Nyquist-Shannon sampling theorem on non infinite signals

Question

Can we sample and recover signals with non infinite "length" using the nyquist shannon theorem? For example if we have a signal

$$ x(t)= u(t+5)-u(t-5) $$ and we know that its sampling period T is less than 10s (T<10). Can we recover it using the nyquist theorem? If not , what could we do to recover it?

Edit: I have taken the fourier transform of this signal and it is : $$X(ω)=2i(\frac{1}{iω}+πδ(ω))sin(5ω)$$ which can lead to finding the period as $$\frac{2π}{5} $$ and actually see that if f>= 5/π it can be recovered. But this may not be possible because i used the theorem although we are on a finite signal

Answer 1

No need to think about "finite length". Even though continuous time signals are defined for infinite time interval, in practice we analyse it in a finite interval only. Your x(t) is defined in the interval [-5 5].

The frequency domain representation of such a signal will be an infinite bandwidth sinc function.

Hence it is not possible define a particular sample rate as per nyquist theorem, to perfectly reconstruct it without losing any information. But you can sample it any definite sample rate, which then implicitly band limits the signal. This sampled signal after reconstruction through DAC and LPF, will not look as perfect as the original one as it would be band limited. It will have a finite transition time for rise and fall.

Answer 2

A different set of basis functions that are time-bounded would work well. For example, decomposing into Haar wavelets. You get many of the benefits of complex exponential bases for certain kinds of signal processing.

There are sampling theorems for wavelet bases (for example, the Haar wavelet basis can represent many functions within a finite range of "frequencies")

I mention this because it is highly relevant to learn about wavelet signal processing if your signals have finite support in the time dimension. For example, for processing 2 D intensities in a photograph - the sharp edges can be resolved with wavelets, often with far fewer terms.