stem/Signal Proc/Fourier Transform.md

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---
tags:
- signals
- maths
---
$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$
$$x(t)=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega t}d\omega$$
## Discrete-Time
$$X(\omega)=\sum_{-\infty}^{\infty}x[n]e^{-j\omega n}$$
$$x[n]=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega n}d\omega$$
## Discrete Fourier Transform
Digital Signal
$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j\omega_{k}n}$$
$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j\omega_{k}n}, n=0,1,\ldots,N-1$$
## Power Spectral Density
PSD
$$P[k]=|X[k]|^2$$
## Spectrogram
- PSD vertically
- Frequency power over time horizontally
- ___Time and frequency resolution inversely proportional___
- Resolution
- Frequency
- $fs/N$
- Time
- $N/fs$
- STFT has fixed resolution depending on window size
- Wider window
- Better frequency res
- Worse time resolution
- Can't tell where stuff changes with big window
- Can't use too wide
- Frequency can change during window
- 20-30ms window of speech usually treated as quasi-stationary
- Overlapping window
- Hop size of 5ms
- Appending windows can cause discontinuities
- Use window function to smooth
- Hann
## Fast-Fourier
FFT
- Faster version of DFT
- Three parts
- Shuffling
- Bit reversal
- Shuffle N-dimensional input into N one-dimensional signals
- N one-point DFTs
- Merge
- N one-point DFTs into one N-point DFT
- Butterfly merging equations
## Short-Time Fourier Transform
STFT
- Short-term
- N-point windowed DFT
- Probably use FFT
$$x[k,m]=\sum_{n=0}^{N-1}x[m\delta+n]w(n)e^{-j\omega_kn}$$
- $\omega$
- Discrete angular frequency
- $m$
- Time-frame index
- $\delta$
- Hop size
- $w(n)$
- Window function
- Hann