Conjugate reciprocal pairs of zeros and poles in FIR design

Assuming the impulse response $h[n]$ of an FIR filter is real for all $n$,

• Why are zeros and poles in FIR design found in reciprocal and conjugate pairs?
• Is the assumption necessary for this phenomenon to take place?

Look up the complex conjugate root theorem which states that:

If all the coefficients of a polynomial are real then its roots are either real or if there is a complex root, then its conjugate is also a root.

This theorem can be applied to the denomerator and numerator of a rational transfer function to judge about its poles and zeros.

"conjugate reciprocal roots" is not a necessary consequence of real coefficients. conjugate reciprocal pole-zeros are considered when we need some specific features such as being all-pass or when designing minimum-phase filters. This is because for instance reflection of a zero to its conjugate reciprocal does not influence on the magnitude response.

If the filter has a finite impulse response (and does not come from a truncated IIR filter, as stated in the comments by Robert), then all its poles must be at the origin. So no point stating that its poles are reciprocal and conjugate pairs: they are all at $z=0$.

If $h(n)$ is real, causal and stable, then its zeros are found in reciprocal and conjugate pairs if and only if the filter has (generalized) linear phase.

• Tendero, you might wanna look up Truncated IIR (TIIR) filters. it's another way to implement FIR. moving sum or moving average, using an accumulator, is the simplest example. TIIR filters are FIR, but internally they have poles that are not at $z=0$. in the I/O transfer function, those poles are canceled by zeros. but if those poles happen to be outside the unit circle, then internally the FIR is not stable. – robert bristow-johnson Jul 15 '17 at 21:28
• @robertbristow-johnson Nice, didn't know about them at all. I just edited my answer, thanks for pointing that out. – Tendero Jul 15 '17 at 21:33
• i can send you an analysis/design document for a truncated biquad, if you want. – robert bristow-johnson Jul 15 '17 at 21:45
• i can think of three ways to implement an FIR. two are in the textbooks. one is the straight-forward "Transversal" implementation (just performing the convolution summation). another is, what we call, "Fast Convolution". the overlap/add or overlap/save techniques using the FFT. those are in textbooks. and then there are TIIRs. – robert bristow-johnson Jul 15 '17 at 21:47
• @robertbristow-johnson Yes, of course, it is always cool to learn new stuff. Send it to me when you can! – Tendero Jul 15 '17 at 22:31