Communication Theory II

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1 Communication Theory II Lecture 17: Conversion of Analog Waveforms into Coded Pulses Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt April 16 th,

2 opulse Modulation Analog Pulse Modulation Digital Pulse Modulation Lecture Outlines osampling Sampling Theorem Practical difficulties ofsampling olpf implementation oaliasing (spectral folding) opractical sampling 2

3 Analog Pulse modulation Pulse Modulation A periodic pulse train is used as a carrier The carrier wave features of each pulse (amplitude PAM, duration PWM, position PPM) are varied in a continuous manner in accordance with the corresponding sample value (basically in analog form), but transmission takes place in discrete times Digital Pulse modulation A message signal is represented in a form that is discrete in both time (PAM) and amplitude (Quantization) Permitting transmission of the message in digital form as a sequence of coded pulses Involve sampling and quantization (PAM and amplitude quantization) Types of analog pulse modulation 3

4 Digital Pulse Modulation (source coding forms) Pulse Code Modulation (PCM) The most favored scheme for transmitting analog information bearing signals (voice and video) Adv: Robust to channel noise efficient regeneration, ease for secure communication, uniform transmission for different baseband signals, ease of multiplexing, uniform transmission format Cost of increased system complexity and increased transmission bandwidth Differential Pulse Code Modulation (PCM) Use of lossy data compression to remove redundancy inherent in message signal to reduce bit rate of the transmitted signal with out serious degradation of overall performance Cost of increased system complexity in exchange of reducing transmission bandwidth of PCM Delta Modulation Simple implementation by oversampling the message signal Cost of increased bandwidth in exchange of reduced complexity 4

5 Before Source Coding Analog signal e.g., voice or video Digital Pulse Modulation Sampling and quantizing using: * PCM * DPCM * DM Source Coding Efficient coding in terms of reducing the average code length using: *Huffman coding * Lempel-Ziv coding Channel Coding Efficient transmission over noisy channel in terms of reducing probability of error using: *Linear block code * Cyclic code Sampling Theorem Source Coding Theorem Channel Coding Theorem 5

6 Sampling Process of converting analog signal to a corresponding sequence of samples that are usually spaced uniformly in time (e.g., every T S secs.) ot s is referred to as the sampling interval of s = 1/T s is called the sampling rate or sampling frequency. othe process is referred to as pulse amplitude modulation PAM and the outcome is a signal with analog (an infinite number of) values 6

7 Sampling Methods othere are 3 sampling methods: Ideal - an impulse at each sampling instant Natural - a pulse of short width with varying amplitude Flattop - sample and hold, like natural but with single amplitude value ohow to choose sampling rate, so that the sequence of samples uniquely defines the original signal? 7

8 Sampling Theorem o A band-limited signal of finite energy that has no frequency components higher than B hertz is completely described by specifying the values of the signal instants of time separated by T s =1/2B seconds o A band-limited signal of finite energy that has no frequency components higher than B hertz is completely recovered from a knowledge of its samples taken at the rate of f s =2B (Nyquist rate) samples per second 8

9 Example: Recovery of a sampled sine wave for different sampling rates T=1/f Each one cycle take two samples 9

10 Sampling Signal and its Fourier Transform Impulse train function δδ TTTT tt = δδ tt nnnnnn = 1 TTTT [1 + 2 cos ww sstt + 2 cos 2 ww ss tt + ] Sampled signal g tt =g tt. δδ TTTT tt = g nnnnnn. δδ(tt nntttt) nn nn = 1 TTTT [g(tt) + 2 g(tt)cos ww sstt + 2 g(tt)cos 2 ww ss tt + ] Ideal Sampling *Fourier of = 1 TTTT nn GG(ww nnnn ss ) 10

11 Spectrum of sampled signal o Spectrum of sampled signal G(ww) consists of GG(ww) repeating periodically with period ww ss = 2ππ TTTT rad/sec or ff ss = 1 TTTT Hz o How to fully reconstruct GG(ww) from G(ww) with no loss of information? This is possible if there is no overlap between successive cycles of G(ww), that is ff ss 2BB or TT ss 1 2BB g(tt) can be recovered from its samples g tt by passing the sampled signal over an ideal low pass filter of bandwidth of B Hz o Minimum sampling rate ff ss = 2BB is called the Nyquist rate = 1 TTTT nn GG(ww nnnn ss ) ff ss = 2BB is the Nyquist rate and TT ss = 1 2BB is the Nyquist interval 11

Reconstruction filter: Interpolation Process o The process of reconstructing a continuous time signal g(t) from its sampling is known as interpolation o The sampled signal contains a component 1 TTTT

12 Reconstruction filter: Interpolation Process o The process of reconstructing a continuous time signal g(t) from its sampling is known as interpolation o The sampled signal contains a component 1 TTTT g(t) o To recover g(t) [or G(w)], the sampled signal must be passed through an ideal low pass filter of bandwidth B Hz and gain 1 TTTT o Interpolation process can be expressed as a filtering operation with an impulse response h(t) = 1 TTTT nn GG(ww nnnn ss ) o In time domain: Each sample, e.g., kth sample g(kts)δδ tt KKKKKK, is convolved by the impulse response h(t) of the filter, its output is g(kts)h(tt KKKKKK) Ex: for h(t)=rect( tt ) the output of kth sample is g(kts) rect(tt KKKKKK) TTTT TTTT 12

13 Ideal Interpolation using an Ideal LPF o g tt =g tt. δδ TTTT tt = kk g kktttt. δδ(tt kktttt) = TTTT rect ww/4ππππ = 2BBTTTT ssssssss 2ππBBtt =ssssssss 2ππBBtt o The output for the input sampled signal g tt is g(t) and can be expressed as: o Note that h tt = 0 for all Nyquist samples instants ± nn 2BB except at tt = 0 13

14 Simple Interpolation using Zero-order Hold Circuit rect( tt KKKKKK TT ss ) Stair-case approximation 14

15 Interpolation using first-order Hold Circuit ( tt KKKKKK 2TTTT ) Tops are connected with straight-line segments Triangle pulse h tt = HH(ww) 15

16 Practical difficulties in signal reconstruction o Ideal Low pass filter unrealizable Solution: using ff ss > 2BB and a LPF with gradual cut-off frequencies o Aliasing (spectral folding) Solution: using an anti-aliasing (LPF) filter BEFORE sampling o Ideal sampling train of impulses unrealizable Solution: using a practical sampling with a train of pulses with finite width 16

Increasing sampling rate over the Nyquist rate o Using ff ss = 2BB Hz leaves no gaps between successive cycles of G(w) and needs an ideal LPF Ideal LPF is unrealizable, and needs infinite time delay

17 Increasing sampling rate over the Nyquist rate o Using ff ss = 2BB Hz leaves no gaps between successive cycles of G(w) and needs an ideal LPF Ideal LPF is unrealizable, and needs infinite time delay units o We can use a LPF with gradual cut offfrequencies but we need ff ss > 2BB The filter is still unrealizable (can not achieve zero frequency response at high frequencies), however it uses smaller delay units The recovered signal approaches the desired signal more closely 17

18 Aliasing (Spectral Folding) o All practical signals are time limited and thus are not frequency band limited (why?) Aliasing (spectral folding): The loss of the tail of G(w) beyond f >f s /2 Hz The reappearance of this tail inverted or folded onto the spectrum o Folding frequency= f s /2 = 1 2TTTT Hz 18

19 A Solution: The Anti-Aliasing Filter o Eliminate (suppress) the signal component of higher frequencies beyond f s /2 o Must performed BEFORE the signal is sampled. Why? o Ideally: performed using an ideal low pass filter with a cut-off frequency of f s /2 o In practice: using a steep-cut-of filter that leaves sharply attenuated residual spectrum beyond the folding frequency f s /2 19

20 Practical Sampling o An impulse train physically non exist, instead a train of pulses with finite width can be used o At a sampling rate ff ss > 2BB, the signal g(t) can be fully reconstructed using a LPF applied to the sampled signal g tt. Why? 20

21 More Complex Sampling Schemes o Second-order sampling: use interlaced sampling trains each at rate of B samples/sec, where B is the signal bandwidth 21

22 Questions 22

Communication Theory II

Communication Theory II Lecture 18: Pulse Code Modulation Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt April 19 th, 2015 1 Lecture Outlines opulse Code Modulation (PCM) Sampling and