stem/Signal Proc/Fourier Transform.md

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