Continuous-Phase Frequency Shift Keying (FSK) Contents

Continuous-Phase Frequency ShiftKeying SK-18FSK-19FSK-20FSK-21IntroductionThe FSK TransmitterThe FSK Transmitter (cont. 1)The FSK Transmitter (cont. 2)The FSK Transmitter (cont. 3)The FSK Transmitter (cont. 4)The FSK Transmitter (cont. 5)Discrete-Time ImplementationTransmitter Implementation (cont.)Switched Oscillator FSKFSK Power Spectral DensityFSK Power Spectrum (cont. 1)FSK Power Spectrum (cont. 2)FSK Power Spectrum (cont. 3)Spectrum with Rectangular PulsesSpecific Examples of FSK SpectraSpectra for M 2Spectra for M 4FSK DemodulationThe Frequency Discriminator

-43FSK-44Discriminator Block DiagramExample of Discriminator OutputAn Approximate DiscriminatorSymbol Clock TrackingSymbol Clock TrackingThe Phase-Locked LoopThe Phase-Locked Loop (cont. 1)The Phase-Locked Loop (cont. 2)The Phase-Locked Loop (cont. 3)The Phase-Locked Loop (cont. 4)The Phase-Locked Loop (cont. 5)Example of PLL OutputDetection by Tone FiltersDetection by Tone Filters (cont. 1)Computing Ik (N ) by IntegrationComputing Ik (N ) by FiltersDiscrete-Time Tone FiltersFilter Implementation (cont. 1)Discrete-Time Filter FrequencyResponseSlide FSK-45 Block Diagram of Receiver UsingTone FiltersSlide FSK-46 Efficient Computation with FIR ToneFiltersSlide FSK-47 Block Diagram of Receiver UsingTone Filters

Slide FSK-48 Recursive Implementation of the ToneFiltersSlide FSK-49 Recursive Implementation (cont. 1)Slide FSK-50 Computation for the RecursiveImplementationSlide FSK-51 Simplification for Binary FSKSlide FSK-52 Simplification for Binary FSK (cont.)Slide FSK-52 Generating a Clock Timing SignalSlide FSK-53 Generating a Clock Signal (cont. 1)Slide FSK-54 Adding a Bandpass Filter to theClock TrackerSlide FSK-55 Adding a Bandpass Filter to theClock Tracker (cont. 1)Slide FSK-56 Adding a Bandpass Filter to theClock Tracker (cont. 2)Slide FSK-57 Detecting the Presence of FSKSlide FSK-58 M 4 Example of Tone FiltersSlide FSK-59 Segment of FSK SignalSlide FSK-60 3400 Hz Filter Output EnvelopeSlide FSK-61 M 4 Example (cont.)Slide FSK-62 Preliminary Clock Tracking SignalSlide FSK-63 Bandpass Filtered Clock SignalSlide FSK-64 Superimposed Preliminary andBandpass Filtered Clock SignalsSlide FSK-65 Error Probabilities for FSKSlide FSK-66 Orthogonal Signal Sets

Slide FSK-67Slide FSK-68Slide FSK-69Slide FSK-70Slide FSK-71Slide FSK-71Slide FSK-72Slide FSK-72Slide FSK-74Slide FSK-75Slide FSK-76Slide FSK-82Slide FSK-84Slide FSK-85Slide FSK-85Slide FSK-87Orthogonal Signal Sets (cont. 1)Orthogonal Signal Sets (cont. 2)Orthogonal Signal Sets (cont. 3)Orthogonal Signal Sets (cont. 4)Experiments for FSKEXP 1. Theoretical SpectraEXP 2. Making a TransmitterEXP 2.1 Handshaking SequenceEXP 2.2 Simulating Customer DataEXP 2.3 Measuring the FSKPower SpectraEXP 3. Making a Receiver Usingan Exact Frequency DiscriminatorEXP 4. Bit-Error Rate TestEXP 5. Making a Receiver Usingan Approximate DiscriminatorEXP 6. Making a Receiver Usinga Phase-Locked LoopEXP 7.1 M 4 Tone FilterReceiverEXP 7.2 Simplified M 2 ToneFilter Receiver

Continuous-Phase FrequencyShift Keying (FSK) Continuous-phase frequency shift keying (FSK) isoften used to transmit digital data reliably overwireline and wireless links at low data rates.Simple receivers with low error probability can bebuilt. Binary FSK is used in most applications,often to send important control information. Early voice-band telephone line modems usedbinary FSK to transmit data at 300 bits persecond or less and were acoustically coupled tothe telephone handset. Teletype machines usedthese modems. Some HF radio systems use FSK to transmitdigital data at low data rates. The 3GPP Cellular Text Telephone Modem(CTM) for use by the hearing impaired overregular cellular speech channels uses four levelFSK. FSK-1

BinaryDataR bits/sec The FSK Transmitter Serial toParallelConverterK. .m(t)D/Afb R/Ksymbols/sec(baud rate)FM M 2KLevelsModulators(t) Carrier fcAt the FSK transmitter input, bits from a binarydata source with a bit-rate of R bits per secondare grouped into successive blocks of K bits bythe “Serial to Parallel Converter.” These blockoccur at the rate of fb R/K symbols per secondwhich is called the baud rate.Each block is used to select one of M 2Kradian frequencies from the setΛk ωc ωd [2k (M 1)] 2π{fc fd [2k (M 1)]}for k 0, 1, . . . , M 1The frequency ωc 2πfc is called the carrierfrequency. FSK-2(1)

The FSK Transmitter (cont. 1)The radian frequenciesΩk ωd [2k (M 1)] 2πfd [2k (M 1)]for k 0, 1, . . . , M 1(2)are the possible frequency deviations from thecarrier frequency during each symbol. The deviations range from ωd (M 1) toωd (M 1) in steps of ω 2ωd . Each selected frequency is sent for Tb 1/fbseconds. The sinusoid transmitted during a blockis called the FSK symbol specified by the block.During the symbol period nTb t (n 1)Tb the“D/A” box uniquely maps each possible inputblock to a possible frequency deviationΩ(n) ωd [2kn (M 1)](3)where kn is the decimal value of the binary inputblock. FSK-3

The FSK Transmitter (cont. 2) The “D/A” block then forms the signalΩ(n)p(t nTb ) where p(t) is the unit height pulseof duration Tb defined as 1for 0 t Tb(4)p(t) 0elsewhereAssuming transmission starts at t 0, thecomplete “D/A” converter output is the staircasesignal XΩ(n)p(t nTb )(5)m(t) n 0This baseband signal is applied to an FMmodulator with carrier frequency ωc andfrequency sensitivity kω 1 to generate the FSKsignal Z ts(t) Ac cos ωc t (6)m(τ ) dτ φ00where Ac is a positive constant and φ0 is arandom initial phase angle of the modulator. FSK-4

The FSK Transmitter (cont. 3)The pre-envelope of s(t) iss (t) Ac ejωc t jeRtm(τ ) dτ jφ0e(7)m(τ ) dτ jφ0(8)0and the complex envelope isx(t) Ac ejRt0eThe phase contributed by the baseband message isZ tZ tX Ω(n)p(τ nTb ) dτθm (t) m(τ ) dτ 0 Xn 00 n 0Ω(n)Ztp(τ nTb ) dτ(9)0Now consider the case when iTb t (i 1)Tb .ThenZ ti 1XΩ(n)Tb Ω(i)θm (t) dτiTbn 0 FSK-5

The FSK Transmitter (cont. 4)θm (t) Tb ωdi 1X [2kn (M 1)] n 0 (t iTb )Tb ωd [2ki (M 1)]Tbi 12ωd Xπ[2kn (M 1)] ωb n 0π2ωd(t iTb )[2ki (M 1)](10)ωbTb The modulation index for an FSK signal isdefined to beh 2ωd ω f ωbωbfb(11)The phase at the start of the ith symbol isθm (iTb ) πhi 1X[2kn (M 1)](12)n 0 FSK-6

The FSK Transmitter (cont. 5) Therefore,θm (t) θm (iTb ) πh [2ki (M 1)]for iTb t (i 1)Tb(t iTb )Tb(13)The phase function θm (t) is continuous andconsists of straight line segments whose slopes areproportional to the frequency deviations.Discrete-Time TransmitterImplementationThe FSK signal sample at time nT is!Z nTs(nT ) cos ωc nT (14)m(τ ) dτ φ00The angle at time nT isZθ(nT ) ωc nT nTm(τ ) dτ φ0(15)0 FSK-7

Transmitter Implementation (cont.) and the angle one sample in the future isθ((n 1)T ) (n 1)TZm(τ ) dτ φ0ωc (n 1)T 0 ωc nT ωc T ZnT0 m(τ ) dτ φ0 (n 1)TZm(τ ) dτnT θ(nT ) ωc T T m(nT )(16) θ(nT ) T Λ(nT )(17)where Λ(nT ) ωc m(nT ) is the total tonefrequency at time nT . The M possible values forT Λ(nT ) can be pre-computed. Thus, the newphase angle can be computed recursively. In yourtransmitter program, make sure that θ(nT ) doesnot get large. FSK-8

Switched Oscillator FSKAnother approach to FSk would be to switchbetween independent tone oscillators. Thisswitched oscillator approach could causediscontinuities in the phase function depending onthe tone frequencies and symbol rate. Thediscontinuities would cause the resulting FSKsignal to have a wider bandwidth than continuousphase FSK. It could also cause problems for aPLL demodulator. FSK-9

Power Spectral Density for FSK The term “power spectrum” will be used for“power spectral density” from here on forsimplicity.Lucky, Salz, and Weldon1 present the solution fora slightly more generalized form of FSK thandescribed above. Let the pulse p(t) to have anarbitrary shape but still be confined to be zerooutside the interval [0, Tb ). The power spectrum,Sxx (ω), of a random process x(t) can be definedas:o1 n2(18)Sxx (ω) lim E Xλ (ω) λ λwhere E{ } denotes statistical expectation andZ λXλ (ω) x(t)e jωt dt(19)01 R.W.Lucky, J. Salz, and E.J. Weldon, Principles ofData Communications, McGraw-Hill, 1968, pp. 203–207and 242–245. FSK-10

Power Spectral Density for FSK(cont. 1) Only formulas for the power spectrum of thecomplex envelope will be presented here since thepower spectrum for the complete FSK signal canbe easily computed as11Sss (ω) Sxx (ω ωc ) Sxx ( ω ωc ) (20)44 The frequency deviation in the complexenvelope during the interval [nTb , (n 1)Tb ) issn (t nTb ) Ω(n)p(t nTb )(21) The phase change caused by this frequencydeviation during the baud when time is takenrelative to the start of the baud isZ tbn (t) Ω(n)p(τ ) dτ for 0 t Tb (22)0 The total phase change over a baud isZ TbBn bn (Tb ) Ω(n)p(τ ) dτ 0FSK-11(23)

Power Spectral Density for FSK(cont. 2) The Fourier transform of a typical modulatedpulse isZ TbFn (ω) ejbn (t) e jωt dt(24)0It is convenient to define the following functions:1. The characteristic function of bn (t)onC(α; t) E ejαbn (t)(25)2. The average transform of a modulated pulseF (ω) E {Fn (ω)}(26)3.nG(ω) E Fn (ω)ejBno(27)4. The average squared magnitude of a pulsetransform (28)P (ω) E Fn (ω) 2 FSK-12

Power Spectral Density for FSK(cont. 3)5.1arg C(1; Tb )(29)TbIn terms of these quantities, the power spectrumis P (ω) 2ℜe [F (ω)G(ω) i jωTb e ; C(1; Tb ) 1 1 C(1;Tb )e jωTb Tb2S(ω) P(ω) F(ω) ωb xx2 Ac X 2 F(γ nω δ (ω γ nωb )b n for C(1; T ) ejγTbb(30)Notice that the spectrum has discrete spectrallines as well as a distributed part when thecharacteristic function has unity magnitude.γ FSK-13

Spectrum for Rectangular FrequencyPulsesThe power spectrum for the case where p(t) is therectangular pulse given by (4) and the frequencydeviations are equally likely reduces to 2F(ω) P (ω) 2ℜe 1 C(1; Tb )e jωTb 2ωd forh not an integer ωb TbSxx (ω) 22P(ω) F(ω) ωb Ac X 2 F(γ nω) δ (ω γ nωb ) b n for h an integer k(31)where 0for k evenγ (32) ωb /2 for k odd FSK-14

Spectrum for Rectangular FrequencyPulses (cont. 1) (ω Ωk )Tb2e j(ω Ωk )Tb /2 (33)(ω Ωk )Tb2sinFn (ω) Tb 2(ω Ωk )TbM 1Tb2 X sin 2P (ω) (ω Ωk )TbMk 02 TbF (ω) MM 1Xk 0(34)(ω Ωk )Tb2e j(ω Ωk )Tb /2(ω Ωk )Tb2(35)sinandM/2sin(M πh)2 Xcos[ωd Tb (2k 1)] C(1; Tb ) MM sin(πh)k 1 (36)FSK-15

Properties of the Spectrum forRectangular Frequency Pulses Fn (ω) has its peak magnitude at the tonefrequency Ωn ωd [2n (M 1)] and zeros atmultiples of the symbol rate, ωb , away from thetone frequency. This is exactly what would beexpected for a burst of duration Tb of a sinusoidat the tone frequency. The term P (ω) is what would result for theswitched oscillator case when the phases of theoscillators are independent random variablesuniformly distributed over [0, 2π). The remaining terms account for the continuousphase property and give a narrower spectrumthan if the the phase were discontinuous. The power spectrum has impulses at the M tonefrequencies when h is an integer. However, theimpulses at other frequencies disappear becausethey are multiplied by the nulls of F (γ nωb ). FSK-16

Specific Examples of FSK SpectraExamples of the power spectral densities forbinary continuous phase and switched oscillatorFSK are shown in Figure 1 for h 0.5, 0.63, 1 and1.5. The spectra become more peaked near theorigin for smaller values of h. They become moreand more peaked near ωd and ωd as happroaches 1 and include impulses at thesefrequencies when h 1. Bell Labs designed itsBell 103 binary FSK modem, released in 1962,with h 2/3 to avoid impulses in the spectrumthat could cause cross talk in the cables.The spectra for M 4 continuous phase andswitched oscillator FSK are shown in Figure 2 forh 0.5, 0.63, 0.9, and 1.5. The CTM with M 4uses a symbol rate of 200 baud with a toneseparation of 200 Hz and, thus, has themodulation index h 1. FSK-17