Lecture 4 Sampling

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Lecture 4 Sampling

• Overview of Sampling Theory

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Sampling Continuous Signals

• Sample Period is T, Frequency is 1/T
• xn xa(n) x(t)tnT
• Samples of x(t) from an infinite discrete sequence

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Continuous-time Sampling

• Delta function d(t)
• Zero everywhere except t0
• Integral of d(t) over any interval including t0
  is 1
• (Not a function but the limit of functions)
• Sifting

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Continuous-time Sampling

• Defining the sequence by multiple sifts
• Equivalently
• Note xa(t) is not defined at tnT and is zero
  for other t

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Reconstruction

• Given a train of samples how to rebuild a
  continuous-time signal?
• In general, Convolve some impluse function with
  the samples
• Imp(t) can be any function with unit integral

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Example

• Linear interpolation
• Integral (0,2) of imp(t) 1
• Imp(t) 0 at t0,2
• Reconstucted function is piecewise-linear
  interpolation of sample values

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DAC Output

• Stair-step output
• DAC needs filtering to reduce excess high
  frequency information

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Sinc(x) Perfect Reconstruction

• Is there an impulse function which needs no
  filtering?
• Why? Remember that Sin(t)/t is Fourier
  Transform of a unit impulse

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Perfect Reconstruction II

• Note Sinc(t) is non-zero for all t
• Implies that all samples (including negative
  time) are needed
• Note that x(t) is defined for all t since
  Sinc(0)1

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Operations on sequences

• Addition
• Scaling
• Modulation
• Windowing is a type of modulation
• Time-Shift
• Up-sampling
• Down-sampling

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Up-sampling
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Down-sampling (Decimation)
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Resampling (Integer Case)

• Suppose we have xn sampled at T1 but want xRn
  sampled at T2L T1

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Sampling Theorem

• Perfect Reconstruction of a continuous-time
  signal with Bandlimit f requires samples no
  longer than 1/2f
• Bandlimit is not Bandwidth but limit of maximum
  frequency
• Any signal beyond f aliases the samples

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Aliasing (Sinusoids)
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Alaising

• For Sinusoid signals (natural bandlimit)
• For Cos(wn), w2pkw0
• Samples for all k are the same!
• Unambiguous if 0ltwltp
• Thus One-half cycle per sample
• So if sampling at T, frequencies of fe1/2T will
  map to frequency e