Time-frequency analysis

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The time-frequency distribution is developed for acquiring the information of a signal in both time and frequency domain simultaneously, and the task of time-frequency analysis is to analyze it. Conventionally, we often use the Fourier transform to obtain the frequency spectrum of a signal. However, it’s not appropriate to analyze a signal which has time-varying frequency components.An example is shown below.

$x(t)=\begin{cases} \cos( \pi t); & t <10 \\ \cos(3 \pi t); & 10 \le t < 20 \\ \cos(2 \pi t); & t > 20 \end{cases}$

1 Time-frequency distribution functions
- 1.1 Ideal TF distribution function
2 Applications

[edit] Time-frequency distribution functions

There are many well-known time-frequency distributions, such as

short-time Fourier transform (including the Gabor transform),
Cohen's class distribution function (Wigner distribution function),
modified Wigner distribution function, Gabor-Wigner distribution function and so on.

More information about the history and the motivation of development of time-frequency distribution can be found in the entry Time-frequency representation.

[edit] Ideal TF distribution function

An ideal time-frequency distribution function roughly requires the following 4 properties:

High clarity makes it easier to be analyzed.
No cross-term avoids confusing us which component is noise or not.
Good mathematical properties benefit to its application.
Lower computational complexity means the time needed to represent a signal on a time-frequency plane.

Here we compare several time-frequency distribution functions.

	Clarity	Cross-term	Good mathematical properties	Computational complexity
Gabor transform	Worse	No	Worse	Low
Wigner distribution function	Best	Yes	Best	High
Gabor-Wigner distribution function	Good	Almost eliminated	Good	High

To analyze the signals well, choosing an appropriate time-frequency distribution function is important. Which time-frequency distribution function should be used depends on what application it applies on. The high clarity of the Wigner distribution function (WDF) is due to the auto-correlation function; however, it also causes the cross-term problem. Therefore, if we want to analyze a single-term signal, using the WDF is better; if the signal is composed of multiple components, the Gabor transform or Gabor-Wigner distribution function may be the better choices.

[edit] Applications

The following applications we are going to be introduced need not only the time-frequency distribution functions but also some operations to the signal. The Linear canonical transform (LCT) is really helpful. By LCTs, the shape and location on the time-frequency plane of a signal can be in the arbitrary form that we want it do be. For example, the LCTs can shift the time-frequency distribution to any location, dilate it in the horizontal and vertical direction without changing its area on the plane, shear (or twist) it, and rotate it (Fractional Fourier transform). This powerful operation, LCT, make it more flexible to analyze and apply the time-frequency distributions. Here we list some applications of time-frequency analysis.

[edit] Finding instantaneous frequency

The definition of instantaneous frequency is the time rate of change of phase, or

$\frac{1}{2 \pi} \frac{d}{dt} \phi (t),$

where $φ(t)$ is the instantaneous phase of a signal. We can know the instantaneous frequency from the time-frequency plane directly if the image is clear enough. Because the high clarity is critical, we often use WDF to analyze it.

[edit] Filter design

The goal of filter design is to remove the undesired component of a signal. Conventionally, we can just filter in the time domain or in the frequency domain individually as shown as below.

The upper methods of filtering can’t work well for every signal which may overlap in the time domain or in the frequency domain. By using the time-frequency distribution function, we can filter in the fractional domain by employing the fractional Fourier transform. An example is shown below.
$Image:filter_fractional.jpg$

The time-frequency analysis in filter design always does with the signals composed of multiple components, so one cannot use WDF due to cross-term. Maybe the Gabor transform, Gabor-Wigner distribution function, and Cohen's class distribution function are better choices.

[edit] Signal decomposition

The concept of signal decomposition is similar to filter design.

[edit] Sampling theory

By Nyquist–Shannon sampling theorem, we can conclude that the minimum number of sampling point without aliasing is equivalent to the area of the time-frequency distribution of a signal (In fact, a little bit of accuracy has been sacrificed because the area of any signal is infinite). Let’s see the example before and after we combine the sampling theory with the time-frequency distribution as follow.

It is obvious that the number of sampling points decreases after we apply the time-frequency distribution.

When we use the WDF, there might be the cross-term problem. On the other hand, using Gabor transform causes clarity problem.

Consequently, when the signal we tend to sample is composed of single component, we use the WDF; however, if the signal consists of more than one component, using the Gabor transform, Gabor-Wigner distribution function, and Cohen's class distribution function are better.

[edit] Modulation and multiplexing

Conventionally, the operation of modulation and multiplexing concentrates in time or in frequency, separately. By taking advantage of the time-frequency distribution, we can make it more efficient to modulate and multiplex. All we have to do is to fill up the time-frequency plane. We present an example as below.

As illustrated in the upper example, using the WDF is not smart since the serious cross-term problem make it difficult to multiplex and modulation.

[edit] Electromagnetic wave propagation

We can represent an electromagnetic wave in the form of a 2 by 1 matrix

$\begin{bmatrix} x \\ y \end{bmatrix},$

which is similar to the time-frequency plane. When electromagnetic wave propagates through free-space, the Fresnel diffraction occurs. We can operate with the 2 by 1 matrix

$\begin{bmatrix} x \\ y \end{bmatrix}$

by LCT with parameter matrix

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}= \begin{bmatrix} 1 & \lambda z \\ 0 & 1 \end{bmatrix},$

where z is the propagation distance and $λ$ is the wavelength. When electromagnetic wave pass through a spherical lens or be reflected by a disk, the parameter matrix should be

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}= \begin{bmatrix} 1 & 0 \\ \frac{-1}{\lambda f} & 1 \end{bmatrix}$

and

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}= \begin{bmatrix} 1 & 0 \\ \frac{1}{\lambda R} & 1 \end{bmatrix}$

respectively, where ƒ is the focal length of the lens and R is the radius of the disk. These corresponding results can be obtained from

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}.$

[edit] Optics

Light is a kind of electromagnetic wave, so we apply the time-frequency analysis to optics in the same way as to electromagnetic wave propagation.

[edit] Signal identification

By Fourier analysis, we can’t recognize the two signals $x 1 (t)$ and $x 2 (t)$ below.

$x_1 (t)=\begin{cases} \cos( \pi t); & t <10 \\ \cos(3 \pi t); & 10 \le t < 20 \\ \cos(2 \pi t); & t > 20 \end{cases}$

$x_2 (t)=\begin{cases} \cos( \pi t); & t <10 \\ \cos(2 \pi t); & 10 \le t < 20 \\ \cos(3 \pi t); & t > 20 \end{cases}$

Thanks to the time-frequency analysis, we can still solve this problem.

[edit] Acoustics

The characteristic of acoustic signals is that its frequency varies really severely with time. Because the acoustic signals usually contain a lot of data, it is suitable to use the Gabor transform to analyze the acoustic signals due to its low computational complexity.

[edit] Biomedical engineering

One can use time-frequency distribution to analyze the electromyography (EMG).