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Unformatted text preview: EE 341 Discrete-Time Linear Systems Week 5 Spring 2011 Page 1 Text reading assignment for this week: Chapter 3, Sec 7 Read problems 5.53 and 5.54 on pages 417 420. Start reading Chapter 5, Sec 0 5 Summary: The Common Types of Fourier Transforms Continuous in Time ( ) x t = Aperiodic in Frequency Discrete in Time [ ] x n = Periodic in Frequency Periodic in Time, = Discrete in Frequency Fourier Series (FS): 1 ( ) jk t k T a x t e d t T ( ) jk t k k x t a e Discrete Time Fourier Series (DTFS) and Discrete Fourier Transform* (DFT): 1 [ ] [ ] ,0 1 N kn N n X k x n W k N 1 1 [ ] [ ] ,0 1 N kn N k x n X k W n N N where 2 N j N W e . Aperiodic in Time, = Continuous in Frequency Fourier Transform (FT): ( ) ( ) ( ) ( ) j t j t X x t e dt x t X e d t Discrete-Time Fourier Transform (DTFT): ( ) [ ] j n n X x n e 2 1 [ ] ( ) 2 j n x n X e d * The fast Fourier transform (FFT) is simply a fast algorithmic implementation of the discrete Fourier transform (DFT). Well cover it and the DTFT soon. EE 341 Discrete-Time Linear Systems Week 5 Spring 2011 Page 2 The Fast Fourier Transform The FFT command that you use in MATLAB implements a Fast Fourier Transform which is simply an efficient way to calculate the discrete Fourier transform (DFT), that we covered last week: 1 [ ] [ ] , 0,1,2,..., 1 N kn n X k x n W k N 2 N complex multiplications and ( 1 ) N N additions are needed to implement this DFT. The approximate number of operations, be it multiply or addition, is commonly referred to as 2 ( ) N . If 10 2 1024 N , then 2 2 6 ( ) ( 2 ) ( 1 ) N , a very large number!...
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