How do I show an impulse response leads to a zero-phase frequency response?

Question

I'm trying to understand how to show that with real coefficients, the phase response of a filter is 0. Here is the impulse response

$h[n] = b_1d[n+1] + b_0d[n] + b_1d[n-1]$

How should I approach showing this?

Hint: look at the symmetry properties of the Fourier Transform, e.g. http://users.aber.ac.uk/ruw/teach/340/ft_symmetry.php — Hilmar, Apr 07 '18 at 12:43

score 1 · Answer 1 · edited Apr 07 '18 at 07:39

1

1) Define $d[n]$ as a sinusoidal

2) Apply angle addition formulas

3) Simplify

4) Interpret the results

Hope this isn't too much. Sounds like a homework problem.

Ced

edited Apr 07 '18 at 07:39

AlexTP

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answered Apr 06 '18 at 22:12

Cedron Dawg

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I'm not sure I expressed myself correctly. h[n] is the impulse response, and so d[n] I'm pretty sure represents impulses ala [1,0,0,...]. I'm supposed to show that it leads to a zero-phase FR when (b0,b1) \in R^2. – Apr 06 '18 at 22:28
@user34931, These two links should help: https://en.wikipedia.org/wiki/Frequency_response and https://dsp.stackexchange.com/questions/536. – Cedron Dawg Apr 06 '18 at 22:39
@user34931, Is this a homework problem or are you trying to learn this stuff on your own? Major hint from the dsp link: "For discrete-time systems, this is possible, because you can write any signal x[n] as a sum of scaled and time-shifted Kronecker delta functions:"
$$ x[n] = \sum_{k=0}^{\infty} x[k] \delta[n - k] $$

From the Wiki article: "It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input." You are asking about the phase. Here is a question and my answer about the magnitude: https://dsp.stackexchange.com/questions/46510.
– Cedron Dawg Apr 06 '18 at 23:15
Yea this is a homework problem I'm trying to tackle.
I understand what an impulse response is, in that a bunch of impulses with scalars, each delayed by different numbers of samples, is essentially an LTI system, (in this case a filter).

The part that confuses me is how knowing that this impulse response has a non phase-shifted response. Is it because the IR is symmetrical?
– Apr 06 '18 at 23:21
@user34931, I suggest you follow my recipe in my answer and see how far you get. Once you have accomplished that, apply the same approach to this definition:
$$ h[n] = b1d[n] + b0d[n-1] + b1*d[n-2] $$

Notice that the phase lag is not zero in this case. Then do a search on "Finite Impulse Response filter with symmetric coefficients"
– Cedron Dawg Apr 06 '18 at 23:34
@user34931, In the "Estimation and plotting" section of the Wiki article, the second bullet point is: "sweeping a constant-amplitude pure tone through the bandwidth of interest and measuring the output level and phase shift relative to the input" That is what the steps I outlined is conceptually doing. – Cedron Dawg Apr 06 '18 at 23:46

score 0 · Answer 2 · answered Apr 07 '18 at 08:56

Given an impulse response of the form $h[n]$ for an LTI system, a zero phase freqency response means that the DTFT (discrete-time Fourier transform) $$H(e^{j\omega}) = \sum_{n=-\infty}^{\infty} h[n]e^{-j\omega n} $$ is real and positive.

If it's real but negative, then that's an easily avoidable $\pi$ phase shift and if it's not real but complex then its either linear phase or nonlinear phase...

How do I show an impulse response leads to a zero-phase frequency response?

2 Answers2