67 lines
1.7 KiB
Markdown
67 lines
1.7 KiB
Markdown
|
$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$
|
||
|
|
$$x(t)=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega t}d\omega$$
|
||
|
|
## 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
|