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 Entry 9, Entry 1, Entry 5, Entry 16, Entry 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 Entry 8, Entry 3, Entry 11, Entry 10, Entry 7, Entry 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 Entry 6, Entry 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 Entry 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 Entry 13. Details, discussions and proofs can be found in the cited references.