This process continues until we get to the last value, y(N

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- I), which is

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If we arrange these equations in matrix form, we have

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Note that the second row of H is formed by circularly shifting the first row to the right by 1. This shift corresponds to a circular shift of the sequence h(n). Similarly, the third row is formed by shifting the second row by I . and so on. Due to this circular property, H is said to be a circulant matrix.

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THE DFT

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[CHAP. 6

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Supplementary Problems

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Discrete Fourier Series

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Find the DFS coefficients for the sequence

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Find the DFS coefficients for the sequence of period N = 8 whose first four values are equal to 1 and the last four are equal to 0. If T(n) is a periodic sequence with a period N ,

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I ( n ) is also periodic with period 3 N . Let f ( k ) denote the DFS coefficient o f i ( n ) when considered to be periodic with a period N , and let R3(k) be the DFS coefficients of i ( n ) when considered to be periodic with a period 3N. Express the DFS coefficients X,(k) in terms of x(k).

If the DFS coefficients of a periodic sequenceI(n) are real, g ( k ) = x*(k), what does this imply about f ( n )

The Discrete Fourier Transform

Find the 10-point DFT of each of the following sequences: (a) x(n) = S(n)

+ 6(n

( b ) x(n) = u(n) - u(n - 6)

Find the 10-point DFT of the sequence

Find the 10-point inverse DFT of

k=O else

Find the N-point DFT of the sequence

where N is an even number.

Find the 16-point inverse DFT of

DFT Properties

If x(n) is a finite-length sequence of length four with a four-point DFT X(k), find the four-point DFT of each of the following sequences in terms of X(k): (a) x(n)

+ S(n)

CHAP. 61

THE DFT

If X(k) is the 10-point DFT of the sequence x(n) = 6(n - I) what sequence, y(n), has a 10-point DFT

+ 26(n - 4) - 6(n - 7)

Y (k) = 2X(k) cos

If the 10-point DFTs of x(n) = S(n) - S(n - 1) and h(n) = u(n) - u(n - 10) are X(k) and H(k), respectively, find the sequence w(n) that corresponds to the 10-point inverse DFT of the product H(k)X(k). Let x(n) be a sequence that is zero outside the interval [O, N y(n) = x ( n )

- 11 with a z-transform X(z).

+ x(N - n)

DFT of x(n)

find the 2N-point DFT of y(n), and express it in terms of X(z). If x(n) is real and x(n) = x ( N If x(n) = 6(n) (4Im[X(k)l.

- n), what can you say about the N-point

+ 2S(n - 2) - S(n - 5 ) has a

10-point DFT X(k), find the inverse DFT of (a) Re[X(k)] and

If x(n) has an N -point DFT X(k), find the N-point DFT of y(n) = cos(2nn/N)x(n).

Find the inverse DFT of Y (k) = IX(k)12 where X(k) is the 10-point DFT of the sequence x(n) = u(n) - u(n - 6). If X(k) is the N-point DFT of x(n), what is the N-point DFT of the sequence y(n) = X(n) Evaluate the sum

when

Sampling the DTFT

The z-transform of the sequence x(n) = u(n) is sampled at five points around the unit circle, X(k) = x(z)I z=e,,,,,, Find the inverse DFT of X(k).

u(n - 7) k = 0, 1 , 2, 3 , 4

Linear Convolution Using the DFT

How many DFTs and inverse DFTs of length N = 128 are necessary to linearly convolve a sequence x(n) of length 1000 with a sequence h(n) of length 64 using the overlap-add method Repeat for the overlap-save method.

A sequence x(n) of length NI = 100 is circularly convolved with a sequence h(n) of length N2 = 64 using DFTs of length N = 128. For what values of n will the circular convolution be equal to the linear convolution

Applications

THE DFT

[CHAP. 6

A continuous-time signal x,(r) is sampled with a sampling frequency o f ly= 2 kHz. I f a 1000-point DFT o f 1000 samples is computed, what is the spacing between the frequency samples X ( k ) in terms o f the analog frequency

Given X(k), the N-point DFT o f x(n), how would you compute the N-point DFT o f the windowed sequence y(n) = w ( n ) x ( n )where w(n) is a Blackman window, 2nn 4nn . w ( n ) = 0.42 - 0.5 cosf0.08~0~N-l N -I' Osn(N-l