Example(a): For the FIR filter of size N = 7 with impulse response h(n) let h(n) = h(N-1-n). Denotes that the filter has a linear phase characteristic. (b) Repeat for N = 8.

Solution (a) For N = 7, the positive symmetry relation h(n) = h(N-1-n) tends to h(n) = h(6-n) which defines that h(0) = h(6), h(1) = h(5), and h(2) = h(4), as given in figure above.

The coefficients, usually, are shown by

where The phase result is obviously k = 0 linear, with slope = -3 = - (N- 1)/2 which seems that the delay is an integer number of samples.

(b) For N = 8, the positive symmetry relation h(n) = h(N-1-n) leads to h(n) = h(7-n), which means h(0) = h(7), h(1) = h(6), h(2) = h(5), and h(3) = h(4) as shown in figure below.

is mainly linear. However, the slope of the phase curve is (-7/2), which is not an integer. The non-integer transfer will cause the values of the sequence to be modified, which, in some cases, can be undesirable.

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