Fundamentals of signals and systems using the web and Matlab1

Fundamentals of signals and systems using the web and Matlab1

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Signals and Systems Textbooks: Fundamentals of signals and systems using the web and Matlab Authors: Edward W. Kamen and Bonnie S. Heck Prentice-Hall International, Inc. Web site: Domestic dealer: Chwa books Corp. Lecture note compiler: Ping-Sung Liao Chapter 1 Fundamental Concepts 1.1 Signals and Systems Signals : x ( t ) is a real-valued, or scalar-valued, function of the time variable t . For instance, f ( t ) = sin ( wt ) The representation of a signal may be described by either a continuous-time signal or a set of sample values. For easily understanding the composition of a signal, the signal may be in terms of the frequency spectrum through Fourier transform. Signal Processing Signal processing plays an important role in either the extraction of the information carried in a signal or the reconstruction of a signal which has been corrupted by spurious signals of noise. Reconstruct the x ( t ) from m ( t ) (estimation/filtering) by canceling the noise n ( t ) m ( t ) = x ( t ) + n ( t ) Systems A system is an interconnection of components with terminals or access ports through which energy and information can be applied or extracted. A mathematical model of a system is usually an idealized representation of the system. There are two types basic of mathematical models: one is input/output representation; the other is state model. Four types of input/output representations are studied here, 1. The input/output differential equation or difference equation, 1 92 08/19/2007
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2. The convolution model, 3. The Fourier transform representation, 4. The transfer function representation. 1.2 Continuous-Time Signals A signal x ( t ) is said to be a continuous-time signal or analog signal when time variable t takes its values from the set of the real number. Step function The unit-step function of u ( t ) is described mathematically by 1, 0 () 0, 0 t ut t = < The magnitude of unit-step function u ( t ) is equal to 1 for all 0 t . Ramp function The unit-ramp function of r ( t ) is described mathematically by Note that for , the slope if r ( t ) is 1. Unit-impulse function δ (t) The unit impulse function, also called the delta function or the Dirac function, is defined in generalized form by , 0 0 tt rt t = < 0 t () 0 0 ( ) 1, for any real number >0 t ε δ =≠ = Periodic signals A continuous-time signal x ( t ) is periodic with period T if x ( t + T ) = x ( t ) , for all t. Note that the fundamental period is the smallest positive number T which satisfies the foregoing definition of periodic signals. Time-Shift signals Given a continuous-time signal, x ( t ), the shifted version of x ( t ) usually is denoted as x ( t-t 1 ) or x ( t+t 1 ) where t 1 > 0. The signal of x ( t-t 1 ) is shifted to the right by t 1 seconds and the signal of x ( t+t 1 ) is shifted to the left by t 1 seconds.
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This note was uploaded on 10/09/2010 for the course ELECTRONIC 56489 taught by Professor Johnes during the Spring '10 term at Tel Aviv Uni..

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Fundamentals of signals and systems using the web and Matlab1

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