Editing Time–frequency analysis (section)

==Applications==

The following applications 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 to 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. The time-frequency analysis have been applied in various applications like, disease detection from biomedical signals and images, vital sign extraction from physiological signals, brain-computer interface from brain signals, machinery fault diagnosis from vibration signals, interference mitigation in spread spectrum communication systems.<ref>{{cite book |last1=Pachori |first1=Ram Bilas |title=Time-Frequency Analysis Techniques and Their Applications |publisher=CRC Press|url=https://www.routledge.com/Time-Frequency-Analysis-Techniques-and-their-Applications/Pachori/p/book/9781032435763?srsltid=AfmBOorjwRC4cJ-ABXieBsYLfFSmwQdGQ3GHNvL_O5pGnBMchjM8x7S8}}</ref><ref>{{cite book |last1=Boashash |first1=Boualem |title=Time-Frequency Signal Analysis and Processing: A Comprehensive Reference |publisher=Elsevier|url=https://www.sciencedirect.com/book/9780123984999/time-frequency-signal-analysis-and-processing}}</ref>

===Instantaneous frequency estimation===

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.

===TF filtering and signal decomposition===

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 below.

[[Image:filter tf.jpg]]

The filtering methods mentioned above 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 Euclidean time–frequency domain or in the fractional domain by employing the [[fractional Fourier transform]]. An example is shown below.

[[Image:filter fractional.jpg]]

Filter design in time–frequency analysis always deals with signals composed of multiple components, so one cannot use WDF due to cross-term. The Gabor transform, Gabor–Wigner distribution function, or Cohen's class distribution function may be better choices.

The concept of signal decomposition relates to the need to separate one component from the others in a signal; this can be achieved through a filtering operation which require a filter design stage. Such filtering is traditionally done in the time domain or in the frequency domain; however, this may not be possible in the case of non-stationary signals that are multicomponent as such components could overlap in both the time domain and also in the frequency domain; as a consequence, the only possible way to achieve component separation and therefore a signal decomposition is to implement a time–frequency filter.

===Sampling theory===

By the [[Nyquist–Shannon sampling theorem]], we can conclude that the minimum number of sampling points without [[aliasing]] is equivalent to the area of the time–frequency distribution of a signal. (This is actually just an approximation, because the TF area of any signal is infinite.)  Below is an example before and after we combine the sampling theory with the time–frequency distribution:

[[Image:sampling.jpg]]

It is noticeable 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 (also called interference). On the other hand, using [[Gabor transform]] causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems.

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, or other reduced interference TFDs may achieve better results.

The [[Balian–Low theorem]] formalizes this, and provides a bound on the minimum number of time–frequency samples needed.

===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 modulate.

===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 [[Linear canonical transformation#Electromagnetic wave propagation|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, acoustics, and biomedicine===
[[Light]] is an electromagnetic wave, so time–frequency analysis applies to optics in the same way as for general electromagnetic wave propagation. 

Similarly, it is a characteristic of acoustic signals, that their frequency components undergo abrupt variations in time and would hence be not well represented by a single frequency component analysis covering their entire durations.

As acoustic signals are used as speech in communication between the human-sender and -receiver, their undelayedly transmission in technical communication systems is crucial, which makes the use of simpler TFDs, such as the Gabor transform, suitable to analyze these signals in real-time by reducing computational complexity. 

If frequency analysis speed is not a limitation, a detailed feature comparison with well defined criteria should be made before selecting a particular TFD. Another approach is to define a signal dependent TFD that is adapted to the data.
In biomedicine, one can use time–frequency distribution to analyze the [[electromyography]] (EMG), [[electroencephalography]] (EEG), [[electrocardiogram]] (ECG) or [[otoacoustic emissions]] (OAEs).