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 [ ], and the transform of the window function, , resulting in a shifted version of the window transform . Assuming is odd, we can show this as follows:

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.*,
.

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