Motions in the time-frequency distribution

Short description: Modification of frequency and time distributions of signals, as used in computer graphics

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. 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.

## Contents

* 1 Shifting
* 1.1 Horizontal shifting
* 1.2 Vertical shifting
* 2 Dilation
* 3 Time stretching
* 4 Shearing
* 4.1 On Vertical axis only (frequency)
* 4.2 On Horizontal axis only (time)
* 5 Generalized Shearing
* 5.1 On Vertical axis only (frequency)
* 5.2 On Horizontal axis only (time)
* 6 Rotation
* 6.1 Clockwise rotation by 90 degrees
* 6.2 Counterclockwise rotation by 90 degrees
* 6.3 Rotation by 180 degrees
* 7 Rotation: Fractional Fourier Transform (FRFTs)
* 8 Twisting: Linear Canonical Transform (LCT)
* 8.1 Transformation Type
* 9 Example
* 10 Applications
* 10.1 Efficient Sampling
* 10.2 Signal Decomposition and Filter Design
* 11 See also
* 12 References

## 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 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−t0)->Sx(t−t0,f)e−j2πft0

WDF:

x(t−t0)->Wx(t−t0,f)

### 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:

ej2πf0tx(t)->Sx(t,f−f0)

WDF:

ej2πf0tx(t)->Wx(t,f−f0)

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 detectors.

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:

1|a|x(ta)->≈Sx(ta,af)

WDF:

1|a|x(ta)->Wx(ta,af)

When this kind of dilation is applied to audio, it causes a chipmunk effect.

## Time stretching

Main article: Audio time stretching and pitch scaling

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:

≈Sx(at,f)

WDF:

≈Wx(at,f)

## 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.

x(t)=ejπat2y(t)

STFT, Gabor:

Sx(t,f)≈Sy(t,f−at)

WDF:

Wx(t,f)=Wy(t,f−at)

### On Horizontal axis only (time)

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

x(t)=ejπt2ay(t)

STFT, Gabor:

Sx(t,f)≈Sy(t−af,f)

WDF:

Wx(t,f)=Wy(t−af,f)

## Generalized Shearing

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 ϕ(t). It is beneficial for implementing higher-order modulation, and furthermore, it reduces bandwidth, allowing for lower sampling rates and decreased white noise through filtering.

### On Vertical axis only (frequency)

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

x(t)=ejϕ(t)y(t)

ϕ(t)=∑k=0naktk

STFT, Gabor:

Sx(t,f)≈Sy(t,f−∑k=1nkak2πtk−1)

WDF:

Wx(t,f)≈Wy(t,f−∑k=1nkak2πtk−1)

### On Horizontal axis only (time)

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

x(t)=h(t)*y(t)

h(t)=IFT(exp(j∑k=0nakfk))

STFT, Gabor:

Sx(t,f)≈Sy(t+∑k=1nkak2πfk−1,f)

WDF:

Wx(t,f)≈Wy(t+∑k=1nkak2πfk−1,f)

## 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.

### 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:

|SX(t,f)|≈|Sx(−f,t)|

Gabor:

GX(t,f)=Gx(−f,t)e−j2πft

WDF:

WX(t,f)=Wx(−f,t)

### 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)]=∫−∞∞x(t)ej2πftdt, then

WX(t,f)=Wx(f,−t)

GX(t,f)=Gx(f,−t)ej2πtf

### 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

WX(t,f)=Wx(−t,−f)

GX(t,f)=Gx(−t,−f)

## Rotation: Fractional Fourier Transform (FRFTs)

Rotating the time-frequency distribution by the angle other than π/2,π,3π/2 and 2π. Compared to Fourier Transform, it transform signal from time domain to fractional domain, domain between time and frequency.

Xϕ(u)=1−jcotϕejπcotϕu2∫−∞∞e−j2πcscϕutejπcotϕ t2x(t)dt,ϕ=0.5aπ

For Fourier Transform, a=1,ϕ=0.5π and cscϕ=1,cotϕ=0

It is equivalent to the clockwise rotation operation with angle ϕ. for Wigner distribution funciton and Gabor transform.

Gabor:

|GXϕ(u,v)|=|Gx(ucosϕ−vsinϕ,usinϕ+vcosϕ)|

WDF:

WXϕ(u,v)=Wx(ucosϕ−vsinϕ,usinϕ+vcosϕ)

## Twisting: Linear Canonical Transform (LCT)

The Linear Canonical Transform makes arbitrary linear and integral transformation of a time-frequency distribution.

F(a,b,c,d)(u)=1jbejπdbu2∫−∞∞e−j2π1bute−jπabt2x(t)dt,b≠0

F(a,0,c,d)(u)=dejπcdu2x(du),b=0

Constraint:

ab−bc=1

### Transformation Type

a. fractional Fourier transform

[abcd]=[cosϕsinϕ−sinϕcosϕ]

* Fourier transform: ϕ=π/2
* identity operation: ϕ=0
* inverse Fourier transform: ϕ=−π/2

b. Fresnel transform (convolution with a chirp)

[abcd]=[1λz01]

c. chirp multiplication

[abcd]=[10τ1]

d. scaling

[abcd]=[1σ00σ]

## Example

If we want the left image to become the right image, we can use the techniques from above to achieve the requirement.

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:

|SX(t,f)|≈|Sx(−f,t)|

Gabor:

GX(t,f)=Gx(−f,t)e−j2πft

WDF:

WX(t,f)=Wx(−f,t)

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

STFT, Gabor:

Sx(t,f)≈Sy(t−13f,f)

WDF:

Wx(t,f)=Wy(t−13f,f)

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

STFT, Gabor:

x(t−t0)->Sx(t−2,f)e−j2πft0

WDF:

x(t−t0)->Wx(t−2,f)

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

STFT, Gabor:

ej2πf0tx(t)->Sx(t,f+1)

WDF:

ej2πf0tx(t)->Wx(t,f+1)

## Applications

### Efficient Sampling

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.

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.

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.

### Signal Decomposition and Filter Design

Signal is decomposed into several components and filter removes the undesired component of a signal. 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:

xo(t)=OF−ϕOFϕ[xit]H(u)
OFϕ(x(t))=1−jcotϕejπcotϕu2∫−∞∞e−j2πcscϕutejπcotϕ t2x(t)dt(FRFTs)

(1) If H(u)=S(−u+u0),H(u)={1 u<u00 u>u0

S(u): Step function

(2) ϕ is determined by the angle of cutoff line and f-axis.

(3) u0 equals the distance from origin to cutoff line.

## 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.

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