Does anyone know of a way to design a filter with arbitrary magnitude and phase response in Matlab?

  • $\begingroup$ also, if your filter is causal, you can't do whatever magnitude and phase you want. you might have to toss is some constant delay or linear-phase term in the phase, just to get a causal impulse response. if it's causal, then $\Re\left\{H[k]\right\}$ and $\Im\left\{H[k]\right\}$ are a Hilbert transform pair. $\endgroup$ May 4, 2015 at 1:22
  • $\begingroup$ What do you actually mean by arbitrary magnitude and phase response, do you mean that you can specify the filter in the frequency domain as a function, or only specify a finite number of points? $\endgroup$
    – fibonatic
    Jun 3, 2015 at 3:04

2 Answers 2


for FIR, the simplest way is to inverse FFT you two-sided complex frequency response:

$$ h[n] = \operatorname{iFFT} \left\{ H[k] \right\} $$

(use a Kaiser window to window the FIR to a length that is acceptable to implement.)

i presume you have access to firpm( ) and/or firls( ).

so first divide your complex frequency response into real and imaginary parts:

$$ H_r[k] = |H[k]| \cos\left(\arg\{H[k]\} \right) $$ $$ H_i[k] = |H[k]| \sin\left(\arg\{H[k]\} \right) $$

send both $H_r[k]$ and $H_i[k]$ to the Parks-McClellan or Least Squares optimal FIR designer, but set ftype to 'hilbert' for $H_i[k]$.

sum the two resulting real-valued FIRs to get your result.

lastly, if you want to design an IIR to an arbitrary magnitude and phase spec, then that's a much bigger problem and consideration. maybe use Prony to design to your impulse response $h[n]$ or use Matt's dissertation that i first learned of now, or use Greg Berchin's FDLS. i dunno. no quick solution for you there.

  • $\begingroup$ I'm trying to find more information on the method that devides the response in real and imaginary parts and designs filters for them separately. It sounds promising for my application but I wonder what this technique is called and why it works. $\endgroup$ Aug 20, 2018 at 11:39
  • $\begingroup$ I wonder the same things. It doesn't work for me at all. $\endgroup$
    – Harry
    Feb 25, 2021 at 20:46
  • $\begingroup$ What did you do? $\endgroup$ Feb 26, 2021 at 13:06

There's a PhD thesis with exactly this phrase in its title (including Matlab programs):

Algorithms for the Constrained Design of Digital Filters with Arbitrary Magnitude and Phase Responses

  • $\begingroup$ at first i thought it was in Deutsche (it's not, thankfully for me) but the next obstacle is reading it without reading it online. do you have it in .pdf, Matt? $\endgroup$ May 4, 2015 at 0:56
  • $\begingroup$ @robertbristow-johnson: If you follow the link in the answer you'll see the abstract, but above it there's a download link for the pdf version. $\endgroup$
    – Matt L.
    May 4, 2015 at 9:56
  • $\begingroup$ I know I'm a bit slow: Matt, I didn't know you were one of Prof. Mecklenbr􏰏äuker's students! Cool! :-) $\endgroup$
    – Peter K.
    Sep 1, 2015 at 11:23
  • $\begingroup$ @PeterK.: I was indeed. You know him? (Maybe this is not the place to discuss such things, but anyway ...) $\endgroup$
    – Matt L.
    Sep 1, 2015 at 12:08
  • 1
    $\begingroup$ @PeterK.: Yes, Franz Hlawatsch was a colleague of mine. As you know he has quite some name in time-frequency analysis, but he's been doing more work in digital communications in the past years. $\endgroup$
    – Matt L.
    Sep 1, 2015 at 12:38

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