Time–frequency analysis

The practical motivation for time–frequency analysis is that classical Fourier analysis assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration.

For example, traditional musical instruments do not produce infinite duration sinusoids, but instead begin with an attack, then gradually decay.

One of the most basic forms of time–frequency analysis is the short-time Fourier transform (STFT), but more sophisticated techniques have been developed, notably wavelets and least-squares spectral analysis methods for unevenly spaced data.

It is a generalization and refinement of Fourier analysis, for the case when the signal frequency characteristics are varying with time.

Whereas the technique of the Fourier transform can be extended to obtain the frequency spectrum of any slowly growing locally integrable signal, this approach requires a complete description of the signal's behavior over all time.

While mathematically elegant, such a technique is not appropriate for analyzing a signal with indeterminate future behavior.

For instance, one must presuppose some degree of indeterminate future behavior in any telecommunications systems to achieve non-zero entropy (if one already knows what the other person will say one cannot learn anything).

To harness the power of a frequency representation without the need of a complete characterization in the time domain, one first obtains a time–frequency distribution of the signal, which represents the signal in both the time and frequency domains simultaneously.

In such a representation the frequency domain will only reflect the behavior of a temporally localized version of the signal.

This enables one to talk sensibly about signals whose component frequencies vary in time.

Therefore, if we want to analyze a single-term signal, using the WDF may be the best approach; if the signal is composed of multiple components, some other methods like the Gabor transform, Gabor-Wigner distribution or Modified B-Distribution functions may be better choices.

: area of the time frequency distribution of the signal The PSD of the white noise is

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 goal of filter design is to remove the undesired component of a signal.

Filter design in time–frequency analysis always deals with signals composed of multiple components, so one cannot use WDF due to cross-term.

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.

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.

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.

We can represent an electromagnetic wave in the form of a 2 by 1 matrix which is similar to the time–frequency plane.

When electromagnetic wave propagates through free-space, the Fresnel diffraction occurs.

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).

The Wigner–Ville distribution (Ville 1948, in a signal processing context) was another foundational step.

An early practical motivation for time–frequency analysis was the development of radar – see ambiguity function.