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# Geometric Representation of Modulation Signals - PowerPoint PPT Presentation

Geometric Representation of Modulation Signals. Digital Modulation involves Choosing a particular signal waveform for transmission for a particular symbol For M possible symbols, the set of all signal waveforms are:.

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Presentation Transcript

• Digital Modulation involves

• Choosing a particular signal waveform for transmission for a particular symbol

• For M possible symbols, the set of all signal waveforms are:

• For binary modulation, each bit is mapped to a signal from a signal set S that has two signals.

• We can view the elements of S as points in vector space.

• Vector space

• We can represent the elements of S as linear combination of basis signals i (t).

• The number of basis signals is the dimension of the vector space.

• Basis signals are orthogonal to each-other.

• Each basis is normalized to have unit energy.

Let {j(t)| j = 1,2,…,N} represent a basis ofSsuch that

(1) Any symbol, si(t) 

si(t)=

(2) Basis signals are orthogonal to each other in time

(3) Each basis signal is normalized to have unit energy

E =

Basis signals  Coordinate system for S

Gram-Schmidt process  systematic way to obtain basis for S

I

Example

Two signal

waveforms to

be used for

transmission

The basis signal

One dimensional

Constellation Diagram

Q

/2

M1 =

I

I

0

3/4

/4

3/2

7/4

54

M2 =

QPSK Constellation Diagram

Rotation by /4 obtain new QPSK signal set

Es = 2Eb

grey coded

QPSK signal

si1

si2

10

7π/4

11

5π/4

01

3π/4

00

π/4

binary symbol

grey coded

QPSK signal

si1

si2

10

3π/2

0

11

π

0

01

π/2

0

00

0

0

si(t) = si1,1(t) + si22(t)

Signal Space Characterization of

QPSK Signal Constellations

ithQPSK signal, based on message points (si1, si2) defined in tables

for i = 1,2 and 0 ≤ t ≤ Ts

= possible states forkfork-1= n/4

= possible states for kfork-1= n/2

I

possible signal transitions

/4 QPSKmodulation

• modulated signal selected from 2 QPSK constellations shifted by /4

• for each symbol  switch between constellations –total of 8 symbols

• states 4 used alternately

• phase shift between each symbol =nk= /4 , n = 1,2,3

• - ensures minimal phase shift, k= /4 between successive symbols

• - enables timing recovery & synchronization

• Properties of Modulation Scheme can be inferred from the Constellation Diagram:

• Bandwidth occupied by the modulation increases as the dimension of the modulated signal increases.

• Bandwidth occupied by the modulation decreases as the signal_points per dimension increases (getting more dense).

• Probability of bit error is proportional to the distance between the closest points in the constellation.

• Euclidean Distance

• Bit error decreases as the distance increases (sparse).

• Digital modulation techniques classified as:

• Linear

• The amplitude of the transmitted signal varies linearly with the modulating digital signal, m(t).

• They usually do not have constant envelope.

• More spectrally efficient.

• Poor power efficiency

• Example: QPSK.

• Non-linear / Constant Envelope

• constant carrier amplitude - regardless of variations in m(t)

• Better immunity to fluctuations due to fading.

• Better random noise immunity.

• improved power efficiency without degrading occupied spectrum

• - use power efficient class Camplifiers (non-linear)

• low out of band radiation (-60dB to -70dB)

• use limiter-discriminator detection

• high immunity against random FM noise & fluctuations

• larger occupied bandwidth than linear modulation

• Frequency Shift Keying

• Minimum Shift Keying

• Gaussian Minimum Shift Keying

• Binary FSK

• Frequency of the constant amplitude carrier is changed according to the message state  high (1) or low (0)

• Discontinuous / Continuous Phase

phase jumps

cos w1t

switch

cos w2t

=

sBFSK(t)= vH(t)

binary 1

binary 0

sBFSK(t)= vL(t)

=

Discontinuous Phase FSK

Switching between 2 independent oscillators for binary 1 & 0

• results in phase discontinuities

• discontinuities causes spectral spreading & spurious transmission

• not suited for tightly designed systems

single carrier that is frequency modulated using m(t)

sBFSK(t) =

=

where (t) =

• m(t) = discontinuous bit stream

• (t) = continuous phase function proportional to integral of m(t)

a0

a1

0

1

VCO

Data

1 1 0 1

FSK

Signal

0 1 1

modulated composite

signal

cos wct

FSK Example

• complex envelope of BFSK is nonlinear function of m(t)

• spectrumevaluation - difficult - performed using actual time

• averaged measurements

• PSD of BFSK consists of discretefrequency components at

• fc

• fc nf , n is an integer

• PSD decay rate (inversely proportional to spectrum)

• PSD decay rate for CP-BFSK 

• PSD decay rate for non CP-BFSK 

• f = frequency offset fromfc

Spectrum & Bandwidth of BFSK Signals

• Transmission Bandwidth of BFSK Signals (from Carson’s Rule)

• B = bandwidth of digital baseband signal

• BT = transmission bandwidth of BFSK signal

• BT= 2f +2B

• assume 1st null bandwidth used for digital signal, B

• - bandwidth for rectangular pulses is given by B = Rb

• - bandwidth of BFSK using rectangular pulse becomes

• BT = 2(f + Rb)

• if RC pulseshaping used, bandwidth reduced to:

• BT = 2f +(1+) Rb

• Two FSK signals, VH(t) and VL(t) are orthogonal if

?

• interference between VH(t) and VL(t) will average to 0 during

• demodulation and integration of received symbol

• received signal will contain VH(t) and VL(t)

• demodulation of VH(t) results in (VH(t) + VL(t))VH(t)

?

vH(t) =

vL(t) =

and

then

and

vH(t)vL(t) =

=

=

=

An FSK signal for 0 ≤ t ≤ Tb

vH(t)vL(t) are orthogonal if Δf sin(4πfcTb) = -fc(sin(4πΔf Tb)

• consider binary CPFSK signal defined over the interval 0 ≤ t ≤ T

s(t) =

• θ(t) = phaseof CPFSK signal

• θ(t) is continuous s(t) is continuous at bit switching times

• θ(t) increases/decreases linearly with t during T

θ(t) = θ(0) ±

‘+’ corresponds to ‘1’ symbol

‘-’ corresponds to ‘0’ symbol

h = deviation ratio of CPFSK

CPFSK Modulation

elimination of phase discontinuity improves

spectral efficiency & noise performance

0 ≤ t ≤ T

2πfct +θ(0) +

= 2πf2 t+θ(0)

f1 =

2πfct +θ(0) -

= 2πf1t+θ(0)

fc=

f2 =

yields

and

thus

h = T(f2 – f1)

To determine fc and h by substitution

• nominal fc= mean of f1 and f2

• h≡f2 – f1 normalized by T

At t = T

θ(T) = θ(0) ± πh

kFSK=

symbol‘1’  θ(T) - θ(0) = πh

symbol‘0’  θ(T) - θ(0) = -πh

• peak frequency deviation F = |fc-fi | =

‘1’ sent  increases phase of s(t) by πh

‘0’ sent  decreases phase of s(t) by πh

• variation ofθ(t)with t follows a path consisting of straight lines

• slope of lines represent changes in frequency

FSK modulation index =kFSK (similar to FM modulation index)

3πh

2πh

πh

0

-πh

-2πh

-3πh

0 T 2T 3T 4T 5T 6T t

Phase Tree

•  depicted from t = 0

• phase transitions across

• interval boundaries of

• incoming bit sequence

• θ(t) - θ(0) = phase of CPFSK signal is even or odd multiple of πh at even or odd multiples of T

θ(t) - (0)

π

0

-2π

-3π

0 T 2T 3T 4T 5T 6T t

θ(t) = θ(0) ±

0 ≤ t ≤ T

Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK

1 0 0 0 0 1 1

•  thus change in phase over T

• is either πor -π

• change in phase of π = change in phase of -π

• e.g. knowing value of bit i doesn’t help to find the value of bit i+1

fi=

nc = fixed integer

assume fi given by as

si(t) =

0 ≤ t ≤ T

for i = 1, 2

si(t) =

0 ≤ t ≤ T

for i = 1, 2

= 0 otherwise

= 0 otherwise

• CPFSK = continuous phase FSK

• phase continuity during inter-bit switching times

1(t) =

2(t) =

0 ≤ t ≤ T

0 ≤ t ≤ T

= 0 otherwise

= 0 otherwise

for i = 1, 2

i(t) =

0 ≤ t ≤ T

= 0 otherwise

BFSK constellation: define two coordinates as

let nc = 2 and T = 1s (1Mbps) then f1= 3MHz,f2 = 4MHz

0 ≤ t ≤ T

s1(t) =

= 0 otherwise

0 ≤ t ≤ T

=

=

s2(t) =

= 0 otherwise

2(t)

0

1

1(t)

BFSK Constellation

cos wct

output

+

-

Decision

Circuit

r(t)

sin wct

Probability of error in coherent FSK receiver given as:

Pe,BFSK =

Coherent BFSK Detector

• 2 correlators fed with local coherent reference signals

• difference in correlator outputs compared with threshold to

• determine binary value

fH

Envelope

Detector

+

-

r(t)

output

Tb

Decision

Circuit

Envelope

Detector

Matched Filter

fL

Pe,BFSK, NC =

Non-coherent Detection of BFSK

• operates in noisy channel without coherent carrier reference

• pair of matched filters followed by envelope detector

• - upper path filter matched to fH (binary 1)

• - lower path filter matched to fL (binary 0)

• envelope detector output sampled at kTb compared to threshold

Average probability of error in non-coherent FSK receiver: