1

T

dt0

output+

-r(t)

Decision Circuit

cos wLt

cos wHt

T

dt0

• 2 correlators fed with local coherent reference signals• difference in correlator outputs compared with threshold to determine binary value

Pe,BFSK =

0N

EQ b

Probability of error in coherent FSK receiver given as:

Coherent BFSK Detector

2

• 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

Pe,BFSK, NC =

02exp

2

1

N

Eb

Average probability of error in non-coherent FSK receiver:

r(t) outputDecision Circuit

+

-

EnvelopeDetector

Matched FilterfL

EnvelopeDetector

Tb

Matched FilterfH

Non-coherent Detection of BFSK

3

Non-coherent Quadrature BFSK Detector

outputDecision Circuit

r(t)T

dt0

+

+

( 2/T) cos wHt

T

dt0

( 2/T) sin wHt

(.)2Z1(T)I-channel

Q-channel

(.)2Z2(T)

+

-T

dt0

Z3(T)I-channel

T

dt0

+

+

( 2/T) cos wLt

( 2/T) sin wLt

(.)2

Q-channel

(.)2Z4(T)

4

Tutorial Derive minimum frequency spacing

(f2 – f1) for

Non-coherent detection (arbitrary phase )

Coherent detection

5

• Type of continuous phase FSK (CPFSK) • Spectrally efficient• Constant envelope• Good BER performance• Self-synchronizing capability• Requires coherent detection

Minimum Shift Keying ( fast FSK)

6

• minimum frequency spacing (bandwidth) for 2 FSK signals to be coherently orthogonal

• minimum bandwidth that allows orthogonal detection

FSK modulation index bR

F2kFSK =

MSK modulation index is kMSK = 0.5 FMSK= b

b

T

R

4

1

4

Minimum Shift Keying

7

MSK can be thought of as special case of OQPSK• uses half-sinusoidal pulses instead of baseband rectangular pulses• arch shaped pulse of period = 2Tb

• modify OQPSK equations using half-sine pulses for N-bit stream

several variations of MSK exist with different basic pulse shapes e.g.

- use only positive ½ sinusoids- use alternating negative & positive ½ sinusoids

• all variations are CPFSK that use different techniques to achieve spectral efficiency

Minimum Shift Keying

8

Transmitted MSK signal (OQPSK variant)

sMSK(t) =

1

0

)(N

iIi tm

1

0

)(N

iQi tm p(t – 2iTb-Tb)sin(2πfct)

p(t) =

elsewhere

TtT

tb

b0

202

cos

½ sine pulse given by

• mIi(t) = ith bit of mI(t), the even bits of m(t)

• mQi(t) = ith bit of mQ(t), the odd bits of m(t)

• mI(t) & mQ(t)are bipolar bit streams (1) that feed I & Q

arms of the modulator - each arm fed at Rb/2

m(t) = ±1 bipolar bit stream

p(t – 2iTb)cos(2πfct) +

9

MSK examplewith ωcT=2.5π; ω1T=2π, ω2T=3π

dQ

Sin

-1 0 1 2 3 4 5 6 7 8

dQ

Sin

Sin

-1 0 1 2 3 4 5 6 7 8

dIC

osC

os

-1 0 1 2 3 4 5 6 7 8MS

Kd

ICo

sd

ata

d0 d1 d2 d3 d4 d5 d6 d7

d0 d2 d4 d6

d1 d3 d5 d7

10

MSK examplewith ωcT=4π; ω1T=3.5π, ω2T=4.5π

dQ

Sin

-1 0 1 2 3 4 5 6 7 8

dQ

Sin

Sin

-1 0 1 2 3 4 5 6 7 8

dIC

osC

os

-1 0 1 2 3 4 5 6 7 8MS

Kd

ICo

sd

ata

d0 d1 d2 d3 d4 d5 d6 d7

d0 d2 d4 d6

d1 d3 d5 d7

11

sMSK(t) =

k

biQiIc

b

b

T

ttmtmtf

T

E 2

)()(2cos2

MSK waveform - as a special case of CPFSK

k = 0 or depending on whether mI(t) = +1 to -1

• sMSK(t) has constant amplitude

• to ensure phase continuity at bit interval select fc = ; n integer4bnR

• phase of MSK varies linearly over Tb

MSK is FSK signal with binary signaling frequencies given by

fc + bT4

1fc -

bT4

1and

12

bi t θ(T) i0 T -π/2 odd

1 T π/2 odd

0 2T 0 even1 2T π even

θ(t) can take on only 2 values at odd or even multiples of T

t = even multiple of T θ(T) - θ(0) = π or 0

t = odd multiple of T θ(T) - θ(0) = ± π/2

0 ≤ t ≤ TtT2

θ(t) = θ(0) ±

h = ½

Phase Continuity of MSK

assuming θ(0) = 0

13

Phase Trellis: path depicts θ(t) corresponding to a binary sequence

• for h = ½ ΔF = Rb/4

• minimum ΔF for two binary FSK signals

to be coherently orthogonal

e.g. if Rb = 100Mbps = ΔF = 25MHzi bi θ(i-1)T θ(iT) i1 1 0 π/2 odd

2 0 π/2 0 even

3 0 0 -π/2 odd4 1 -π/2 π even

5 1 π π/2 odd

6 1 π/2 π even

7 0 π -π/2 odd8 1 -π/2 π even

θ(t) - (0)

π

π/2

0

-π/2

-π

0 2T 4T 6T t

1 0 0 1 1 1 0

bi t θ(T) i0 T -π/2 odd

1 T π/2 odd

0 2T 0 even1 2T π even

14

Orthonormal basis for MSK as

1(t) = tftTT c

2cos2

cos2

0 ≤ t ≤ T

2(t) = tftTT c

2sin2

sin2

0 ≤ t ≤ T

s1

π/20‘1’π/2π ‘0’-π/2π ‘1’-π/20‘0’

s2θ(T)θ(0)bi

bE

bE

bE

bE

bE

bEbE

bE

s(t) = s1(t)1(t) + s2(t)2(t)then

T

dttts2

0

1 )()( s1 = = )θ(Eb 0cos -T ≤ t ≤ T

T

T

dttts )()( 2s2 = = )(sin TEb 0 ≤ t ≤ 2T

with

15

MSK Power Spectrum

RF power spectrum obtained by frequency shifting |F{p(t)}|2

F{} = fourier transform

p(t) = MSK baseband pulse shaping function (1/2 sin wave)

p(t) =

elsewhere

TtT

tb

b

0

||2

cos

PMSK(f) = 2

222

2

222 16.1

)(2cos16

16.1

)(2cos16

b

bc

b

bc

Tf

Tff

Tf

Tff

Normalized PSD for MSK is given as

16

MSK spectrum (1) has lower side lobes than QPSK (amplitude)

(2) has wider side lobes than QPSK (frequency)

• 99% MSK power is within bandwidth B = 1.2/Tb

• 99% QPSK power is within bandwidth B = 8/Tb

norm

aliz

ed P

SD

(dB

) QPSK, OQPSKMSK

PSD of MSK & QPSK signals

fc fc+0.5Rb fc+Rb fc+1.5Rb fc+2Rb

100

-10

-20

-30-40

-50

-60

17

MSK QPSK signaling is bandwidth efficient,

achieving 2 bps per Hz of channel bandwidth. However, the abrupt changes results in large side lobes. Away from the main lobe of the signal band, the power spectral distribution falls off only as ω-2 .

MSK achieves the same bandwidth efficiency. With constant envelope (no discontinuity in phase), the power spectral distribution falls off as ω-4 away from the main signal band.

18

• MSK has faster roll-off due to smoother pulse function

• Spectrum of MSK main lobe > QPSK main lobe- using 1st null bandwidth MSK is spectrally less efficient

• MSK has no abrupt phase shifts at bit transitions - bandlimiting MSK signal doesn’t cause envelop to cross zero - envelope is constant after bandlimiting

• small variations in envelope removed using hardlimiting - does not raise out of band radiation levels

• constant amplitude non-linear amplifiers can be used

• continuous phase is desirable for highly reactive loads

• simple modulation and demodulation circuits

MSK spectrum

19

MSK Transmitter

(i) cos(2fct) cos(t/2T) 2 phase coherent signals at fc ¼R

(ii) Separate 2 signals with narrow bandpass filters

(iii) Combined to form I & Q carrier components x(t), y(t)

(iv) Mix and sum to yield SMSK(t) = x(t) mI(t) + y(t) mQ(t)

mI(t) & mQ(t) = even & odd bit streams

x(t)

y(t)

SMSK(t)

mQ(t)

_

+

+

+

mI(t)

+

+

cos(2fct)

cos(t/2T)

bc Tf 4/1

bc Tf 4/1

20

Coherent MSK Receiver

(i) SMSK(t) split & multiplied by locally generated

x(t) & y(t) (I & Q carriers)

(ii) mixer outputs are integrated over 2T & dumped

(iii) integrate & dump output fed to decision circuit every 2T• input signal level compared to threshold decide 1 or 0

• output data streams correspond to mI(t) & mQ(t)

• mI(t) & mQ(t) are offset & combined to obtain demodulated signal

*assumes ideal channel – no noise, interference

21

Coherent MSK Receiver

x(t)

y(t)SMSK(t)

T

dt2

0

)(

t = 2(k+1)T

t = 2(k+1)T

T

dt2

0

)(Threshold

Device

Threshold Device

mQ(t)

mI(t)

22

Gaussian MSK

• Gaussian pulse shaping to MSK- smoothens phase trajectory of MSK signal

over time, stabilizes instantaneous frequency

variations

- results in significant additional reduction of

sidelobe levels

• GMSK detection can be coherent (like MSK)

or noncoherent (like FSK)

23

• premodulation pulse shaping filter used to filter NRZ data

- converts full response message signal into partial

response scheme

full response baseband symbols occupy Tb

partial response transmitted symbols span several Tb

- pulse shaping doesn’t cause pattern’s averaged phase

trajectory to deviate from simple MSK trajectory

Gaussian MSK

24

GMSKs main advantages are• power efficiency - from constant envelope (non-linear amplifiers)• excellent spectral efficiency

GMSK filter can be completely defined from B3dB Tb

- customary to define GMSK by B3dBTb

• pre-modulation filtering introduces ISI into transmitted signal

• if B3db Tb > 0.5 degradation is not severe

B3dB = 3dB bandwidth of Gaussian Pulse Shaping Filter

Tb = bit duration = baseband symbol duration

• irreducible BER caused by partial response signaling is the cost for spectral efficiency & constant envelope

Gaussian MSK

25

Impulse response of pre-modulation Gaussian filter :

hG(t) =

2

2exp

is related to B3dB by

BB

5887.0

2

2ln =

transfer function of pre-modulation Gaussian Filter is given by

HG(f) = 22exp f

Gaussian MSK

26

(i) Reducing B3dBTb : spectrum becomes more compact (spectral efficiency)

• causes sidelobes of GMSK to fall off rapidly

B3dBTb = 0.5 2nd lobe peak is 30dB below main lobe

MSK 2nd peak lobe is 20dB below main lobe

• MSK GMSK with B3dBTb =

(ii) increases irreducible error rate (IER) due to ISI• ISI degradation caused by pulse shaping increases• however - mobile channels induce IER due to mobile’s velocity

• if GMSK IER < mobile channel IER no penalty for using GMSK

Impact of B3dBTb

27

PSD of GMSK signals

0 0.5 1.0 1.5 2.0 (f-fc)T

0

-10

-20

-30-40

-50

-60

BTb = (MSK)

BTb = 1.0BTb = 0.5BTb = 0.2

Increasing BTb

• reduces signal spectrum• results in temporal spreading and distortion

28

BTb 90% 99% 99.9% 99.99% 0.2 GMSK 0.52 0.79 0.99 1.22

0.25 GMSK 0.57 0.86 1.09 1.370.5 GMSK 0.69 1.04 1.33 2.08

MSK 0.78 1.20 2.76 6.00

RF bandwidth containing % power as fraction of Rb

[Ish81] BER degradation from ISI caused by GMSK filtering is minimal at B3dBTb = 0.5887

• degradation in required Eb/N0 = 0.14dB compared to case of no ISI

e.g. for BT = 0.2 99% of the power is in the bandwidth of 1.22Rb

29

• [Mur81] shown to perform within 1dB of optimal MSK with B3dBTb = 0.25

• since pulse shaping causes ISI Pe is function of B3dBTb

Pe =

0

2

N

EQ b

Pe = bit error probability

is constant related to B3dBTb

• B3dBTb = 0.25 = 0.68

• B3dBTb = = 0.85 (MSK)

BER of GMSK for AWGN channel

30

(i) pass mNRZ(t) through Gaussian base band filter (see figure

below)

- mNRZ(t) = NRZ bit stream

• output of Gaussian filter passed to FM modulator

• used in digital implementation for- Global System for Mobile (GSM)- US Cellular Digital Packet Data (CDPD)

(ii) alternate approach is to use standard I/Q modulator

GMSK Transmitter Block Diagram

NRZ bits RF GMSK OutputGaussianLPF

FM Transmitter

GMSK Transmitter

31

GMSK Receiver

RF GMSK signal can be detected using

(i) orthogonal coherent detectors (block diagram)

(ii) simple non-coherent detectors (e.g. standard FM discriminators)

(i) GMSK Receiver Block Diagram-orthogonal coherent detectors

loop filter

modulated IF input signal

/2

IF LO

clockrecovery

/2

demodulated signal

I

Q

32

carrier recovery using De Budas method for (similar to Costas loop)

S’(t) = output of frequency doubler that contains 2 discrete frequencycomponents

- divide S’(t) by four: S’(t) /4

- equivalent to PLL with frequency doubler

33

demodulated signal

clock recovery

loop filter

VCO

D Q

C

D

C Q

D Q

Cmodulated IFinput signal

D Q

C

D Q

C

D Q

C

Logic Circuit for GMSK demodulation

De Budas method implemented using digital logic

• 2 D flip flops (DFF) act as quadrature product demodulator • XORs act as based band multipliers• mutually orthogonal reference carriers generated using 2 DFFs

• VCO center frequency set to 4 fc ( fc = carrier center frequency)

34

e.g.

Assume 0.25GMSK: B3dbTb = 0.25 & Rb = 270kbpsthen

Tb = Rb-1 = 3.7us

B3dB = 0.25/Tb = 67.567kHz

Occupied Spectrum - 90% power 0.57Rb = 153.9kHz

- use table

Detecting GMSK signal by sampling output of FM demodulatoris a non-optimal, effective method