1

Minimum Shift Keying [MSK] Modulation---a special form of FSK

• MSK is generated as follows

g1(t)g1(t)ai

g2(t)g2(t)bi

sinωct

DD

cosωct

Delay=T2

1R

a1 a2 a3 a4 Input bits

1/R 1/R

T

•Similar to generation of OQPSK--- except for different I/Q baseband filters•Linear modulation ---> can equalize•Bandwidth determined by g1(t) and g2(t)

Si (t)

a1cosπt/T a2cosπt/T

b1sinπt/T b2sinπt/T

-T/2 0 T/2Time

In-phase terms

Quadrature offset

-T/2 0 TTime

g1(t)

g2(t)

Detect as SQAM signalb

a

Q

I

Not a strictly bandlimited signal---since time limited pulses are used for shaping

2

MSK is an FSK Signal

1 if

1 if 0 cos)(

1 ,1 cos

1 ,1 cos)(

−==

==

+−=

±=−=

+=

±==

=

i

iji

ci

iic

iici

a

aT

tbatts

baTt

t

baT

ttts

πθ

θθπ

ω

ππ

ω

πω

m

m

•The MSK signal can be written as

•The MSK signal has a constant envelope•As shown below, MSK may be interpreted as a form of FSK

( )( )

421

2or ,/ e wher

1 if 0 cos

1 if 0 cos )(

R

TfT

batt

battts

iic

iici

==∆

=∆=∆

==+∆−=

−==+∆+=

πω

πω

θθωωθθωω

3

Why Is It 'Minimum' Shift Keyed?• For the synchronous or coherent FSK detector shown below, using an integrator (over a symbol

period) as the linear filter, it can be shown that the closest that the two frequencies can be spaced(for the signals to be unambiguously resolved in the absence of noise) is R/4

• If the tranmitted signal is the lower frequency, then, to a first approximation, the output of thebottom integrator is

• For this integral to be zero, we have the condition

dt)tcos(1/2dtt(t)cos 2

1/R

0 12

1/R

0ω−ω=ω ∫∫ cf

Consider

∫∫ ∆∆ππ∆∆ππ

====ρρ

ππ==

ππ==

••==++

T21

T1nff

22

11

ft2ftsin2

dt(t)s(t)s

tfcos2T2

(t)S

tfcos2T2

(t)S

21

+

-

Low-passfilter

Low-passfilter

Low-passfilter

Low-passfilter

cosω1t

cosω2t

fc(t) Binaryoutput

ρρ

1/2MSK Optimal FSK

∆∆fT = .715~2 dB penalty antipodal(difficult to synchronize)

ρ ρ = inner product between s1(t) and s2(t); measuresthe degree of "orthogonality" between the 2 signals

∆∆fT

= 0sin 2∆ω∆ωR

⇒⇒ ∆ ∆f = R/4

∆∆f = f2 - f1

4

POWER SPECTRA FOR MSK AND QPSK

Spectrum

Frequency

MSK

QPSK

1.00

0.20

0.05

Tb = bit rate

0.5/Tb 1/Tb

fFraction of out-of-band power

QPSKMSK

MSK has lower side lobes than rectangularly filtered QPSK, as shown, and the out-of-band power is significantly lower. A measure ofthe compactness of a modulation spectrum is the bandwidth which contains 99% of the total power of the signal, 1.2/Tb for MSK.

5

GMSK Modulator: Gaussian Filter Precedes g1(t) and g2(t)and reduces the bandwidth [relative to MSK]

• Gaussian filter produces an even faster drop off in high frequency content than MSK

• No longer a linear modulation [like MSK]: since the baseband pulse is not a cos/sine timefunction

• Good approximation for MSK as a linear signal, so can equalize

Non Linearcos[ωωt+φφ(t,ααn)]

cosωωt

-sinωωt

GaussianFilter

GaussianFilter

CosCos

SinSin

φφ(t,ααn)

∫∫ dt∫∫ dt

Phase Pulse Shaping

Pulse Shaping

cos[φφ(t,ααn)]

sin[φφ(t,ααn)]

0.3RB

Gaussian

-3

dB

f

FrequencyPulse Shape

6

GMSK Coherent ReceiverLaurent, IEEE Trans. On Communications, February 1986

• LPF eliminates 2f0 terms

• Filtering is accomplished in BPF

cos 2ππf0t

- sin 2ππf0t

GaussianBPF

GaussianBPFxGMSK(t)

cos φ(t)

In-PhaseChannel

QuadratureChannel

sin φ(t)

Delay2T

Delay2T

Delay2T

Delay2T

DetectorDetector

DetectorDetector

LPFLPF

LPFLPF

SampleEvery 2T

SampleEvery 2T

0, 2T,…

T, 3T,…Bi

Can Detect GMSK as a Pair of Antipodal Quadrature Signals ----process as linear modulation: good approximation