Properties of FIR digital filters The sinusoidal steady state transfer method of a digital filter is periodic in the sampling frequency. We get

in which h(n) shows the terms of the unit pulse response. The above relation may be composed into imaginary and real elementsby writing

where the imaginary parts and real of the transfer function are provided by

These relation for HR(ω) and HI(ω) describe that

1.   HR(ω) is an even method of frequency and HI(ω) is an odd method of frequency.

2.   If h(n) is an even sequence, the imaginary phase of the transfer method, HI(ω), will be zero. (The even sequence, h(n), calculated by the odd sequence sin ωn will provided an odd sequence. An odd sequence summed over symmetric limits provides zero.) In that case

3.   Similarly, if h(n) is an odd number sequence, the real phase of the transfer method,

HR(ω), will be zero

Thus an even unit pulse response provides a real-valued transfer method and an odd unit pulse response provides on imaginary-valued transfer method. Recall that a real transfer method has a phase shift of 0 or ± p radians, while an imaginary transfer method has a phase shift of ±Π/2 radians as given in figures below. So, by creating the unit pulse response either even or odd, we may calculate a transfer function that is either imaginary or real.

Two kinds of applications In designing digital filters we are normally interested in one of the given two situations:

1.   Filtering We are used in the amplitude response of the filter (e.g., low pass, band pass, etc.) without phase distortion. That is derived by using a real valued transfer method, i.e.,

H (e^jω) = H_R(ω), with H_I(ω) = 0.

2.   Filtering plus quadrature phase shift These applications adds integrators, Hilbert transform devices and differentiators. For all of these the given transfer function is  not real, i.e.,

H (e^jω) = j H_I(ω), with H_R(ω) = 0

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