# Does a linear phase FIR filter shifted in frequency preserve linear phase?

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?

• i think you answered your own question, if the coefficients are symmetric – user28715 Nov 26 '18 at 17:15
• But they won't be symmetric after multiplication with e^jwn/N – Albin Stigo Nov 26 '18 at 17:38

## 1 Answer

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.

• @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... – Fat32 Nov 26 '18 at 17:47
• 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... – Albin Stigo Nov 26 '18 at 18:52
• 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. – Fat32 Nov 26 '18 at 19:20