When the sequence of coefficients is symmetrical (b0 = bM, b1 = bM−1, etc.), the frequency...

When the sequence of coefficients is symmetrical (b0 = bM, b1 = bM−1, etc.), the frequency response can be manipulated as in Example 1. Following the style of that example, show that the frequency response of an FIR filter with coefficients {bk}= {1, −2, 4, −2, 1} can be expressed as

Example 1

Formula for the Frequency Response

Consider an LTI system for which the difference equation coefficients ar {bk} = {1, 2, 1}. Substituting into (6.3) gives

To obtain formulas for the magnitude and phase of the frequency response of this FIR filter, we can manipulate the equation as follows:

Since for frequencies , the magnitudes is and the phase is .