Linear phase

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. Approximations can be achieved with infinite impulse response (IIR) designs, which are more computationally efficient. Several techniques are:

• a Bessel transfer function which has a maximally flat group delay approximation function
• a phase equalizer

Definition

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:

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

where:

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

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

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

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.

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

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

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

Examples

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

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

where:

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

It follows that a complex exponential function:

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

is transformed into:

$A(\omega )\cdot e^{i(\omega (t-\tau )+\theta )}=e^{i(\omega t+\theta )}\cdot A(\omega )e^{-i\omega \tau }$

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 $\omega \tau ,$  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.  A necessary but not sufficient condition is:

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

for some $\alpha ,\beta \in \mathbb {R}$.

Generalized linear phase

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

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

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

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.

See also

• Minimum phase