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Signals occur in a wide range of physical phenomenon. They might be human speech, blood pressure variations with time, seismic waves, radar and sonar signals, pictures or images, stress and strain signals in a building structure, stock market prices, a city's population, or temperature across a plate. These signals are often modeled or represented by a real or complex valued mathematical function of one or more variables. For example, speech is modeled by a function representing air pressure varying with time. The function is acting as a mathematical analogy to the speech signal and, therefore, is called an analog signal. For these signals, the independent variable is time and it changes continuously so that the term continuous-time signal is also used. In our discussion, we talk of the mathematical function as the signal even though it is really a model or representation of the physical signal.
The description of signals in terms of their sinusoidal frequency content has proven to be one of the most powerful tools of continuous and discrete-time signal description, analysis, and processing. For that reason, we will start the discussion of signals with a development of Fourier transform methods. We will first review the continuous-time methods of the Fourier series (FS), the Fourier transform or integral (FT), and the Laplace transform (LT). Next the discrete-time methods will be developed in more detail with the discrete Fourier transform (DFT) applied to finite length signals followed by the discrete-time Fourier transform (DTFT) for infinitely long signals and ending with the Z-transform which allows the powerful tools of complex variable theory to be applied.
More recently, a new tool has been developed for the analysis of signals. Wavelets and wavelet transforms [9], [1], [5], [16], [15] are another more flexible expansion system that also can describe continuous and discrete-time, finite or infinite duration signals. We will very briefly introduce the ideas behind wavelet-based signal analysis.
The problem of expanding a finite length signal in a trigonometric series was posed and studied in the late 1700's by renowned mathematicians such as Bernoulli, d'Alembert, Euler, Lagrange, and Gauss. Indeed, what we now call the Fourier series and the formulas for the coefficients were used by Euler in 1780. However, it was the presentation in 1807 and the paper in 1822 by Fourier stating that an arbitrary function could be represented by a series of sines and cosines that brought the problem to everyone's attention and started serious theoretical investigations and practical applications that continue to this day [8], [3], [11], [10], [7], [12]. The theoretical work has been at the center of analysis and the practical applications have been of major significance in virtually every field of quantitative science and technology. For these reasons and others, the Fourier series is worth our serious attention in a study of signal processing.
We assume that the signal
where
and
where
and
where
the squared error is
which is minimized over all
It follows that if
A useful condition [6], [11] states that if
The form of the Fourier series using both sines and cosines makes determination of the peak value or of the location of a particular frequency term difficult. A different form that explicitly gives the peak value of the sinusoid of that frequency and the location or phase shift of that sinusoid is given by
and, using Euler's relation and the usual electrical
engineering notation of
the complex exponential form is obtained as
where
The coefficient equation is
The coefficients in these three forms are related by
and
It is easier to evaluate a signal in terms of
Although the function to be expanded is defined only over a specific finite region, the series converges to a function that is defined over the real line and is periodic. It is equal to the original function over the region of definition and is a periodic extension outside of the region. Indeed, one could artificially extend the given function at the outset and then the expansion would converge everywhere.
It can be very helpful to develop a geometric view of the Fourier series
where
The properties of the Fourier series are important in applying it to signal
analysis and to interpreting it. The main properties are given here
using the notation that the Fourier series of a real valued function
| even | 0 | even | 0 | even | 0 |
| odd | 0 | 0 | odd | even | 0 |
| 0 | even | 0 | even | even | |
| 0 | odd | odd | 0 | even |
Note the derivative of a triangle wave is a square wave. Examine the series coefficients to see this. There are many books and web sites on the Fourier series that give insight through examples and demos.
Four of the most important theorems in the theory of Fourier analysis are the inversion theorem, the convolution theorem, the differentiation theorem, and Parseval's theorem [4].
All of these are based on the orthogonality of the basis function of the
Fourier series and integral and all require knowledge of the convergence
of the sums and integrals. The practical and theoretical use of Fourier
analysis is greatly expanded if use is made of distributions or
generalized functions (e.g. Dirac delta functions,
The following theorems and results concern the existence and convergence of the Fourier series and the discrete-time Fourier transform [13]. Details, discussions and proofs can be found in the cited references.
The Fourier series expansion results in transforming a periodic, continuous
time function,
Many practical problems in signal analysis involve either infinitely long
or very long signals where the Fourier series is not appropriate.
For these cases, the Fourier transform (FT) and its inverse (IFT) have
been developed. This transform has been used with great success in
virtually all quantitative areas of science and technology where the
concept of frequency is important. While the Fourier series was used before
Fourier worked on it, the Fourier transform seems to be his original idea.
It can be derived as an extension of the Fourier series by letting the
length or period
The Fourier transform (FT) of a real-valued (or complex) function of the
real-variable
giving a complex valued function of the real variable
Because of the infinite limits on both integrals, the question of convergence is important. There are useful practical signals that do not have Fourier transforms if only classical functions are allowed because of problems with convergence. The use of delta functions (distributions) in both the time and frequency domains allows a much larger class of signals to be represented [14].
The properties of the Fourier transform are somewhat parallel to those
of the Fourier series and are important in applying it to
signal analysis and interpreting it. The main properties are given
here using the notation that the FT of a real valued function
| even | 0 | even | 0 | even | 0 |
| odd | 0 | 0 | odd | even | 0 |
| 0 | even | 0 | even | even | |
| 0 | odd | odd | 0 | even |
Deriving a few basic transforms and using the properties allows a large class of signals to be easily studied. Examples of modulation, sampling, and others will be given.
Note the Fourier transform takes a function of continuous time into a function of continuous frequency, neither function being periodic. If “distribution" or “delta functions" are allowed, the Fourier transform of a periodic function will be a infinitely long string of delta functions with weights that are the Fourier series coefficients.
The Laplace transform can be thought of as a generalization of the Fourier
transform in order to include a larger class of functions, to allow the
use of complex variable theory, to solve initial value differential
equations, and to give a tool for input-output description of linear
systems. Its use in system and signal analysis became popular in the
1950's and remains as the central tool for much of continuous time system
theory. The question of convergence becomes still more complicated and
depends on complex values of
The definition of the Laplace transform (LT) of a real valued function defined
over all positive time
and the inverse transform (ILT) is given by the complex contour integral
where
Notice that the Laplace transform becomes the Fourier transform on the
imaginary axis, for
There is a considerable literature on the Laplace transform and its use in continuous-time system theory. We will develop most of these ideas for the discrete-time system in terms of the z-transform later in this chapter and will only briefly consider only the more important properties here.
The unilateral Laplace transform cannot be used if useful parts of the signal exists for negative time. It does not reduce to the Fourier transform for signals that exist for negative time, but if the negative time part of a signal can be neglected, the unilateral transform will converge for a much larger class of function that the bilateral transform will. It also makes the solution of linear, constant coefficient differential equations with initial conditions much easier.
Many of the properties of the Laplace transform are similar to those for Fourier transform [2], [14], however, the basis functions for the Laplace transform are not orthogonal. Some of the more important ones are:
Examples can be found in [14], [2] and are similar to those of the z-transform presented later in these notes. Indeed, note the parallals and differences in the Fourier series, Fourier transform, and Z-transform.
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Last edited by Daniel Williamson on Jun 25, 2008 3:13 pm GMT-5.
