Sampling Theory

Convert between continuous and discrete-time signals.

Nyquist-Shannon Sampling Theorem

A bandlimited signal with maximum frequency $f_{max}$ can be perfectly reconstructed if:

$$ f_s \geq 2f_{max} $$

Reconstruction

Ideal (Sinc Interpolation)

$$ x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot \text{sinc}\left(\frac{t - nT_s}{T_s}\right) $$

Where $\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$.

Practical (Linear Interpolation)

1from scipy import interpolate
2
3# Linear interpolation
4f = interpolate.interp1d(t_samples, x_samples, kind='linear')
5x_interp = f(t_new)
6
7# Cubic spline
8f = interpolate.interp1d(t_samples, x_samples, kind='cubic')

Upsampling & Downsampling

1from scipy import signal
2
3# Upsample by factor of L
4x_up = signal.resample(x, len(x) * L)
5
6# Downsample by factor of M (with anti-aliasing)
7x_down = signal.decimate(x, M)

Further Reading

• Sampling Theorem - Wikipedia

Related Snippets

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• Window Functions
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