While going through Proakis's Digital Signal Processing (page 21) ,he stated that if a continuous time signal $~x(t)~$ that has been sampled each $~T~$ seconds to produce a discrete time signal $~x(n)~$ then the relationship between the variables $t$ and $n$ is :

$$ t=nT \tag($$

Question : in the LHS we have a continuous variable whereas in the RHS we have a variable that can only take step sizes of $T$ , so clearly $t$ and $nT$ do not span the same range , then how is the formula above justified ?


It's about the relationship between the discrete time signal $x_d[n]$ and the continuous-time signal $x_c(t)$:


So you formally replace the variable $t$ by $nT$ but this just means that you sample the continuous-time signal at sample instants $t_n=nT$. So $t=nT$ is only true for the values of $t$ that we're interested in, and these are the discrete values $t_n=nT$.


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