Several techniques can be used to move signals in the time-frequency distribution.
Similar to computer graphic techniques, signals can be subjected to horizontal shifting, vertical shifting, dilation (scaling), shearing, rotation, and twisting.
These techniques can help to save the bandwidth with proper motions apply on the signals.
As a coincident, the following transformations happen to have the motion properties in the time-frequency distribution.
If t0 is greater than 0, we would be shifting the signal to the right on time axis.
If f0 is greater than 0, we would be shifting the signal to the upward on frequency axis.
Such an effect is typically implemented using heterodyning Dilation is like doing scaling on one of the axis and area is the same after the process.
When this kind of dilation is applied to audio, it causes a chipmunk effect.
(the most common case), it's narrowing on the time axis, reducing the area.
Shearing by definition is moving the side of the signal on one direction.
It's shearing on frequency axis, since this only changes the phase.
Transforming the time-frequency distribution from a band-like pattern to a curved shape requires the use of polynomials of order three or higher with respect to
It is beneficial for implementing higher-order modulation, and furthermore, it reduces bandwidth, allowing for lower sampling rates and decreased white noise through filtering.
It's shearing on frequency axis, since this only changes the phase.
Many transforms has the property of rotations, like Gabor-Wigner, Ambiguity function (counterclockwise), modified Wigner, Cohen's class distribution.
Changing the sign of both time and frequency would be like flipping twice on both axis, and it ends up like doing 180 degrees rotation.
It is equivalent to the clockwise rotation operation with angle
for Wigner distribution function and Gabor transform.
First, we apply clockwise rotation of 90 degree by using one of the transform.
Second, we set a = 1/3, and perform a horizontal shearing on t-axis.
Third, we shift the signal 2 to the right on t-axis by setting t0 = 2 STFT, Gabor:
Finally, we shift the signal 1 to the left on f-axis by setting f0 = -1 STFT, Gabor:
As mentioned in the introduction, the above techniques can be used to save the bandwidth or the filter cost.
After some operations like the above example, the signal turn into the position like this.
As a result, the bandwidth was saved, since the area became smaller.
The Fourier transform is suitable to filter out the noise that is a combination of sinusoid functions.
If signal are not separable in both time and frequency domains, using the fractional Fourier transform (FRFTs) is suitable to filter out the noise that is a combination of higher order exponential functions.
Fiter designed by the fractional Fourier transform: (1) If
is determined by the angle of cutoff line and f-axis.
equals the distance from origin to cutoff line.