dsp-mathematics

name: dsp-mathematics description: Mathematical foundations of digital signal processing. Use when understanding DSP theory, deriving filter coefficients, analyzing frequency response, or working with complex numbers, Fourier transforms, and Z-transforms.

DSP Mathematics

Core Insight

Sinusoids are eigenfunctions of LTI systems. When you put a sinusoid into a linear time-invariant system, you get the same sinusoid out (possibly scaled and phase-shifted). This is why frequency-domain analysis works.

Euler's Formula

The foundation of DSP mathematics:

e^(jθ) = cos(θ) + j·sin(θ)

A complex exponential e^(jωt) is a rotating phasor. Its real part is a cosine, imaginary part is a sine.

Quick Reference

Topic	Reference
Complex numbers, Euler's formula, phasors	complex-numbers.md
DFT, frequency bins, basis functions	fourier-theory.md
Nyquist, aliasing, reconstruction	sampling-theory.md
LTI systems, superposition, impulse response	linear-systems.md
Convolution theorem, properties	convolution.md
Z-domain, poles, zeros, stability	z-transform.md
H(z), frequency response derivation	transfer-functions.md
Group delay, linear phase, unwrapping	phase-analysis.md
Essential trig identities for DSP	trig-identities.md

Key Relationships

Time Domain              Frequency Domain
────────────────────────────────────────
x[n]            ←─DFT─→  X[k]
h[n]            ←─DFT─→  H[k]
y[n] = x[n] * h[n]  ←→   Y[k] = X[k] · H[k]

Convolution in time  =  Multiplication in frequency
Multiplication in time  =  Convolution in frequency

Notation Conventions

Symbol	Meaning
`j`	Imaginary unit (√-1), preferred over `i` in DSP
`ω`	Angular frequency (rad/s or rad/sample)
`Ω`	Normalized angular frequency (ω/fs)
`z`	Complex variable in Z-transform
`e^(jω)`	Point on unit circle in Z-plane
`H(z)`	Transfer function
`H(e^jω)`	Frequency response (H evaluated on unit circle)
`*`	Convolution operator
`·`	Multiplication

Fundamental Formulas

Angular frequency:     ω = 2πf
Normalized frequency:  Ω = 2πf/fs = ωT  (where T = 1/fs)

DFT:      X[k] = Σ x[n] · e^(-j2πkn/N)    for n = 0 to N-1
IDFT:     x[n] = (1/N) Σ X[k] · e^(j2πkn/N)    for k = 0 to N-1

Z-transform:  X(z) = Σ x[n] · z^(-n)
Frequency response:  H(e^jω) = |H(e^jω)| · e^(j∠H(e^jω))
                               ─────────   ──────────────
                               magnitude      phase