I know that if I sampling with impulse train so I get in the frequency plane X(f)*h(f) (when x(f) is my signal, * means convolution and h(f) is fourier transform of impulse train).

what the difference between this way to x(t=n*Ts) sampling way?

If I can get explenation in the "frequency plane" it will be great.


1 Answer 1


I cannot recall a meaningful difference between the ideal impulse train sampling and the uniform sampling relation indicated by the expression $t_n = n T_s$.

Given a continuous-time bandlimited signal $x_c(t)$, when you sample this with an ideal impulse train $\delta_{T_s}(t) = \sum_k \delta(t - k T_s)$, then the relation between the obtained discrete-time sequence $x[n]$ and the continuous-time signal will be given by :

$$ x[n] = x_c(t_n) = x_c(n T_s)$$

Hence, the latter expression is a consequence of the former operation.

  • $\begingroup$ I think the only difference is that using deltas allows one to write the interpolator as a low-pass filter: if the filter impulse response is $\text{sinc}(t/T)$, and the input is $\sum x(nT) \delta(t-nT)$, then the Shannon interpolation formula follows immediately. $\endgroup$
    – MBaz
    Nov 21, 2019 at 1:16

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