Frequency response

Realization of linear phase FIR filters A basic and important special subset of FIR filters has a linear phase characteristic. Linear phase gives if the impulse response is symmetric is related to its center. For a causal filter whose impulse response starts at 0 and ends at N-1, that symmetry is shown thus

Even:  h(n) = h(N-1-n),         for n = 0, 1,..., (N-1) - a total of N points

Odd:    h(n) = - h(N-1-n),      for n = 0, 1,..., (N-1) - a total of N points

That symmetry gives the transfer function to be written so that only half the value of multiplications is needed for the resulting realization.

Linear phase - phase and delay distortion Take a low pass filter with frequency response

H (e^jw ) shown as

where k is an integer. That is a linear phase filter with the tangent slope of the phase "curve" in the pass band being -k. Let X (e ^jw) shows the Fourier transform of an input sequence x(n). Then the change of the output sequence y(n) is provided by Y (e^jw ) = X (e^jw ) . H (e^jw) . If X (e^jw ) is purely within the pass band of H (e^jw) then 

Y (e^jw) = X (e^jw) . e^-jwk

So the output function y(n) may be obtained as the inverse F-transform of Y (e^jw ) as y(n) = x(n-k),  a old version of x(n)

Thus the linear phase filter could not change the shape of the original signal, normally translated (delayed) it by k samples. If the phase response will not be linear, the output signal could have been a translated version of x(n).

It may be define that a causal IIR filter may not provide a linear phase characteristic and that only simple forms of causal FIR filters may provide linear phase.

Theorem If h(n) shows the impulse response of a discrete time system, a sufficient and necessary condition for linear phase is that h(n) have a finite duration N, and that it be symmetric about to its midpoint.

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