A filter is called a linear phase filter if the phase component of the frequency response is a linear function of frequency. 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:
A is a real-valued function with 2π periodicity.
k is an integer, and k/2 is the group delay in units of samples.
is a Fourier series that can also be expressed in terms of the Z-transform of the filter impulse response. I.e.: where the notation distinguishes the Z-transform from the Fourier transform.
Examples
When a sinusoid passes through a filter with constant group delay 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 of the filter, where |A| has relatively large values. Therefore, both magnitude and phase graphs are customarily used to examine a filter's linearity. A "linear" phase graph may contain discontinuities of π and/or 2π radians. The smaller ones happen where A changes sign. Since |A| cannot be negative, the changes are reflected in the phase plot. The 2π discontinuities happen because of plotting the principal value of instead of the actual value. In discrete-time applications, one only examines the region of frequencies between 0 and the Nyquist frequency, because of periodicity and symmetry. Depending on the frequency units, the Nyquist frequency may be 0.5, 1.0, π, or ½ of the actual sample-rate. Some examples of linear and non-linear phase are shown below. vs normalized frequency A discrete-time filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric. A necessary but not sufficient condition is: for some.
Generalized linear phase
Systems with generalized linear phase have an additional frequency-independent constant added to the phase. 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.