1. 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

2. Bandlimited Waveforms
Definition: A waveform w(t) is (Absolutely Bandlimited) to B hertz if
W(f) = ℑ [w(t)] = for |f| ≥ B
Bandlimited
|W(f)| f -B B
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.

3. 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.

4. 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.

5. Sampling Theorem

6. 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

7. 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 ???

8. 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.

9. 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

10. 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.

11. 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.
Notice the distortion on the reconstructed signal due to undersampling
Low Pass Filter for Reconstruction

12. Undersampling and Aliasing
Effect of Practical Sampling
Notice that the spectral components have decreasing magnitude

13. 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.

14. Reconstruction Using a Zero-order Hold
Basic System
Anti-imaging Filter is used to correct distortions occurring due to non ideal reconstruction

15. Discrete Processing of Continuous-time Signals
Basic System
Equivalent Continuous Time System

16. Sample Quiz

17. Quiz Continued