Objectives:
This chapter introduces principles of the finite impulse response (FIR) filter design and investigates the design methods such as the Fourier transform method, window method, frequency sampling method design, and optimal design method. Then the chapter illustrates how to apply the designed FIR filters to solve real-world problems such as noise reduction and digital crossover for audio applications. The major topics discussed in this chapter are included in the following outline.
In this chapter, we describe techniques of designing finite impulse response (FIR) filters. An FIR filter is completely specified by the following input-output relationship:
| (7.1) |  |
where b i represents FIR filter coefficients and K + 1 denotes the FIR filter length. Applying the z-transform on both sides of Equation (7.1) leads to
| (7.2) |  |
Factoring out X( z) on the right-hand side of Equation (7.2) and then dividing X( z) on both sides, we have the transfer function, which depicts the FIR filter, as
| (7.3) |  |
The following example serves to illustrate the notations used in Equations (7.1) and (7.3) numerically.
Given the following FIR filter:
Determine the transfer function, filter length, nonzero coefficients, and impulse response.
Solution:
Applying z-transform on both sides of the difference equation yields
Then the transfer function is found to be
The filter length is K + 1 = 3, and the identified coefficients are
Taking the inverse z-transform of the transfer function, we have
This FIR filter impulse...
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