Fourier Analysis of DiscreteTime Signals

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Fourier Analysis of Discrete-Time Signals

• Discrete Frequency Analysis of periodic signals
  (DTFS)
• Chapter 10.1
• Lecture 8

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Types of Fourier functions
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Cont-time Fourier Series review

• Cont-time, periodic signal can be represented as
  a trigonometric Fourier Series and that form is
• ej0t, ej?t, ej2?t, ej3?t SCkejk?t
• and Ck

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Adjust for Discrete, Periodic Signals

• We can also do this for Discrete signals
• fk ej0k, ejOk, ej2Ok, ej3Ok ejrOk
  SDrejOkr
• where Dr

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• Example fk 3cos(p/4k), note the notation
  changes
• Find fk and Dr
• Find Period

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0 1 2 3 4 5 6 7
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• Example x2n 63cos(p/4n),

0 1 2 3 4 5 6 7
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• Example x3n 63cos(p/4n)2cos(p/2n)

0 1 2 3 4 5 6 7
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• Ex x4n 63cos(p/4n)2cos(p/2n)4cos(3p/4n)
  

0 1 2 3 4 5 6 7
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• Ex x5n 63cos(p/4n)2cos(p/2n)4cos(3p/4n)
  5cos(pn)

0 1 2 3 4 5 6 7
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• Ex What happens if N 9
• x6n 3cos(p/4n)

0 1 2 3 4 5 6 7 9
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More DTFS

• Recall
• Some reminders
• Dr is a _______________________
• ___ and ________ are an integers

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Even more DTFS

• Dr is periodic in ____
• Dr S xn e-j(2p/N)(rN)n
• S xn e-j(2p/N)rn e-j
  
• DTFS samples the spectrum from 0, 2p in
  increments of ______
• Dr 1/N S xn e-j n

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Yet another example

• As before, determine Dr, but this time plot
• Dr 1/ S xn e j n
• D0 1/ S xn e j n
• D1 1/ S xn e j n

0
1
2
3
4
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n/r
Xn
Dr
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