0. Chapter 4: Sampling of Continuous-Time Signals
Department of Computer Eng.
Sharif University of Technology
Discrete-time signal processing
Chapter 4: Sampling of Continuous-Time Signals
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, © Prentice Hall Inc.

1. Chapter 4: Sampling of Continuous-Time Signals
4.1 Periodic Sampling
In this method x[n] obtained from xc(t) according to the relation :
The sampling operation is generally not invertible i.e., given the output x[n] it is not possible in general to reconstruct xc(t). Although we remove this ambiguity by restricting xc(t).
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2. Sampling with a Periodic Impulse Train
Figure(a) is not a representation of any physical circuits, but it is convenient for gaining insight in both the time and frequency domain.
(a) Overall system
(b) xs(t) for two sampling rates
(c) Output for two sampling rates
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3. 4.2 Frequency Domain Representation of Sampling
Let us now consider the Fourier transform of xs(t):
If and
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4. Frequency Domain Representation of Sampling
By applying the continuous-time Fourier transform to equation
We obtain
consequently
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5. Exact Recovery of Continuous-Time from Its Samples
(a) represents a band limited Fourier transform of xc(t) Whose highest nonzero frequency is
(b) represents a periodic impulse train with frequency.
(c) shows the output of impulse modulator in the case
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6. Exact Recovery of Continuous-Time from Its Samples
In this case don’t overlap
therefore xc(t) can be recovered from xs(t) with an ideal low pass filter with gain T and cutoff frequency
It means =
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7. Chapter 4: Sampling of Continuous-Time Signals
Aliasing Distortion
(a) represents a band limited Fourier transform of xc(t) Whose highest nonzero frequency is
(b) represents a periodic impulse train with frequency.
(c) shows the output of impulse modulator in the case
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8. Chapter 4: Sampling of Continuous-Time Signals
Aliasing Distortion
In this case the copies of overlap and is not longer recoverable by lowpass filtering therefore the reconstructed signal is related to original continuous-time signal through a distortion referred to as aliasing distortion.
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9. Example: The effect of aliasing in the sampling of cosine signal
Suppose
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10. Nyquist Sampling Theorem
Sampling theorem describes precisely how much information is retained when a function is sampled, or whether a band-limited function can be exactly reconstructed from its samples.
Sampling Theorem: Suppose that is band-limited to a frequency interval , i.e.,
Then xc(t) can be exactly reconstructed from equidistant samples
where is the sampling period, is the sampling frequency (samples/second), is for radians/second.
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11. Chapter 4: Sampling of Continuous-Time Signals
Oversampled
Suppose that is band-limited:
Then if is sufficiently small, appears as:
Condition:
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12. Chapter 4: Sampling of Continuous-Time Signals
Critically Sampled
Critically sampled:
According to the Sampling Theorem, in general the signal cannot be reconstructed from samples at the rate
This is because of errors will occur if , the folded
frequencies will add at
Consider the case:
and note that for
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13. Undersampled (aliased)
If sampling theorem condition is not satisfied
The frequencies are folded - summed. This changes the shape of the spectrum. There is no process whereby the added frequencies can be discriminated - so the process is not reversible.
Thus, the original (continuous) signal cannot be reconstructed exactly. Information is lost, and false (alias) information is created.
If a signal is not strictly band-limited, sampling can still be done at twice the effective band-limited.
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14. 4.3 Reconstruction of a Bandlimited Signal from Its Samples
Figure(a) represents an ideal reconstruction system.
Ideal reconstruction filter has the gain of T and cutoff frequency
we choice
This choice is appropriate for any relationship between and
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15. Reconstruction of a Bandlimited Signal from Its Samples
Therefore
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16. Reconstruction of a Bandlimited Signal from Its Samples
For all integer values of m. independent from the sampling period T.
Therefore the resulting signal is an exact reconstruction of xc(t) at the sampling times. the fact that, if there is no aliasing, the low pass filter interpolates the correct reconstruction between the samples, and if there is aliasing, it can’t interpolate them correctly.
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17. Chapter 4: Sampling of Continuous-Time Signals
Ideal D/C Converter
The properties of the ideal D/C converter are most easily seen in the frequency domain.
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18. 4.4 Discrete-Time Processing of Continuous-Time Signals
A major application of discrete-time systems is in the processing of continuous-time signals.
We know from the previous sections
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19. 4.4.1 LTI Discrete-Time systems
In the LTI systems we have
IF then
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20. 4.4.1 LTI Discrete-Time systems
In general if the discrete-time system is LTI and if the sampling frequency is above the Nyquist rate associated with the band width of the input xc(t), then the overall system will be equivalent to a LTI continuous-time system with an effective frequency response given by:
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21. Chapter 4: Sampling of Continuous-Time Signals
Example: Ideal Continuous-Time Lowpass Filtering Using a Discrete-Time Lowpass Filter
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22. Chapter 4: Sampling of Continuous-Time Signals
Example: Ideal Continuous-Time Lowpass Filtering Using a Discrete-Time Lowpass Filter
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23. Chapter 4: Sampling of Continuous-Time Signals
Example: Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator
The ideal continuous-time differentiator system is
For processing bandlimited signals, it is sufficient that
Therefore the corresponding discrete-time system has frequency response
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24. Chapter 4: Sampling of Continuous-Time Signals
Example: Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator
If this system has the input
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25. Chapter 4: Sampling of Continuous-Time Signals
Impulse Invariance
If the desired continuous-time system has bandlimited frequency response then how to choose so that
In this case the discrete-time system is said to be an impulse-invariant version of the continuous time system.
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26. Example: A discrete-time lowpass filter obtained by impulse invariance
We want to obtain an ideal lowpass discrete-time filter with cutoff frequency we can do this by sampling a continuous-time ideal lowpass filter with cutoff frequency
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27. 4.6 Changing the sampling rate using discrete-time processing
We have seen that a continuous-time signal can be represented by a discrete-time signal.
It is often necessary to change the sampling rate of x[n] and obtain a new discrete-time signal such that
One approach is to reconstruct and then resample it with period , but it is of interest to consider methods that involve only discrete time operations.
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28. 4.6.1 Sampling rate reduction by an integer factor
Discrete-time sampler or compressor
If then is an exact representation of
Downsampling: the operation of reducing the sampling rate (including any filtering).
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29. Chapter 4: Sampling of Continuous-Time Signals
Frequency domain relation between the input and output of the compressor
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30. Downsampling without Aliasing
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31. Downsampling with aliasing
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32. Downsampling with prefiltering to avoid aliasing
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33. 4.6.2 Increasing the sampling rate by an integer factor
We will refer to the operation of increasing the sampling rate upsampling
The system on the left is called a sampling rate expander. Its output is
The system on the right is a lowpass discrete-time filter with cutoff frequency and gain L.
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34. Increasing the sampling rate by an integer factor
This system is an interpolator because of it fills in the missing samples.
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35. Increasing the Sampling Rate By an Integer Factor
If the input sequence was obtained by sampling without aliasing then is correct for all n, And is obtained by oversampling of
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36. Chapter 4: Sampling of Continuous-Time Signals
Linear Interpolation
In practice ideal lowpass filters can not be implemented exactly. In some cases, simple interpolation procedure are adequate. Since linear interpolation is often used.
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37. 4.6.3 Changing the Sampling Rate by a Noninteger Factor
By combining decimation and interpolation it is possible to change the sampling rate by a noninteger factor.
The interpolation and decimation filter can be combined together.
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38. Changing the Sampling Rate by a Noninteger Factor
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