Symmetry and linear phase

In digital signal processing, the system is often required to have linear phase. For example, in the filter design, the linear phase characteristics of the FIR filter make it favored.

From the mathematical concept, the linear phase is the requirement that the phase frequency characteristic of the system is a straight line. And from the mathematical formula, it is easy to prove that if the system impulse response satisfies the condition of symmetry, whether it is odd symmetry or even symmetry, the system must have linear phase. How to understand it?

The physical meaning represented by the linear phase is that the system produces the same delay for all frequency signals. For a causal system, the best case is that the output is only relevant to the current input, and the delay is zero. A general cause-and-effect system can produce delays.

The Fourier analysis theory shows that under certain conditions, arbitrary signals can be decomposed into the superposition of sinusoidal signals. If a system does not have a linear phase, the system may distort or deform the input signal. For example, a square wave signal through a system, if the system has a linear phase, then through the system is still a square wave, only a delay in time. This is because all frequencies are delayed by the same time. If the system does not have linear phase, the output is no longer the standard square wave, the rising edge is no longer so steep, but there will be a more obvious transition zone. And there will be some ripples at the top of the square wave. This also allows you to understand the importance of linear phase from one side.

Also by Fourier analysis theory, we know that a point on the frequency domain represents a sinusoidal signal in the time domain. For example, in the frequency domain, there is a value at the F=0.1FS, we know that in the time domain must have a frequency of 0.1fs sine signal, for the convenience of description, with a complex signal: exp (j*2*pi*0.1*n). It is known from the principle of reciprocity that a point in the time domain represents a sinusoidal signal in the frequency domain. For example, in the time domain, there is a value in the N=n0, in the frequency domain must correspond to a sine signal: exp (-j*w*n0), where w represents the digital frequency. This is also well understood by the time delay nature of the DFT. This means that if the system impulse response H (n) Time domain N=n0 has a value, it will cause the input signal n0*ts time delay. Similarly, if H (n) also has a value in N=n2, and the value of the same size as H (n0), the delay is N2, if N0 and N2 about N1 symmetry, then the delay time to take n0 and N2 average, that is, the two points caused by the system delay is N1. Similarly, if the other points in the H (n) sequence are symmetric about N1, the result of each point force acting on N1 symmetry is to delay the input signal n1*ts seconds. Then the total system delay is n1*ts seconds.

Based on the principle of reciprocity, we can easily understand the relationship between symmetry and linear phase, and can easily calculate the delay time of the system.

Symmetry and linear phase