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

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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 ω = 2 kπ/N if and only its DFT (or spectrum) S contains all zeros except for S [ k ] 6 = 0. Linearity property of the DFT: s = α x + β y DFT ←→ S = α X + β 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 0 <k<N/ 2 | S [ k ] | cos 2 πkn N + S [ k ] + 1 N S [ N/ 2]( - 1) n The second term (corresponding to frequency ω = π ) 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 ω = 4(2 π/ 8) = π . Specifically, s = 1 8 v (4) = 1 8 £ 1 - 1 1 - 1 1 - 1 1 - 1 / T since s [ n ] = 1 8 e jπn = ( - 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 ω = 3(2 π/ 8) = 3 π/ 4, and is given by s [ n ] = 1 8 e j (3 π/ 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
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We now introduce a second nonzero entry in S , i.e., a second sinusoidal component in s . Letting S = £ 0 0 0 1 0 1 0 0 / T we have s = v (3) + v (5) 8 which is the scaled sum of two columns of V . The corresponding frequencies are ω = 3 π/ 4 (as before); and ω = 5 π/ 4, which is the same frequency as ω = - 3 π/ 4 for complex sinusoids. Since e jωn + e
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