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

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

10

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

11

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

12

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

13

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

14

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

15

• 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