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EEN – 321 Digital Communications

Dr. Omer Mohsin Mubarak

Assistant Professor

Department of Electronics Engineering

Iqra University, Islamabad

[email protected]

Course Overview

� Pre-requisite:

� EEN-312: Communications Systems

� Tentative course outline

� Sampling

� Quantization


Fall 20132

� Quantization

� Probability and Random Process

� Channels and their Models

� Bandpass Signal Representation

� Digital Modulation Techniques

� Source Coding

� Channel Coding

IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction

Signals

� Continuous/Analog Signals

� Represented by a continuous variable e.g. x(t) for all values of t

� Discrete time signals

� Continuous variable defined only at fixed time intervals e.g. x[n]= x(nT) for integer values of n and T is the sampling period


Fall 20134

� Digital signals

� Variable can only take one out of fixed set of values at fixed time intervals e.g. x[n]=map(x(nT)) map functions rounds value to nearest value from predefined set. Set=[1 2 3 4 5], s=[1.2 2.3 2.8 4.3], map(s)=[1 2 3 4]

� Examples: Binary sequence, English text (only can be one of pre-defined alphabets)

Signals

Continuous signal Discrete time signal

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Amplitude

2 Hz Signal

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Amplitude

2 Hz Signal sampled at 100 Hz


Fall 20135

Digital Signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

Time (sec)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

Time (sec)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

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1

Time (sec)

Amplitude

2 Hz Signal sampled at 100 Hz with 11 quantization leves

Communication system

� Purpose:

� To transmit signal from a source to a destination using a channel

� Examples:

� In radio transmission microphone is source which converts voice to electrical signal, speakers in radio are destination and space is


Fall 20136

channel

� Similarly, in television transmission video camera and microphone are sources, picture tube and speakers are destination and channel is again space

� In telephone, microphone is source and destination is speaker at other set. Channel is wire line in telephone and it is space in cellular phones

Communication systems

� Storage devices can also be considered as channels

� For example, CDs, disks, tapes

� Source of a signal can be analog or digital

� A digital communication system is a communication system with a digital source – signal from analog source


Fall 20137

system with a digital source – signal from analog source (e.g., microphone) is converted to digital signal before being sent through a digital communication system

� Analog to digital conversion is performed using A/D converters – sampling & quantization are performed

� Every digital signal can be represented using binary strings

Communication channel

� Examples: wireline, fiber optic cable, space, storage medium, underwater acoustic

� Channel induces distortion in a signal

� Distortion effect can be removed by inversing the channel effect if the channel is deterministic, i.e., distortion


Fall 20138

effect if the channel is deterministic, i.e., distortion induced is known and is same every time a signal passes through that channel

� However, real life channels are not deterministic as they have random noise – can not be predicted


� Noise is approximated using its statistics found by performing experiments – different models have been proposed

� Most significant noise is thermal noise and usually modelled as Gaussian noise


Fall 20139

� Additive white Gaussain noise (AWGN) channel model is mostly used

� White noise mean that noise is uncorrelated – noise samples at two different time instances does not have any correlation


� AWGN channel implies that signal is passed through a deterministic function and added with white Gaussian noise

Deterministic channel

Transmitted signal

Received signal


Fall 201310

channelsignal signal

White Gaussian noise

Telegraphy

� Telegraph is an example of historic digital communication system

� Message is sent using Morse codes – alphabets (A-Z), digits (0-9), “.”, “,” and “?” were represented using “.” (dots), “-” dashes and space


Fall 201311

� Only pre-defined characters can be sent and thus it is considered as a digital system

� Signal was manually transmitted by generating these pulses

Telegraphy

� Dot is represented by small duration pulse, dash is represented using larger duration pulses and space by absence of any pulse

� Characters are NOT encoded using fixed length sequences of dots, dashes and spaces


Fall 201312

� Frequent characters are encoded using short sequences and rare characters are encoded using longer sequences

� This results in reducing the average length of coded sequence – the concept is also used in modern digital communication

Digital communication system

Digital modulator


Fall 201313

Digital demodulator


� Digital source: Generates digital signal for transmission

� Analog source can be converted to digital source using sampling and quantization

A/D ConverterAnalog


Fall 201314

Analog source Sampling Quantization

Digital source


� Quantization induces error – sufficient number of bits are used so to confine error within specified limits

� Source encoder: Aims to reduce number of bits by removing redundancy – reducing number of bits reduces bandwidth required for transmission


Fall 201315

� Source encoding compression techniques are based on following:

� Two symbols may not be equally probable (e.g. If there is high probability of one occurring as compared to the other)

� There may be correlation between consecutive bits (e.g. Pixel values in an image)


� Channel encoder: Designed depending on the channel used. It is aimed for error correction, thus depends on how noisy channel is and what types of errors can be introduced – extra redundant bits are added for error detection or correction. If error is introduced in the channel, receiver may detect or even correct that error


Fall 201316

channel, receiver may detect or even correct that error using these extra bits

� Digital modulation: Channels are mostly analog, thus digital modulation is used as an interface between digital data and analog channel. Output of modulator is an analogsignal.


� Modulated signal is received at the receiver after passing through a channel

� Digital demodulator: Analog signal is input and estimated digital data is output

� Channel decoder: Demodulator may have detected some


Fall 201317

� Channel decoder: Demodulator may have detected some bits wrongly due to channel noise. Channel decoder aims to correct these errors using error correcting bits

� Source decoder: Aims to reproduce the data generated by digital source

Why digital?

� Cheaper and easy to design digital communication system, as compared to analog communication system

� Digital devices are often programmable and thus digital systems offer much more flexibility

� Channel capacity defined as maximum rate of transmission


Fall 201318

� Channel capacity defined as maximum rate of transmission through a channel, is becoming almost equal for digital and analog communication systems due to advancement of technology

� Thus, most modernized communication systems are digital

Channel constraints

� Spectrum available for transmission is limited because:

� Frequency response of the channel, i.e., some frequencies may be much more attenuated than others

� Sharing with other users

� Transmitted power is limited because:


Fall 201319

� Higher transmission power consumes more battery in mobile users and thus required to be recharged more frequently

� Higher transmission power causes stronger interference for other users sharing the channel

Channel constraints

� Noise variance in a channel is also an important parameter – variance is a parameter to determine how much the signal values are spread

� Higher channel noise corrupts the transmitted signal and increases probability of erroneous bits at receiver.


Fall 201320

� Amplifying a signal increases noise and signal by the same amount, hence noise variance is usually represented as noise variance under normalized channel gain, i.e., for channel gain of unity

SamplingSamplingSamplingSamplingSamplingSamplingSamplingSampling

Sampling

� Sampling is used to convert an analog signal x(t) to discrete time signal x[n] =x(nT)

� Where T is the sampling period (fs=1/T is the sampling frequency)

� Consider x(t) as a band limited signal, which implies that


Fall 201322

� Consider x(t) as a band limited signal, which implies that its Fourier transform X(f) = 0 for |f|>w, for a given value of w

� Effect of sampling in frequency domain is creation of replicas of X(f) at integer multiples of fs (equivalent to 2πradians in DTFT)

Sampling

� In order to recover original signal from sampled signal, these replicas are removed using a low pass filter


Fall 201323

Aliasing

� Nyquist sampling theorem states that original signal can be recovered from sampled signal only if sampling frequency is at least twice the maximum frequency in the signal, i.e., fs≥2w

� This ensures that replicas of original spectrum do not


Fall 201324

overlap with each other and hence original signal can be recovered by passing sampled signal through low pass filter

� When sampling frequency is less than 2w, replicas of original signal spectrum overlap with each other making it impossible to recover original signal – this effect is called aliasing

Aliasing


Fall 201325

Reconstructing analog signal

� Provided sampling frequency meets Nyquist critera, analog signal can be reconstructed from discrete time signal by estimated intermediate values between samples, i.e., interpolation

� Sampled signal x[n] can be represented as an analog signal:∞


Fall 201326

� This signal is low pass filtered to obtain intermediate values between sampled values

� Let h(t) be the impulse response of the low pass filter

∑∞

−∞=

−=n

nTtnxtz )(][)( δ


� The output of low pass filter is given as:

∑∑∞

−∞=

∞

−∞=

−=−=

=

nn

nTthnxnTtnxth

tzthty

)(][)(][*)(

)(*)()(

δ


Fall 201327

� Let h(t) be a rectangular pulse of duration Ts

� In this case, reconstructed signal will hold the sampled value till the next sample. In other words, reconstructed signal will stay constant for the duration Ts


� The spectrum of this filter is like a sinc, which results in


Fall 201328

� The spectrum of this filter is like a sinc, which results in attenuating the side images of the sampled signal but does not remove them completely. This is the reason for not getting the original input signal back.

� Note that only one sample of the sampled signal contributes to the reconstructed signal at any instant of time


� Now consider h(t) as a triangular pulse extending from –Ts

to Ts with maximum of 1 at t=0

� In this case, y(t) will be equivalent to joining samples of z(t) by straight lines – first derivative of y(t) between two samples is constant


Fall 201329

� The spectrum of this filter is like a sinc2, which results in attenuating the side images more than the attenuation provided earlier when h(t) spectrum is equal to a sinc but again not removing side images completely



Fall 201330

� Note that due to the shape of the impulse response of this reconstruction filter, two successive samples contribute to the output reconstructed signal at any instant of time

� Reconstruction is relatively better than with rectangular h(t), but reconstruction is still not perfect


� In order to obtain perfect reconstruction, h(t) should be a sinc function – inverse Fourier transform of ideal low pass filter

( )( )

∞

== s

s

s tfctf

tfth sin

sin)(

π

π


Fall 201331

� Fourier transform of a sinc function is a rectangular pulse and Fourier transform of a sinc function time shifted by nTs

is e-j2πnTs times a rectangular pulse

( ) ( )( )∑∞

−∞=

−=n

sss nTtfcnTxty sin)(


� Thus, Fourier transform of y(t) is given as:

( )

( )∑

∑∞

−∞=

−

∞

−∞=

−

=

=

n

fnTj

s

n

fnTj

s

s

s

enTxfU

fUenTxfY

π

π

2

2

)(

)()(


Fall 201332

� Where U(f) is an ideal low pass filter with unity gain from –fs/2 to fs/2 and zero for all other frequencies

� The summation term in the Y(f) equation is DTFT of x[n] with replicas of x(t) spectrum

−∞=n


� Thus, Y(f) is equal to X(f) for f within Nyquist range – this implies perfect reconstruction

� Since sinc function extends to infinity, this filter requires infinite samples of sampled signal to compute y(t)

wfforfXfY s 2)()( >=


Fall 201333

infinite samples of sampled signal to compute y(t)

� Spectrum of this filter is rectangular (ideal low pass filter), which results in completely removing all the adjacent spectrum images. This is the only filter that provides perfect reconstruction of the original signal.



Fall 201334

� Perfect reconstruction

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