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cos w H t. output. + -. Decision Circuit. . r(t). cos w L t. Probability of error in coherent FSK receiver given as:. P e,BFSK =. Coherent BFSK Detector. 2 correlators fed with local coherent reference signals difference in correlator outputs compared with threshold to

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Output

cos wHt

output

+

-

Decision

Circuit

r(t)

cos wLt

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


Output

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:


Output

I-channel

Z1(T)

(.)2

( 2/T) cos wHt

+

+

( 2/T) sin wHt

Z2(T)

(.)2

+

-

output

r(t)

Decision

Circuit

Q-channel

Z3(T)

I-channel

(.)2

( 2/T) cos wLt

+

+

( 2/T) sin wLt

Z4(T)

(.)2

Q-channel

Non-coherent Quadrature BFSK Detector


Tutorial

Tutorial

  • Derive minimum frequency spacing

    (f2 – f1) for 

    • Non-coherent detection

      (arbitrary phase )

    • Coherent detection


Output

  • Type of continuous phase FSK (CPFSK)

  • Spectrally efficient

  • Constant envelope

  • Good BER performance

  • Self-synchronizing capability

  • Requires coherent detection

Minimum Shift Keying ( fast FSK)


Output

FSK modulation index

  • minimum frequency spacing (bandwidth) for 2 FSK signals

  • to be coherently orthogonal

  • minimum bandwidth that allows orthogonal detection

kFSK=

MSK modulation index is kMSK= 0.5 

FMSK=

Minimum Shift Keying


Output

Minimum Shift Keying

  • 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


Output

sMSK(t) =

  • mI(t) & mQ(t)are bipolar bit streams (1) that feedI& Q

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

  • 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)

½ sine pulse given by

p(t) =

m(t) = ±1 bipolar bit stream

p(t – 2iTb-Tb)sin(2πfct)

Transmitted MSK signal

(OQPSK variant)

p(t – 2iTb)cos(2πfct) +


Msk example with c t 2 5 1 t 2 2 t 3

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

d0 d1 d2 d3 d4 d5 d6 d7

d0 d2 d4 d6

d1 d3 d5 d7


Msk example with c t 4 1 t 3 5 2 t 4 5

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

d0 d1 d2 d3 d4 d5 d6 d7

d0 d2 d4 d6

d1 d3 d5 d7


Output

sMSK(t) =

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

  • sMSK(t) has constant amplitude

  • to ensure phase continuity at bit interval  selectfc = ; n integer

  • fc -

and

  • fc +

MSK waveform

- as a special case of CPFSK

MSK is FSK signal with binary signaling frequencies given by

  • phase of MSK varies linearly over Tb


Output

Phase Continuity of MSK

h = ½

θ(t) = θ(0) ±

0 ≤ t ≤ T

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

t =even multiple of T θ(T) - θ(0)= πor 0

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

assuming θ(0) = 0


Output

π

π/2

0

-π/2

θ(t) - (0)

1 0 0 1 1 1 0

0 2T 4T 6T t

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 = 25MHz


Output

Orthonormal basis for MSK as

1(t) =

2(t) =

0 ≤ t ≤ T

0 ≤ t ≤ T

bi

θ(0)

θ(T)

s1

s2

then

s(t) = s1(t)1(t) + s2(t)2(t)

with

‘0’

0

-π/2

‘1’

π

-π/2

s1=

-T ≤ t ≤ T

‘0’

π

π/2

‘1’

0

π/2

s2=

=

0 ≤ t ≤ 2T

=


Output

p(t) =

PMSK(f) =

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)

Normalized PSD for MSK is given as


Output

PSD of MSK & QPSK signals

10

0

-10

-20

-30

-40

-50

-60

QPSK, OQPSK

MSK

normalized PSD (dB)

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

  • MSK spectrum

    • (1) has lower side lobes than QPSK (amplitude)

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

      • 99% MSK power is within bandwidthB = 1.2/Tb

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


Output

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.


Output

MSK spectrum

  • 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


Output

mI(t)

_

+

x(t)

SMSK(t)

+

+

y(t)

mQ(t)

+

+

cos(2fct)

cos(t/2T)

MSK Transmitter

(i) cos(2fct)cos(t/2T) 2 phase coherent signals atfc ¼R

(ii) Separate 2 signals with narrow bandpass filters

(iii) Combined to formI & 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


Output

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


Output

SMSK(t)

Coherent MSK Receiver

Threshold

Device

mI(t)

t = 2(k+1)T

x(t)

y(t)

Threshold

Device

mQ(t)

t = 2(k+1)T


Output

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)


Output

Gaussian MSK

  • 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


Output

Gaussian MSK

  • GMSKs main advantages are

    • power efficiency - from constant envelope (non-linear amplifiers)

    • excellent spectral efficiency

  • 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

  • GMSK filter can be completely defined from B3dB Tb

    • - customary to define GMSK by B3dBTb


Output

Impulse responseof pre-modulation Gaussian filter :

hG(t) =

transfer function of pre-modulation Gaussian Filter is given by

HG(f) =

 is related toB3dB by

 =

Gaussian MSK


Output

Impact of B3dBTb

  • (i) ReducingB3dBTb : spectrum becomes more compact (spectral efficiency)

    • causes sidelobes of GMSK to fall off rapidly

      • B3dBTb = 0.5  2nd lobe peak is 30dB below main lobe

      • MSK 2nd peak lobe is 20dB below main lobe

    • MSK  GMSK with B3dBTb = 

  • (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


Output

0

-10

-20

-30

-40

-50

-60

BTb =  (MSK)

BTb = 1.0

BTb = 0.5

BTb = 0.2

0 0.5 1.0 1.5 2.0 (f-fc)T

PSD of GMSK signals

  • Increasing BTb

  • reduces signal spectrum

  • results in temporal spreading and distortion


Output

RF bandwidth

containing % power as fraction of Rb

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

  • [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


Output

Pe =

BER of GMSK for AWGN channel

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

  • since pulse shaping causes ISI  Peis function of B3dBTb

Pe= bit error probability

 is constant related to B3dBTb

  • B3dBTb = 0.25   = 0.68

  • B3dBTb =    = 0.85 (MSK)


Output

Gaussian

LPF

FM

Transmitter

RF GMSK Output

NRZ bits

GMSK Transmitter Block Diagram

GMSK Transmitter

  • (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


Output

demodulated

signal

 /2

modulated IF

input signal

loop

filter

Q

IF LO

I

 /2

clock

recovery

  • 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


Output

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

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

      • components

    • - divide S’(t) by four: S’(t) /4

    • - equivalent to PLL with frequency doubler


Output

D Q

C

D Q

C

D Q

C

D Q

C

modulated IF

input signal

D Q

C

clock

recovery

demodulated

signal

D

C Q

loop

filter

VCO

  • De Budasmethod 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

  • VCOcenter frequency set to 4  fc ( fc = carrier center frequency)

Logic Circuit for GMSK demodulation


Output

Detecting GMSK signal by sampling output of FM demodulator

is a non-optimal, effective method

  • e.g.

  • Assume 0.25GMSK: B3dbTb= 0.25 & Rb = 270kbps

  • then

    • Tb = Rb-1= 3.7us

    • B3dB = 0.25/Tb = 67.567kHz

    • Occupied Spectrum - 90% power 0.57Rb = 153.9kHz

      • - use table


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