04 The Continuous-Time Fourier Transfor

04 The Continuous-Time Fourier Transfor - ELEC211: Signals...

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1 AKW Fall 2009 Chapter 4 ELEC211: Signals and Systems Chapter 4: Continuous-time Fourier Transform Frequency representation of aperiodic signals: continuous-time Fourier transform Fourier transform for periodic signals Properties of CT Fourier transform Convolution and multiplication properties Amplitude Modulation and Multiplexing Examples
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2 AKW Fall 2009 Chapter 4 From Fourier Series to Fourier transform - CT Example: Consider again the periodic square wave x ( t ) with period T : < < < = 2 / | | 0 | | 1 ) ( 1 1 T t T T t t x - T 1 T 1 T - T t T k T k a o o k ω ) sin( 2 1 = T o π 2 = Recall from eq. (3.44) that the Fourier Series (FS) Coefficients of x ( t ) are eq.(4.1) & eq.(3.44) ) ( sinc 2 sin ) ( sinc 1 0 1 T k T T a x x x k = = x ( t ) Expressing in the form of a normalized sinc function (if so preferred):
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3 AKW Fall 2009 Chapter 4 More on the Sinc Function (Normalized Form) ; 09 . 1 ) ( sinc ; 1 ) ( sinc 1 - - = dx x dx x ± ± = = = = ,... 2 , 1 0 0 1 sin ) ( sinc x x x x x π We will come back to the sinc function later. Reason for the Gibb’s phenomena
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4 AKW Fall 2009 Chapter 4 What happens if we increase T while keeping T 1 fixed? From eq.(4.1), moving T to the LHS: Which means that Ta k are simply sample values of an envelope function: eq.(4.2) The fundamental frequency is decreased when T is increased. o o k k T k Ta ω ) sin( 2 1 = T o π 2 = 0 1 ) sin( 2 ωω k T Ta k = = sampled at ω= k 0
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5 AKW Fall 2009 Chapter 4 T o π ω 2 = 1 T ω= 1 2 T 1 3 T 1 2 T 1 2 T 1 2 T Plot of Ta k ) sin( 2 1 T Fixed envelop T =4 T 1 T =8 T 1 T =16 T 1 From F.S. to F.T. - Continued
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6 AKW Fall 2009 Chapter 4 For fixed T 1 , the envelope is independent of T . Ta k are the sample values on this envelop: As T increases, the samples are closer and closer, since becomes smaller As , the square wave approaches a single rectangular pulse (aperiodic), and k approaches the continuous envelope function. In other words, in the limit that x ( t ) becomes aperiodic, the F.S. representation of x ( t ) becomes a continuous function we have the Fourier Transform instead of Fourier Series ! ω ) sin( 2 1 T T o π 2 = T In Summary o k k T Ta ωω = = ) sin( 2 1
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7 AKW Fall 2009 Chapter 4 Definition of the Fourier Transform Now, consider an aperiodic x ( t ) that is of finite duration . That is, there is some T 1 such that x ( t ) = 0 for | t |> T 1 . We can construct , a periodic version of x ( t ), such that for ) ( ~ t x - T 1 T 1 T - T t ) ( ~ t x - T 1 T 1 t ) ( t x ) ( ) ( ~ t x t x = ) ( ) ( ) ( ) ( ~ kT t x kT t t x t x k k = = −∞ = −∞ = δ 2 / 2 / T t T This is the Poisson Sum. Recall that a Poisson Sum is always periodic.
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8 AKW Fall 2009 Chapter 4 Recall that the F.S. (Fourier Series) synthesis and analysis equations for the periodic function are : The analysis equation can be rewritten as: We define the following integral as the Fourier Transform of x ( t ) We see that the F.S. Coeffs of are dt e t x T a T T t jk k = 2 / 2 / 0 ) ( ~ 1 ω t jk k k e a t x 0 ) ( ~ −∞ = = ) ( ~ t x dt e t x T dt e t x T a t jk T T t jk k = = 0 0 ) ( 1 ) ( 1 2 / 2 / dt e t x j X t j = ) ( ) ( ) ( 1 0 jk X T a k =
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This note was uploaded on 09/24/2010 for the course ELEC 211 taught by Professor Albertk.wong during the Fall '09 term at HKUST.

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04 The Continuous-Time Fourier Transfor - ELEC211: Signals...

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