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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Spectrum Analysis of Sinusoids
       Optimal Peak-Finding in the Spectrum
          Least Squares Sinusoidal Parameter Estimation
             Sinusoidal Amplitude Estimation


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Sinusoidal Amplitude Estimation

If the sinusoidal frequency $ \omega_0$ and phase $ \phi$ happen to be known, we obtain a simple linear least squares problem for the amplitude $ A$. That is, the error signal

$\displaystyle x(n)-\hat{{\cal A}}e^{j\hat{\omega}_0n} = x(n)-{\hat A}e^{j(\omega_0 n+\phi)} $

becomes linear in the unknown parameter $ {\hat A}$. As a result, the sum of squared errors

$\displaystyle J({\hat A}) \isdef \sum_{n=0}^{N-1}\left\vert x(n)-{\hat A}e^{j(\omega_0 n+\phi)}\right\vert^2 \protect$ (5.11)

becomes a simple quadratic (parabola) over the real line. Quadratic forms in any number of dimensions are easy to minimize. For example, the ``bottom of the bowl'' can be reached in one step of Newton's method. From another point of view, the optimal parameter $ {\hat A}$ can be obtained as the coefficient of orthogonal projection of the data $ x(n)$ onto the space spanned by all values of $ {\hat A}$ in the linear model $ {\hat A}e^{j(\omega_0 n+\phi)}$.

Yet a third way to minimize (4.11) is the method taught in elementary calculus: differentiate $ J({\hat A})$ with respect to $ {\hat A}$, equate it to zero, and solve for $ {\hat A}$. In preparation for this, it is helpful to write (4.11) as

\begin{eqnarray*} J({\hat A}) &\isdef & \sum_{n=0}^{N-1} \left[x(n)-{\hat A}e^{... ...rline{{\hat A}} e^{-j(\omega_0 n+\phi)}\right\} + N {\hat A}^2. \end{eqnarray*}

Differentiating with respect to $ {\hat A}$ and equating to zero yields

$\displaystyle 0 = \frac{d J({\hat A})}{d{\hat A}} = - 2$re$\displaystyle \left\{\sum_{n=0}^{N-1} x(n) e^{-j(\omega_0 n+\phi)}\right\} + 2N{\hat A}. $

Solving this for $ {\hat A}$ gives the optimal least-squares amplitude estimate

$\displaystyle {\hat A}= \frac{1}{N}$re$\displaystyle \left\{\sum_{n=0}^{N-1} x(n) e^{-j(\omega_0 n+\phi)}\right\} = \frac{1}{N}$re$\displaystyle \left\{\hbox{\sc DTFT}_{\omega_0 }\left[e^{-j\phi} x\right]\right\}. \protect$ (5.12)

That is, the optimal least-squares amplitude estimate may be found by the following steps:
  1. Multiply the data $ x(n)$ by $ e^{-j\phi}$ to zero the known phase $ \phi$.
  2. Take the DFT of the $ N$ samples of $ x$, suitably zero padded to approximate the DTFT, and evaluate it at the known frequency $ \omega_0$.
  3. Discard any imaginary part since it can only contain noise, by (4.12).
  4. Divide by $ N$ to obtain a properly normalized coefficient of projection [248] onto the sinusoid

    $\displaystyle s_{\omega_0 }(n)\isdef e^{j\omega_0 n}. $

Previous: Least Squares Sinusoidal Parameter Estimation
Next: Sinusoidal Amplitude and Phase Estimation

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About the Author: 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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