Signal Processing

From time domain to frequency domain and back.

Fourier Series

Any periodic function can be decomposed into sinusoids:

\[f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right]\]

Where:

\[a_n = \frac{2}{T} \int_0^T f(t) \cos(n\omega_0 t) \, dt\]
\[b_n = \frac{2}{T} \int_0^T f(t) \sin(n\omega_0 t) \, dt\]

Fourier Transform

Continuous

For non-periodic signals:

\[F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} \, dt\]

Inverse:

\[f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} \, d\omega\]

Discrete (DFT)

For sampled signals:

\[X[k] = \sum_{n=0}^{N-1} x[n] e^{-i 2\pi kn/N}\]

Fast Fourier Transform (FFT)

Cooley-Tukey algorithm reduces complexity from \(O(N^2)\) to \(O(N \log N)\).

For audio at 44.1 kHz, real-time spectrum analysis becomes practical.

Sampling Theory

Nyquist-Shannon Theorem

A signal can be perfectly reconstructed from samples if:

\[f_s > 2 f_{max}\]

Where \(f_s\) is sample rate and \(f_{max}\) is highest frequency component.

Aliasing

Sampling below Nyquist causes frequency folding:

  • 22 kHz signal sampled at 44.1 kHz → correctly captured

  • 30 kHz signal sampled at 44.1 kHz → aliases to 14.1 kHz

Anti-aliasing filters remove frequencies above \(f_s/2\) before sampling.

Convolution

Definition

\[(f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \, d\tau\]

Convolution Theorem

Convolution in time domain = multiplication in frequency domain:

\[\mathcal{F}\{f * g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}\]

This is why FFT is fundamental: filters become simple multiplications.

Digital Filters

FIR (Finite Impulse Response)

\[y[n] = \sum_{k=0}^{M} b_k x[n-k]\]

  • Always stable

  • Linear phase possible

  • Higher order needed for sharp cutoffs

IIR (Infinite Impulse Response)

\[y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{j=1}^{N} a_j y[n-j]\]

  • More efficient (fewer coefficients)

  • Can be unstable

  • Nonlinear phase

Common Types

Filter Purpose

Low-pass

Remove high frequencies (anti-aliasing, smoothing)

High-pass

Remove low frequencies (DC removal, bass cut)

Band-pass

Isolate frequency range (radio tuning)

Notch

Remove specific frequency (hum removal)

Z-Transform

Digital equivalent of Laplace transform:

\[X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}\]

Transfer Function

For LTI systems:

\[H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{j=1}^{N} a_j z^{-j}}\]

Poles inside unit circle → stable system.

Applications

Audio

  • Compression: MP3, AAC use psychoacoustic models + transform coding

  • Effects: Reverb, chorus, EQ via convolution/filtering

  • Analysis: Pitch detection, spectrograms

Communications

  • Modulation: Shifting signals to carrier frequencies

  • Coding: Error correction, compression

  • Equalization: Compensating channel effects

Image Processing

2D Fourier transform for:

  • Filtering (blur, sharpen, edge detection)

  • Compression (JPEG uses DCT)

  • Analysis (pattern recognition)

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