Motions in the time-frequency distribution

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:

${\displaystyle \frac{1}{\sqrt{|a|}}x(\frac{t}{a})\to <!--\; -->\approx S_x(\frac{t}{a},af)}$

WDF:

${\displaystyle \frac{1}{\sqrt{|a|}}x(\frac{t}{a})\to W_x(\frac{t}{a},af)}$

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

Time stretching

Time stretching is doing scaling only on the time axis, leaving frequencies the same. When ${\displaystyle a<1}$ (the most common case), it's narrowing on the time axis, reducing the area.

STFT, Gabor:

${\displaystyle \approx S_x(at,f)}$

WDF:

${\displaystyle \approx W_x(at,f)}$

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.

${\displaystyle x(t)=e^{j\pi a{t}^2}y(t)}$

STFT, Gabor:

${\displaystyle S_x(t,f)\approx S_y(t,f-at)}$

WDF:

${\displaystyle W_x(t,f)=W_y(t,f-at)}$

On Horizontal axis only (time)

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

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

STFT, Gabor:

${\displaystyle S_x(t,f)\approx S_y(t-af,f)}$

WDF:

${\displaystyle W_x(t,f)=W_y(t-af,f)}$

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 ${\displaystyle \varphi (t)}$. 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.

${\displaystyle x(t)=e^{j\varphi (t)}y(t)}$

${\displaystyle \varphi (t)=\sum _{k=0}^n{a}_k{t}^k}$

STFT, Gabor:

${\displaystyle S_x(t,f)\approx S_y(t,f-\sum _{k=1}^n\frac{k{a}_k}{2\pi }{t}^{k-1})}$

WDF:

${\displaystyle W_x(t,f)\approx W_y(t,f-\sum _{k=1}^n\frac{k{a}_k}{2\pi }{t}^{k-1})}$

On Horizontal axis only (time)

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

${\displaystyle x(t)=h(t)\ast y(t)}$

${\displaystyle h(t)=IFT(exp(j\sum _{k=0}^n{a}_k{f}^k))}$

STFT, Gabor:

${\displaystyle S_x(t,f)\approx S_y(t+\sum _{k=1}^n\frac{k{a}_k}{2\pi }{f}^{k-1},f)}$

WDF:

${\displaystyle W_x(t,f)\approx W_y(t+\sum _{k=1}^n\frac{k{a}_k}{2\pi }{f}^{k-1},f)}$

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.

${\displaystyle X(f)=FT(x(t))}$

STFT:

${\displaystyle |S_X(t,f)|\approx |S_x(-f,t)|}$

Gabor:

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

WDF:

${\displaystyle W_X(t,f)=W_x(-f,t)}$

Counterclockwise rotation by 90 degrees

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

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

${\displaystyle W_X(t,f)=W_x(f,-t)}$

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

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 ${\displaystyle X(f)=x(-t)}$, then

${\displaystyle W_X(t,f)=W_x(-t,-f)}$

${\displaystyle G_X(t,f)=G_x(-t,-f)}$

Rotation: Fractional Fourier Transform (FRFTs)

Rotating the time-frequency distribution by the angle other than ${\displaystyle \pi /2,\pi ,3\pi /2}$ and ${\displaystyle 2\pi }$. Compared to Fourier Transform, it transform signal from time domain to fractional domain, domain between time and frequency.

${\displaystyle X_{\varphi }(u)=\sqrt{1-jcot\varphi }{e}^{j\pi cot\varphi u^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-j2\pi csc\varphi ut}{e}^{j\pi cot\varphi t^2}x(t)dt,\varphi =0.5a\pi }$

For Fourier Transform, ${\displaystyle a=1,\varphi =0.5\pi }$ and ${\displaystyle csc\varphi =1,cot\varphi =0}$

It is equivalent to the clockwise rotation operation with angle ${\displaystyle \varphi }$. for Wigner distribution function and Gabor transform.

Gabor:

${\displaystyle |G_{X_{\varphi }}(u,v)|=|G_x(ucos\varphi -vsin\varphi ,usin\varphi +vcos\varphi )|}$

WDF:

${\displaystyle W_{X_{\varphi }}(u,v)=W_x(ucos\varphi -vsin\varphi ,usin\varphi +vcos\varphi )}$

Twisting: Linear Canonical Transform (LCT)

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

${\displaystyle F_{(a,b,c,d)}(u)=\sqrt{\frac{1}{jb}}{e}^{j\pi \frac{d}{b}{u}^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-j2\pi \frac{1}{b}ut}{e}^{-j\pi \frac{a}{b}{t}^2}x(t)dt,b\ne 0}$

${\displaystyle F_{(a,0,c,d)}(u)=\sqrt{d}{e}^{j\pi cd{u}^2}x(du),b=0}$

Constraint:

${\displaystyle ab-bc=1}$

Transformation Type

a. fractional Fourier transform

${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}cos\varphi & sin\varphi \\ -sin\varphi & cos\varphi \end{array}\right]}$

• Fourier transform: ${\displaystyle \varphi =\pi /2}$
• identity operation: ${\displaystyle \varphi =0}$
• inverse Fourier transform: ${\displaystyle \varphi =-\pi /2}$

b. Fresnel transform (convolution with a chirp)

${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}1& \lambda z\\ 0& 1\end{array}\right]}$

c. chirp multiplication

${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ \tau & 1\end{array}\right]}$

d. scaling

${\displaystyle \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\sigma }& 0\\ 0& \sigma \end{array}\right]}$

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:

${\displaystyle |S_X(t,f)|\approx |S_x(-f,t)|}$

Gabor:

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

WDF:

${\displaystyle W_X(t,f)=W_x(-f,t)}$

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

STFT, Gabor:

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

WDF:

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

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

STFT, Gabor:

${\displaystyle x(t-t_0)\to S_x(t-2,f)e^{-j2\pi f{t}_0}}$

WDF:

${\displaystyle x(t-t_0)\to W_x(t-2,f)}$

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

STFT, Gabor:

${\displaystyle e^{j2\pi f_{0}t}x(t)\to S_x(t,f+1)}$

WDF:

${\displaystyle e^{j2\pi f_{0}t}x(t)\to W_x(t,f+1)}$

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:

${\displaystyle x_o(t)=O_F^{-\varphi }{O}_F^{\varphi }[x_{i}t]H(u)}$

${\displaystyle O_F^{\varphi }(x(t))=\sqrt{1-jcot\varphi }{e}^{j\pi cot\varphi u^2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-j2\pi csc\varphi ut}{e}^{j\pi cot\varphi t^2}x(t)dt(FRFTs)}$

(1) If ${\displaystyle H(u)=S(-u+u_0),H(u)=\{\begin{array}{l}1u<u_0\\ 0u>u_0\end{array}}$

${\displaystyle S(u)}$: Step function

(2) ${\displaystyle \varphi }$ is determined by the angle of cutoff line and f-axis.

(3) ${\displaystyle u_0}$ equals the distance from origin to cutoff line.