Comm 502: Communication Theory - GUC...Comm 502: Communication Theory Lecture 2 Transmission of...

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Dr. Ahmed El-Mahdy

Communications DepartmentComm 502: Communication Theory

Comm 502: Communication Theory

Lecture 2

Transmission of Baseband Signals

- Sampling theorem,Pulse Amplitude Modulation

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Dr. Ahmed El-Mahdy


Types of Communication Channels

Wireless Communication

channelsWire Communication

channels

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Dr. Ahmed El-Mahdy


Transmission of Baseband Signals

Baseband channel: Coaxial cable or

pair of wiresConversion from a bit stream to a

Sequence of pulse waveform

PAM PCM

Pulse Code Modulation Block diagram

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Dr. Ahmed El-Mahdy


Transmission of Baseband Signals Block diagram (Cont.)

• Data already in a digital format would bypass the formatting function.

•Textual information is transformed into binary digits by use of a coder (for ex. ASCII code(American Standard Code for Information Interchange).

• Analog information is formatted using three separate processes:

- Sampling- Quantization- Coding

In all cases, the formatting step results in a sequence of binary digits.

•These digits are transmitted through a baseband channel after transforming the digits to waveforms (pulses).

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Dr. Ahmed El-Mahdy


Example: coding the word: THINK using 6-bit ASCII code

Formatting Text

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Dr. Ahmed El-Mahdy


Formatting Text (Cont.)

ASCII Code

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Dr. Ahmed El-Mahdy


Formatting Text (cont.)• Message: Text message consists of a sequence of

alphanumeric Characters.

• Bit stream: When digitally transmitted, the characters are

first encoded to a sequence of bits called bit

stream.

• Symbols: group of k bits can be combined to form a

symbol, from a finite symbol set of

symbols.

A system using symbol set of size M is called M-ary system

For k=1 (i.e. M=2), the system is called Binary (only two

waveforms is used). For k=2, (i.e. M=4), the system is

called 4-ary system.

kM 2

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Dr. Ahmed El-Mahdy


Formatting Analog Information

• Analog information is formatted using three separate

processes:

- Sampling

- Quantization

- Coding

The process of transforming an analog waveform into a form that is compatible with a digital communication system starts with sampling the waveform to produce a discrete pulse amplitude modulated (PAM) waveform.

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Dr. Ahmed El-Mahdy


Sampling• Analog signal is sampled every TS seconds.

• Ts is referred to as the sampling interval.

• fs = 1/Ts is called the sampling rate or sampling frequency.

• There are 3 sampling methods:

– Impulse (ideal) sampling - an impulse at each sampling instant

– Natural sampling - a pulse of short width with varying amplitude

– Flat-top sampling - sample and hold, like natural but with single amplitude value.

• The process is referred to as pulse amplitude modulation PAM .

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Dr. Ahmed El-Mahdy


Impulse (Ideal) SamplingFrequency DomainTime Domain

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Dr. Ahmed El-Mahdy


Spectrum of Impulse Sampling

The spectrum of the impulse

sampled waveform

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Dr. Ahmed El-Mahdy


Impulse Samplingwith increasing sampling frequency

Sampled waveform

0

1 201

Sampled waveform

0

1 201

Sampled waveform

0

1 201

Sampled waveform

0

1 201

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Dr. Ahmed El-Mahdy


Natural sampling• Natural sampling is performed by

multiplying w(t) by a train of pulses:

where

• Natural sampling takes a slice

of the waveform and the top of

the slice preserves the shape of

the waveform.

PAM using Natural sampling

k

skTtPts

)(

)()()( tstwtws

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Dr. Ahmed El-Mahdy


Spectrum of Natural Sampling

• The spectrum (F.T) of the natural sampled signal can be derived as follows:

k

ss

s

s

τ

kTtPF.T*W(f)(f)W

tsF.T*W(f)(f)W

:Transform rierTaking Fou

tw(t)s(t)w

ProofReading

ONLY

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Dr. Ahmed El-Mahdy


Spectrum of Natural Sampling (Cont.)

From signals and systems, the Fourier transform of

the periodic pulse is

Then we have:

n

s

ss

n

s

ss

s

nffδ*W(f)T

nτsinc

T

τ

nffδT

nτsinc

T

τ*W(f)(f)W

][

ProofReading

ONLY

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Dr. Ahmed El-Mahdy



n

s

ss

s nffWT

nτsinc

T

τ(f)W

tionDelta Functhe ofproperty shifting the Using

Constant for each n

(not depend on the f)&

discrete

Periodic

W(f)

ProofReading

ONLY

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Dr. Ahmed El-Mahdy



-10 0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

f

Mag

nit

ud

e o

f W

(f)

MAGNITUDE SPECTRUM of analog signal

-10 0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

f

Mag

nit

ud

e o

f W

s(f

)

MAGNITUDE SPECTRUM of NATURAL SAMPLED PAM signal

n

s

ss

s nffWT

nτsinc

T

τ(f)W

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Dr. Ahmed El-Mahdy


Flat-Top Sampling

• Easy to implement using sample and hold operation.

• Flat top sampling takes a slice of the waveform but cuts off the top of the slice horizontally.

The top of the slice does not preserve the shape

of the waveform.

Pulse Amplitude Modulation using

Flat Top sampling

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Dr. Ahmed El-Mahdy


Spectrum of Flat Top Sampling

k

s

s

s )kf-W(fT

(f)W

fsinc

Variable

continuously

with f

Periodic

W(f)

Proof in AppendixReading

ONLY

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Dr. Ahmed El-Mahdy


-10 0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

f

Mag

nit

ud

e o

f W

(f)

MAGNITUDE SPECTRUM of analog signal

-10 0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

f

Mag

nit

ud

e o

f W

s(f

)

MAGNITUDE SPECTRUM of Flat-topped SAMPLED PAM signal

Spectrum of Flat Top Sampling

k

s

s

s )kf-W(fT

(f)W

fsinc

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Dr. Ahmed El-Mahdy


Difference between Ideal, Flat-Top Sampling and Natural Sampling

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Dr. Ahmed El-Mahdy


Pulse Amplitude Modulation• PAM is an engineering term that is used to describe the

conversion of the analog signal to a pulse type signal in which the amplitude of the pulse represents the analog information.

• Because it uses pulses, the BW of the PAM is wider than of the analog waveform.

Pulse Amplitude Modulation

Pulse Amplitude modulation that

uses flat-top sampling

Pulse Amplitude modulation that

uses natural sampling

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Dr. Ahmed El-Mahdy


Comparison between Natural Sampling and Flat-Top Sampling

- In flat top sampling the electronic circuit required for sampling are less complicated as compared to the one used in natural sampling because

it is easier to generate rectangular pulses with fixed amplitude (sample and hold).

- if message reconstruction by lowpass filtering: Natural samples results in no distortion but thereis a distortion when flat top pulses are involved because is multiplied by another function of

frequency ( )

)kf-W(f s

ff sinc)Q(

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Dr. Ahmed El-Mahdy


Why Flat-top sampling is so common in communication although it generates spectral distortion

The reasons for using flat top sampling in communications are:

(1) The rectangular pulse (in Time Domain) is an in easy

shape to generate.

(2) The distortion in the spectrum can be corrected in the

receiver by using a low pass filter followed by an

equalizing filter as shown in the fig:

(f) W(t)

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Dr. Ahmed El-Mahdy


• Questions

– What should be the sampling rate?

– Can we reconstruct the original signal after the sampling process?

Effect of Sampling on Frequency Content of Signals

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Effect of Sampling on Frequency Content of Signals

m(t)

t (sec.)

Representation of analog signal m(t) in

time domain

f (Hz)

M(f)

0 W-W

• Let assume that the frequency content of analog signal in the frequency domain is confined with W

• Define TS as the sampling interval

• Define fS as the sampling frequency

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Dr. Ahmed El-Mahdy


fS>2Wm(t)

f (Hz)

M(f)

W-W

TS=1/fSt (sec)

0 fS-fS fS-W fS+W-fS+W-fS-W

• By using a LPF at the receiver, it is possible to reconstruct the original signal from received samples

LPF

fcutoff

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Dr. Ahmed El-Mahdy


fS=2Wm(t)

f (Hz)

M(f)

W-W

TS=1/f

S

t (sec)

0 fS=2W-fS=-2W 3W-3W

• By using a LPF with fcutoff=W at the receiver, it is possible to reconstruct the original signal from received samples

LPF

fcutoff

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Dr. Ahmed El-Mahdy


fS<2Wm(t)

f (Hz)

M(f)

W-W

TS=1/f

S

t (sec)

0 fS-fS fS+W-fS-W

• It is no longer possible to reconstruct the original signal from received samples

-fS+W fS-W

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Dr. Ahmed El-Mahdy


Sampling Theorem

• Sampling Theorem states that

A band-limited signal of finite energy which has no frequency components higher than W Hz may be completely recovered from knowledge of its samples taken at the rate of fs=2W samples per second.

• fS=2W is called the Nyquist Rate

• Ts=1/2W is called the Nyquist interval

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Dr. Ahmed El-Mahdy


According to the Nyquist theorem, the

sampling rate must be

at least 2 times the highest frequency

contained in the signal.

Then

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Dr. Ahmed El-Mahdy


Practical Sampling Rates

• Speech:» Telephone quality speech has a bandwidth of 4

kHz

» Most telephone systems sample at 8000 samples/sec

• Audio:» The highest frequency the human ear can hear

is approximately 15 kHz

» CDs sample at rate 44,000 samples/sec

• Video:» The human eye requires samples at a rate of at

least 20 frames/sec to achieve smooth motion

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Dr. Ahmed El-Mahdy


A complex low-pass signal has a bandwidth of 200 kHz. What is the minimum

sampling frequency for this signal? If the signal is multiplied by a a cos with

frequency fc=1 MHz What is the minimum sampling frequency in this case?

SolutionThe bandwidth of a low-pass signal is between 0 and f,

where f is the maximum frequency in the signal.

Therefore, we can sample this signal at 2 times the

highest frequency (200 kHz). The sampling rate is

therefore 400 KHz.

-The maximum frequency in the signal now is 1200 kHz,

then the minimum sampling rate is: 2400 kHz

Example

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Dr. Ahmed El-Mahdy


-200 kHz 200 kHz

1200 kHz800 kHz-1200 kHz

-1000 kHz

-800 kHz

1000 kHz

Low Pass signal

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Dr. Ahmed El-Mahdy


Appendix: Proof of Spectrum of Flat-top Sampling Waveform

The output of the sample and hold circuit generates the sampled

signal:

:

2/||,0

2/||,1)(

:

)(

)

issignalresultingtheofTransformFourier

t

ttpth

isshapepulsethesamplingtopflatfor

andshapepulsesamplingthedenotesthwhere

k

sss kT-h(t )w(kT(t)w

ProofReading

ONLY

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Dr. Ahmed El-Mahdy


Proof of Spectrum of Flat-top Sampling Waveform

ProofReading

ONLY

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Dr. Ahmed El-Mahdy


Thank You

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