Outline
Introduction
Derivation
Example
Conclusion
Introduction Hello! My name is Ryan Johnson! You might be wondering was 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 and a continuous time sampling of that original signal. We will also take a look at how this relationship translates to the frequency domain.
Derivation F = fourier transform
$ \begin{align} comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ \end{align} $ \\ Explanation 1 $ \begin{align} X_s(f) &= F(x_s(t)) &= F(comb_T)\\ &= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ &= \frac{1}{T}X(f)*P_\frac{1}{T}(f)\\ &= \frac{1}{T}rep_\frac{1}{T}X(f)\\ \end{align} $
Example
Conclusion So the relationship between X(f) and Xs(f) is that Xs(f) is a a rep of X(f) in frequency domain with period of and magnitude scaled by . the relationship between X(f) and Xd(ω) is that Xd(ω) is also a a rep of X(f) in frequency domain with period 2π and magnitude is also scaled by , but the frequency is scaled by 2πT
