Gabor systems are applied in numerous engineering applications, many of them without obvious connection to the traditional field of time-frequency analysis for deterministic signals.

Any countable set of test functions in a Hilbert space conveys a linear mapping between function spaces and sequence spaces. In one direction scalar products are taken (analysis mapping), and in the other direction the members from the sequence space are used as coefficient sequences in a series of the form (synthesis mapping). In our concrete context the analysis mapping is given by the Gabor transform (0.5) (i.e., the sampled short time Fourier transform) and the synthesis mapping is given by the Gabor expansion (0.6). In principle there exists two basic setups for the use of Gabor systems, which pervade most applications:

- The overall system acts on the sequence space (where is some LCA group) by (i) Gabor synthesis, (ii) (desired or undesired and mostly linear but probably nonlinear) modification, (iii) Gabor analysis. This setup underlies e.g. the so-called multicarrier modulation schemes in digital communication, see Fig.1, but also applies to system identification and radar tracking procedures.
- The overall system acts on the function space by (i) Gabor analysis, (ii) (desired or undesired and mostly nonlinear but also linear) modification, (iii) Gabor synthesis. Typical tasks where one encounters this setup (illustrated in Fig.1 include signal enhancement, denoising or image compression.

**Speech signal analysis:**

Speech signals are one of the classical applications of linear
time-frequency representations. In fact, the analysis
of speech signals was the driving force that led to the invention
of (a nondigital filter bank realization of) the
*spectrogram*. The advent of the FFT made the STFT up to now
to the standard tool of speech analysts.

**Time-Varying Spectral Estimation:**

Time-invariant spectral estimation, a standard procedure for time series analysis, is
of fundamental relevance in most areas of science and
technology. The basic principle is to model the time series
as a realization of a stationary random process whose power spectrum
can be characterized by a typically lower dimensional data set
(uniformly spaced frequency samples in a nonparametric setup
or, e.g., poles and zeros of an ARMA model).
There exists a large variety of robust and efficient
procedures for power spectrum estimation. The natural
generalization of the stationary framework to
nonstationary environments leads to a longstanding theoretical problem:
What is the proper definition of
a time-varying power spectrum of a nonstationary process?
Classical definitions are Priestley's evolutionary spectrum
and the Wigner-Ville spectrum
which are closely related to the operator symbols of Weyl and
Kohn-Nirenberg.
Whether or not one trusts either definition
it is clear that a time-varying power spectrum is
(i) time-frequency parametrized and (ii) a quadratic form
of the data, hence the magnitude-squared Gabor coefficients
establish the basis of any nonparametric
estimation procedure.

**Representation and Identification of Linear Systems:**

The theory of linear time-invariant (LTI) systems and in particular
the symbolic calculus of transfer functions is a standard tool
in all areas of mechanical and electrical engineering.
Strict translation invariance is however almost always
a pragmatical modeling assumption which establishes a more or less
accurate approximation to the true physical system. Hence
it is a problem of longstanding interest to generalize the transfer
function concept from LTI to linear time-varying (LTV) systems. Such a
time-varying transfer function was suggested by Zadeh in 1950.
It is formally equivalent to the Weyl-Heisenberg operator symbol of
Kohn and Nirenberg.
Pseudodifferential operators are the classical way to establish a symbol
classification that keeps some of the conceptual power which the Fourier
transform has for LTI systems. Recently Gabor frames have turned out to
be a useful tool to the analysis of pseudodifferential
operators.

**Digital Communication:**

Digital communication systems transmit sequences of
binary data over a continuous time physical channel.
An ideal physical channel is bandlimited without in-band distortions.
Under this idealized assumption, digital communication systems
can be implemented by selecting a Gabor-structured
orthonormal system, transmitting a linear combination of the
elementary signals, weighted by the binary coefficients (Gabor synthesis)
and the receiver recovers the coefficients by computing
inner products with the known basis functions (matched filter
receiver = Gabor analysis), see Fig..
However, in wireless communication systems
which is one of the challenging research areas, the physical
channel is subject to severe linear distortions and hundreds of
users communicate over the same frequency band at the same time.
Traditional OFDM (orthogonal frequency division multiplex) systems
can be interpreted as orthonormal Gabor systems with critical sampling
and come therefore with the well-known bad time-frequency
localization properties of the building blocks. Since completeness is not
a concern here, recent works suggest the use of a coarser grid ,
together with good TF-localized atoms to obtain
more robustness.

**Image representation and biological vision:**

Gabor functions were successfully applied to model the response of simple
cells in the *visual cortex*.
In our notation, each pair of adjacent cells in the visual cortex
represents the real and imaginary part of one coefficient
corresponding to . Clearly the Gabor model cannot
capture the variety and complexity of the visual system, but it seems
to be a key in further understanding of biological vision.

Among the people who paved the way for the use of Gabor analysis in pattern recognition and computer vision one certainly has to mention Zeevi, M. Porat, and their coworkers and Daugman. Motivated by biological findings, Daugman and Zeevi and Porat proposed the use of Gabor functions for image processing applications, such as image analysis and image compression.

Since techniques from signal processing are of increasing importance in medical diagnostics, we mention a few applications of Gabor analysis in medical signal processing. The Gabor transform has been used for the analysis of brain function, such as for detection of epileptic seizures in EEG signals, study of sleep spindles. The role of the Heisenberg group in magnetic resonance imaging has been recently analyzed by Schempp.