lesson_34 - EEL 3135: Signals and Systems Dr. Fred J....

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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Fourier Transforms Lesson Number: 34 [Section 13-5 to 13-8] Background: The Fourier transform is intrinsic to the study of signals and systems. Recall that the continuous-time Fourier transform (CTFT) pair is given by the analysis and synthesis equations, - - = dt e t x X t j ϖ ) ( ) (j , - = π d e X t x t j ) (j 2 1 ) ( 1 A modification of the Fourier transform, called the continuous-time Fourier series (CTFS), simplified the computational problem by restricting the study to only periodic continuous-time signals. The CTFS series analysis and synthesis equations are: - = T t jn p n dt e t x T c 0 0 0 ) ( 1 , -∞ = = n t jn n p e c t x 0 ) ( 2 where 0 is called the fundamental frequency (r/s) and is given by 0 =2 π / T 0 , T 0 is the signals fundamental period, c n is the n th harmonic and corresponds to the signal being located at frequency f n = n / T 0 , or 0 =2 π n / T 0 . Discrete-Time Fourier Transform (DTFT) The discrete-time equivalent of a Fourier transform is the discrete-time Fourier transform (DTFT). The DTFT defines a method of computing a Fourier representation of a discrete-time signal x [ k ] . The DTFT analysis and synthesis equations are: = - = k jk j e k x e X ω ] [ ) ( , - = d e X k x k j ) ( 2 1 ] [ 3 where [– π , π ) and is called the normalized digital frequency defined with respect to the sampling frequency f s . If the actual analog frequency is , then the normalized frequency is: s s T f = = . 4 It should be noted that Equation 3 is periodic in the frequency domain with period f s , or in terms of the normalized frequency , that is X ( e j ω )= X ( e j( ω +2k π ) ). For example, the DTFT of a discrete- time rectangular pulse r [ k ], symmetrically distributed about k =0, is shown in Figure 1. 1
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor Figure 1: DTFT of a symmetric rectangular pulse. Discrete-Time Fourier Series (DTFS) The DTFT of a time-series of period N , can be expressed in terms of a modification of Equation 3 which exploits known properties of a periodic signal. A discrete-time periodic signal can be expanded in terms of the complex exponential " W " operator, a shorthand notational convenience. Specifically, let: N m j m N e W / 2 π - = . 5 where: ± ± - = - = - otherwise. 0 } , 2 , , 0 { if 1 0 ) ( N N n m N W N k k n m N 6 For illustration purposes, W 8 i is interpreted in Figure 2. Figure 2: W 8 i distribution. The discrete-time Fourier series (DTFS) synthesis and analysis equations are given by: W 8 0 =1.0 W 8 7 = W 8 -1 =e -j π /2 W 8 1 =e j π /2 W 8 2 =e j π W 8 3 =e j3 π /2 W 8 4 =-1.0 W 8 5 = W 8 -3 =e -j3 π /2 W 8 6 = W 8 -2 =e -j π 45 0 2 ω
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EEL 3135: Signals and Systems Dr. Fred J. Taylor, Professor - = - = 1 0 ] [ N n nk N n W c k x , - = = 1 0 ] [ 1 N k kn N n W k x N c 7 for n {0,1,2,…, N –1} . A periodic, and therefore non-causal signal, having a period of N samples would repeat every T 0 seconds, where s s f N NT T = = 0 . 8 The frequency f 0 =1/ T 0 , or ϖ 0 =2 π f 0 , is called the fundamental or first harmonic frequency. The frequency k 0 =2 kf 0 is called the k th harmonic. The highest harmonic would be the ( N –1) th
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lesson_34 - EEL 3135: Signals and Systems Dr. Fred J....

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