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Several techniques can be used to move signals in the [[Time-frequency analysis|time-frequency distribution]]. Similar to computer graphic techniques, signals can be subjected to horizontal shifting, vertical shifting, dilation (scaling), shearing, and rotation. These techniques can help to save the bandwidth with proper motions apply on the signals. Moreover, filters with proper motion transformation can save the hardware cost without additional filters.
 
The following examples assume time in the horizontal axis versus frequency in the vertical axis. As a coincident, the following transformations happen to have the motion properties in the time-frequency distribution.
 
==Shifting==
Shifting on time axis is like horizontal shifting in time-frequency distribution. On another hand, shifting on the frequency axis would be vertical shifting in time-frequency distribution.
 
===Horizontal shifting===
If t<sub>0</sub> is greater than 0, we would be shifting the signal to the right on time axis. (negative would be left)
 
STFT, Gabor:
:<math>x(t-t_0) \rightarrow S_x(t-t_0,f)e^{-j2 \pi ft_0}</math>
 
WDF:
:<math>x(t-t_0) \rightarrow W_x(t-t_0,f)\,</math>
 
[[Image:TFA shift horizontal.jpg]]
 
===Vertical shifting===
If f<sub>0</sub> is greater than 0, we would be shifting the signal to the upward on frequency axis. (negative would be downward)
 
STFT, Gabor:
:<math>e^{j2 \pi f_0t}x(t) \rightarrow S_x(t,f-f_0)</math>
 
WDF:
:<math>e^{j2 \pi f_0t}x(t) \rightarrow W_x(t,f-f_0)</math>
 
[[Image:TFA shift vertical.jpg]]
 
This results in an [[amplitude modulation]] signal.
This sort of shift is also used in a [[frequency extender]].
This sort of shift is also used in most [[bat detector]]s.
 
Such an effect is typically implemented using [[heterodyning]]
 
==Dilation==
Dilation is like doing scaling on one of the axis and area is the same after the process. When a > 1, it's expanding on time axis, and narrowing on frequency axis ;vice versa when a < 1.
 
STFT, Gabor:
:<math>\frac{1}{\sqrt{|a|}}x(\frac{t}{a}) \rightarrow \approx S_x(\frac{t}{a},af)</math>
 
WDF:
:<math>\frac{1}{\sqrt{|a|}}x(\frac{t}{a}) \rightarrow W_x(\frac{t}{a},af)</math>
 
[[Image:TFA dilation ag1.jpg]]  [[Image:TFA dilation as1.jpg]]
 
When this kind of dilation is applied to audio, it causes a [[Alvin_and_the_Chipmunks#Recording_technique|chipmunk effect]].
 
== time stretching ==
{{main|time stretching}}
 
Time stretching is doing scaling only on the time axis, leaving frequencies the same.
When a < 1 (the most common case), it's narrowing on the time axis, reducing the area.
 
STFT, Gabor:
:<math>\approx S_x(at,f)</math>
 
WDF:
:<math>\approx W_x(at,f)</math>
 
==Shearing==
Shearing by definition is moving the side of the signal on one direction. Vertical and Horizontal shearing is introduced here.
 
===On Vertical axis only (frequency)===
It's shearing on frequency axis, since this only changes the phase.  
 
<math>x(t) = e^{j \pi at^2}y(t) \, </math>
 
STFT, Gabor:
:<math>S_x(t,f) \approx S_y(t,f-at) \, </math>
 
WDF:
:<math>W_x(t,f) = W_y(t,f-at) \, </math>
 
[[Image:TFA shear vertical.jpg]]  [[Image:TFA shear vertical as1.jpg]]
 
===On Horizontal axis only (time)===
It's shearing on time axis, since this only changes the time.  
 
<math>x(t) = e^{j \pi \frac{t^2}{a}}y(t) \, </math>
 
STFT, Gabor:
:<math>S_x(t,f) \approx S_y(t-af,f) \, </math>
 
WDF:
:<math>W_x(t,f) = W_y(t-af,f) \, </math>
 
[[Image:TFA shear horizontal.jpg]]  [[Image:TFA shear horizontal as1.jpg]]
 
==Rotation==
Many transforms has the property of rotations, like Gabor-Wigner, [[Ambiguity function]] (counterclockwise), modified Wigner, [[Cohen's class distribution function|Cohen's class distribution]].
 
[[Short-time Fourier transform|STFT]], Gabor, and [[Wigner distribution function|WDF]] is introduced in here.
 
===Clockwise rotation by 90 degrees===
By switching the time and negative frequency to frequency and time would act like rotating 90 degrees clockwise.
 
<math>X(f) = FT(x(t)) \, </math>
 
STFT:
:<math>|S_X(t,f)| \approx |S_x(-f,t)| \, </math>
 
Gabor:
:<math>G_X(t,f) = G_x(-f,t)e^{-j2 \pi ft} \, </math>
 
WDF:
:<math>W_X(t,f) = W_x(-f,t) \, </math>
 
[[Image:TFA rotate c90.jpg]]
 
===Counterclockwise rotation by 90 degrees===
By switching the negative time and frequency to frequency and time would act like rotating 90 degrees counterclockwise.
 
If <math>X(f) = IFT[x(t)] = \int_{-\infty}^{\infty} x(t)e^{j2 \pi ft} \, dt </math>, then
 
:<math>W_X(t,f) = W_x(f,-t) \, </math>
 
:<math>G_X(t,f) = G_x(f,-t)e^{j2 \pi tf} \, </math>
 
[[Image:TFA rotate cc90.jpg]]
 
===Rotation by 180 degrees===
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.
 
If <math> X(f) = x(-t) \, </math>, then
 
:<math>W_X(t,f) = W_x(-t,-f) \, </math>
 
:<math>G_X(t,f) = G_x(-t,-f) \, </math>
 
[[Image:TFA rotate 180.jpg]]
 
==Example==
If we want the left image to become the right image, we can use the techniques from above to achieve the requirement.
 
[[Image:Hm3pro5.jpg]]
 
There are several ways to solve this problem, this is one of the possible solutions.
 
First, we apply clockwise rotation of 90 degree by using one of the transform.
 
STFT:
:<math>|S_X(t,f)| \approx |S_x(-f,t)| \, </math>
 
Gabor:
:<math>G_X(t,f) = G_x(-f,t)e^{-j2 \pi ft} \, </math>
 
WDF:
:<math>W_X(t,f) = W_x(-f,t) \, </math>
[[Image:Hm3pro5 1step.jpg]]
 
Second, we set a = 1/3, and perform a horizontal shearing on t-axis.
 
STFT, Gabor:
:<math>S_x(t,f) \approx S_y(t- \frac{1}{3} f,f) \, </math>
 
WDF:
:<math>W_x(t,f) = W_y(t- \frac{1}{3} f,f) \, </math>
[[Image:Hm3pro5 2step.jpg]]
 
Third, we shift the signal 2 to the right on t-axis by setting t<sub>0</sub> = 2
 
STFT, Gabor:
:<math>x(t-t_0) \rightarrow S_x(t-2,f)e^{-j2 \pi ft_0}</math>
 
WDF:
:<math>x(t-t_0) \rightarrow W_x(t-2,f)\,</math>
 
[[Image:Hm3pro5 2 1step.jpg]]
 
Finally, we shift the signal 1 to the left on f-axis by setting f<sub>0</sub> = -1
 
STFT, Gabor:
:<math>e^{j2 \pi f_0t}x(t) \rightarrow S_x(t,f+1)</math>
 
WDF:
:<math>e^{j2 \pi f_0t}x(t) \rightarrow W_x(t,f+1)</math>
 
[[Image:Hm3pro5 3step.jpg]]
 
==Applications==
As mentioned in the introduction, the above techniques can be used to save the bandwidth or the filter cost.
 
Assume the signal look like this.
 
[[Image:TFA application 1.jpg]].
 
The dashed box is the filter, and the area of the dashed box would be the bandwidth required.
 
After some operations like the above example, the signal turn into the position like this.
 
[[Image:TFA application 2.jpg]]
 
As a result, the bandwidth was saved, since the area became smaller. Moreover, only a lowpass filter is required to recover the signal, instead of a bandpass filter.
 
==See also==
Other time-frequency transforms:
*[[Short-time Fourier transform]]
*[[Wigner distribution function]]
*[[Ambiguity function]]
*[[Cohen's class distribution function]]
 
==References==
* J.J. Ding, "time-frequency analysis and wavelet transform course note," the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
* J.J. Ding, "time-frequency analysis and wavelet transform homework 3," the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
 
[[Category:Integral transforms]]
[[Category:Fourier analysis]]

Revision as of 17:58, 27 February 2014

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