Aliasing in Time We conclude that an original discrete time sequence with

Aliasing in time we conclude that an original

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Aliasing in Time We conclude that an original discrete time sequence with finite duration L can be exactly recovered from its spectrum samples at frequencies ω k = 2 π k N if N L . C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 20 / 193
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Aliasing in Time Example (7.1) Consider the following signal x ( n ) = a n u ( n ) , 0 < a < 1 (16) Sample the spectrum at ω k = 2 π k N . Look at the effect of time domain aliasing as N varies C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 21 / 193
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Aliasing in Time Example (7.1) Recall that the DTFT of x ( n ) is written as X ( ω ) = 1 1 - ae j ω (17) Sampling the frequency domain at ω k = 2 π k N yields X ( k ) = X ( ω ) | ω = 2 π k N = 1 1 - ae - j 2 π k N (18) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 22 / 193
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Aliasing in Time Example (7.1) The periodic inverse of X ( k ) , ˜ x ( n ) , is written as ˜ x ( n ) = X s = -∞ x ( n - sN ) = X s = -∞ a n - sN u ( n - sN ) (19) It is clear that u ( n - sN ) = 0 for s > n / N and u ( n - sN ) = 1 for s n / N . C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 23 / 193
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Aliasing in Time Example (7.1) We can rewrite (19) as ˜ x ( n ) = 0 X s = -∞ a n - sN = a n 0 X s = -∞ a - sN = a n X s = 0 a sN = a n 1 - a N , 0 n N - 1 (20) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 24 / 193
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Aliasing in Time Example (7.1) Where the factor 1 1 - a N represents the effect of aliasing. Below, we show the change in the aliasing factor for various values of N , a = 0 . 8. N = 5 N = 10 N = 50 N = 100 1 1 - a N 1 . 4874 1 . 1203 1 . 000014 1 (21) Notice that the effect of aliasing tends toward zero as N → ∞ . C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 25 / 193
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The DFT Consider again a finite sequence x ( n ) with non-zero values at 0 n L - 1 Let ˜ x ( n ) be a periodic repetition of x ( n ) with period N L such that ˜ x ( n ) = X s = -∞ x ( n - sN ) (22) over a single period of ˜ x ( n ) we can write ˜ x ( n ) = x ( n ) , 0 n L - 1 0 , L n N - 1 (23) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 26 / 193
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(25) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 27 / 193
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The DFT We can redefine x ( n ) ˜ x ( n ) over a single period This will require that the original signal of length L is padded by N - L zeros We can then rewrite the DFT as X ( k ) = N - 1 X n = 0 x ( n ) e - j 2 π kn N , k = 0 , 1 , . . . , N - 1 (26) where increasing the length does not change the information contained in X ( k ) because x ( n ) = 0 for n L This is call the Discrete Fourier Transform (DFT) C. Williams & W. Alexander (NCSU) THE DISCRETE FOURIER TRANSFORM ECE 513, Fall 2019 28 / 193
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