Coherent BFSK Detector 2 correlators fed with local coherent reference signals difference in correlator outputs compared with threshold to determine binary value output + - r(t) Decision Circuit cos wLt cos wHt  Pe,BFSK = Probability of error in coherent FSK receiver given as:

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 r(t) output Decision Circuit + -  Envelope Detector Matched Filter fL Tb fH Average probability of error in non-coherent FSK receiver: Pe,BFSK, NC =

Non-coherent Quadrature BFSK Detector output Decision Circuit r(t) + ( 2/T) cos wHt  ( 2/T) sin wHt (.)2 Z1(T) I-channel Q-channel Z2(T) - Z3(T) ( 2/T) cos wLt ( 2/T) sin wLt Z4(T)

Tutorial Derive minimum frequency spacing (f2 – f1) for  Non-coherent detection (arbitrary phase ) Coherent detection

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

Minimum Shift Keying kFSK = FSK modulation index minimum frequency spacing (bandwidth) for 2 FSK signals to be coherently orthogonal minimum bandwidth that allows orthogonal detection FSK modulation index kFSK = MSK modulation index is kMSK = 0.5  FMSK=

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

Transmitted MSK signal (OQPSK variant) sMSK(t) = p(t – 2iTb-Tb)sin(2πfct) p(t – 2iTb)cos(2πfct) + p(t) = 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

MSK example with 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 ωcT=4π; ω1T=3.5π, ω2T=4.5π d0 d1 d2 d3 d4 d5 d6 d7 d0 d2 d4 d6 d1 d3 d5 d7

- as a special case of CPFSK MSK waveform - as a special case of CPFSK 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  select fc = ; n integer MSK is FSK signal with binary signaling frequencies given by fc + fc - and phase of MSK varies linearly over Tb

Phase Continuity of MSK 0 ≤ t ≤ T θ(t) = θ(0) ± h = ½ Phase Continuity of MSK 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 bi t θ(T) i T -π/2 odd 1 π/2 2T even π assuming θ(0) = 0

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 bi t θ(T) i T -π/2 odd 1 π/2 2T even π θ(t) - (0) π π/2 -π/2 -π 0 2T 4T 6T t 1 0 0 1 1 1 0 i bi θ(i-1)T θ(iT) 1 π/2 odd 2 even 3 -π/2 4 π 5 6 7 8

Orthonormal basis for MSK as 0 ≤ t ≤ T 2(t) = s1 = = -T ≤ t ≤ T s2 = 0 ≤ t ≤ 2T with s1 π/2 ‘1’ π ‘0’ -π/2 s2 θ(T) θ(0) bi s(t) = s1(t)1(t) + s2(t)2(t) then

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) = Normalized PSD for MSK is given as PMSK(f) =

MSK spectrum PSD of MSK & QPSK signals QPSK, OQPSK MSK (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 normalized PSD (dB) QPSK, OQPSK MSK PSD of MSK & QPSK signals fc fc+0.5Rb fc+Rb fc+1.5Rb fc+2Rb 10 -10 -20 -30 -40 -50 -60

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.

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

MSK Transmitter mI(t)  cos(2fct) cos(t/2T) mQ(t) (i) cos(2fct) 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(2fct) cos(t/2T)

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

Coherent MSK Receiver Threshold Device mI(t) x(t) SMSK(t) y(t) t = 2(k+1)T x(t) y(t) Threshold Device mQ(t) t = 2(k+1)T

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)

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

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

Gaussian MSK Impulse response of pre-modulation Gaussian filter : hG(t) =  is related to B3dB by  = transfer function of pre-modulation Gaussian Filter is given by HG(f) =

Impact of B3dBTb (i) Reducing B3dB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

PSD of GMSK signals BTb = 1.0 BTb = 0.5 BTb = 0.2 Increasing BTb 0 0.5 1.0 1.5 2.0 (f-fc)T -10 -20 -30 -40 -50 -60 BTb =  (MSK) BTb = 1.0 BTb = 0.5 BTb = 0.2 Increasing BTb reduces signal spectrum results in temporal spreading and distortion

containing % power as fraction of Rb RF bandwidth containing % power as fraction of Rb 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 1.37 0.5 GMSK 0.69 1.04 1.33 2.08 MSK 0.78 1.20 2.76 6.00 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

BER of GMSK for AWGN channel [Mur81] shown to perform within 1dB of optimal MSK with B3dBTb = 0.25 since pulse shaping causes ISI  Pe is function of B3dBTb Pe = Pe = bit error probability  is constant related to B3dBTb B3dBTb = 0.25   = 0.68 B3dBTb =    = 0.85 (MSK)

GMSK Transmitter GMSK Transmitter Block Diagram NRZ bits (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 Output Gaussian LPF FM Transmitter

RF GMSK signal can be detected using 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 clock recovery  demodulated signal I Q

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

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) demodulated signal clock recovery loop filter VCO D Q C D C Q modulated IF input signal Logic Circuit for GMSK demodulation

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