In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount (the slope of the linear function), which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another. Zero-phase filters are a special case of linear-phase filters where the group delay is zero. In non-real-time digital signal processing, this can be obtained from any linear-phase filter simply by shifting the filtered output of a linear-phase filter backwards in time by the linear-phase filter's group delay, so sometimes all linear-phase filters are loosely referred to as zero-phase filters. 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 ( ω ) = A ( ω ) e − j ω τ , {\displaystyle H(\omega )=A(\omega )\ e^{-j\omega \tau },}
where A(ω) is a real-valued function, and τ {\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
H 2 π ( ω ) = A ( ω ) e − j ω k / 2 , {\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.
H 2 π ( ω ) {\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:
H 2 π ( ω ) = H ^ ( z ) | z = e j ω = H ^ ( e j ω ) , {\displaystyle H_{2\pi }(\omega )=\left.{\widehat {H}}(z)\,\right|_{z=e^{j\omega }}={\widehat {H}}(e^{j\omega }),}
where the H ^ {\displaystyle {\widehat {H}}} notation distinguishes the Z-transform from the Fourier transform.
Examples When a sinusoid s i n ( ω t + θ ) {\displaystyle sin(\omega t+\theta )} passes through a filter with constant (frequency-independent) group delay τ , {\displaystyle \tau ,} the result is
A ( ω ) sin [ ω ( t − τ ) + θ ] = A ( ω ) sin ( ω t + θ − ω τ ) , {\displaystyle A(\omega )\sin[\omega (t-\tau )+\theta ]=A(\omega )\sin(\omega t+\theta -\omega \tau ),}
where A ( ω ) {\displaystyle A(\omega )} is a frequency-dependent amplitude multiplier, the phase shift ω τ {\displaystyle \omega \tau } is a linear function of angular frequency ω {\displaystyle \omega } , and − τ {\displaystyle -\tau } is the slope. It follows that a complex exponential function
e i ( ω t + θ ) = cos ( ω t + θ ) + i sin ( ω t + θ ) {\displaystyle e^{i(\omega t+\theta )}=\cos(\omega t+\theta )+i\sin(\omega t+\theta )}
is transformed into
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