Difference between revisions of "Slecture Topic 7" - Rhea

Outline

1. Introduction
2. Derivation
3. Example
4. Conclusion

Introduction

This slecture will discuss the frequency domain view of the relationship between a signal, and a sampling of that signal. Essentially, given a signal x(t), we are going to take a look at the similarities and differences in X(f) and X_s(f). X_s(f) is the Fourier Transform of the sampling, x_s(t), of x(t).

Derivation

Given an arbitrary signal x(t), its Fourier Transform is X(f)

The sampling of signal x(t), is the comb of x(t), which is equivalent to multiplying x(t) by the impulse train p_T(t).

So, x_s(t) = x(t) x p_T(t) = comb_T(x(t))

F(x_s(t)) = X_s(f) = (1/T) * rep_1/T(X(f))

Analytically, we can see that the frequency domain view of the sampling's amplitude is scaled by a factor of 1/T. The shape of the frequency responses are the same, but it is repeated every 1/T in the sampling's frequency response. In the following example we will see the relationship graphically.