SP_V_P01_4p - Introduction to Signal Processing 1...

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Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha III. Time-Domain Analysis of Discrete- Time Systems A. Introduction A discrete-time signal consists of a sequence of numbers. Arise as a result of sampling continuous-time data systems. Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha Notation Discrete-time signals are often denoted as [] x n . The argument n in the square braces denotes the index of the signal value. Definition Systems whose input and output are discrete-time signals are called discrete-time systems . When a discrete-time signal is the result of sampling of a continuous-time signal it can also be denoted by () x nT . Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Here T is the uniform interval between successive samples. n is the index of the sample number. The figure below is an example of a discrete-time signal. Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha -5 0 5 10 15 0 0.5 1 1.5 2 2.5 t or n x(nT) or x[n] or tn x n x nT or
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Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha Example Consider a continuous time signal () t x te = . If x t is sampled at intervals of time 0.1sec T = , then the sampled, discrete-time signal can be represented as, 0.1 nT n x nT e e −− == . Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha In practice, digital filters process continuous-time signals ( x t ) by operating on their sampled, discrete-time versions ( [] x n ). After filtering, the discrete-time output ( yn ) is again converted to a continuous-time signal ( yt ). Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha Size of a Discrete-Time Signal The “size” of a discrete-time signal is measured by its energy x E . The equivalent of the continuous-time definition of energy of a signal for a discrete-time signal is x n , 2 x n E xn =−∞ = . This definition holds for both real and complex discrete- time signals.
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Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha A necessary condition for the energy of a discrete-time signal to exist is that the signal amplitude must goes to zero as the sample index n goes to infinity: [ ] 0 xn as n →∞ . Signals with finite values of energy: x E <∞ are classified energy signals . Introduction to Signal Processing 1 0 Copyright © 2005-2009 – Hayder Radha For signals with indefinite x E , we evaluate the time average of their energy, or power x P defined as. 2 1 lim [ ] 21 N x N N Px n N →∞ = + Similar to the continuous-time case, the power of a signal is used for characterizing periodic signals (who intrinsically have indefinite energy). Signals may be neither energy nor power signals.
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

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SP_V_P01_4p - Introduction to Signal Processing 1...

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