Time-frequency analysis
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Time-frequency analysis is a body of techniques for characterizing and manipulating signals whose [[Fourier analysis|component frequencies]] vary in time, such as transient signals.
Whereas the technique of the [[Fourier transform]] can be used to obtain the frequency spectrum of a signal whose component frequencies do not change with time, such a technique is not appropriate for analyzing a signal that has time-varying frequency components, such as the example below.<br /><br />
: <math>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}</math>
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In order to analyze a signal whose component frequencies vary in time, one first obtains a time-frequency distribution of the signal, which represents the signal in both the time and frequency domains simultaneously. The techniques of time-frequency analysis may then be applied to the signal in order to extract information from the signal, to separate the signal from noise or interfering signals, etc.
==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]].
===Ideal TF distribution function===
An ideal time-frequency distribution function roughly requires the following 4 properties:<br />
#'''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.
{| class="wikitable"
|-
|
| '''Clarity'''
| '''Cross-term'''
| '''Good mathematical properties'''
| '''Computational complexity'''
|-
| '''Gabor transform'''
| Worst
| No
| Worst
| 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.
==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.
===Finding instantaneous frequency===
The definition of [[instantaneous frequency]] is the time rate of change of phase, or
: <math>\frac{1}{2 \pi} \frac{d}{dt} \phi (t), </math>
where <math>\phi (t)</math> 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.
===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.<br />
[[Image:filter_tf.jpg]]<br />
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.<br />
[[Image:filter_fractional.jpg]]<br />
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.
===Signal decomposition===
The concept of signal decomposition is similar to filter design.
===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.<br />
[[Image:sampling.jpg]]
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.
===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.<br />
[[Image:mul_mod.jpg]]
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.
===Electromagnetic wave propagation===
We can represent an electromagnetic wave in the form of a 2 by 1 matrix
: <math>\begin{bmatrix}
x \\
y
\end{bmatrix},</math>
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
: <math>\begin{bmatrix}
x \\
y
\end{bmatrix}</math>
by LCT with parameter matrix
: <math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}=
\begin{bmatrix}
1 & \lambda z \\
0 & 1
\end{bmatrix},
</math>
where ''z'' is the propagation distance and <math>\lambda </math> is the wavelength. When electromagnetic wave pass through a spherical lens or be reflected by a disk, the parameter matrix should be
: <math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}=
\begin{bmatrix}
1 & 0 \\
\frac{-1}{\lambda f} & 1
\end{bmatrix}
</math>
and
: <math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}=
\begin{bmatrix}
1 & 0 \\
\frac{1}{\lambda R} & 1
\end{bmatrix}
</math>
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
: <math>\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}.
</math>
===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.
===Signal identification===
By Fourier analysis, we can’t recognize the two signals <math>x_1 (t)</math> and <math>x_2 (t)</math> below.
: <math>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}</math>
: <math>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}</math>
Thanks to the time-frequency analysis, we can still solve this problem.
===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.
===Biomedical engineering===
One can use time-frequency distribution to analyze the [[electromyography]] (EMG).
[[Category:Signal processing]]
[[Category:Digital signal processing]]
[[Category:Computational science]]
[[Category:Mathematical analysis]]
[[Category:Applied mathematics]]
[[Category:Time series analysis]]
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