SP_IV_P01_4p - Introduction to Signal Processing 1...

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Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha IV. Continuous-Time Signal Analysis: The Fourier Transform As we have seen, the Fourier series provided a useful frequency-domain analytical tool for the representation of periodic signals (that have finite power). We now focus on the frequency-domain representation of aperioidc signals, which are much more common in many practical systems. Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha A. Aperiodic Signal Representation by the Fourier Integral The various forms of the Fourier series we studied earlier provided us with the means for representing periodic signals with period 0 T as the sum of sinusoidal functions. Here, we will build on the Fourier series by studying an equivalent method for the representation of a general aperiodic signal () x t . Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha We can utilize the Fourier series by modifying an aperiodic signal into a periodic one. Hence, let us construct a new signal 0 T x t by repeating x t with a time period 0 T . Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha If we let 0 T →∞ and, hence, make the interval duration infinite, we get back the original signal x t . 0 0 lim ( ) ( ) T T x tx t →∞ =
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Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha Thus, the Fourier series of 0 () T x t will represent x t in the limit 0 T →∞ . Recall that the exponential Fourier series of 0 T x t will be, 0 0 j nt Tn n x tD e ω =−∞ = Where, 0 0 0 0 2 0 2 1 T j nT T Dx t e d t T = a n d : 0 0 2 T π = . Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha Note that within the limits of the integration 00 , 22 TT ⎡⎤ −+ ⎢⎥ ⎣⎦ , we have 0 T x tx t = . Hence: 0 0 0 2 0 2 1 T j n T t e d t T = Note also that integrating x t from [ ] , −∞ ∞ is the same as integrating it between the original limits , . Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha Hence, 0 0 1 j n t e d t T −∞ = Similar to the Fourier series, this expression (i.e. n D ) is a function of the frequency domain through the ( discrete ) frequency harmonics : 0 n = . In other words, in the frequency domain, we are only able to define the signal over discrete samples of frequencies ( 0 n ). Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha Moving forward, we are interested in the following: ¾ We are interested in observing the behavior (or generating a representation) of continuous-time signals over a continuum of frequencies . Hence, as a step in this direction, we will consider a new frequency-domain function that is defined over the (general) frequency variable : ( ) j t X xte d t −∞ =
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Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha ¾ We are also interested in observing the effect of changing the value of 0 T on the overall behavior of the frequency-domain representation of the signal.
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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_IV_P01_4p - Introduction to Signal Processing 1...

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