Linear phase

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:

${\displaystyle H(\omega )=A(\omega )\ e^{-j\omega \tau },}$

where:

• A(ω) is a real-valued function.
• ${\displaystyle \tau }$ is the group delay.

For a discrete-time application, the discrete-time Fourier transform of the linear phase impulse response has the form:

${\displaystyle H_{2\pi }(\omega )=A(\omega )\ e^{-j\omega k/2},}$

where:

• A(ω) is a real-valued function with 2π periodicity.
• k is an integer, and k/2 is the group delay in units of samples.

${\displaystyle H_{2\pi }(\omega )}$ is a Fourier series that can also be expressed in terms of the Z-transform of the filter impulse response. I.e.:

${\displaystyle H_{2\pi }(\omega )=\left.{\widehat {H}}(z)\,\right|_{z=e^{j\omega }}={\widehat {H}}(e^{j\omega }),}$

where the ${\displaystyle {\widehat {H}}}$ notation distinguishes the Z-transform from the Fourier transform.

When a sinusoid${\displaystyle ,\ \sin(\omega t+\theta ),}$  passes through a filter with constant (frequency-independent) group delay ${\displaystyle \tau ,}$  the result is:

${\displaystyle A(\omega )\cdot \sin(\omega (t-\tau )+\theta )=A(\omega )\cdot \sin(\omega t+\theta -\omega \tau ),}$

where:

• ${\displaystyle A(\omega )}$ is a frequency-dependent amplitude multiplier.
• The phase shift ${\displaystyle \omega \tau }$ is a linear function of angular frequency ${\displaystyle \omega }$, and ${\displaystyle -\tau }$ is the slope.

It follows that a complex exponential function:

${\displaystyle e^{i(\omega t+\theta )}=\cos(\omega t+\theta )+i\cdot \sin(\omega t+\theta ),}$

is transformed into:

${\displaystyle A(\omega )\cdot e^{i(\omega (t-\tau )+\theta )}=e^{i(\omega t+\theta )}\cdot A(\omega )e^{-i\omega \tau }}$[note 1]

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. A "linear" phase graph may contain discontinuities of π and/or 2π radians. The smaller ones happen where A(ω) changes sign. Since |A(ω)| cannot be negative, the changes are reflected in the phase plot. The 2π discontinuities happen because of plotting the principal value of ${\displaystyle \omega \tau ,}$  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.

phase response vs normalized frequency (ω/π)

Bode plots. Phase discontinuities are π radians, indicating a sign reversal.

Phase discontinuities are removed by allowing negative amplitude.

Two depictions of the frequency response of a simple FIR filter

A discrete-time filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric.[2]  A necessary but not sufficient condition is:

${\displaystyle \sum _{n=-\infty }^{\infty }h[n]\cdot \sin(\omega \cdot (n-\alpha )+\beta )=0}$

for some ${\displaystyle \alpha ,\beta \in \mathbb {R} }$.[3]

Systems with generalized linear phase have an additional frequency-independent constant ${\displaystyle \beta }$ added to the phase. In the discrete-time case, for example, the frequency response has the form:

${\displaystyle H_{2\pi }(\omega )=A(\omega )\ e^{-j\omega k/2+j\beta },}$

${\displaystyle \arg \left[H_{2\pi }(\omega )\right]=\beta -\omega k/2}$ for ${\displaystyle -\pi <\omega <\pi }$

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.[4]

• Minimum phase