DFT, Complex Exponent, Gabor Wavelet.
This applet shows the Spectrum of the complex signal
f(t) = ejw1t = cos(w1t) + jsin(w1t).
One of the features of the spectrum is that there are
lines at w1 and
ws + w1
but none at ws - w1.
Because of the periodic spectrum there are lines at
nws + w1
but none at
nws - w1.
To see why this happens, the applet shows the real and imaginary
parts of the DTFT spectrum of the cosine(blue, brown) and the sine
(yellow green, pink magenta). The lines are made easier to see because
the real parts are shifted up and the imaginary parts are shifted down.
(3.0, NA, 0.01, 3.78, 10, NA, 20, 375, 15.0, 0.2) here
the imaginary parts of the spectrum of the sine and cosine are zero
and the real parts cancel at ws - w1. The real part of the real cosine
spectrum has even symmetry while that of the imaginary sine has
(3.0, NA, 0.52, 4.29, 10, NA, 20, 375, 15.0, 0.2) here
the real parts of the spectrum of the sine and cosine are zero
and the imaginary parts cancel at ws - w1. This is because the change in the
start of the sampling time exchanges the sine and the cosine.
(3.0, NA, 0.29, 4.06, 10, NA, 20, 375, 15.0, 0.2) here
both the real and imaginary parts of the spectrum of the sine and cosine cancel at ws - w1. Because of the shifting of the spectra it may not look that way.
The Gabor wavelet is useful for analysis of signals which contain
bursts of sinusoidal waveforms. The wavelet is characterised by 3
parameters the width
and the scale factor s.
s and w has the same effect. Increasing
them widens the wavelet and reduces the frequency of the oscillations
in the time domain. In the frequency domain this corresponds to a
reduction of the bandwidth and center frequency. u changes the frequency of the oscillations in
the wavelet without affecting its width. In the frequency domain this
moves the center frequency without affecting the bandwidth.
Since this is a complex wavelet with a cos + j sin function, then
the spectrum does not contain any component between
ws/2 and ws.
Unlike Fourier Transforms which uses functions that have infinite
width in the time domain and zero width in the frequency domain,
wavelets have finite width in both the time and frequency domains.
They may therefore be suited for representation of signals with
finite duration in both domains and which has varying frequency.