School of Electrical, Electronics and Computer Engineering University of Newcastle-upon-Tyne

School of Electrical, Electronics andComputer Engineering

University of Newcastle-upon-Tyne

Baseband Digital Modulation Baseband Digital Modulation

Prof. Rolando CarrascoProf. Rolando Carrasco

Lecture Notes University of Newcastle-upon-Tyne

2007

Baseband digital informationBaseband digital information

Bit-rate, Baud-rate and Bit-rate, Baud-rate and BandwidthBandwidth

BB1

denotes the duration of the 1 bitHence Bit rate =

bits per second

All the forms of the base band signalling shown transfer data at the same bit rate.

E denotes the duration of the shortest signalling element.Baud rate is defined as the reciprocal of the duration of the shortest signalling element .

Baud Rate = E1

baud

In general Baud Rate ≠ Bit Rate

For NRZ : Baud Rate = Bit Rate

RZ : Baud Rate = 2 x Bit Rate

Bi-Phase: Baud Rate = 2 x Bit Rate

AMI: Baud Rate = Bit Rate

Non Return to Zero (NRZ)Non Return to Zero (NRZ)

The highest frequency occurs when the data is 1010101010…….i.e.

This sequence produces a square wave with periodic time E 2

Fourier series for a square wave,

If we pass this signal through a LPF then the maximum bandwidth would be 1/T Hz, i.e. to just allow the fundamental (1st harmonic) to pass.

Non Return to Zero (NRZ) Non Return to Zero (NRZ) (Cont’d)(Cont’d)

The data sequence 1010…… could then be completely recovered

Hence the minimum channel bandwidth

RateBaudSinceRateBaud

TB

EE

1

22

11min

Return to Zero (RZ)Return to Zero (RZ)

Considering RZ signals, the max frequency occurs when continuous 1’s are transmitted.

This produces a square wave with periodic time E 2

2min

RateBaudfB U

If the sequence was continuous 0’s, the signal would be –V continuously, hence

''DCfL

.

Bi-PhaseBi-Phase

Maximum frequency occurs when continuous 1’s or 0’s transmitted.

E1

2min

RateBaudfB U

This is similar to RZ with Baud Rate = = 2 x Bit rate

The minimum frequency occurs when the sequence is 10101010…….e.g.

B E

2min

RateBaudfB L

In this case =

Baud Rate = Bit rate

Digital Modulation and Digital Modulation and NoiseNoise

The performance of Digital Data Systems is dependent on the bit error rate, BER, i.e. probability of a bit being in error.

NasNbitsTotal

EErrorsofNoP

Digital Modulation

There are four basic ways of sending digital data

The BER (P) depends on several factors• the modulation type, ASK FSK or PSK• the demodulation method• the noise in the system• the signal to noise ratio

Prob. of Error or BER,


Amplitude Shift Keying ASK


Frequency Shift Keying FSK


Phase Shift Keying PSK

System Block diagram for System Block diagram for AnalysisAnalysis

DEMODULATOR – DETECTOR – DECISION DEMODULATOR – DETECTOR – DECISION

For ASK and PSK

Demodulator-Detector-DecisionDemodulator-Detector-Decision

FOR FSK

DemodulatorDemodulator

Demodulator Cont’d)Demodulator Cont’d)

TRCdesignHence

dtVRC

V INout

1

Detector-DecisionDetector-Decision

1V 0V - is the voltage difference between a ‘1’ and ‘0’.

)22

( 21 VVVREF

Detector-Decision (Cont’d)Detector-Decision (Cont’d)

ND is the noise at the Detector input.

Probability of Error,

DNerf

221

2

1

Hence

0 v1v0 v

0-

P(v0)

vn

Probability density of binary signalProbability density of binary signal

v0v1

2

210

2

)(

02

1

2

1)(

vv

n evP

)(1 nvP

vn

n

vv

vve dveP

n2

20

10

2

)(

2

12

1

Using the change of variable2

0vvx n

Probability density function of noiseProbability density function of noise

(*)

DN2

22

1

01

21

vv

dxxe eP

dxezerfcz

x

22

)(

222

1 011

vverfcPe

This becomes

The incomplete integral cannot be evaluated analytically but can be recast as a complimentary error function, erfc(x), defined by

Equations (*) and (**) become

n

vvvv

e

e

dveP

vverfP

zerfzerfc

n2

21

10

2

)(2

0

011

2

1

221

2

1

)(1)(

(**)

It is clear from the symmetry of this problem that Pe0 is identical to Pe1 and the probability of error Pe, irrespective of whether a ‘one’ or ‘zero’ was transmitted, can be rewritten in terms of v = v1 – v0

22

12

1

v

erfPe

for unipolar signalling (0 and v)

for polar signalling (symbol represented by voltage 2

v


PSKFSKASKOptimumFor

PRK

N

SerfPSK

N

SerfFSK

OOK

N

SerfASK

IN

INe

IN

INe

IN

INe

,,

12

1

21

2

1

41

2

1

dB/10in SNR10

ePePeP

SNR in wattASK FSK PSK

000.002415.848912

00.00080.012710.0010

0.00020.0060.03796.30968

0.00240.0230.07913.98116

0.01250.05650.13122.51194

0.03750.1040.18671.58492

0.07860.15870.23981.000

Linear gainSNR in dB

000.002415.848912

00.00080.012710.0010

0.00020.0060.03796.30968

0.00240.0230.07913.98116

0.01250.05650.13122.51194

0.03750.1040.18671.58492

0.07860.15870.23981.000

Linear gainSNR in dB

Probability of Symbol Error

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 2 4 6 8 10 12 14

SNR in dB

Pro

bab

ility

of

Sym

bo

l Err

or

ASK

FSK

PSK


FM/ FSK Demodulation

One form of FM/FSK demodulator is shown below

In general VIN (t) will be

tCosVtV INcIN )(

IN ININ f 2Where is the input frequency (rad/sec)

ttCosttCosV

V

BACosBACosCosBCosASince

tCosVtCosVV

tVtVV

ININININc

x

INcINcx

ININx

2

2

1

)(.

2

FM/ FSK Demodulation (Cont’d)

INININc

x

ININININININc

x

CostCosV

V

ttCosttCosV

V

22

22

2

)2(

2

)1(222

2

2

tCosV

and

tCosV

INc

INc

i.e

Thus there are two components

Component (1) is at frequency 2 fIN Hz and component (2) is effectively a ‘DC’ voltage if

IN is constant.

The cut-off frequency for the LPF is designed so that component (1) is removed and component (2) is passed to the output.

tCosV

V INc

OUT 2

2


The V/F characteristics and inputs are shown belowAnalogue FM

ccDCc

mmDCout

mmDCIN

DCIN

INout

mc

fTVf

ftCosVVfei

tCosVVV

tmVV

fVf

cxmy

Vf

1,

..

)(

0

0

Modulation Index m

m

m

c

f

V

f

f


tnCosJVtVFM mcn

ncs

1

)()(

The spectrum of the analogue FM signal depends on and is given by

Digital FSK

ccDCc

DC

DC

DCIN

DCIN

DCIN

INout

fTVf

sforfVVf

sforfVVf

sforVVV

sforVVV

tmVV

fVf

cxmy

1,

'0

'1

'0

'1

)(

000

011

0

1

0

Normalized frequency Deviation ratio

0101 .. ffModulusei

R

ffh

b

The spectrum of FSK depends on h

Digital FSK (Cont’d)

FM/ FSK Demodulation (Cont’d)FM/ FSK Demodulation (Cont’d)

Consider again the output from the demodulator INc

OUT CosV

V2

2

4cT

cc f

T1

cfThe delay is set to where and is the nominal carrier frequency

c

INcOUT f

fCos

VV

4

2

2

2 Hence

c

INcOUT f

fCos

VV

22

2

FM/ FSK Demodulation (Cont’d)FM/ FSK Demodulation (Cont’d)

The curve shows the demodulator F/V characteristics which in this case is non linear.

Practical realization of F/V processPractical realization of F/V process

The comparator is LIMITER – which is a zero crossing detector to give a ‘digital’ input to the first gate.

This is form of ‘delay and multiply’ circuit where the delay is set by C and R with

= CR

Practical realization of F/V process (Cont’d)Practical realization of F/V process (Cont’d)


INf cf

Consider now

≠


c

INOUT f

fAEV

4 Plotting Vout versus

INf (Assuming A=1)

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School of Electrical, Electronics and Computer Engineering University of Newcastle-upon-Tyne