Transcript Chapter2_Lect7.ppt

Chapter 2
Ideal Sampling and Nyquist Theorem
Topics:
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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
for |f| ≥ B
W(f) = ℑ [w(t)] = 0
Bandlimited
|W(f)|
-B
0
B
f
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 -∞ < t < ∞ 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 Ts=1/ 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.
(f s )Min =2B
 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
Waveform
Impulse sampled 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 fs<2B, 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.
Low Pass Filter for
Reconstruction
Notice the distortion on
the reconstructed signal
due to undersampling
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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
N=2BT0
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
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