OutlineTransmitters (week 1 and 2)Digital Communication System:Transmitter TopicsIncreasing Noise ImmunityIncreasing bandwidth EfficiencyQAM modulationSlide 8Slide 9QAM waveformsSlide 11QAM signal spaceEuclidean distance between codesSlide 14Rectangular QAM signal spaceSlide 16Slide 17Channel ModelingBaseband DemodulationBandwidth required of QAMSlide 21Slide 22Slide 23Actual QAM bandwidthSlide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Channel BandwidthOutline•Transmitters (Chapters 3 and 4, Source Coding and Modulation) (week 1 and 2)•Receivers (Chapter 5) (week 3 and 4) •Received Signal Synchronization (Chapter 6) (week 5)•Channel Capacity (Chapter 7) (week 6)•Error Correction Codes (Chapter 8) (week 7 and 8)•Equalization (Bandwidth Constrained Channels) (Chapter 10) (week 9)•Adaptive Equalization (Chapter 11) (week 10 and 11)•Spread Spectrum (Chapter 13) (week 12)•Fading and multi path (Chapter 14) (week 12)Transmitters (week 1 and 2)•Information Measures•Vector Quantization•Delta Modulation•QAMDigital Communication System:TransmitterReceiverInformation per bit increasesnoise immunity increasesBandwidth efficiency increasesTransmitter Topics•Increasing information per bit•Increasing noise immunity •Increasing bandwidth efficiencyIncreasing Noise Immunity •Coding (Chapter 8, weeks 7 and 8)Increasing bandwidth Efficiency•Modulation of digital data into analog waveforms–Impact of Modulation on Bandwidth efficiencyQAM modulation•Quadrature Amplitude Modulation–Really Quadrature Phase Amplitude modulation TtMmtftgVtftgAtftgAetgjAAtsmcmcmscmctfjmsmcmc0,,2,1)2cos()(2sin)(2cos)()()(Re)(2Amplitude and Phase modulationg(t) is a pulse waveform to control the spectrum, e.g., raised cosineQAM waveforms•To construct the wave forms we need to know fc, g(t), Amc, and Ams•However, we can write sm(t) as an linear combination of orthonormal waveforms:)()()(2211tfstfstsmmmQAM waveforms•QAM orthonormal waveforms:tftgtftftgtfcgcg2sin)(2)(2cos)(2)(21)()()(2211tfstfstsmmm 2221gmsgmcmmmAAsss dttgg)(2QAM signal space•QAM wave form can be represented by just the vector sm –(still need fc, g(t), and g to make actual waveforms)•Signal space Constellation determines all of the code vectors sm1sm2Euclidean distance between codesnmnmmnmmmnnmnmnmnmnnmmnnmmnmnmnmemnssssssssssssdsssssssss)Re()Re(2222)()(222222222211121222211)(Is the Energy of the signalIs the cross correlation of the signalsEuclidean distance between codes•Signals of similar energy and highly cross correlated have a small Euclidean separation•Euclidean separation of adjacent signal vectors is thus a good measure of the ability of one signal to be mistaken for the other and cause error•Choose constellations with max space between vectors for min error probabilityRectangular QAM signal space•Minimum Euclidean distance between the M codes is?sm1sm2Rectangular QAM signal space•Euclidean distance between the M codes is:))()(())()((2,2,1,,2,1,,,2,1,,2,1,,2222)(nmnmgnsmsncmcgnmemnnnnnsnncmmmmsmmcjjiidAAAAdMjMidjAdiAMjMidjAdiAssRectangular QAM signal space•Minimum euclidean distance between the M codes is:sm1sm2gemneddd2min)()(mingd2Channel Modeling•Noise–Additive–White–GaussianContaminated baseband signalBaseband Demodulation•Correlative receiver•Matched filter receiver64-QAM Demodulated DataBandwidth required of QAM•If k bits of information is encoded in the amplitude and phase combinations then the data rate:TkR /Where 1/T = Symbol Rate = R/kBandwidth required of QAM•Can show that bandwidth W needed is approximately 1/T for Optimal ReceiverMRkRTW2log1Where M = number of symbols(k = number of bits per symbol)Bandwidth required of QAM•Bandwidth efficiency of QAM is thus:MWR2logBandwidth required of QAMActual QAM bandwidth•Consider Power Spectra of QAM   ))(()(21)()(Re)()(Re)(22cvvcvvssfjvvsstfjfffffeetvtsccBand-pass signals can be expressedAutocorrelation function isFourier Transform yields Power spectrum inTerms of the low pass signal v(t) Power spectrumActual QAM bandwidth•Power Spectra of QAMnsncnnnnnjAAIInTtgItv}{)()(For linear digital mod signalsSequence of symbols isFor QAMActual QAM bandwidth)()(1)()()(1)()()()()()(][21)(2_***ffGTfmTmTmTnTtgnTtgmmTtgnTtgIIEiivvmggiivvmmiin mmnvv Where][21)(*mnniiIIEmTime averaging this:Fourier Transform:Assume stationary symbolsActual QAM bandwidth)()(1)(2ffGTfiivvG(f) is Fourier transform of g(t))( fiiis power spectrum of symbolsActual QAM bandwidthG(f) is Fourier transform of g(t)Actual QAM bandwidthG(f) is Fourier transform of g(t)Actual QAM bandwidth)( fiipower spectrum of symbols•Determined by what data you send•Very random data gives broad spectrumActual QAM bandwidth ))(()(21)(cvvcvvssfffff )()(1)(2ffGTfiivvWhite noise for randomSymbol stream and QAM?Channel Bandwidth•3-dB bandwidth•Or your definition and justification•g(t) = Modulated 64-QAM