I. Previously on IET. © Tallal Elshabrawy 2 Basic Blocks of Digital Communications Source of continuous- time (i.e., analog) message signal Low pass Filter

I. Previously on IET

© Tallal Elshabrawy 2

Basic Blocks of Digital Communications

Source of continuous-time

(i.e., analog) message signal

Low pass Filter

Sampler Quantizer Encoder Band Pass Modulated Signal

Analog-to-Digital Converter

m-ary SymbolEncoder

Transmitting Filter

Modulation


Time-Limited Signal = Frequency Unlimited Spectrum

TS0

Fourier Transfor

m

It is desirable for transmitted signals to

be band-limited (limited frequency

spectrum)

Guarantee completely

orthogonal channels for pass-band signals

WHY?

Square Pulse is a Time-Limited Signal

1/TS 2/TS 3/TS-1/TS-2/TS-3/TS

0


Inter-symbol Interference (ISI)

Sampling Instants

yk(t) yk(iTS)

Frequency Limited Spectrum=Time-Unlimited Signals

A time unlimited signal means inter-symbol interference (ISI)

Neighboring symbols affect the measured value and the corresponding decision at sampling instants


Nyquist Criterion for No ISI

For a given symbol transmitted at iTS

S

A k=1z kT

0 o/w

Transmitting Filter g(t)

sk(t)

+

wk(t)

Receiving Filter g (TS-

t)

xk(t)yk(t)

yk(TS)ku

Sample at t=TS

Assume AWGN Noise wk(t) is negligible

Transmitting Filter g(t)

Receiving Filter g (TS-

t)

yk(t) yk(TS)

Sample at t=TS

z(t)=g(t)* g(TS-t)

ku


z(t): Impulse

Response

Z(f): Spectrum(Transfer Function)

T: symbol interval

RS: symbol rater: roll-off factor

Z(f)

Raised Cosine Filter Bandwidth = RS(1+r)/2

Pulse-shaping with Raised-Cosine Filter


Examples

An analog signal of bandwidth 100 KHz is sampled according to the Nyquist sampling and then quantized and represented by 64 quantization levels. A 4-ary encoder is adopted and a Raised cosine filter is used with roll off factor of 0.5 for base band transmission. Calculate the minimum channel bandwidth to transfer the PCM wave

An analog signal of bandwidth 56 KHz is sampled, quantized and encoded using a quaternary PCM system with raised-cosine spectrum. The rolloff factor is 0.6. If the total available channel bandwidth is 2048 KHz and the channel can support up to 10 users, calculate the number of representation levels of the Quantizer.


Phase Shift Keying (PSK) Modulation

1 0 1 1 0 1Base band Signal

X(t)

Band Pass SignalY(t)

X(t) x Y(t)

cos(2πfct)c cj2πf t j2πf te e

2


PSK Demodulation

X(t)cos(2πfct)x

2cos(2πfct)

X(t)[2cos2(2πfct)]

Low Pass Filter X(t)

X(t)[2cos2(2πfct)] =X(t)[1+cos(4πfct)]

X(t)[2cos2(2πfct)]=X(t) +X(t)cos(4πfct)]

Base band Signal(i.e., low frequency content)

High frequency content


Orthogonality of sin and cos Functions

X(t)cos(2πfct)

x

2sin(2πfct)

X(t)[2sin(2πfct)cos(2πfct)]

Low Pass Filter 0

X(t)[2sin(2πfct)cos(2πfct)]=X(t) sin(4πfct)]

High frequency content

© Tallal Elshabrawy

XI(t)

x

cos(2πfct)

XI(t)cos(2πfct)

XQ(t)

x

sin(2πfct)

XQ(t)sin(2πfct)

+Y(t)

Serial-to-

Parallel

X(t)

Quadrature- PSK Modulation (QPSK)

11


QPSK Demodulation

x Low Pass FilterXI(t)

xXQ(t)

Y(t) Parallel-to-Serial

X(t)

Low Pass Filter

2cos(2πfct)

2sin(2πfct)


Modulation in Time-Limited Communications

13

BinaryEncoder

Transmitting Filter

CosineModulation

1

0

TS

Time Representation

Frequency Representation

1 ES=(1)2×1=1

cS

2Cos 2πf t

TBinary

Symbols

Time Representation


Rectangular Filter In Phase Modulation

0

TS

f 0 ffc-fc

TS

-1

0

-TS

f

0

f

fc-fc

S

S S

2T

S cS0

T T2

S c cS S0 0

2E Cos 2πf t dt

T

2 2 1E Cos 2πf t dt 1 Cos 4πf t dt

T T 2

ES=(-1)2×1

S

2

T

S

2

T

S

2

T

S

2

T

ST

2

ST

2


Modeling of In phase Modulation

BinaryEncoder

Transmitting Filter

CosineModulation

cS

2A Cos 2πf t

T

A-A

cS

2A Cos 2πf t

T

ES=A2


Modulation in Band-Limited Communications

15

BinaryEncoder

Transmitting Filter

CosineModulation

1

0

Time Representation

Frequency Representation 1/RS

ES=(1)2×1=1

Binary Symbols

Time Representation


Raised Cosine Filter

0fc-fc

ES=(-1)2×1

1

t

fRS/2-RS/2 fc+ RS/2-fc- RS/2

f

SS

12 T

2R

In Phase Modulation

t

S

2

T

t

-1

cS

2Cos 2πf t

T

S

2

T

t

fc- RS/2-fc- RS/2f

fc fc+ RS/2-fc -fc+ RS/2 0

SS

12 T

2R

0

0-RS/2 RS/2

-1/RS

2

S

S S2S S

E G f df

1 2E 2R 1

4R T

Bit Rate = RS

Bandwidth = RS

1 b/s/Hz

-fc+ RS/2 fc- RS/2



BinaryEncoder

Transmitting Filter

CosineModulation

( )tfπ2CosT2

×A cS

A-A

cS

2A Cos 2πf t

T

ES=A2


Modulation in Time-Limited Communications

17

BinaryEncoder

Transmitting Filter

SineModulation

1

0

TS

Time Representation


1 ES=(1)2×1=1

( )tfπ2SinT2

× cS

Binary Symbols

Time Representation


Rectangular Filter In Quadrature Modulation

0

TS

f

0

f

fc

-fc

TS

-1

0

-TS

f

S

2T

S cS0

2E Sin 2πf t dt

T

ES=(-1)2×1

-fc

fc

f0

ST2

S

2

T

ST2

S

2

T

S

2j

T

S

2j

T

ST2

j

S

2j

T



BinaryEncoder

Transmitting Filter

SineModulation

( )tfπ2SinT2

×A cS

jA

-jA cS

2A Sin 2πf t

T

ES=A2


Modulation in Band-Limited Communications

19

BinaryEncoder

Transmitting Filter

SineModulation

1

0

Time Representation

Frequency Representation 1/RS

ES=(1)2×1=1

Binary Symbols

Time Representation


Raised Cosine Filter

0

fc

-fc

ES=(-1)2×1

1

t

fRS/2-RS/2

fc+ RS/2

-fc- RS/2f

SS

T2×R21

j

t

ST2

t

-1S

2

T

t

fc- RS/2

-fc- RS/2f

fc fc+ RS/2

-fc -fc+ RS/2 0

SS

1j 2 T2R

0

0-RS/2 RS/2

-1/RS

2

S

S S2S S

E G f df

1 2E 2R 1

4R T

Bit Rate = RS

Bandwidth = RS

1 b/s/Hz

fc- RS/2

-fc+ RS/2

SS

1j 2 T2R

SS

T2×R21

j

( )tfπ2SinT2

× cS

In Quadrature Modulation



BinaryEncoder

Transmitting Filter

SineModulation

( )tfπ2SinT2

×A cS

jA

-jA cS

2A Sin 2πf t

T

ES=A2


Modulation Constellations

21

BPSK QPSK

8-QPSK 16 QAM

1 b/s/Hz 2 b/s/Hz

4 b/s/Hz3 b/s/Hz

SE

( )2/E,2/E SS S SE / 2, E / 2

S SE / 2, E / 2 S SE / 2, E / 2

( )0,ES SE ,0

( )2/E,2/E SS S SE / 2, E / 2

S SE / 2, E / 2 S SE / 2, E / 2

( )SE,0

S0, E

SE


Basic Communication Model in AWGN

Detection Performance: Correct Detection

S = S*

Erroneous Detection S ≠ S*

+S

N

Channel Model

R=S+N

TXR

RX DetectionS*


BPSK Modulation over AWGN Channels

( )tfπ2CosT2

×E cS

S S cS

2E Cos 2πf t

T

ES Energy per Symbol S SS E , E

SESE



Gaussian Noise

2

2

n

2σN 2

1f n e

2πσ

N ,

0


Received signal distribution given transmitted


R , SE

0

2S

2

S

r E

2σR E 2

1f r e

2πσ

SESE



Error Calculation given transmitted] [∞,∞R -∈SE

0

SESE

S

Se EP Pr r 0 Pr n E

2S

2

S

E n

2σe E 2

1P e dn

2πσ

2z

2

x

1Q x e dz

2π

2 2S

2 2

S

E n n

2σ 2σ

2 2E

1 1e dn = e dn

2πσ 2πσ

Symmetry of Gaussian Distribution

Let S

S

Enz ,n E z ,dn σdz

σ σ

2 2

S

S S

z z22 2

Se E 2E σ E σ

σ 1P e dz e dz Q E σ

2π2πσ

1Q x erfc x 2

2

S

2Se E

1P erfc E 2σ

2

2S

2

S

r E

2σR E 2

1f r e

2πσ


Received signal distribution given transmitted


SE-

0

2S

2

S

r E

2σR E 2

1f r e

2πσ

SESE

] [∞,∞R -∈



Error Calculation given transmittedSE-

0

SESE-

S

Se EP Pr r 0 Pr n E

2

2

S

S

n

2σe - E 2

E

1P e dn

2πσ

Let S

S

Enz ,n E z ,dn σdz

σ σ

2 2

S

S S

z z22 2

Se - E 2E σ E σ

σ 1P e dz e dz Q E σ

2π2πσ

S

2Se E

1P erfc E 2σ

2

2S

2

S

r E

2σR E 2

1f r e

2πσ

] [∞,∞R -∈

2z

2

x

1Q x e dz

2π

1Q x erfc x 2

2



Signal Power & Symbol Error Performance

SE SE

0

S eE P



Signal Power & Symbol Error Performance

SE SE

0

2eσ P


BER of PSK over AWGN Channels

Notes: Define N0 Total Noise Power

N0/2 Noise Power over Cosine axis, i.e., σ2=N0/2

Each symbol corresponds to a single bit Eb = ES

Pb = Pe

S S

e S Se E e E

2 2 2e S S S

P Pr E P Pr E P

1 1 1 1 1P erfc E 2σ erfc E 2σ erfc E 2σ

2 2 2 2 2

( )0bb NEerfc21

=P

Documents

I. Previously on IET. © Tallal Elshabrawy 2 Basic Blocks of Digital Communications Source of continuous- time (i.e., analog) message signal Low pass Filter