Question

Show what happens when the symmetry is not imposed in FIR filters. what happens to the linear phase?

Answer 1

Linear Phase

What is the association between FIR filters and “linear-phase?”

Most FIRs are linear-phase filters; when a linear-phase filter is desired, a FIR is usually used.

What is a linear phase filter?

“Linear Phase” refers to the condition where the phase response of the filter is a linear (straight-line) function of frequency (excluding phase wraps at +/- 180 degrees). This results in the delay through the filter being the same at all frequencies. Therefore, the filter does not cause “phase distortion” or “delay distortion”. The lack of phase/delay distortion can be a critical advantage of FIR filters over IIR and analog filters in certain systems, for example, in digital data modems.

What is the condition for linear phase?

FIR filters are usually designed to be linear-phase (but they don’t have to be.) A FIR filter is linear-phase if (and only if) its coefficients are symmetrical around the center coefficient, that is, the first coefficient is the same as the last; the second is the same as the next-to-last, etc. (A linear-phase FIR filter having an odd number of coefficients will have a single coefficient in the center which has no mate.)

What is the delay of a linear-phase FIR?

The formula is simple: given a FIR filter which has N taps, the delay is: (N – 1) / (2 * Fs), where Fs is the sampling frequency. So, for example, a 21 tap linear-phase FIR filter operating at a 1 kHz rate has delay: (21 – 1) / (2 * 1 kHz)=10 milliseconds.

What is the alternative to linear phase?

Non-linear phase, of course. Actually, the most popular alternative is “minimum phase”. Minimum-phase filters (which might better be called “minimum delay” filters) have less delay than linear-phase filters with the same amplitude response, at the cost of a non-linear phase characteristic, a.k.a. “phase distortion”.

A lowpass FIR filter has its largest-magnitude coefficients in the center of the impulse response. In comparison, the largest-magnitude coefficients of a minimum-phase filter are nearer to the beginning.