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Signal Proc
Fourier Transform
X(ω)=∫−∞∞x(t)e−jωtdtX(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt

x(t)=12π∫2πX(ω)ejωtdωx(t)=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega t}d\omega

Discrete-Time

X(ω)=∑−∞∞x[n]e−jωnX(\omega)=\sum_{-\infty}^{\infty}x[n]e^{-j\omega n}

x[n]=12π∫2πX(ω)ejωndωx[n]=\frac{1}{2\pi}\int_{2\pi}X(\omega)e^{j\omega n}d\omega

Discrete Fourier Transform

Digital Signal

X[k]=∑n=0N−1x[n]e−jωknX[k]=\sum_{n=0}^{N-1}x[n]e^{-j\omega_{k}n}

x[n]=1N∑k=0N−1X[k]ejωkn,n=0,1,…,N−1x[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]∣2P[k]=|X[k]|^2

Spectrogram

  • PSD vertically
  • Frequency power over time horizontally
  • Time and frequency resolution inversely proportional
  • Resolution
    • Frequency
      • fs/Nfs/N
    • Time
      • N/fsN/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]=∑n=0N−1x[mδ+n]w(n)e−jωknx[k,m]=\sum_{n=0}^{N-1}x[m\delta+n]w(n)e^{-j\omega_kn}
  • ω\omega
    • Discrete angular frequency
  • mm
    • Time-frame index
  • δ\delta
    • Hop size
  • w(n)w(n)
    • Window function
    • Hann
ConvolutionImage Processing