1 combined linear & constant envelope modulation m-ary modulation: digital baseband data sent by...

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Combined Linear & Constant Envelope Modulation

M-ary modulation: digital baseband data sent by varying RF carrier’s

(i) envelope ( eg. MASK)

(ii) phase /frequency ( eg. MPSK, MFSK)

(iii) envelope & phase offer 2 degrees of freedom ( eg. MQAM)

(i) n bits encoded into 1 of M symbols, M 2n

(iii) a signal, si(t) , sent during each symbol period, Ts = n.Tb

(ii) each symbol mapped to signal si(t), M possible signals:

s1(t),…,sM(t)

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M-ary modulation is useful in bandlimited channels

• greater B log2M

• significantly higher BER

- smaller distances in constellation

- sensitive to timing jitter

MPSK

MQAM

MFSK

OFDM

Combined Linear & Constant Envelope Modulation

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Mary Phase Shift Keying

Carrier phase takes 1 of M possible values – amplitude constant

i = 2(i-1)/M, i = 1,2,…M

Modulated waveform:

si(t) =

)1(

22cos

2i

Mtf

T

Ec

s

s 0 t Ts, i = 1,2,…M

Es = log2MEb energy per symbol

Ts = log2MTb symbol period

written in quadrature form as:

si(t) = tfM

iEtfM

iE cscs 2sin

2)1(sin2cos

2)1(cos

for i = 1,2,…M

2Ts

Basis Signal ?

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1(t) = tfT c

s2cos

2

defined over 0 t Ts

2(t) = tfT c

s2sin

2

Orthogonal basis signals

sMPSK(t) =

)(

2)1(sin)(

2)1(cos 21 tiEtiE ss

i = 1,2,…M

MPSK signal can be expressed as


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• MPSK basis has 2 signals 2 dimensional constellation

• M-ary message points equally spaced on circle with radius

• MPSK is constant envelope when no pulse shaping is used

sE

MEs

sin2

2(t)

1(t)

sE

MPSK signal can be• coherently detected

= Arctan(Y/X)•Minimum | I - |

• non-coherent detected with differential encoding


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Probability of symbol error in AWGN channel – using

distance between adjacent symbols as

MEs

sin2

Pe

MN

MEQ b

sinlog2

20

2

Pe

When differentially encoded & non-coherently detected, Pe estimated for M 4 as:

Pe = average symbol error probability in AWGN channel


2 Q 4 Es

No

sin 2M

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Power Spectrum of MPSK

Ts = Tblog2M

- Ts = symbol duration

- Tb = bit duration

PMPSK(f) =

22

)(

)(sin

)(

)(sin

2 sc

sc

sc

scs

Tff

Tff

Tff

TffE

2

2

22

2

22

log)(2

log)(2sin

log)(2

log)(2sin

2

log

MTff

MTff

MTff

MTffME

bc

bc

bc

bcb

PMPSK(f) =

Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) }

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Increase in M with Rb held constant

• Bnull decreases B increases • denser constellation higher BERno

rmal

ized

PS

D (

dB)

fc-½Rb fc-¼Rb fc fc+¼Rb fc+½Rb

0

-10

-20

-30

-40

-50

-60

fc-⅔Rb fc-⅓Rb fc+⅓Rb fc+⅔Rb

PSD for M = 8 & M = 16

rect pulses

RCF

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M 2 4 8 16 32 64

B = Rb/Bnull 0.5 1.0 1.5 2 2.5 3

Eb/N0 (dB) 10.5 10.5 14.0 18.5 23.4 28.5

• B = bandwidth efficiency• Rb = bit rate• Bnull = 1st null bandwidth• Eb/N0 for BER = 10-6

MPSK Bandwidth Efficiency vs Power Efficiency

bandwidth efficiency & power efficiency assume • Ideal Nyquist Pulse Shaping (RC filters)• AWGN channel without timing jitter or fading

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Advantages:

Bandwidth efficiency increases with M

Drawbacks:

Jitter & fading cause large increase in BER as M increases

EMI & multipath alter instantaneous phase of

signal

– cause error at detector

Receiver design also impacts BER

Power efficiency reduces for higher M

MPSK in mobile channels require Pilot Symbols or Equalization


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• allows amplitude & phase to vary• general form of M-ary QAM signal given by

Emin = energy of signal with lowest amplitude

ai, bi = independent integers related to location of signal point

Ts = symbol period• energy per symbol / distance between adj. symbols isn’t constant probability of correct symbol detection is not same for all symbols

• Pilot tones used to estimate channel effects

0 t Ts i = 1,2,…M

si(t) = tfbT

Etfa

T

Eci

sci

s 2sin

22cos

2 minmin

Mary- Quadrature Amplitude Modulation

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Assuming rectangular pulses - basis functions given by

1(t) = tfT c

s2cos

20 t Ts

2(t) = tfT c

s2sin

20 t Ts

(ai, bi) = element in L2 matrix, where L = M

coordinates of ith message point = minEai minEbiand

ai1(t) + bi2(t)minE minE si(t) = 0 t Ts i = 1,2,…M

QAM signal given by:


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)3,3()3,1()3,1()3,3(

)1,3()1,1()1,1()1,3(

)1,3()1,1()1,1()1,3(

)3,3()3,1()3,1()3,3(

{ai,bi} =

e.g. let M = 16, then {ai,bi} given based on

ai1(t) + bi2(t)minE minE

1(t) + 2(t)minE minE s11(t) = -3 3 0 t Ts

1(t) + 2(t)minE minE s21(t) = -3 0 t Ts


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QAM: modulated signal is hybrid of phase & amplitude modulation

• each message point corresponds to a quadbit

•Es is not constant – requires

linear channel

-1.5 -0.5 0.5 1.5

2(t)

1(t)

1.5

0.5

0

-0.5

-1.5

1011

1010

0001

0011

1001

1000

0000

0010

1110

1100

0100

0101

1111

1101

0110

0111

16 ary- Quadrature Amplitude Modulation

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ai1(t) + bi2(t)minE minE

)1,1(...)1,3()1,1(

............

)3,1(...)3,3()3,1(

)1,1(...)1,3()1,1(

LLLLLL

LLLLLL

LLLLLL{ai,bi} =

In general, for any M = L2


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Pe

0)1(

3114

NM

EQ

Mav

In terms of average energy, Eav

Power Spectrum & Bandwidth Efficiency of QAM = MPSK

Power Efficiency of QAM is better than MPSK

The average error probability, Pe for M-ary QAM is approximated by

Pe

0

min2114

N

EQ

M

• assuming coherent detection • AWGN channel • no fading, timing jitter


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28

5

1024

33.52418.51510.5Eb/N0 (BER = 10-6)

64321B = Rb/Bnull

409625664164M

M-ary QAM - Bandwidth Efficiency & Power Efficiency• Assume Optimum RC filters in AWGN • Does not consider fading, jitter, - overly optimistic


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MFSK - transmitted signals defined as

0 t Ts, i = 1,2,…M si(t) =

tin

TT

Ec

ss

s )(cos2

• fc = nc/2Ts

• nc = fixed integer

Each of M signals have • equal energy • equal duration

• adjacent sub carrier frequencies separated by 1/2Ts Hz

• sub carriers are orthogonal to each other

0 t Ts, i = 1,2,…M si(t) =

t

T

if

T

E

sc

s

s

22cos

2

Mary Frequency Shift Keying

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MFSK coherent detection - optimum receiver • receiver has bank of M correlators or matched filters• each correlator tuned to 1 of M distinct carrier frequencies

• average probability of error, Pe (based on union bound)

Pe

0

2log1

N

MEQM b


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MFSK non-coherent detection • using matched filters followed by envelope detectors

• average probability of error, Pe

Pe =

0

1

1

1

)1(exp

1

1

)1(

Nk

kE

k

M

ks

M

k

k

Pe

02exp

2

1

N

EM s

bound Pe use leading terms of binomial expansion


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MFSK Channel Bandwidth

Coherent detectionM

MRb

2log2

)3( B =

Impact of increasing M on MFSK performance

bandwidth efficiency (B) of MFSK decreases

• MFSK signals are bandwidth inefficient (unlike MPSK)

power efficiency (P) increases

• with M orthogonal signals signal space is not crowded

• power efficient non-linear amplifiers can be used without performance degradation

Non-coherent detectionM

MRb

2log2B =

22

7.50.2932

6.98.29.310.813.5Eb/N0 (BER = 10-6)0.180.420.550.570.4B = Rb/Bnull

6416842M

Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency

28

5

1024

33.52418.51510.5Eb/N0 (BER = 10-6)

64321B = Rb/Bnull

409625664164M

M-ary QAM - Bandwidth Efficiency & Power Efficiency

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Summary of M-ary modulation in AWGN Channel

0 4 8 16 32 64 M

B

32.5

21.5

10.5

0

MPSK/QAM

Coherent MFSK

0 4 8 16 32 64 M

30

25

20

15

10

5

0

MPSK

QAM

Coherent MFSK

EB/N

0

(BE

R =

10-6

)

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Shannon Limit:

• Most schemes are away from Eb/N0 of –1.6 dB by 4dB or more

• FEC helps to get closer to Shannon limit • FSK allows exchange of BW efficiency for power efficiency

log10 Eb/ N0

-3 -2 -1 0 1 2 3 log2 C/B

15

12

9

6

3Error Free Region

16 PSK

16 QAM

4 PSK/QAM BPSK

4 FSK

16 FSK

BFSK

-1.6dB

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Power Efficiency Eb/N0 = energy used by a bit for detection

B =

BN

CE

B

C b

02 1log

Bandwidth Efficiency

Power & BW Efficiency

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if C ≤ B log2(1+ S/N) error free communication is possible

if C > B log2(1+ S/N) some errors will occur

• assumes only AWGN (ok if BW << channel center frequency)• in practice < 3dB (50%) is feasible

S = EbC is the average signal power (measured @ receiver)

N = BN0 is the average noise power

Eb = STb is the average received bit energy at receiver

N0 = kT (Watts Hz–1) is the noise power density (Watts/Hz),

- thermal noise in 1Hz bandwidth in any transmission line

Power & BW Efficiency )()()(

0

dBR

BdB

N

EdB

N

S b

1 combined linear & constant envelope modulation m-ary modulation: digital baseband data sent by...

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