Описание:Course description:
A central goal in signal analysis is to extract information from signals that are
related to real-world phenomena. Examples are the analysis of speech, images,
and signals in medical or geophysical applications. One reason for analyzing
such signals is to achieve better understanding of the underlying physical
phenomena. Another is to find compact representations of signals which
allow compact storage or efficient transmission of signals through real-world
environments. The methods of analyzing signals are wide spread and range
from classical Fourier analysis to various types of linear time-frequency trans-
forms and model-based and non-linear approaches. This course concentrates on
transforms, but also gives a brief introduction to linear estimation theory and
related signal analysis methods. The course is self-contained for listeners with
some background in system theory and linear algebra, as typically
gained in undergraduate courses in electrical and computer engineering.
The first part of this course cover the classical concepts of signal
representation, including integral and discrete transforms. Lecture 1 contains
an introduction to signals and signal spaces. It explains the basic tools
for classifying signals and describing their properties. Lecture 2 gives an
introduction to integral signal representation. Examples are the Fourier,
Hartley and Hilbert transforms. Lecture 3 discusses the concepts and tools
for discrete signal representation. Examples of discrete transforms are given
in Lecture 4. Some of the latter are studied comprehensively, while others are
only briefly introduced, to a level required in the later chapters. Lecture 5 is
dedicated to the processing of stochastic processes using discrete transforms
and model-based approaches. It explains the Karhunen-Loeve transform and
the whitening transform, gives an introduction to linear estimation theory
and optimal filtering, and discusses methods of estimating autocorrelation
sequences and power spectra.
The final part of this course are dedicated to transforms that
provide time-frequency signal representations. In Lecture 6 multirate filter
banks are considered. They form the discrete-time variant of time-frequency
transforms. The chapter gives an introduction to the field and provides an
overview of filter design methods. The classical method of time-frequency
analysis is the short-time Fourier transform, which is discussed in Lecture 7.
This transform was introduced by Gabor in 1946 and is used in many appli-
cations, especially in the form of spectrograms. The most prominent example
of linear transforms with time-frequency localization is the wavelet transform.
This transform attracts researchers from almost any field of science, becauseit has many useful features: a time-frequency resolution that is matched to
many real-world phenomena, a multiscale representation, and a very efficient
implementation based on multirate filter banks. Lecture 8 discusses the
continuous wavelet transform, the discrete wavelet transform, and the wavelet
transform of discrete-time signals. Finally, Lecture 9 is dedicated to quadratic
time-frequency analysis tools like the Wigner distribution, the distributions
of Cohen’s class, and the Wigner Ville spectrum.