102_1_lecture9

# 102_1_lecture9 - UCLA Fall 2010 Systems and Signals Lecture...

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UCLA Fall 2010 Systems and Signals Lecture 9: Discrete Time Fourier Series October 25, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1

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Introduction Today’s topics: Review of Continuous Time Fourier Series Discrete Time Fourier Series Filtering EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2
Review Many continuous time, periodic signals can be represented as sums of complex exponentials. Formally, if the signal is periodic in T ( x ( t ) = x ( t + T ) ), and has a fundamental frequency ω 0 = 2 π T , we can write: x ( t ) = X k = -∞ a k e jkω 0 t This is the synthesis equation. Complex exponentials are summed to produce the time-domain signal. Complex exponentials are scaled by { a k } - Fourier series coeﬃcients . Notice that some signals require an inﬁnite number of Fourier coeﬃcients! EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3

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Notice also that the frequency of complex exponentials included in the equation is an integer multiple of fundamental frequency. That is, ω = 0 . If we included other frequencies, the signal would not be periodic in T! Fourier series coeﬃcients (how much each complex exponentials contributes to the signal) are determined using the analysis equation: a k = 1 T Z T x ( t ) e - jkω 0 t dt The integral is over one period of x ( t ) . Can integrate from 0 to T , - ± to T - ± , - T/ 2 to 2 , etc – all are equivalent. In practice choose simpler limits. Memorize these equations. Be very clear on the sign and variables in the exponent. Remember 1 T factor in the analysis equation! EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4
Representing signals in terms of their frequency content allows us to talk about two domains (time and frequency). For example, last class we’ve seen that a square wave signal: x(t) -T −20 −15 −10 −5 0 5 10 15 20 −0.05 0 0.05 0.1 0.15 0.2 T 1 /T = 1/8 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5

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Examples. 1. Given signal has a 0 6 = 0 . Can this signal be odd? 2. Find the Fourier coeﬃcients a k of x ( t ) = 1 + cos(2 πt ) , periodic in T = 1 . 3. Find the time-domain representation of x ( t ) , if its single nonzero Fourier coeﬃcient is a 2 = 1 . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6
We usually use shorthand notation for computing Fourier series: x ( t ) FS --→ a k Last class, introduced many properties of Fourier series: here, x ( t ) a k , y ( t ) b k , z ( t ) c k . Linearity z ( t ) = Ax ( t ) + By ( t ) c k = Aa k + Bb k Time Shifting y ( t ) = x ( t - t 0 ) b k = a k e - jkω 0 t 0 Time Reversal y ( t ) = x ( - t ) EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7

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b k = a - k Conjugation y ( t ) = ± x ( t ) ² * b k = a * - k Parseval’s relation 1 T Z T | x ( t ) | 2 dt = X k = -∞ | a k | 2 Ability to apply these properties will save you signiﬁcant time. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 8
Example. Suppose a signal is periodic and even: x ( t ) = x ( t + T ) and x ( t ) = x ( - t ) . Suppose further that x ( t ) is real.

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102_1_lecture9 - UCLA Fall 2010 Systems and Signals Lecture...

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