Outline
Introduction
Derivation
Example
Conclusion
References
Introduction
Hello! My name is Ryan Johnson! You might be wondering what a slecture is! A slecture is a student lecture that gives a brief overview about a particular topic! In this slecture, I will discuss the relationship between an original signal, x(t), and a sampling of that original signal, x_s(t). We will also take a look at how this relationship translates to the frequency domain, (X(f) & X_s(f)).
Derivation
F = Fourier transform
$ \begin{align} comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ \end{align} $ The comb of a signal is equal to the signal multiplied by an impulse train which is equal to the sampled signal. Essentially, the comb is grabbing points on the graph x(t) at a set interval, T. $ \begin{align} X_s(f) &= F(x_s(t)) = F(comb_T(x(t))\\ &= F(x(t)p_T(t))\\ &= X(f) * F(p_T(f)\\ \end{align} $ Multiplication in time is equal to convolution in frequency.
$ \begin{align} X_s(f)&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ \end{align} $ definition of the Fourier transform of an impulse train.
$ \begin{align} &= \frac{1}{T}X(f)*p_\frac{1}{T}(f)\\ &= \frac{1}{T}rep_\frac{1}{T}X(f)\\ \end{align} $ The result is a repetition of the Fourier transformed signal.
Example
Conclusion
Xs(f) is a rep of X(f) in the frequency domain with amplitude of 1/T and period of 1/T.
References
