Signal Sampling

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.
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Topics

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

$\tilde{f}(t)=f(t){\delta }_T(T)=\sum _{k=-\infty }^{\infty }f(t-kT)$

• ${\delta }_T(t)$ is the impulse train

$\tilde{F}(j\omega )=\mathcal{F}[f(t){\delta }_T(t)]=\frac{1}{2\pi }F(j\omega )\ast {\omega }_0{\delta }_{{\omega }_0}(\omega )=\frac{1}{T}\sum _{k=-\infty }^{\infty }F(j(\omega -k{\omega }_0))$

• Fourier transform of sampled function
• Essentially a periodic repetition of $F(j\omega )$ spaced every ${\omega }_0$