# Sampling of frequency response

Let's consider any physical quantity depending on the frequency. For example, the impedance of a certain electrical component: $$Z(f)$$.

Now, imagine to measure it in a continuous interval of frequencies. You get a graph. Now, let's take some samples separated by $$∆f$$ (uniform sampling).

My questions are:

1. If I compute the Fourier Transform of $$Z(f)$$, what will I get? I think it is a signal like $$Z(t)$$, but it seems strange to me that from a frequency dependent signal, it will appear a time - dependence only by calculating the Fourier Transform.
2. Which is the mathematical condition to apply to $$∆f$$ to avoid aliasing? I'd say that it should be greater of the maximum frequency of $$Z(f)$$, which is a graph on frequency. Is this frequency variation of Z with respect to frequency related to time?
• Re 1.: "Fourier transform" (not: transformer) of a frequency signal pretty much is a time signal, with a twist (just write down the Fourier Transform formula for a time signal $x(t)$, and the Fourier Transform of that, and you'll notice how the exponent doesn't fully cancel). Why is that surprising? That's exactly what a Fourier Transform does. (also, there's actually plenty of questions here dealing with "Fourier Transform of Fourier Transform of signal", do a bit of searching around here :) ) – Marcus Müller Oct 6 '19 at 9:46
• Re 2: Your statement makes no sense, please draw a spectrum of a signal with a maximum frequency $f_\text{max}$, and then sample that in the spectral domain with a resolution $\Delta f > f_\text{max}$! – Marcus Müller Oct 6 '19 at 9:48
• Yes, I have added now an example of spectrum – Kinka-Byo Oct 6 '19 at 14:00

$$\Delta f < 1/L$$, where L is the length of the impulse response. The sampling theorem is the same in both domains: in order to sample in time the signal needs to be limited in frequency and in order to sample in frequency the signal needs to be limited in time. Note: if $$\Delta f$$ is too big you get time domain aliasing, not frequency domain aliasing.
• No. At least not for a causal system. In the frequency domain we typically select the valid frequency range from $[-f_N/2,+f_N/2]$ where $f_N$ is the Nyquist frequency. That's why the total bandwidth is $f_N$ and NOT $f_N/2$. This is of more a convention of convenience: the signals are periodic anyway, so technically you can set the start and end point wherever you want as long as you cover one entire period – Hilmar Oct 6 '19 at 15:30