Sinusoidal Peak Finding
For each sinusoidal component of a signal, we need to determine its
frequency, amplitude, and phase (when needed). As a starting point,
consider the windowed complex sinusoid with complex amplitude
and frequency
:
![]() |
(11.20) |
As discussed in Chapter 5, the transform (DTFT) of this windowed signal is the convolution of a frequency domain delta function at





![\begin{eqnarray*}
X_w(\omega) &=& \sum_{n=-\infty}^{\infty}[w(n)x(n)]e^{ -j\omega nT}
\qquad\hbox{(DTFT($x_w$))} \\
&=& \sum_{n=-(M-1)/2}^{(M-1)/2} \left[w(n){\cal A}_xe^{j\omega_xnT}\right]e^{ -j\omega nT}\\
&=& {\cal A}_x\sum_n w(n) e^{-j(\omega-\omega_x)nT} \\
&=& \zbox {{\cal A}_xW(\omega-\omega_x)}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1822.png)
Hence,

At
, we have

If we scale the window to have a dc gain of 1, then the peak magnitude
equals the amplitude of the sinusoid, i.e.,
, as shown in Fig.10.8.
If we use a zero-phase (even) window, the phase at the peak equals the
phase of the sinusoid, i.e.,
.
Next Section:
Tracking Sinusoidal Peaks in a Sequence of FFTs
Previous Section:
Following Spectral Peaks