# Conceptual clarification of Sampling theorem

I am new to signal processing.

We were introduced to the sampling theorem and intuitively I am struggling to make sense of it:

There are 3 points I am struggling to comprehend:

1. The analog signal: the theorem says that you multiply it by a train of time shifted impulse functions. Is the analog signal assumed to be known and therefore this is not an instantaneous function that happens in real time? Is it sampled for say 1 second and then the resulting impulse train, now stretching for 1 second will just have the amplitude of the analog signal? If you then Fourier transform those impulses, how do you possibly get the frequency spectrum for the original signal?

2. The resources I have looked at show a real time analog signal and then show the frequency spectrum for the signal. How did they obtain that in real time? They then convolve the frequency response of the real time signal to get several copies of those frequency responses at frequency multiples of the sampling time.

3. This is the Fourier transform of the time shifted impulse function

$$\begin{array}{l} \mathcal{F}\{f( t)\} =\mathcal{F}\{\delta ( t-a)\} \ =e^{i2\pi fa} =\cos( 2\pi fa) +i\sin( 2\pi fa) \end{array}$$

How does this Euler's function produce an impulse in the frequency domain?

Thank you.

• in your third point, the impulse is in the time-domain; t is time, right? Oct 6, 2022 at 13:03
• @Fat32 yes, I was just trying to workout what a time shifted impulse in the time domain, looks like in the frequency domain.
– SS1
Oct 6, 2022 at 13:06
• You say "sampling theorem" but it does not sound as if you are speaking of the Nyquist-Shannon sampling theorem. I'm not sure that the technique of modeling the sampling process as multiplication by impulses is called "the sampling theorem" anywhere in the literature. Oct 6, 2022 at 14:39
• @TimWescott It may not be called "the sampling theorem" but generally the multiplication with impulse train is one of the approaches to mathematics of sampling. Oct 6, 2022 at 14:53
• I was asking another question about sampling and, in that question, pretty clearly spell out the sampling theorem. It might be useful. Oct 6, 2022 at 15:37

The technique of modeling the sampling process as multiplying a continuous-time signal by a series of time-shifted, amplitude-1 Dirac delta functionals is for modeling.

The reason this is done, as far as I can tell, is because doing so unifies the Laplace domain and the $$z$$ domain, and it unifies the four Fourier transforms. This means that instead of having these separate-but-similar entities for sampled- and continuous-time (and sampled- and continuous-frequency, in the case of the Fourier transforms), you have the Fourier domain and the Laplace/z domain, you have a whole bunch of rules about how they're common, and a few rules about how they're different.

Multiplying by $$\delta(t)$$ is not what's done in practice.

Multiplying by $$\delta(t)$$ is a nice mathematical dodge that takes a continuous-time signal and coughs up a signal that is technically continuous-time, but contains an obvious discrete-time sequence of samples. Unlike the real world, these samples are multiplied by $$\delta(t - n T_s)$$ -- but other than that, it's very close. In the real world, you just operate on that discrete-time sequence of samples.

Basically -- you really only care about the impulses for the purpose of analysis. Having those $$\delta(t - n T_s)$$ is what will, as you get further in your studies, allow you to see that the Laplace and $$z$$ domains are twins, and that the four Fourier domains are closely related.

In practice there's a plethora of different ways that one might acquire a series of measurements taken of the real world at regular intervals. Over 99% of the time, the point at which the measurements turn from continuous-time into some form of sampled time is inside of a monolithic analog to digital converter. In the residue, it's done with shaft encoders, or resolver-to-digital converters, or some junior employee with a measuring tape and a clipboard, or any of the other methods people have dreamed up over the years.

In none of these methods is a physical signal multiplied by another signal that goes to infinity (because that would be impossible) and stays at infinity for an infinitesimally short length of time (because that would be impossible).

So put it out of your mind that this multiplication by $$\delta(t)$$ is an actual physical process.

Having established that framework:

Is the analog signal assumed to be known and therefore this is not an instantaneous function that happens in real time?

It can be a real time function, with future values unknown. Else, how would it be useful in the real world?

If you then Fourier transform those impulses, how do you possibly get the frequency spectrum for the original signal?

You don't. The "why" is complicated. You'll have to trust me on this, but when you get there you'll find that the explanation may well be couched in the properties of $$\delta(t)$$ and the Fourier transform of two signals that have been multiplied together. It doesn't have to be -- you can use logic, hand-waving, and trigonometry to do the job. But, if you use $$\delta(t)$$ then (again, trust me) you'll get a more unified view of the subject than if you pull out a random rule involving $$\cos \omega t$$, $$\sin \omega t$$, or $$e^{j \omega t}$$.

The resources I have looked at show a real time analog signal and then show the frequency spectrum for the signal. How did they obtain that in real time?

They don't. You're in theory land at this point. For the purposes of introducing things to you slowly enough that your brain does not turn to mush, they're starting with impossible situations (like, a signal which is known for all time, past and future, and thus has a Fourier transform). Later, through hand-waving or graduate classes, you can progress to how to use these tools that seem to only work on unreal situations to deal with real situations.

How does this Euler's function produce an impulse in the frequency domain?

Uh -- it does, but I'm going to hope that someone else gives you a good demonstration of this.

• Thank you for taking the time to explain this.
– SS1
Oct 6, 2022 at 15:51