Chapter 2 Ideal Sampling and Nyquist Theorem

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Chapter 2 Ideal Sampling and Nyquist Theorem

• Topics
• Impulse Sampling and Digital Signal Processing
  (DSP)
• Ideal Sampling and Reconstruction
• Nyquist Rate and Aliasing Problem
• Dimensionality Theorem
• Reconstruction Using Zero Order Hold.

Huseyin Bilgekul Eeng360 Communication Systems
I Department of Electrical and Electronic
Engineering Eastern Mediterranean University
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Bandlimited Waveforms
Definition A waveform w(t) is (Absolutely
Bandlimited) to B hertz if
W(f) I w(t) 0 for f B
Bandlimited
W(f)
f
-B
B
0
Definition A waveform w(t) is (Absolutely Time
Limited) if
w(t) 0, for t T
Theorem An absolutely bandlimited waveform
cannot be absolutely time limited, and
vice versa.
A physical waveform that is time limited, may not
be absolutely bandlimited, but it may be
bandlimited for all practical purposes in the
sense that the amplitude spectrum has a
negligible level above a certain frequency.
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Sampling Theorem
Sampling Theorem Any physical waveform may be
represented over the interval -8 lt t lt 8 by
The fs is a parameter called the Sampling
Frequency . Furthermore, if w(t) is bandlimited
to B Hertz and fs 2B, then above expansion
becomes the sampling function representation,
where an w(n/fs)
That is, for fs 2B, the orthogonal series
coefficients are simply the values of the
waveform that are obtained when the waveform is
sampled with a Sampling Period of Ts1/ fs
seconds.
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Sampling Theorem

• The MINIMUM SAMPLING RATE allowed for
  reconstruction without error is called the
  NYQUIST FREQUENCY or the Nyquist Rate.
• Suppose we are interested in reproducing the
  waveform over a T0-sec interval, the minimum
  number of samples that are needed to reconstruct
  the waveform is

• There are N orthogonal functions in the
  reconstruction algorithm. We can say that N is
  the Number of Dimensions needed to reconstruct
  the T0-second approximation of the waveform.
• The sample values may be saved, for example in
  the memory of a digital computer, so that the
  waveform may be reconstructed later, or the
  values may be transmitted over a communication
  system for waveform reconstruction at the
  receiving end.

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Sampling Theorem
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Impulse Sampling and Digital Signal Processing

• The impulse-sampled series is another orthogonal
  series.
• It is obtained when the (sin x) / x orthogonal
  functions of the sampling theorem are replaced by
  an orthogonal set of delta (impulse) functions.
• The impulse-sampled series is identical to the
  impulse-sampled waveform ws(t)
• both can be obtained by multiplying the
  unsampled waveform by a unit-weight impulse
  train, yielding

Impulse sampled waveform
Waveform
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Reconstruction of a Sampled Waveform

• The sampled signal is converted back to a
  continuous signal by using a reconstruction
  system such as a low pass filter.

x(nTs)

• Same sample values may give different
  reconstructed signals x1(t) and x2(t). Why ???

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Ideal Reconstruction in the Time Domain
x(nTs)

• The sampled signal is converted back to a
  continuous signal by using a reconstruction
  system such as a low pass filter having a Sa
  function impulse response

The impulse response of the ideal low pass
reconstruction filter.
Ideal low pass reconstruction filter output.
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Spectrum of a Impulse Sampled Waveform
The spectrum for the impulse-sampled waveform
ws(t) can be evaluated by substituting the
Fourier series of the (periodic) impulse train
into above Eq. to get
By taking the Fourier transform of both sides of
this equation, we get
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Spectrum of a Impulse Sampled Waveform

• The spectrum of the impulse sampled signal is
  the spectrum of the unsampled signal that is
  repeated every fs Hz, where fs is the sampling
  frequency (samples/sec).
• This is quite significant for digital signal
  processing (DSP).
• This technique of impulse sampling maybe be used
  to translate the spectrum of a signal to another
  frequency band that is centered on some harmonic
  of the sampling frequency.

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Undersampling and Aliasing

• If fslt2B, The waveform is Undersampled. The
  spectrum of ws(t) will consist of overlapped,
  replicated spectra of w(t). The spectral overlap
  or tail inversion, is called aliasing or spectral
  folding. The low-pass filtered version of ws(t)
  will not be exactly w(t). The recovered w(t) will
  be distorted because of the aliasing.

Notice the distortion on the reconstructed signal
due to undersampling
Low Pass Filter for Reconstruction
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Undersampling and Aliasing
Effect of Practical Sampling
Notice that the spectral components have
decreasing magnitude
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Dimensionality Theorem

• THEOREM When BT0 is large, a real waveform may
  be completely specified by
• N2BT0
• independent pieces of information that will
  describe the waveform over a T0 interval. N is
  said to be the number of dimensions required to
  specify the waveform, and B is the absolute
  bandwidth of the waveform.
• The information which can be conveyed by a
  bandlimited waveform or a bandlimited
  communication system is proportional to the
  product of the bandwidth of that system and the
  time allowed for transmission of the information.
• The dimensionality theorem has profound
  implications in the design and performance of all
  types of communication systems.

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Reconstruction Using a Zero-order Hold
Basic System
Anti-imaging Filter is used to correct
distortions occurring due to non ideal
reconstruction
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Discrete Processing of Continuous-time Signals
Basic System
Equivalent Continuous Time System
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Sample Quiz
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Quiz Continued