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Chapters

Spectral Audio Signal Processing
    Wavelet Filter Banks
       Geometric Signal Theory
          Example Basis Signals

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Example Basis Signals

  • ``Natural'' Basis:
    $\displaystyle \varphi_k$ $\displaystyle \isdef$ $\displaystyle [\ldots, 0,\underbrace{1}_{k^{\hbox{\tiny th}}},0,\ldots], \quad \hbox{\textit{i.e.},}$  
    $\displaystyle \varphi_k (n)$ $\displaystyle \isdef$ $\displaystyle \delta(n-k)$  


    $\displaystyle \implies\quad \left<\varphi_k ,x\right>$ $\displaystyle =$ $\displaystyle x(k)$  
    $\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}x(k) \varphi_k (n)$  
      $\displaystyle =$ $\displaystyle \sum_{k=-\infty}^{\infty}x(k) \delta(n-k) \;\isdef \; (x \ast \delta)(n)$  

  • Normalized DFT Basis for $ {\bf C}^N$:

    $\displaystyle \varphi_k (n) \isdef e^{j\omega_k n}/\sqrt{N},\quad \omega_k \isdef 2\pi k/N, \quad n,k \in [0,N-1] $


    $\displaystyle \implies\quad \left<\varphi_k ,x\right>$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} x(n) e^{-j\omega_k n}$  
      $\displaystyle \isdef$ $\displaystyle \hbox{DFT}_{N,k}(x)/\sqrt{N} \isdef X(\omega_k )/\sqrt{N}$  
           
    $\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} \left<\varphi_k ,x\right> \varphi_k (n)$  
      $\displaystyle =$ $\displaystyle \frac{1}{N} \sum_{k=0}^{N-1} X(k)e^{j\omega_k n}$  
      $\displaystyle \isdef$ $\displaystyle \hbox{DFT}_{N,n}^{-1}(X)$  

  • Normalized Fourier Transform Basis:

    $\displaystyle \varphi _\omega (t) \isdef e^{j\omega t}/\sqrt{2\pi},\quad \omega , t\in (-\infty,\infty) $


    $\displaystyle \displaystyle \implies\quad \left<\varphi _\omega ,x\right>$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$  
      $\displaystyle \isdef$ $\displaystyle \hbox{FT}_\omega (x)/\sqrt{2\pi} \isdef X(\omega )/\sqrt{2\pi}$  
           
    $\displaystyle x(t)$ $\displaystyle =$ $\displaystyle \int_{-\infty}^{\infty} \left<\varphi _\omega ,x\right> \varphi _\omega (t) d\omega$  
      $\displaystyle =$ $\displaystyle \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega )e^{j\omega t} d\omega$  
      $\displaystyle \isdef$ $\displaystyle \hbox{FT}_t^{-1}(X)$  

  • Normalized DTFT Basis:

    $\displaystyle \varphi _\omega (n) \isdef e^{j\omega n}/\sqrt{2\pi},\quad \omega \in (-\pi,\pi], \quad n\in (-\infty,\infty) $

    (Find inner product $ \left<\varphi _\omega ,x\right>$ and reconstruction of $ x(n)$ in terms of $ \{\varphi _\omega \}$ as an exercise.)

  • Normalized STFT Basis:

    $\displaystyle \varphi_{mk}(n) \isdef \frac{w(n-mR)e^{j\omega_k n}}{\left\Vert\... ...cdot)}\,\right\Vert} = \frac{w(n-mR) e^{j\omega_k n}}{\sqrt{\sum_n{w^2(n)}}}, $

    $\displaystyle \omega_k = 2\pi k/N, \; k \in [0,N-1], \; n\in (-\infty,\infty),\; w(n)\in{\cal R} $

    • Overcomplete in general
    • Orthonormal when
      $ R=M=N$
          (Hop size = Window length = DFT length)
      $ w = $ Rectangular Window $ w_R$
      $ \implies\varphi_{mk}= \hbox{\sc Shift}_{mN}\left[\hbox{\sc ZeroPad}_\infty\left(\varphi_k ^{\hbox{\tiny DFT}}\right)\right]$, i.e., $ \varphi_{mk}(n) = e^{j\omega_k n}/\sqrt{N}$ for $ mN \leq n \leq (m+1)N-1$, and 0 otherwise
      In this case,
      $\displaystyle \displaystyle \left<\varphi_{mk},x\right>$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{N}} \sum_{n=-\infty}^{\infty} x(n) w_R(n-mN) e^{-j\omega_k n}$  
        $\displaystyle \isdef$ $\displaystyle \hbox{STFT}_{N,m,k}(x)/\sqrt{N} \isdef X_m(\omega_k )/\sqrt{N}$  


      $\displaystyle \displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty}\sum_{k=0}^{N-1} \left<\varphi_{mk},x\right> \varphi_{mk}(n)$  
        $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty} w_R(n-mN)\frac{1}{N}\sum_{k=0}^{N-1} X_m(\omega_k )e^{j\omega_k n}$  
        $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty} \hbox{\sc Shift}_{mN,n}\left\{\hbox{\sc ZeroPad}_\infty\left[\hbox{DFT}_N^{-1}(X_m)\right]\right\}$  
        $\displaystyle \isdef$ $\displaystyle \hbox{STFT}_{N,n}^{-1}(X)$  

  • Continuous Wavelet Transform Basis:
    $\displaystyle \displaystyle \varphi_{s\tau}(t)$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{\vert s\vert}} f^\ast\left(\frac{\tau-t}{s}\right),$  
           
    $\displaystyle \tau,s,t$ $\displaystyle \in$ $\displaystyle (-\infty,\infty)$  
           
    $\displaystyle X(s,\tau)$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{\vert s\vert}} \int_{-\infty}^{\infty} x(t) f\left(\frac{t-\tau}{s}\right) dt$  

    • $ s = $ scale parameter
    • $ 1/\sqrt{\vert s\vert}$ maintains energy invariance
    • $ X(s,\tau) = $ wavelet coefficients
    • $ f(t) = $ mother wavelet
    • $ f(t)$ typically bandpass
    • A qualitative example is shown in Fig.12.2.
      Figure: Qualitative appearance of first three wavelets when the scale parameter is $ s=2$.
      \includegraphics[scale=0.8]{eps/wavelets}

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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