2
$\begingroup$

Let's say I have a symmetric (and therefore linear phase) FIR low pass filter with real coefficients.

If I then shift this filter in some direction in frequency by multiplying its coefficients with a complex exponential (to get a band pass filter with complex coefficients), will it still be linear phase?

$\endgroup$
  • $\begingroup$ i think you answered your own question, if the coefficients are symmetric $\endgroup$ – Stanley Pawlukiewicz Nov 26 '18 at 17:15
  • $\begingroup$ But they won't be symmetric after multiplication with e^jwn/N $\endgroup$ – Albin Stigo Nov 26 '18 at 17:38
4
$\begingroup$

The generalized linear phase FIR filter has the following frequency response:

$$H(\omega) = A(\omega)~e^{j (\alpha \omega + \beta)}$$ for some constant $\alpha$ and $\beta$ and $A(\omega)$ being real. The group delay of this filter is independent of frequency: $$\tau = -\frac{d (\alpha \omega + \beta) } {d \omega} = - \alpha $$

Now if you linearly shift this filters frequency response then you get the new filter's frequency response as :

$$H_2(\omega) = H(\omega-\omega_0) = A(\omega-\omega_0)~e^{j(\alpha (\omega-\omega_0) + \beta)} = B(\omega)~e^{ j ( \alpha \omega - \alpha \omega_0 + \beta}) $$

which also shows the generalized linear phase property. The new group delay is:

$$\tau = -\frac{d (\alpha \omega - \alpha \omega_0 + \beta) } {d \omega} = - \alpha $$

the same as the previous filter. And is also independent of frequency $\omega$.

Recall to page 295 of Discrete-Time Signal Processing 2E by A.Oppenheim for a more detailed discussion of the generalized linear phase systems.

$\endgroup$
  • $\begingroup$ @AlbinStigo your welcome! It's a common practice to design baseband (lowpass) linear phase FIR filters and then modulate them to the frequency of operational interest to achieve a bandpass one. Though most typically done by cosine modulation to get real coefficients unless you deliberately want an analytic filter... $\endgroup$ – Fat32 Nov 26 '18 at 17:47
  • $\begingroup$ I was doing this and didn't really mind that my coefficients were complex since I was doing FFT convolution of an analytical signal (SDR application)... But are you telling me now that I might be able to translate a base band filter and still get real coefficients? Kind of had a gut feeling that it was still linear phase but wasn't sure at all... $\endgroup$ – Albin Stigo Nov 26 '18 at 18:52
  • $\begingroup$ yes you can. If $h[n]$ is your prototype linear phase FIR lowpass filter with real coefficeints, then the filter $h_2[n] = \cos(\omega_0 n) h[n]$ will be a linear phase bandpass filter centered around $\omega = \pm \omega_0$ with real coefficeints... Note that this filter has a conjugate symmetric frequency response unlike an analytic one that's created by the $h_3[n] = e^{j \omega_0 n} h[n]$ which is analytic with a single lobe and having complex coefficients. $\endgroup$ – Fat32 Nov 26 '18 at 19:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.