# Why do FIR linear interpolators have linear phase?

I wonder why linear interpolation is a linear-phase FIR filter.

In fact, I am reading Dr. Rice's great book "Digital Communications; A Discrete-Time Approach," and he mentioned on page 466 as the 3rd property of interpolating filters: "interpolating filter is linear phase FIR filter."

I'd be really appreciative if somebody could help me with that.

• ah, to explain this most shortly, it would be helpful to know: Do you know why time-symmetrical filter are linear phase? Oct 25 at 16:25
• Sure, but an FIR filter with $\mu(k)$ and $1-\mu(k)$ as its taps will be linear phase iff $\mu(k)=0.5$ and NOT always!
– Ali
Oct 26 at 6:00
• hm, no, that's not true; it needs to be symmetrical, that's all. No requirements on values are made. Oct 26 at 7:51
• (you can time-shift a linear phase filter as much as you want, and it stays linear phase. That's usually how people derive that symmetrical filters are linear phase: by shifting a zero-phase filter and seeing that the phase indeed stays linear for any shift) Oct 26 at 8:34
• but afaik FIR filter is a digital filter and a two-tap FIR with [0.5, 0.5] is different from [.3, .7] no matter where on the time/index axis
– Ali
Oct 26 at 13:34

Linear interpolation is equivalent to a triangular filter kernel, scaled/sampled such that only two non-zero samples are present.

A triangular analog prototype is linear phase because it is symmetric.

-k

For tapped-delay line FIR filters, if you want to know "What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?", see the following web page: https://www.dsprelated.com/showarticle/808.php

They Don't.

To address this question, linear interpolation is NOT a linear phase filter except for when we are interpolating to half a sample. The necessity for a linear phase filter is that the filter coefficients be complex conjugate symmetric or anti-symmetric. Linear interpolation as an FIR filter is done with two coefficients $$a$$ and $$b$$ such that:

$$y[n] = a x[n] + b x[n-1]$$

with $$a+b = 1$$

and the fractional interpolation in samples given by $$a/b$$

Given the first statement on what makes a filter linear phase, this can only occur when $$a= b= 0.5$$. For any other values of $$a$$ and $$b$$, the result will not be linear phase.

The reference to interpolating filters in Dr. Rice's book are likely referring to higher order polynomial interpolation and not "linear interpolation"; using a multitap filter with more than 2 coefficients rather than the "linear interpolation" between two samples that the OP is considering (given the comment under the originating question). Higher order polynomial interpolation can and typically do have linear phase given an implementation with symmetric coefficients. This is the typical but not necessary condition as we can make interpolators that are not linear phase as well. Consider a typical interpolation process of doing a zero-insert followed by filtering to remove the images the zero-insert creates. Certainly a linear phase filter would be preferred to for the filter to minimize distortion and be closest to the ideal Sinc function interpolator, but this is not necessary to achieve interpolation within target requirements. Any realizable interpolation approach linear phase or otherwise will have a finite distortion regardless since perfect interpolation requires an infinitely long filter.

To see that this filter is linear phase, note that the coefficients are symmetric about the center point of the filter that is defined by $$\mu(k) = 1/2$$.