Motions in the time-frequency distribution

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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, 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 propeties in the time-frequency distribution.

Contents

[edit] 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.

[edit] Horizontal shifting

If t0 is greater than 0, we would be shifting the signal to the right on time axis. (negative would be left)

STFT, Gabor:

x(t-t_0) \rightarrow S_x(t-t_0,f)e^{-j2 \pi ft_0}

WDF:

x(t-t_0) \rightarrow W_x(t-t_0,f)\,

Image:TFA shift horizontal.jpg

[edit] Vertical shifting

If f0 is greater than 0, we would be shifting the signal to the upward on frequency axis. (negative would be downward)

STFT, Gabor:

e^{j2 \pi f_0t}x(t) \rightarrow S_x(t,f-f_0)

WDF:

e^{j2 \pi f_0t}x(t) \rightarrow W_x(t,f-f_0)

Image:TFA shift vertical.jpg

[edit] 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:

\frac{1}{\sqrt{|a|}}x(\frac{t}{a}) \rightarrow \approx S_x(\frac{t}{a},af)

WDF:

\frac{1}{\sqrt{|a|}}x(\frac{t}{a}) \rightarrow W_x(\frac{t}{a},af)

Image:TFA dilation ag1.jpg Image:TFA dilation as1.jpg

[edit] Shearing

Shearing by definition is moving the side of the signal on one direction. Vertical and Horizontal shearing is introduced here.

[edit] On Vertical axis only (frequency)

It's shearing on frequency axis, since this only changes the phase.

x(t) = e^{j \pi at^2}y(t) \,

STFT, Gabor:

S_x(t,f) \approx S_y(t,f-at) \,

WDF:

W_x(t,f) = W_y(t,f-at) \,

Image:TFA shear vertical.jpg Image:TFA shear vertical as1.jpg

[edit] On Horizontal axis only (time)

It's shearing on time axis, since this only changes the time.

x(t) = e^{j \pi \frac{t^2}{a}}y(t) \,

STFT, Gabor:

S_x(t,f) \approx S_y(t-af,f) \,

WDF:

W_x(t,f) = W_y(t-af,f) \,

Image:TFA shear horizontal.jpg Image:TFA shear horizontal as1.jpg

[edit] Rotation

Many transforms has the property of rotations, like Gabor-Wigner, Ambiguity function (counterclockwise), modified Wigner, Cohen's class distribution.

STFT, Gabor, and WDF is introduced in here.

[edit] Clockwise rotation by 90 degrees

By switching the time and negative frequency to frequency and time would act like rotating 90 degrees clockwise.

X(f) = FT(x(t)) \,

STFT:

|S_X(t,f)| \approx |S_x(-f,t)| \,

Gabor:

G_X(t,f) = G_x(-f,t)e^{-j2 \pi ft} \,

WDF:

W_X(t,f) = W_x(-f,t) \,


Image:TFA rotate c90.jpg

[edit] Counterclockwise rotation by 90 degrees

By switching the negative time and frequency to frequency and time would act like rotating 90 degrees counterclockwise.

If X(f) = IFT[x(t)] = \int_{-\infty}^{\infty} x(t)e^{j2 \pi ft} \, dt , then

W_X(t,f) = W_x(f,-t) \,
G_X(t,f) = G_x(f,-t)e^{j2 \pi tf} \,

Image:TFA rotate cc90.jpg

[edit] 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 X(f) = x(-t) \, ,then

W_X(t,f) = W_x(-t,-f) \,
G_X(t,f) = G_x(-t,-f) \,

Image:TFA rotate 180.jpg

[edit] 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:

|S_X(t,f)| \approx |S_x(-f,t)| \,

Gabor:

G_X(t,f) = G_x(-f,t)e^{-j2 \pi ft} \,

WDF:

W_X(t,f) = W_x(-f,t) \,

Image:Hm3pro5 1step.jpg

Second, we set a = 1/3, and perform a horizontal shearing on t-axis.

STFT, Gabor:

S_x(t,f) \approx S_y(t- \frac{1}{3} f,f) \,

WDF:

W_x(t,f) = W_y(t- \frac{1}{3} f,f) \,

Image:Hm3pro5 2step.jpg


Third, we shift the signal 2 to the right on t-axis by setting t0 = 2

STFT, Gabor:

x(t-t_0) \rightarrow S_x(t-2,f)e^{-j2 \pi ft_0}

WDF:

x(t-t_0) \rightarrow W_x(t-2,f)\,

Image:Hm3pro5 2 1step.jpg

Finally, we shift the signal 1 to the left on f-axis by setting f0 = -1

STFT, Gabor:

e^{j2 \pi f_0t}x(t) \rightarrow S_x(t,f+1)

WDF:

e^{j2 \pi f_0t}x(t) \rightarrow W_x(t,f+1)

Image:Hm3pro5 3step.jpg

[edit] 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, instead of a bandwidth filter is needed, only a low pass filter is required to recover the signal.

[edit] See also

Other time-frequency transforms:

[edit] 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.