Help with Signal Processing JohnDCook.com/DSP

The (j, k) entry of the DFT matrix is ω^jk where ω = exp(-2π i/N), indexes j and k running from 0 to N-1.

Amplitude modulation and beats johndcook.com/blog/2016/03/1…

A finite-order, causal, LTI filter is stable if and only if the poles of its transfer function are inside the unit circle.

A bandwitdh-limited signal can be recovered from (infinitely many) evenly spaced samples, if sampled above the Nyquist frequency.

How to compute cepstrum: signal -> DFT -> magnitude -> log -> inverse DFT

LTI operators commute johndcook.com/blog/2023/05/0…

A function's Fourier series coefficients are its Fourier transform evaluated at integers.

Psychoacoustics is the study of how humans perceive sound.

At a jump discontinuity, the partial sums of a function's Fourier series converge to the average of the values on either side of the jump.

Cepstral alanysis vocabulary johndcook.com/blog/2016/05/2…

Let f_n(x) = H_n(x) φ(x) where H_n = nth Hermite polynomial and φ = standard normal PDF. The Fourier transform of f_n is (-i)^n f_n.

The standard normal probability density is its own Fourier transform.

Making high frequencies from low frequencies johndcook.com/blog/2020/05/0…

A linear time-invariant (LTI) system is determined by its impulse response, i.e. how it transforms δ[n].

The imaginary part of the Fourier transform of a real signal is an odd function.

Nyquist sampling theorem: A continuous bandlimited signal is uniquely determined by samples at a frequency of twice the band limit.

Why does FM sound better than AM? johndcook.com/blog/2024/10/1…

In the context of DSP, D/C stands for discrete-to-continuous and C/D stands for continuous-to-discrete.

A lowpass filter cuts out high frequencies and lets low frequencies pass through.