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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 17 Tue 04/14 Lecture Topics: zero-padded extension of a vector; detection of sinusoids using the DFT Textbook References: sections 3.7, 3.8 Key Points: • If y is formed by appending ( M − 1) L zeros to the L-point vector s , then the DFT Y will contain the entries of S at positions 0 ,M,... , ( L − 1) M . • The frequency ω of a discrete-time complex sinusoidal vector s [ n ] = e jω n , n = 0 : L − 1 can be determined within 2 π/N radians/sample, with N > L , by zero-padding s to length N and locating the maximum in the resulting amplitude spectrum. • The same technique can be used to estimate the frequency of a real-valued sinusoidal vector reliably. It can be also applied to a noisy sum of sinusoids provided a sufficiently large number of samples is taken. Theory and Examples: 1 . Another extension of interest is zero-padding, i.e, appending N − L zeros to s to obtain a vector y of length N , as illustrated below. y 0 10 35 s 0 9 There is no easy way of obtaining the DFT vector Y from S . However, in the special case N = ML , it is possible to proceed in the reverse direction—i.e., obtain S from Y . 2 . Again, the ( kM ) th Fourier sinusoid for length ML is the same as the k th one for length L , only periodically repeated. This periodic repetition has no effect on the inner product with y , since the nonzero portion of y is limited to indices 0 : L − 1, and is identical to s . This means that, with N = ML , the two inner products Y [ kM ] = N- 1 summationdisplay n =0 y [ n ] e- jkM 2 π N n = N- 1 summationdisplay n =0 y [ n ] e- jk 2 π L n and S [ k ] = L- 1 summationdisplay n =0 s [ n ] e- jk 2 π L n are sums of the same nonzero terms, and are therefore equal....
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- Spring '08
- DFT, Amplitude Spectrum, Fourier frequency