R13 - ENEE 241 02 * READING ASSIGNMENT 13 Thu 03/26 Lecture...

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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 13 Thu 03/26 Lecture Topic: examples of the DFT and its inverse; DFT of a real-valued signal Textbook References: sections 3.2, 3.3 Key Points: The N-point vector s is a complex sinusoid of frequency = 2 k/N if and only its DFT (or spectrum) S contains all zeros except for S [ k ] 6 = 0. Linearity property of the DFT: s = x + y DFT S = X + Y The DFT S of a N-point real-valued signal s exhibits circular conjugate symmetry: S [0] = S * [0] and S [ N- k ] = S * [ k ] , k = 1 : N- 1 ( DFT 2 ) Every real-valued signal s can be expressed as s [ n ] = 1 N S [0] + 2 N X <k<N/ 2 | S [ k ] | cos 2 kn N + S [ k ] + 1 N S [ N/ 2](- 1) n The second term (corresponding to frequency = ) is present only when N is even. Theory and Examples: 1 . In the previous assignment, we considered DFT vectors (or spectra) with a single nonzero entry. The corresponding time domain-signals are Fourier sinusoids. Thus: If S = 0 0 0 0 1 0 0 0 / T then the time-domain signal s is a sinusoid of frequency = 4(2 / 8) = . Specifically, s = 1 8 v (4) = 1 8 1- 1 1- 1 1- 1 1- 1 / T since s [ n ] = 1 8 e jn = (- 1) n 8 , n = 0 : 7 If S = 0 0 0 1 0 0 0 0 / T then s = v (3) / 8. This is a sinusoid of frequency = 3(2 / 8) = 3 / 4, and is given by s [ n ] = 1 8 e j (3 / 4) n , n = 0 : 7 In other words, s = 1 8 h 1- 2 2 + j 2 2- j 2 2 + j 2 2- 1 2 2- j 2 2 j- 2 2- j 2 2 i T 1 We now introduce a second nonzero entry in S , i.e., a second sinusoidal component in s ....
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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R13 - ENEE 241 02 * READING ASSIGNMENT 13 Thu 03/26 Lecture...

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