[DSP] Calculating the frequency response from filter coefficients

Question

I can't find any understandable information on this topic. On the dutch wikipedia, I found that you can apply a z-transform which produces a formula in this form:
www.music.mcgill.ca/~gary/618/week1/img15.gif

This FIR filter is used as an example:
upload.wikimedia.org/math/b/9/e/b9e2ed5184f98621922f716e5216f33d.png

With the z-transform:
upload.wikimedia.org/math/4/d/6/4d6621be8fabf4db8816c12f34ed9877.png

And in that example, e^it (the natural logarithm raised to the imaginary unit, and t = theta) is substituted for z: upload.wikimedia.org/math/0/6/e/06eada8fedfb492bd63bb50491b042aa.png

Then a plot of that function is used and considered as the frequency response. I though that this method was an easy way to calculate the frequency response of a filter. However, is this method valid? When I thought about a small delay (which 'blocks' the original signal) it occurs to me that the frequency response should be 1 for every frequency, since the signal isn't changed, only delayed, but with this method, I calculated that the frequency response would be:

y(n) = 0*x(n) + 1*x(n-1)  

Z-Transform

H(z) = 0 + 1z^-1

Substituting e^it (with t=theta):

H(e^it) = 0 + 1 * e^-it

As this produces a sine wave as frequency response, I must be doing something wrong, or misunderstood something. I would be very glad if someone could help me!

Answer 1

One easy method is graphical in nature: you can use this as a basis for an algorithm or you can use it manually to graph out the frequency response, and it's also helpful to get a quick idea of the response "by eye". This works for both FIR and IIR filters.

First plot your poles and zeroes on graph, along with the unit circle. Then for for any given frequency for which you want to calculate the magnitude of the frequency response:

• Draw lines from all the zeroes to the corresponding point on the unit circle and calculate their lengths.
• Do the same for the poles.
• Multiply all the zero line lengths to get a product N.
• Do the same for pole line lengths and call this D.
• The magnitude will then be N / D.

Obviously you will want to repeat the above operation for a number of points on the unit circle.

Answer 2

As per rwong's comment, the system function H gives you the phase and magnitude response of the system at a particular frequency. This means if the input to the system is cos[ωn] = cos[2πfn], the output will be a(f)cos[2πfn + Φ(f)], where a(f) = |H(f)| and Φ(f) = phase(H(f)). In your case the magnitude is 1 since the signal isn't being scaled in any way, only shifted in time. And the phase shift is -ω, where ω is the angular frequency of the sinusoidal input to the system.

I hope the following isn't too rudimentary for Stack Overflow, but maybe going over the basic basics of time series analysis will be helpful to minibear and others.

If you have a system with an impulse response of h[n] = δ[n-1] (where δ[n] is a delta function), as in your example, this means you're delaying the input by 1 time step. Think about what this means in terms of the phase of a sinusoid. The fastest changing sinusoid has a digital frequency of 0.5 (i.e. a period of 2 samples) -- e.g. cos[πn]. This is the series [1,-1,...]. If you delay this signal by 1 you get the series [-1,1,...], i.e. cos[πn - π] = cos[π(n - 1)], i.e. the input signal phase shifted by -π radians (-180 degrees). Look at a longer period signal with a digital frequency of 0.25 (i.e. a period of 4 samples) -- e.g. cos[0.5πn]. This is the series [1,0,-1,0,...]. A unit delay yields the series [0,1,0,-1,...], i.e. cos[0.5πn - 0.5π] = cos[0.5π(n - 1)], i.e. the input signal phase shifted by -π/2 radians (-90 degrees). Similarly, you can work out that an input of cos[0.25πn] yields an output of cos[0.25πn - 0.25π] = cos[0.25π(n - 1)], i.e. the input phase shifted by -π/4 radians (-45 degrees), etc, etc.

It's pretty clear that if the input angular frequency is ω (e.g. 0.5π), the output will be phase shifted by Φ = -ω. Think about the signal as a train going around the unit circle on a counterclockwise route, with its time series values corresponding to stops on this route. An angular frequency of 0.5π means it makes 4 stops at the following radian values: 0, 0.5π, π, 1.5π. Then it returns to 0 and repeats the cycle over and over. If this train gets delayed by a stop, that corresponds to a shift of -0.5π radians on the scheduled route.

Getting back to H(f), I hope it makes sense why it equals exp(-i2πf) = exp(-iω). Similarly, if your system has a delay of 2, then h[n] = δ[n-2] and H(f) = exp(-i4πf) = exp(-2iω) -- which is a delay of 2 stops on the unit circle. That's all the frequency response of a system/filter tells you, i.e. how much a system scales and delays each input sinusoid as a function of frequency.

FIR systems (i.e. finite impulse response, corresponding to a moving average model [MA]) are the simplest since they're just a sum of delta (i.e. scale and delay) functions on the feed forward path. IIR systems (i.e. infinite impulse response, corresponding to an autoregressive model [AR]) are more interesting to analyze since they have a feedback path.