Lec-1 [Introduction_ Shannon_ FT]

7/27/2019 Lec-1 [Introduction_ Shannon_ FT]

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Digital Communications

EE-6322

Dr. Sarmad SohaibAssistant Professor

Department of Electrical Engineering

University of Engineering and Technology, Taxila

[email protected]

Contact Details

Dr. Sarmad Sohaib

Room-16, Electrical Engineering Department

051 9047 558

Office hours: after class or by appointment

[email protected]

8/30/2012Digital Communication (Dr. Sarmad Sohaib)

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Assessment and Grading

Quizzes 20%

No make-up quizzes

Mostly un announced so revise the last lec before coming

N-1 quizzes will be counted towards the final grading

Midterm exam 40%

Final exam 40%

Assignment

Will be given time to time for your own practice

Solve as many end of chapter problems as possible

University attendance policy will be adhered (85%)

8/30/2012 3Digital Communication (Dr. Sarmad Sohaib)

Course Objective

To study the design and implementation of

digital communication systems


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Big Picture


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Information

source and input

transducer

Source

encoder

Channel

encoder

Digital

modulator

Channel

Output

transducer

Source

decoder

Channel

decoder

Digital

demodulator

Source analog (e.g., audio, video) or digital

(e.g., output of a computer).

Source Coding (Data Compression) The process of efficiently converting the output of

either an analog or digital source into a sequenceof binary digits.

Sequence of binary digits from the source encoderis called information sequence


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Channel Encoder

Introduce redundancy in controlled manner.

Used at the receiver to overcome the effects of

noise and interference encountered in thetransmission of the signal through a channel.

Increases the reliability of the received data.

Mapping k-bit sequence into a unique n-bit

sequence, called a code word.

Code rate: k/n


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Digital Modulator

Interface to the communication channel

Maps the binary sequence into signal waveforms.

E.g. binary digit 0 is mapped in to a waveform s0(t)

binary digit 1 is mapped in to a waveform s1(t)

b coded information bits can be transmitted using M=2b distinct

waveforms si(t), i=0,1,,M-1, one waveform for each of the 2b

possible b-bit sequences. (Also calledM-ary modulation)


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Channel

Physical medium that is used to send the signal fro the

transmitter to the receiver

E.g. atmosphere (wireless), optical fiber cables,

microwave radio e.t.c.

It corrupts the transmitted signal in random manner

Additive thermal noise generated by electronic devices

Man-made noise, e.g., automobile ignition noise

Atmosphere noise, e.g., electrical lightning discharge duringthunderstorms


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Digital demodulator

Reduce the waveforms to a sequence of numbers

that represent estimates of the transmitted data

symbols (binary or M-ary)

Channel decoder

Attempts to reconstruct the original information

sequence from knowledge of the code used by the

channel encoder and redundancy contained in the

received data


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Source decoder

Accepts the output sequence from the channel

decoder

Attempts to reconstruct the original signal that wastransmitted from the source


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Measure of distortion introduced by

Digital Communication System (DCS)

Probability of a bit-error

It is a measure of how well the demodulator anddecoder perform

It is a function of

code characteristics

type of waveform used for information transmission

transmit power

characteristics of channel

method of demodulation and decoding

Bit error rate (BER)

Worst possible BER ?


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Mathematical Models for

Communication Channels

Reflects the most important characteristics of

the transmission medium

Helps in designing the channel encoder and

modulator at transmitter and the demodulator

and channel decoder at the receiver


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The additive noise channel

Simplest mathematical model

Additive noise process may arise from electronic components

and amplifiers at the receiver of the communication system or

from interference encountered in transmission

If the noise has Gaussian distribution it is called additive white

Gaussian noise (AWGN)


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+s(t)

n(t)

r(n)=s(t)+n(t)

Channel

Linear Filter Channel

Filter used to ensure that the transmitted signals do not exceed

specified bandwidth limitations and thus do not interfere with

one another.

Such channels are generally characterized mathematically as

linear filter channels with additive noise


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+s(t)

n(t)

r(n) = h(t) * s(t) + n(t)

h(t) is the impulse response of the li near filter

* denotes convolution

Channel

Linear

Filter

h(t)

Linear Time-variant Filter Channel

Characterizes time-invariant multipath propagation of the transmitted signal

Such filters are characterized by a time-variant channel impulse response

h(;t), where h(;t) is the response of the channel at time tdue to an impulse

applied at time t-

represents the age (elapsed-time) variable


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+s(t)

n(t)

Channel

Linear

Filter

h(;t)

( ) ( ) ( ; ) ( )

( ; ) ( ) ( )

r t s t h t n t

h t s t d n t

= +

= +


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3 key parameters for reliable

communication


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+s(t) r(t)

H(f)

n(t)

Signal power P [W]

Bandwidth W[Hz]

one-sided

Noise power spectral density N0 [W/Hz]

2

0

The 3 combine to give signal-to-noise ratio at Rx ( )P

SNR H f N W

=

Thinking in 20s (Before Shannon)

Nyquist [1924]: The maximum symbol rate is Wcomplex symbols/second

Complex symbol send as : real x cos (.) imaginary x sin(.)

Only has to do with sampling th eorem

Deterministic idea, as there is no noise.

If the symbols alphabet isA, then each symbol conveys log2|A| bits.

Rb=Wlog2|A| (W=symbol rate)

A={-1,1}(one bit per sample) Rb=W (BPSK)

A={1j} (two bits per sample) Rb=2W (4-QAM)

A={1,2,3,} Rb= (Problem with this alphabet)

Hartley [1928]: With amplitude constraint A, and determinist resolution

, the maximum bit rate is:


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2logb

AR W

+ =

What will be the size of alphabet as a

function ofA and?


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+A

-A

Deal with

the power

constraint

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Point separation for

detecting the

element at receiver

(i.e. dealing withnoise)

Modulation was instantaneous

i.e. you map a bit to a symbol or a collection of

bits to a symbol

Now we take block of 10,000 bits and map itto a block of 5000 or 10,000 symbols.

Increase signal noise or power, or live with

inevitable noise.


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Turning Point: Claude Shannon, 1948

The father of information theory and modern communication theory

1948 paper A (The) Mathematical Theory of Communication

Quantified information; separation theorem; noisy channel theorem

Two forms of the channel capacity (speed limit) for H(f) -- constant over

bandwidth W

Given W, P, the maximum bit rate is:

Give W, Rb, the minimum SNR is:

The speed limit:

a modem achieving bit rateRb with no errorsRb < C

How to get close?


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2

0

log 1P

C WN W

= +

/

0

2 1bR WP

N W=

Review: Deterministic Signal

Processing


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Notation


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Examples

Energy ofs(t) = cos(200t)?

Energy ofx(t) = e^{-t}u(t)?


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Some Important Signals-I


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1 | | 1/ 2

( ) 1/ 2 1/ 2

0 o therwise

t

rect t t


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F.T. Example-1

8/30/2012Digital Communication (Dr. Sarmad Sohaib) 33

sin( )( ) ( ) sinc( )

t f Tx t rect X f T f T T

T f T

= = =

sin(2 )( ) ( ) 2 sinc(2 ) 2

2 2

f W tG f rect g t W W t W

W W t

= = =

T/2-T/2

1x(t)

W-W

1G(f)

t

f

f0

1/T

2/T-1/T

T

X(f)

t0

1/(2W)

2/(2W)

-1/(2W)

2W

x(t)

F.T. Example-2


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02

0( ) ( ) ( )

j f tx t e X f f f

= =

Delay in time domain phase shift in frequency domain

AND VICE VERSA

02

0( ) ( ) ( )

j f tx t t t X f e

= =

F.T. Example-3


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0 02 2

0

0 0

1 1( ) cos(2 )

2 2

1 1

( ) ( ) ( )2 2

j f t j f tx t f t e e

X f f f f f

= = +

= + +

c

0 f0-f0 f

1/21/2

F.T. Example-4


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0 02 2

0

0 0

1 1( ) sin(2 )

2 2

1 1

( ) ( ) ( )2 2

j f t j f tx t f t e e

j j

X f f f f fj j

= =

= +

c

0 f0

-f0

f

1/(2j)

-1/(2j)


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F.T. Example-5


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( ) 1t

t0

( )t

f0

1

1 ( )f

f0

( )f

t0

1

Filtering


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Baseband versus Passband


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Lec-1 [Introduction_ Shannon_ FT]