Consider the rectangular sequence r[n] = u[n] – u[n – 8] of length L = 8, as sketched below: 1 ¤[n] 7 According to page 215 of the book, the magnitude of the DTFT of this sequence is sin(4ô) |X(ej®)|=| sin(0.5ô) as sketched below: 8 During a bluejeans session, one of your colleagues takes the N-point DFT of { x[1], ... x[7]} with an unspecified value of N. They meant to screen share but Can you guess what value of N they used? You have only three clues: • They tell you that N> 8. • They tell you that NE{8, 16, 32, 64, 128, ..} is a power of 2. • They tell you that between 10% and 11% of the N-point DFT coefficients a precisely zero. Explain!

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PROB. 9.5.* Consider the rectangular sequence a[n] = u[n] – u[n – 8] of length L = 8,
as sketched below:
x[n]
n
7
According to page 215 of the book, the magnitude of the DTFT of this sequence is:
|X(ej@)| = |
sin (4ô)
sin (0.5ô)
as sketched below:
8
During a bluejeans session, one of your colleagues takes the N-point DFT of {x[0],
x[1], ... x[7]} with an unspecified value of N. They meant to screen share but forgot.
Can you guess what value of N they used? You have only three clues:
• They tell
• They tell you that NE{8, 16, 32, 64, 128, ...} is a power of 2.
(a)
you
that N> 8.
• They tell you that between 10% and 11% of the N-point DFT coefficients are
precisely zero.
(b)
Explain!
Transcribed Image Text:PROB. 9.5.* Consider the rectangular sequence a[n] = u[n] – u[n – 8] of length L = 8, as sketched below: x[n] n 7 According to page 215 of the book, the magnitude of the DTFT of this sequence is: |X(ej@)| = | sin (4ô) sin (0.5ô) as sketched below: 8 During a bluejeans session, one of your colleagues takes the N-point DFT of {x[0], x[1], ... x[7]} with an unspecified value of N. They meant to screen share but forgot. Can you guess what value of N they used? You have only three clues: • They tell • They tell you that NE{8, 16, 32, 64, 128, ...} is a power of 2. (a) you that N> 8. • They tell you that between 10% and 11% of the N-point DFT coefficients are precisely zero. (b) Explain!
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