1

Elements of a Digital Communication System

• Block diagram of a communication system:

Information source and

input transducer

Channel decoder

Output transducer

Channel

Digital modulator

Channel encoder

Source encoder

Source decoder

Digital demodulator

2

Mathematical Models for Communication Channels

• Additive Noise Channel:– In presence of attenuation:

+

S(t)

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

n(t)

channel

)()()()( tntsttr

3

Mathematical Models for Communication Channels

• The Linear filter channel:

)()()()()(*)()( tndtsctntctstr

Linear filter c(t) +

channel

s(t)

n(t)

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

4

Mathematical Models for Communication Channels

• Linear Time-Variant Filter Channel:– Are charachterized by a time-variant channel impulse response

);( tc

Linear time_variant filter +

channel

s(t)

n(t)

r(t)

5

Representation of Band-Pass Signals and Systems

• Representation of Band-Pass Signals:

Energy of the signal:

• Representation of Linear Band-Pass Systems:

• Response of a Band-Pass System to a Band-Pass Signal:

)2sin()()2cos()()( tftytftxts cc

dttsl2)(

2

1

tfjl

tfjl

cc ethethth 22 )()()(

tfjl

cetrtr 2)(Re)(

6

Orthogonal Expansion of Signals

• We can express M orthonormal signals as a Linear combination of basis functions and hence can be defined as

• Linear digitally modulated signals can be expanded in terms of two orthonormal basis functions given by:

and

)(tsn )(tf n

Mktfsts n

N

nknk ,..,2,1),()(

1

tfT

tf c2cos2

)(1 tfT

tf c2sin2

)(2

7

Representation of Digitally Modulated Signals

• Pulse-amplitude-modulated Signals (PAM):

• Phase-modulated signals (PSK):

• Quadrature amplitude modulation (QAM):

tfjmm

cetgAts 2)(Re)( m=1,2,…M

tfmM

tgtfmM

tgts ccm 2sin)1(2

sin)(2cos)1(2

cos)()(

tfjmsmcm

cetgjAAts 2)()(Re)( m=1,2,..,M,

tftgAtftgA cmscmc 2sin)(2cos)(

8

Representation of Digitally Modulated Signals

• Orthogonal multidimensional signals:

• Biorthogonal signals:• Simplex signals:• m=1, 2,…, M..

• Signal waveforms from binary codes:

cTt 0

mNmmm cccC ...21

,' sss mm

mNmm ccC ....1

tf

TtSc c

c

cmjmj 2cos

2)(1

tf

TtSc c

c

cmjmj 2cos

2)(0 cTt 0

9

Optimum Receivers Corrupted by additive White Gaussian Noise- I

• General Receiver:

Receiver is subdivided into:– 1. Demodulator.

• (a) Correlation Demodulator.

• (b) Matched Filter Demodulator.

– 2. Detector.

+

Sm(t)

r(t) = sm(t) + n(t)

n(t)

channel

)0(),()()( Tttntstr m

10

Optimum Receivers Corrupted by additive White Gaussian Noise- II

• Correlation Demodulator:– Decomposes the received signal and noise into a series of

linearly weighted orthonormal basis functions.

• Equations for correlation demodulator:

dttftntsdttftrrk

T T

mkk )()()()()(0 0 Nk ,...2,1

,)()(0

dttftss k

T

mmk

,)()(0

dttftnn k

T

km

11

Optimum Receivers Corrupted by additive White Gaussian Noise- III

• Matched Filter Demodulator:– Equation of a matched filter:

• Output of the matched filter is given by:

k=1,2, …N

),()( tTfth kk Tt 0

dthtrty k

T

k )()()(0

dtTftr k

T)()(

0

12

Optimum Receivers Corrupted by additive White Gaussian Noise- IV

• Optimum Detector:– The optimum detector should make a decision on the

transmitted signal in each signal interval based on the observed vector.

• Optimum detector is defined by:

m=1,2,… M

or

N

nmnmn

N

nn

N

nnm ssrrsrD

1

2

11

2 2),(

,222

mm ssrr

),( msrD2

2 mm ssr

13

OFDM

• It is a block modulation scheme where data symbols are transmitted in parallel by employing a large number of orthogonal sub-carriers.

• Equation of complex envelope of the OFDM signal:

• where

n

nxnTtbAts ),()(

1

0

)21

(2exp)(),(

N

knan T

tN

kjxthxtb

k

14

General FFT based OFDM system-I

• Block diagram of FFT based OFDM transmitter :

• Equations at the transmmitter end:

)(2

exp)(1

0

tuNT

ktjxAts r

s

N

kk

,2

exp)(1

0

s

N

kksn N

knjxAnTsX

X0

XN-1XN-2...X1X0

IFFT

XN-1XN-2...X1

Insert Cyclic prefix

D/A

D/A

nX gnX

gInX

gQnX

)(~ tsl

)(~ tsQ

kX

15

General FFT based OFDM system-II

• Block diagram of FFT based OFDM receiver:

• At the demodulator:

1

0

21 N

n

N

nij

ni eRN

Z

R0

ZN-1ZN-2...Z1Z0

FFT

RN-1RN-2...R1

Serial metric computer

nZ

)( mv

Remove cyclic prefixA/D

A/D)(~ trl

)(~ trQ

nkR ,

QnR

L

m

N

mij

mi eg0

2

iii AxZ

16

General FFT based OFDM system-II

• Merits of OFDM:

– 1. the modulation and the demodulation can be achieved in the frequency-domain by using a DFT.

– 2. the effects of ISI can be eliminated with the introduction of the guard interval.

17

IMPLEMENTATION OF OFDM SYSTEM-I

• Basic implementation of OFDM system:

iii AxZ

1

0

21 N

n

N

nij

ni eRN

Z

L

m

N

mij

mi eg0

2

Serial To Parallel

Converter

BPSK

Detector

BPSK

BPSK

+

+

+

Bits

0

1

127

R0

R127

R1

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X0

X1

X127

n0

n1

n127

18

SIMULATION RESULTS.

• Perfomance charachteristics were obtained for the simulated OFDM system.

10 15 20 25 30 35 40 45 5010

-6

10-5

10-4

10-3

10-2

10-1

Non Fading ChannelFading Channel

19

Conclusion.

• 1. OFDM communication system exhibits better Pe Vs SNR curves in case of Non-Fading channel as compared to the Fading channel.

• 2. As the value of the SNR is increased the value of Pe gradually decreases.

• 3. Perfomance charachteristics of simulated OFDM communication system are consistent with the performance charachteristics of the general OFDM communication system.