Linear phase

Consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR) filter by having coefficients which are symmetric or anti-symmetric.

[1] Approximations can be achieved with infinite impulse response (IIR) designs, which are more computationally efficient.

For a continuous-time application, the frequency response of the filter is the Fourier transform of the filter's impulse response, and a linear phase version has the form: where: For a discrete-time application, the discrete-time Fourier transform of the linear phase impulse response has the form: where:

is a Fourier series that can also be expressed in terms of the Z-transform of the filter impulse response.

the result is: where: It follows that a complex exponential function: is transformed into: For approximately linear phase, it is sufficient to have that property only in the passband(s) of the filter, where |A(ω)| has relatively large values.

Therefore, both magnitude and phase graphs (Bode plots) are customarily used to examine a filter's linearity.

[3] Systems with generalized linear phase have an additional frequency-independent constant

In the discrete-time case, for example, the frequency response has the form: Because of this constant, the phase of the system is not a strictly linear function of frequency, but it retains many of the useful properties of linear phase systems.