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D R . T A R E K T U T U N J I
P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N
2 0 1 4
Fourier Series and Fourier Transforms
Fourier Series
The frequency representation of periodic and aperiodic signals indicates how their power or energy is allocated to different frequencies. Such a distribution over frequency is called the spectrum of the signal. For a periodic signal the spectrum is discrete, as its power is concentrated at
frequencies multiples of a so-called fundamental frequency, directly related to the period of the signal. On the other hand, the spectrum of an aperiodic signal is a continuous function of frequency.
The Fourier representation is also useful in finding the frequency response of linear time-invariant systems, which is related to the transfer function obtained with the Laplace transform.
The frequency response of a system indicates how an LTI system responds to sinusoids of different frequencies. Such a response characterizes the system and permits easy computation of its steady-state response, and will be equally important in the synthesis of systems.
Fourier analysis is in the steady state, while Laplace analysis considers both transient and steady state
Frequency Response
RC Example
Consider the RC circuit.
Let the voltage source be vs(t)=4 cos(t+pi/4) volts, the resistor be R = 1Ohm, and the capacitor C =1 F.
Find the steady-state voltage across the capacitor.
RC Example
Fourier Series
Fourier Series
The Fourier series is a representation of a periodic signal x(t) in terms of complex exponentials or sinusoids of frequency multiples of the fundamental frequency of x(t) The advantage of using the Fourier series to represent
periodic signals is not only the spectral characterization obtained, but in finding the response for these signals when applied to LTI systems.
The Fourier series can be used to represent signals by a combination of different sines and cosines
Fourier Series: Trigonometric
Example
Example
Power Spectrum
Symmetry
Example
Example
Basic Properties of Fourier Series
Fourier Transform
Fourier Transform and Laplace
Notes
If x(t) has a finite time support and in that support x(t) is finite, its Fourier transform exists. To find it use the integral definition or the Laplace transform of x(t)
If x(t) has a Laplace transform X(s) with a region of convergence including the jW, its Fourier transform is X(s) at s = jW•.
If x(t) is periodic of infinite energy but finite power, its Fourier transform is obtained from its Fourier series using delta functions.
Consider the Laplace transform if the interest is in transients and
steady state, and the Fourier transform if steady-state behavior is of interest.
Represent periodic signals by their Fourier series before considering their Fourier transforms.
Attempt other methods before performing integration to find the Fourier transform.
Example
Example
Example
Fourier Transform of Periodic Signals
Example
Example
Fourier Transform Example
Fourier Transform Example
Basic Properties
Convolution and Filtering
Conclusion
The Fourier series can be used to represent a signal by a combination of different sin and cos signals
Fourier Transforms are used to provide information about the frequency behavior of signals.
The frequency analysis provide the fundamentals for filter design.
There is a well-established relationship between the Fourier and Laplace Transforms.
Reference
Chapter 4 and 5, Signals and Systems using MATLAB by Luis Chaparro. Elsevier Publisher 2011