />

The methods discussed in this chapter will not handle these cases where the coefficients are time varying.<br />

26.3 Digital Filter Design Problem<br />

Design Specification<br />

A digital filter is usually designed so that its output has a desired frequency content, i.e., the frequency<br />

response is frequency selective. The filter coefficients are then optimized so that the frequency response<br />

H(ω)<br />

will best approximate an ideal frequency response I(ω)<br />

. The ideal response varies for different<br />

applications.<br />

Frequency selective filter: The ideal frequency response is either one or zero.<br />

(ω) =<br />

⎧ 1, ω<br />

⎪<br />

in the pass band<br />

⎨ 0, ω in the stop bands<br />

⎪<br />

⎩ don’t care, ω in the transition bands<br />

The frequency selective filter is designed so that the actual frequency response H(ω)<br />

is close to 1<br />

in the passband and nearly 0 in the stopband. An example of a frequency selective filter is shown<br />

in Fig. 26.1.

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