Time-frequency analysis 857897 224894663 2008-07-10T21:53:13Z Cwkmail 4154028 rewrote sentence for clarity 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> <!-- Image with unknown copyright status removed: [[Image:ft_vs_gt.jpg]] --> 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 &fnof; 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]] [[zh:時頻分析]]