DFT, Complex Exponent, Gabor Wavelet.

Cuthbert Nyack

This applet shows the Spectrum of the complex signal
f(t) = e^jw1t = cos(w1t) + jsin(w1t).
One of the features of the spectrum is that there are lines at w1 and w_s + w1 but none at w_s - w1.
Because of the periodic spectrum there are lines at nw_s + w1 but none at nw_s - 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.

eg parameters:-
(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 w_s - w1. The real part of the real cosine spectrum has even symmetry while that of the imaginary sine has odd symmetry.

eg parameters:-
(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 w_s - w1. This is because the change in the start of the sampling time exchanges the sine and the cosine.

eg parameters:-
(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 w_s - 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 w the frequency u 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 w_s/2 and w_s.

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.