Signal Sampling

#Math
The main takeaway is that a signal that is discrete in time is periodic in spectrum (and vice versa).

• This means we can sample a signal, takes its Fourier Transform, apply a lowpass filter, and then apply an Inverse Fourier Transform to recover the original signal.

Topics

• Nyquist Rate
• Nyquist Frequency
• Nyquist Interval
• Sampling Frequency
• Whittaker-Shannon Interpolation

$\displaystyle \tilde{f}(t)=f(t)\delta_{T}(T)=\sum_{k = -\infty}^{\infty}f(t-kT)$

• $\displaystyle \delta_{T}(t)$ is the impulse train

$\displaystyle \tilde{F}(j\omega)=\mathcal{F}[f(t)\delta_{T}(t)]=\frac{1}{2\pi}F(j\omega)* {\omega}{0}\delta{{\omega}{0}}(\omega)=\frac{1}{T}\sum{k = -\infty}^{\infty}F(j(\omega-k{\omega}_{0}))$

• Fourier transform of sampled function
• Essentially a periodic repetition of $\displaystyle F(j\omega)$ spaced every $\displaystyle {\omega}_{0}$