Andy Pack
f29c435494
Affected files: .obsidian/graph.json .obsidian/workspace.json Gaming/Steam controllers.md Gaming/Ubisoft.md STEM/Signal Proc/Convolution.md STEM/Signal Proc/Fourier Transform.md STEM/Signal Proc/Pole-Zero.md STEM/Signal Proc/System Classes.md STEM/Signal Proc/Transfer Function.md STEM/Speech/Linguistics/Consonants.md STEM/Speech/Linguistics/Linguistics.md STEM/Speech/Linguistics/Terms.md STEM/Speech/Linguistics/Vowels.md STEM/Speech/Literature.md STEM/Speech/NLP/Jargon.md STEM/Speech/NLP/NLP.md STEM/Speech/NLP/Recognition.md STEM/Speech/Perception/Perception.md STEM/Speech/Speech Processing/Applications.md STEM/Speech/Speech Processing/Source-Filter.md STEM/Speech/Speech Processing/Vocal Tract.md Work/Applications/Anthropic/Cover letter.md Work/Applications/Anthropic/In line with values.md Work/Applications/Anthropic/Why Work.md Work/Companies.md Work/Freelancing.md Work/Products.md Work/Tech.md
72 lines
1.7 KiB
Markdown
72 lines
1.7 KiB
Markdown
---
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tags:
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- signals
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- maths
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---
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$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$
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$$x(t)=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega t}d\omega$$
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## Discrete-Time
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$$X(\omega)=\sum_{-\infty}^{\infty}x[n]e^{-j\omega n}$$
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$$x[n]=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega n}d\omega$$
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## Discrete Fourier Transform
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Digital Signal
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$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j\omega_{k}n}$$
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$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j\omega_{k}n}, n=0,1,\ldots,N-1$$
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## Power Spectral Density
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PSD
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$$P[k]=|X[k]|^2$$
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## Spectrogram
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- PSD vertically
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- Frequency power over time horizontally
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- ___Time and frequency resolution inversely proportional___
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- Resolution
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- Frequency
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- $fs/N$
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- Time
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- $N/fs$
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- STFT has fixed resolution depending on window size
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- Wider window
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- Better frequency res
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- Worse time resolution
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- Can't tell where stuff changes with big window
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- Can't use too wide
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- Frequency can change during window
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- 20-30ms window of speech usually treated as quasi-stationary
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- Overlapping window
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- Hop size of 5ms
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- Appending windows can cause discontinuities
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- Use window function to smooth
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- Hann
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## Fast-Fourier
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FFT
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- Faster version of DFT
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- Three parts
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- Shuffling
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- Bit reversal
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- Shuffle N-dimensional input into N one-dimensional signals
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- N one-point DFTs
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- Merge
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- N one-point DFTs into one N-point DFT
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- Butterfly merging equations
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## Short-Time Fourier Transform
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STFT
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- Short-term
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- N-point windowed DFT
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- Probably use FFT
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$$x[k,m]=\sum_{n=0}^{N-1}x[m\delta+n]w(n)e^{-j\omega_kn}$$
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- $\omega$
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- Discrete angular frequency
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- $m$
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- Time-frame index
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- $\delta$
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- Hop size
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- $w(n)$
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- Window function
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- Hann
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