# Is it possible to discretely sample the function

I had a few questions on sampling(I'm quite new witht his), I tried to answer them, I think that I did the first one correct , but not sure about the 2 other: . given the next functions,Is it possible to discretely sample the function? if so what is the maximal allowed distance between the samples? otherwise explain why?

1. $$f(x)=\sin(\alpha x)$$

For this one I said it is possible using Nyquist theorem, assumin $$\alpha =2\pi f$$ and $$T=\frac{1}{f}=\frac{2\pi}{\omega}$$ then the allowed distance is $$\frac{T}{2}=\frac{\pi}{\omega}$$

(hopefully i got this allright).

now from the second question I'm not sure.

1. $$f(x) = \left \{ \begin{array}{cl} 1, & \text{-1\leq x \leq 1} \\ 0, & \text{else} \end{array} \right .$$

for this function I'm pretty sure the answer is it is not possible, since it is a straight line, so I'm not sure.. (please help me with this one).

1. $$f(x)= \mbox{convolution between function of question 1 , function of question 2.}$$

In this case I'm guessing it is possible since its the same as sampling first question function (only that it is sliced).

• One can discretely sample the second one, albeit not in the Nyquist framework Oct 3, 2018 at 19:18

1. yes, with spacing between samples closer than $$\frac{\pi}{\alpha}$$.
2. no. it's not sufficiently "bandlimited"
3. yes, the convolution of a sinusoid with anything is a sinusoid of the same frequency. so it's "yes" for the same reason as 1.
• Can you please explain what does not sufficiently bandlimited mean? I’m quite new with this , so it would really help me;) also In question 3 it is a sinusoid from -1 to 1, in the rest it’s zero, it’s not an issue? Oct 2, 2018 at 20:02
• @robert: thanks for your answer but, when you say "not sufficiently bandlimited", is that the same as saying that there's no periodic component ? Oct 2, 2018 at 20:58
• no, i am saying that function of $f(x)$ in 2. (which is three straight lines with two discontinuities) is not bandlimited at all. you can sample it, if you want, but you cannot reconstruct the original from the samples. Oct 2, 2018 at 21:09

Traditional periodic sampling (Nyquist/Shannon) requires band-limited signals. And when signals are band-limited, you get a minimal sampling frequency that is sufficient to preserve all the information, and recover the continuous signal from the discrete samples only.

Your first sine $$x_1$$ is band-limited, so OK for 1). Knowing that a (continuous) convolution of $$x_1$$ and $$x_2$$ in time is equivalent to a product of Fourier transforms, it suffices to know if one of them is band-limited to conclude that the convolution is band-limited as well. So OK for 3) too.

The second case is more interesting. The signal takes two values, 0 and 1. And it changes at times 0 and 1. So this opens the discussion toward discretization in time AND amplitude. And there are non-Nyquist frameworks to discretize such signals: level-crossing, compressive sampling, finite-rate-of-innovation, for instance. Under a model milder than Nyquist/Shannon (eg: for binary signals, being zero at $$-\infty$$, record only times of changes), the second signal can be discretized, with only two values (timings at 0 and 1). And one could discuss whether a sine with irrational quantities could be discretized in values (and I believe, not).

To wrap it up:

1. From a standard teaching perspective, probably positive answers to 1) and 3) are expected
2. From a more advanced one, on time only, 2) is valid too
3. From a complete "discrete" need, only 2) and perhaps 3) as a limiting case
• you're right that practically, if you sample the rectangular pulse enough, a decent practical reconstruction can be made if you don't mind gibbs phenomenon. describing the pulse parametrically with 2 or 3 numbers is not the same as "sampling". Oct 4, 2018 at 7:49
• Let me still believe that 'sampling' means: taking samples, as pairs of $(t,s(t)$ within some discretization or precision. Traditional Nyquist sampling often forget about value discretization, and both are important. Oct 6, 2018 at 15:40
• yeah, but the issue isn't about amplitude quantization. that's when you get into quantization noise and dithering and noiseshaping. the issue i was trying to simple-mindedly focus on was uniform sampling. so the only thing we know about the $t$-axis is the sampling period $T$. we don't really have ordered pairs of $(t, x(t))$. uniform sampling is $t=nT$ for integer $n$ and we have a single list of values $x(nT)$. Oct 7, 2018 at 3:03