Geometric representation of modulation signals
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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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Geometric Representation of Modulation Signals

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Geometric representation of modulation signals

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:

  • 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.


Geometric representation of modulation signals1

Geometric Representation of Modulation Signals

  • 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.


Geometric representation of modulation signals

Geometric Representation of Modulation Signals

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


Example

Q

I

Example

Two signal

waveforms to

be used for

transmission

The basis signal

One dimensional

Constellation Diagram


Geometric representation of modulation signals

Q

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


Geometric representation of modulation signals

binary symbol

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


Geometric representation of modulation signals

Q

= 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


Constellation diagram

Constellation Diagram

  • 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).


Linear modulation techniques

Linear Modulation Techniques

  • 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 envelope modulation

Constant Envelope Modulation

  • 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

    • - simplified receiver design

    • high immunity against random FM noise & fluctuations

    • from Rayleigh Fading

  • larger occupied bandwidth than linear modulation


Constant envelope modulation1

Constant Envelope Modulation

  • Frequency Shift Keying

  • Minimum Shift Keying

  • Gaussian Minimum Shift Keying


Frequency shift keying fsk

Frequency Shift Keying (FSK)

  • Binary FSK

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

  • Discontinuous / Continuous Phase


Geometric representation of modulation signals

input data

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


Geometric representation of modulation signals

Continuous Phase FSK

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)


Fsk example

x

a0

a1

0

1

VCO

Data

1 1 0 1

FSK

Signal

0 1 1

modulated composite

signal

cos wct

FSK Example


Geometric representation of modulation signals

  • 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


Geometric representation of modulation signals

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


Geometric representation of modulation signals

General FSK signal and orthogonality

  • 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)

?


Geometric representation of modulation signals

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)


Geometric representation of modulation signals

  • 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


Geometric representation of modulation signals

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


Geometric representation of modulation signals

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)


Geometric representation of modulation signals

θ(t) - (0) rads

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


Geometric representation of modulation signals

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


Geometric representation of modulation signals

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


Geometric representation of modulation signals

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


Geometric representation of modulation signals

0 ≤ t ≤ T

s1(t) =

= 0 otherwise

0 ≤ t ≤ T

=

=

s2(t) =

= 0 otherwise

2(t)

0

1

1(t)

BFSK Constellation


Geometric representation of modulation signals

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


Geometric representation of modulation signals

Matched Filter

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:


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