Discrete Sinusoids

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Discrete Sinusoids

• Chapter 8.1 8.2
• Lecture 3

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Periodicity (see pg 547-548)Refers to the fact
that multiple discrete sinewaves look the same

• An Example
• Let xn xnN
• Sample a Cos every T-sec
• x(t) cos(?ot)
• x(t) xn cos(?onT)
• xn cos(nN ?oT)
• cos(n?oT N?oT)

• n and N are integers
• T sampling interval
• ?o freq of cont. signal
• This is the sample of the cont. signal
• If the discrete signal (xn) is periodic
• ?o has a factor of 2p

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Periodicity

• N?oT (2p)k
• NT(2p/To) (2p)k
• .

• Due to the 2p in ?o
• Also, ?o 2p/To
• To is the period of the Cos
• Recall k and N are integers Therefore, T/To must
  be a ratio of integers for xn to be periodic

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from Lathi (eq 8.9a)

• Recall
• And
• Also
• So

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Examples

• Assume T0.03
• Let x(t) cos(20pt)
• Using Lathis approach

• Sampling period
• Continuous signal

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Examples

• Let x(t) cos(20t), T 0.03s
• Let ____ cos(3pk/7), from book (observe
  Lecture3_b.m)
• Let ____ cos(10k/7), from book

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Nonuniqueness of Discrete Sinusoidssee pg 550

• gt1 equation produces the same disc. Sin wave
• Not true for a continuous signal
• Lets see an example
• So, will cos(7.4pn) or cos(8.4pn)
  cos(0.4pn)?
• Recall cos(Ok) cos(Ok2pk), k is an INTEGER

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Other discrete signals

• Unit Impulse
• Unit Step
• Exponential