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Sampling Theorem

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Book says 'IF a signal is Band limited to frequency F, Fs = 2F' ... may be VERY large, but as long as it is determinable, we don't care how big ... – PowerPoint PPT presentation

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Title: Sampling Theorem


1
Sampling Theorem
  • Chapter 5.1
  • Lecture 1

2
Shannons Sampling Theorem
  • Used to create an accurate discrete-time sample
    from a continuous-time signal
  • We have a continuous SIGNAL with frequency F
  • We want a discrete SAMPLE
  • We have to take samples at some frequency Fs
  • Book says IF a signal is Band limited to
    frequency F, Fs 2F
  • Note that limited does not mean a low
    frequency. Rather, it means that there is an
    upper bound
  • The bound may be VERY large, but as long as it is
    determinable, we dont care how big
  • Our signal frequency, F, is the noted in the book
    as B
  • WHAT DOES ALL THAT MEAN?

3
Test Signal
  • Consider this signal f(t)
  • With a freq. spectrum F(?)

4
To Sample this (Mathematically)
  • Start with Signal, f(t)
  • Multiply by Impulse Train (impulses every T
    seconds, T 1/Fs)

5
To Sample this (Mathematically)
  • This gives us

6
To Sample this (Mathematically)
  • d(t)
  • ?s
  • f(t) f(t) d(t)
  • F(?)

7
Recap (graphs)
  • Start with f(t) ?
  • Sample by multiplying by d(t) to get f(t) ?
  • Determine F(?)
  • Determine F(?) (which is F(?) repeated assume
    ?s gt 2 ?max)

8
Sketch this spectrum
  • If ?s 2 ?max
  • If ?s lt ?max

9
Example 1
  • Observe a 1kHz sinusoid in the freq-domain
  • Sample at 10kHz (F ___, Fs ____)

10
Example 1, cont
  • Sample at 2kHz (F ___, Fs ___)
  • Sample at 700Hz (F ___, Fs ___)
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