the mobile communications handbook

Mobile Communications Contents PARTIBasicPrinciples 1 Complex Envelope Representations for Modulated Signals Leon W. Couch, II 2 Sampling Hwei P. Hsu 3 Pulse Code Modulation Leon W. Couch, II 4 Baseband Sig nalling and Pulse Shaping Michael L. Honig and Melbourne Barton 5 Channel Equalization John G. Proakis 6 Line Coding Joseph L. LoCicero and Bhasker P. Patel 7 Echo Cancellation Giovanni Cherubini 8 Pseudonoise Sequences Tor Helleseth and P. Vijay Kumar 9 Optimum Receivers Geoffrey C. Orsak 10 Forward Error Correction Coding V.K. Bhargava and I.J. Fair 11 Spread Spectrum Communications Laurence B. Milstein and Marvin K. Simon 12 Diversity Arogyaswami J. Paulraj 13 Digital Communication System Performance Bernard Sklar 14 Telecommunications Standardization Spiros Dimolitsas and Michael Onufry PARTIIWireless 15 Wireless Personal Communications: A Perspective Donald C. Cox 16 Modulation Methods Gordon L. St¨uber 17 Access Methods Bernd-Peter Paris 18 Rayleigh Fading Channels Bernard Sklar 19 Space-Time Processing Arogyaswami J. Paulraj 20 Location Strategies for Personal Communications Services Rav i Jain, Yi-Bing Lin, and Seshadri Mohan 1 21 Cell Design Principles Michel Daoud Yacoub 22 Microcellular Radio Communications Raymond Steele 23 Fixed and Dynamic Channel Assignment Bijan Jabbari 24 Radiolocation Techniques Gordon L. St¨uber and James J. Caffery, Jr. 25 Power Control Roman Pichna and Qiang Wang 26 Enhancements in Second Generation Systems Marc Delprat and Vinod Kumar 27 The Pan-European Cellular System Lajos Hanzo 28 Speech and Channel Coding for North American TDMA Cellular Systems Paul Mermelstein 29 The Br itish Cordless Telephone Standard: CT-2 Lajos Hanzo 30 Half-Rate Standards Wai-Yip Chan, Ira Gerson, and Toshio Miki 31 Wireless Video Communications Madhukar Budagavi and Raj Talluri 32 Wireless LANs Suresh Singh 33 Wireless Data Allen H. Levesque and Kaveh Pahlavan 34 Wireless ATM: Interworking Aspects Melbourne Barton, Matthew Cheng, and Li Fung Chang c  1999 by CRC Press LLC 35 Wireless ATM: QoS and Mobility Management Bala Rajagopalan and Daniel Reininger 36 An Overview of cdma2000, WCDMA, and EDGE Tero Ojanper¨a and Steven D. Gray c  1999 by CRC Press LLC Couch, II, L.W. “Complex Envelope Representations for Modulated Signals” Mobile Communications Handbook Ed. Suthan S. Suthersan Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC ComplexEnvelopeRepresentations forModulatedSignals 1 LeonW.Couch,II UniversityofFlorida 1.1 Introduction 1.2 ComplexEnvelopeRepresentation 1.3 RepresentationofModulatedSignals 1.4 GeneralizedTransmittersandReceivers 1.5 SpectrumandPowerofBandpassSignals 1.6 AmplitudeModulation 1.7 PhaseandFrequencyModulation 1.8 QPSKSignalling DefiningTerms References FurtherInformation 1.1 Introduction Whatisageneralrepresentationforbandpassdigitalandanalogsignals?Howdowerepresenta modulatedsignal?Howdoweevaluatethespectrumandthepowerofthesesignals?Thesearesome ofthequestionsthatareansweredinthischapter. Abasebandwaveformhasaspectralmagnitudethatisnonzeroforfrequenciesinthevicinityof theorigin(i.e.,f=0)andnegligibleelsewhere.Abandpasswaveformhasaspectralmagnitudethat isnonzeroforfrequenciesinsomebandconcentratedaboutafrequencyf=±f c (wheref c 0), andthespectralmagnitudeisnegligibleelsewhere.f c iscalledthecarrierfrequency.Thevalueof f c maybearbitrarilyassignedformathematicalconvenienceinsomeproblems.Inothers,namely, modulationproblems,f c isthefrequencyofanoscillatorysignalinthetransmittercircuitandisthe assignedfrequencyofthetransmitter,suchas850kHzforanAMbroadcastingstation. Incommunicationproblems,theinformationsourcesignalisusuallyabasebandsignal—for example,atransistor-transistorlogic(TTL)waveformfromadigitalcircuitoranaudio(analog) signalfromamicrophone.Thecommunicationengineerhasthejobofbuildingasystemthatwill transfertheinformationfromthissourcesignaltothedesireddestination.AsshowninFig.1.1,this 1 Source:Couch,LeonW.,II.1997.DigitalandAnalogCommunicationSystems,5thed.,PrenticeHall,UpperSaddleRiver, NJ. c  1999byCRCPressLLC usuallyrequirestheuseofabandpasssignal,s(t),whichhasabandpassspectrumthatisconcentrated at±f c wheref c isselectedsothats(t)willpropagateacrossthecommunicationchannel(eithera wireorawirelesschannel). FIGURE1.1:Bandpasscommunicationsystem.Source:Couch,L.W.,II.1997.DigitalandAnalog CommunicationSystems,5thed.,PrenticeHall,UpperSaddleRiver,NJ,p.227.Withpermission. Modulationistheprocessofimpartingthesourceinformationontoabandpasssignalwithacarrier frequencyf c bytheintroductionofamplitudeand/orphaseperturbations.Thisbandpasssignal iscalledthemodulatedsignals(t),andthebasebandsourcesignaliscalledthemodulatingsignal m(t).Examplesofexactlyhowmodulationisaccomplishedaregivenlaterinthischapter.This definitionindicatesthatmodulationmaybevisualizedasamappingoperationthatmapsthesource informationontothebandpasssignals(t)thatwillbetransmittedoverthechannel. Asthemodulatedsignalpassesthroughthechannel,noisecorruptsit.Theresultisabandpass signal-plus-noisewaveformthatisavailableatthereceiverinput,r(t),asillustratedinFig.1.1.The receiverhasthejoboftryingtorecovertheinformationthatwassentfromthesource;˜mdenotesthe corruptedversionofm. 1.2 ComplexEnvelopeRepresentation Allbandpasswaveforms,whethertheyarisefromamodulatedsignal,interferingsignals,ornoise, mayberepresentedinaconvenientformgivenbythefollowingtheorem.v(t)willbeusedtodenote thebandpasswaveformcanonically.Thatis,v(t)canrepresentthesignalwhens(t)≡v(t),the noisewhenn(t)≡v(t),thefilteredsignalplusnoiseatthechanneloutputwhenr(t)≡v(t),orany othertypeofbandpasswaveform 2 . THEOREM1.1 Anyphysicalbandpasswaveformcanberepresentedby v(t)=Re  g(t)e jω c t  (1.1a) Re{·}denotestherealpartof{·}.g(t)iscalledthecomplexenvelopeofv(t),andf c istheassociated carrierfrequency(hertz)whereω c =2πf c .Furthermore,twootherequivalentrepresentationsare 2 Thesymbol≡denotesanequivalenceandthesymbol  =denotesadefinition. c  1999byCRCPressLLC v(t)=R(t)cos [ ω c t+θ(t) ] (1.1b) and v(t)=x(t)cosω c t−y(t)sinω c tψ (1.1c) where g(t)=x(t)+jy(t)=|g(t)|e j  g(t) ≡R(t)e jθ(t) (1.2) x(t)= Re{g(t)}≡R(t)cosθ(t)ψ (1.3a) y(x)= Im{g(t)}≡R(t)sinθ(t)ψ (1.3b) R(t)  =|g(t)|≡  x 2 (t)+y 2 (t)ψ (1.4a) θ(t)  =  g(t)=tan −1  y(t) x(t)  (1.4b) Thewaveformsg(t),x(t),y(t),R(t),andθ(t)areallbasebandwaveforms,and,exceptforg(t), theyareallrealwaveforms.R(t)isanonnegativerealwaveform.Equation(1.1a–1.1c)isalow-pass- to-bandpasstransformation.Thee jω c t factorin(1.1a)shifts(i.e.,translates)thespectrumofthe basebandsignalg(t)frombasebanduptothecarrierfrequencyf c .Incommunicationsterminology thefrequenciesinthebasebandsignalg(t)aresaidtobeheterodyneduptof c .Thecomplexenvelope, g(t),isusuallyacomplexfunctionoftimeanditisthegeneralizationofthephasorconcept.That is,ifg(t)happenstobeacomplexconstant,thenv(t)isapuresinewaveoffrequencyf c andthis complexconstantisthephasorrepresentingthesinewave.Ifg(t)isnotaconstant,thenv(t)isnot apuresinewavebecausetheamplitudeandphaseofv(t)varieswithtime,causedbythevariations ofg(t). RepresentingthecomplexenvelopeintermsoftworealfunctionsinCartesiancoordinates,we have g(x)≡x(t)+jy(t)ψ (1.5) wherex(t)=Re{g(t)}andy(t)=Im{g(t)}.x(t)issaidtobethein-phasemodulationassociated withv(t),andy(t)issaidtobethequadraturemodulationassociatedwithv(t).Alternatively,the polarformofg(t),representedbyR(t)andθ(t),isgivenby(1.2),wheretheidentitiesbetween Cartesianandpolarcoordinatesaregivenby(1.3a–1.3b)and(1.4a–1.4b).R(t)andθ(t)arereal waveformsand,inaddition,R(t)isalwaysnonnegative.R(t)issaidtobetheamplitudemodulation (AM)onv(t),andθ(t)issaidtobethephasemodulation(PM)onv(t). Theusefulnessofthecomplexenveloperepresentationforbandpasswaveformscannotbeoverem- phasized.Inmoderncommunicationsystems,thebandpasssignalisoftenpartitionedintotwochan- nels,oneforx(t)calledtheI(in-phase)channelandonefory(t)calledtheQ(quadrature-phase) channel.Indigitalcomputersimulationsofbandpasssignals,thesamplingrateusedinthesimu- lationcanbeminimizedbyworkingwiththecomplexenvelope,g(t),insteadofwiththebandpass signal,v(t),becauseg(t)isthebasebandequivalentofthebandpasssignal[1]. c  1999byCRCPressLLC 1.3 Representation of Modulated Signals Modulation is the process of encoding the source information m(t) (modulating signal) into a band- pass signal s(t) (modulated signal). Consequently, the modulated signal is just a special application of the bandpass representation. The modulated signal is given by s(t) = Re  g(t)e jω c t  (1.6) where ω c = 2πf c . f c is the carrier frequency. The complex envelope g(t) is a function of the modulating signal m(t). That is, g(t) = g[m(t)] (1.7) Thus g[·] performs a mapping operation on m(t). This was shown in Fig. 1.1. Table 1.1 givesanoverviewofthebig picture for themodulationproblem. Examples of the mapping function g[m]are given for amplitude modulation (AM), double-sideband suppressed carrier (DSB- SC), phase modulation (PM), frequency modulation (FM), single-sideband AM suppressed carrier (SSB-AM-SC), single-sideband PM (SSB-PM), single-sideband FM (SSB-FM), single-sideband en- velope detectable (SSB-EV), single-sideband square-law detectable (SSB-SQ), and quadrature mod- ulation (QM). For each g[m], Table 1.1 also shows the corresponding x(t) and y(t) quadrature modulation components, and the corresponding R(t) and θ(t) amplitude and phase modulation components. Digitally modulated bandpass signals are obtained when m(t) is a digital baseband signal—for example, the output of a transistor transistor logic (TTL) circuit. Obviously, it is possible to use other g[m] functions that are not listed in Table 1.1. The question is: Are they useful? g[m]functions are desired that are easy to implement and that will give desirable spectral properties. Furthermore, in the receiver the inverse function m[g] is required. The inverse should be single valued over the range used and should be easily implemented. The inverse mapping should suppress as much noise as possible so that m(t) can be recovered with little corruption. 1.4 Generalized Transmitters and Receivers A more detailed description of transmitters and receivers as first shown in Fig. 1.1 will now be illustrated. There are two canonical forms for the generalized transmitter, as indicated by (1.1b) and (1.1c). Equation (1.1b)describes anAM-PMt ype circuitas showninFig. 1.2. The basebandsignalprocessing circuit generates R(t) and θ(t)from m(t).TheR and θ are functions of the modulating signal m(t), as given in Table 1.1, for the particular modulation type desired. The signal processing may be implemented either by using nonlinear analog circuits or a digital computer that incorporates the R and θ algorithms under software program control. In the implementation using a digital computer, one analog-to-digital converter (ADC) will be needed at the input of the baseband signal processor and two digital-to-analog converters (DACs) w ill be needed at the output. The remainder of the AM-PM canonical form requires radio frequency (RF) circuits, as indicated in the figure. Figure 1.3 illustrates the second canonical form for the generalized transmitter. This uses in-phase and quadrature-phase (IQ) processing. Similarly, the formulas relating x(t) and y(t) to m(t ) are shown in Table 1.1, and the baseband signal processing may be implemented by using either analog hardware or digital hardware with software. The remainder of the canonical form uses RF circuits as indicated. Analogous to the transmitter realizations, there are two canonical forms of receiver. Each one consistsof RFcarriercircuitsfollowedbybasebandsignal processingasillustrated inFig.1.1. Typically c  1999 by CRC Press LLC TABLE 1.1 Complex Envelope Functions for Various Types of Modulation a Corresponding Quadrature Modulation Type of Mapping Functions Modulation g(m) x(t) y(t) AM A c [1 +m(t)] A c [1 +m(t)] 0 DSB-SC A c m(t) A c m(t) 0 PM A c e jD p m(t) A c cos[D p m(t)] A c sin[D p m(t)] FM A c e jD f  t −∞ m(σ )dσ A c cos  D f  t −∞ m(σ )dσ  A c sin  D f  t −∞ m(σ )dσ  SSB-AM-SC b A c [m(t) ±j ˆm(t)] A c m(t) ±A c ˆm(t) SSB-PM b A c e jD p [m(t)±j ˆm(t)] A c e ∓D p ˆm(t) cos[D p m(t)] A c e ∓D p ˆm(t) sin[D p m(t)] SSB-FM b A c e jD f  t −∞ [m(σ )±j ˆm(σ)]dσ A c e ∓D f  t −∞ ˆm(σ)dσ cos  D f  t −∞ m(σ )dσ  A c e ∓D f  t −∞ ˆm(σ)dσ sin  D f  t −∞ m(σ )dσ  SSB-EV b A c e {ln[1+m(t)]±j ˆ ln|1+m(t)|} A c [1 +m(t)]cos{ ˆ ln[1 +m(t)]} ±A c [1 +m(t)]sin{ ˆ ln[1 +m(t)]} SSB-SQ b A c e (1/2){ln[1+m(t)]±j ˆ ln|1+m(t)|} A c √ 1 +m(t) cos{ 1 2 ˆ ln[1 +m(t)]} ±A c √ 1 +m(t) sin{ 1 2 ˆ ln[1 +m(t)]} QM A c [m 1 (t) +jm 2 (t)] A c m 1 (t) A c m 2 (t) c  1999 by CRC Press LLC TABLE 1.1 Complex Envelope Functions for Various Types of Modulation a (Continued) Corresponding Amplitude and Phase Modulation Type of Modulation R(t) θ(t) Linearity Remarks AM A c |1 +m(t)|  0, m(t) > −1 180 ◦ , m(t) < −1 L c m(t) > −1 required for envelope de- tection DSB-SC A c |m(t)|  0, m(t) > 0 180 ◦ , m(t) < 0 L Coherent detection required PM A c D p m(t) NL D p is the phase deviation constant (rad/volt) FM A c D f  t −∞ m(σ )dσ NL D f is the frequency deviation con- stant (rad/volt-sec) SSB-AM-SC b A c  [m(t)] 2 +[ˆm(t)] 2 tan −1 [± ˆm(t)/m(t )] L Coherent detection required SSB-PM b A c e ±D p ˆm(t) D p m(t) NL SSB-FM b A c e ±D f  t −∞ ˆm(σ)dσ D f  t −∞ m(σ )dσ NL SSB-EV b A c |1 +m(t)|± ˆ ln[1 +m(t)] NL m(t) > −1 is required so that the ln(·) will have a real value SSB-SQ b A c √ 1 +m(t) ± 1 2 ˆ ln[1 +m(t)] NL m(t) > −1 is required so that the ln(·) will have a real value QM A c  m 2 1 (t) +m 2 2 (t) tan −1 [m 2 (t)/m 1 (t)] L Used in NTSC color television; re- quires coherent detection Source: Couch, L.W., II, 1997, Digital and Analog Communication Systems, 5th ed., Prentice Hall, Upper Saddle River, NJ, pp. 231-232. With permission. a A c > 0 is a constant that sets the power level of the signal as evaluated by use of (1.11); L, linear; NL, nonlinear; and [ˆ·] is the Hilbert transform (a −90 ◦ phase-shifted version of [·]). For example, ˆm(t) = m(t ) ∗ 1 πt = 1 π  ∞ −∞ m(λ) t−λ dλ. b Use upper signs for upper sideband signals and lower signals for lower sideband signals. c In the strict sense, AM signals are not linear because the carr ier term does not satisfy the linearity (superposition) condition. FIGURE 1.2: Generalized t ransmitter using the AM-PM generation technique. Source: Couch, L.W., II. 1997. Digital and Analog Communication Systems, 5th ed., Prentice Hall, Upper Saddle River, NJ, p. 278. With permission. the carrier circuits are of the superheterodyne-receiver t ype which consist of an RF amplifier, a down converter (mixer plus local oscillator) to some intermediate frequency (IF), an IF amplifier and then detector circuits [1]. In the first canonical form of the receiver, the carrier circuits have amplitude and phase detectors that output ˜ R(t) and ˜ θ(t), respectively. This pair, ˜ R(t) and ˜ θ(t), describe the polar form of the received complex envelope, ˜g(t). ˜ R(t) and ˜ θ(t) are then fed into the signal processor c  1999 by CRC Press LLC FIGURE1.3:Generalizedtransmitterusingthequadraturegenerationtechnique.Source:Couch, L.W.,II.1997.DigitalandAnalogCommunicationSystems,5thed.,PrenticeHall,UpperSaddleRiver, NJ,p.278.Withpermission. whichusestheinversefunctionsofTable1.1togeneratetherecoveredmodulation,˜m(t).Thesecond canonicalformofthereceiverusesquadratureproductdetectorsinthecarriercircuitstoproduce theCartesianformofthereceivedcomplexenvelope,˜x(t)and˜y(t).˜x(t)and˜y(t)aretheninputted tothesignalprocessorwhichgenerates˜m(t)atitsoutput. Onceagain,itisstressedthatanytypeofsignalmodulation(seeTable1.1)maybegenerated (transmitted)ordetected(received)byusingeitherofthesetwocanonicalforms.Bothoftheseforms convenientlyseparatebasebandprocessingfromRFprocessing.Digitaltechniquesareespecially usefultorealizethebasebandprocessingportion.Furthermore,ifdigitalcomputingcircuitsare used,anydesiredmodulationtypecanberealizedbyselectingtheappropriatesoftwarealgorithm. 1.5 SpectrumandPowerofBandpassSignals Thespectrumofthebandpasssignalisthetranslationofthespectrumofitscomplexenvelope. TakingtheFouriertransformof(1.1a),thespectrumofthebandpasswaveformis[1] V(f)= 1 2  G ( f−f c ) +G ∗ ( −f−f c )  (1.8) whereG(f)istheFouriertransformofg(t), G(f)=  ∞ −∞ g(t)e −j2πft dt, andtheasterisksuperscriptdenotesthecomplexconjugateoperation.Thepowerspectradensity (PSD)ofthebandpasswaveformis[1] P v (f)= 1 4  P g ( f−f c ) +P g ( −f−f c )  (1.9) whereP g (f)isthePSDofg(t). TheaveragepowerdissipatedinaresistiveloadisV 2 rms /R L orI 2 rms R L whereV rms istherms valueofthevoltagewaveformacrosstheloadandI rms isthermsvalueofthecurrentthroughthe c  1999byCRCPressLLC [...]... Sampling Theorem in the Frequency Domain The sampling theorem expressed in Eq (2.4) is the time-domain sampling theorem There is a dual to this time-domain sampling theorem, i.e., the sampling theorem in the frequency domain Time-limited signals: A continuous-time signal m(t) is called time limited if m(t) = 0 for |t| > |T0 | (2.25) Frequency-domain sampling theorem: The frequency-domain sampling theorem... energized and a binary counter is started The output of the ramp generator is continuously compared to the sample value; when the value of the ramp becomes equal to the sample value, the binary value of the counter is read This count is taken to be the PCM word The binary counter and the ramp generator are then reset to zero and are ready to be reenergized at the next sampling time This technique c 1999... proportional to the divider reference voltage multiplied by the value of the digital word The Motorola MC1408 and the National Semiconductor DAC0808 8-b DAC chips are examples of this technique The DAC chip outputs samples of the quantized analog signal that approximates the analog sample values This may be smoothed by a low-pass reconstruction filter to produce the analog output The Communications Handbook. .. but the speed of this type of ADC is usually limited by the speed of the counter The Maxim ICL7126 CMOS ADC integrated circuit uses this technique The serial encoder compares the value of the sample with trial quantized values Successive trials depend on whether the past comparator outputs are positive or negative The trial values are chosen first in large steps and then in small steps so that the process... example, if the data rate of the baseband information source is 9600 bits/sec, then the null-to-null bandwidth of the QPSK signal would be 9.6 Hz since = 2 Referring to Fig 1.6, it is seen that the sidelobes of the spectrum are relatively large so, in practice, the sidelobes of the spectrum are filtered off to prevent interference to the adjacent channels This filtering rounds off the edges of the rectangular... regenerated at the output of each repeater, where the input consists of a noisy PCM waveform The noise at the input, however, may cause bit errors in the regenerated PCM output signal • The noise performance of a digital system can be superior to that of an analog system In addition, the probability of error for the system output can be reduced even further by the use of appropriate coding techniques These... of the steps are of equal size Since we are approximating the analog sample values by using a finite number of levels (M = 8 in this illustration), error is introduced into the recovered output analog signal because of the quantizing effect The error waveform is illustrated in Fig 3.2c The quantizing error consists of the difference between the analog signal at the sampler input and the output of the. .. nTs ) (2.3) which is known as the Nyquist–Shannon interpolation formula and it is also sometimes called the cardinal series The sampling interval Ts = 1/(2fM )is called the Nyquist interval and the minimum rate fs = 1/Ts = 2fM is known as the Nyquist rate Illustration of the instantaneous sampling process and the sampling theorem is shown in Fig 2.1 The Fourier transform of the unit impulse train is given... digital transmission, the speech is normally sampled at the rate fs = 8 kHz The guard band is then fs − 2fM = 1.4 kHz The use of a sampling rate higher than the Nyquist rate also has the desirable effect of making it somewhat easier to design the low-pass reconstruction filter so as to recover the original signal from the sampled signal 2.4 Sampling of Sinusoidal Signals A special case is the sampling of... The trial voltages are generated by a series of voltage dividers that are configured by (on-off) switches These switches are controlled by digital logic After the process converges, the value of the switch settings is read out as the PCM word This technique requires more precision components (for the voltage dividers) than the ramp technique The speed of the feedback ADC technique is determined by the . SpectrumandPowerofBandpassSignals Thespectrumofthebandpasssignalisthetranslationofthespectrumofitscomplexenvelope. TakingtheFouriertransformof(1.1a),thespectrumofthebandpasswaveformis[1] V(f)= 1 2  G ( f−f c ) +G ∗ ( −f−f c )  (1.8) whereG(f)istheFouriertransformofg(t), G(f)=  ∞ −∞ g(t)e −j2πft dt, andtheasterisksuperscriptdenotesthecomplexconjugateoperation.Thepowerspectradensity (PSD)ofthebandpasswaveformis[1] P v (f)= 1 4  P g ( f−f c ) +P g ( −f−f c )  (1.9) whereP g (f)isthePSDofg(t). TheaveragepowerdissipatedinaresistiveloadisV 2 rms /R L orI 2 rms R L whereV rms istherms valueofthevoltagewaveformacrosstheloadandI rms isthermsvalueofthecurrentthroughthe c  1999byCRCPressLLC load. For bandpass waveforms, Equation (1.1a–1.1c) may represent either the. Introduction Totransmitanalogmessagesignals,suchasspeechsignalsorvideosignals,bydigitalmeans,thesignal hastobeconvertedintodigitalform.Thisprocessisknownasanalog-to-digitalconversion .The samplingprocessisthefirstprocessperformedinthisconversion,anditconvertsacontinuous-time signalintoadiscrete-timesignalorasequenceofnumbers.Digitaltransmissionofanalogsignalsis possiblebyvirtueofthesamplingtheorem,andthesamplingoperationisperformedinaccordance withthesamplingtheorem. Inthischapter,usingtheFouriertransformtechnique,wepresentthisremarkablesamplingthe- oremanddiscusstheoperationofsamplingandpracticalaspectsofsampling. 2.2. (1.4a) θ(t)  =  g(t)=tan −1  y(t) x(t)  (1.4b) Thewaveformsg(t),x(t),y(t),R(t),andθ(t)areallbasebandwaveforms,and,exceptforg(t), theyareallrealwaveforms.R(t)isanonnegativerealwaveform.Equation(1.1a–1.1c)isalow-pass- to-bandpasstransformation.Thee jω c t factorin(1.1a)shifts(i.e.,translates)thespectrumofthe basebandsignalg(t)frombasebanduptothecarrierfrequencyf c .Incommunicationsterminology thefrequenciesinthebasebandsignalg(t)aresaidtobeheterodyneduptof c .Thecomplexenvelope, g(t),isusuallyacomplexfunctionoftimeanditisthegeneralizationofthephasorconcept.That is,ifg(t)happenstobeacomplexconstant,thenv(t)isapuresinewaveoffrequencyf c andthis complexconstantisthephasorrepresentingthesinewave.Ifg(t)isnotaconstant,thenv(t)isnot apuresinewavebecausetheamplitudeandphaseofv(t)varieswithtime,causedbythevariations ofg(t). RepresentingthecomplexenvelopeintermsoftworealfunctionsinCartesiancoordinates,we have g(x)≡x(t)+jy(t)ψ