Quadrature Amplitude Modulation (QAM) TransmitterIntroductionAmplitude Modulation by CosineAmplitude Modulation by SineBaseband Digital QAM TransmitterPhase Shift by 90 DegreesHilbert TransformerDiscrete-Time Hilbert TransformerSlide 9Performance Analysis of PAMSlide 11Performance Analysis of QAMPerformance Analysis of 16-QAMSlide 14Slide 15Average Power AnalysisEE445S Real-Time Digital Signal Processing Lab Fall 2014 Lecture 15 http://www.ece.utexas.edu/~bevans/courses/realtimeQuadrature Amplitude Modulation (QAM) TransmitterProf. Brian L. EvansDept. of Electrical and Computer EngineeringThe University of Texas at Austin15 - 2Introduction•Digital Pulse Amplitude Modulation (PAM)Modulates digital information onto amplitude of pulseMay be later upconverted (e.g. to radio frequency)•Digital Quadrature Amplitude Modulation (QAM)Two-dimensional extension of digital PAMBaseband signal requires sinusoidal amplitude modulationMay be later upconverted (e.g. to radio frequency)•Digital QAM modulates digital information onto pulses that are modulated ontoAmplitudes of a sine and a cosine, or equivalentlyAmplitude and phase of single sinusoidAmplitude Modulation by Cosine•y1(t) = x1(t) cos(c t)Assume x1(t) is an ideal lowpass signal with bandwidth 1Assume 1 << cY1() is real-valued if X1() is real-valued•Demodulation: modulation then lowpass filtering01-X1()0Y1()½-c - -c + cc - c + c½X1c½X1cReview     ccXXY1112121Baseband signal Upconverted signal15 - 3Amplitude Modulation by Sine•y2(t) = x2(t) sin(c t)Assume x2(t) is an ideal lowpass signal with bandwidth 2Assume 2 << cY2() is imaginary-valued if X2() is real-valued•Demodulation: modulation then lowpass filteringY2()j ½-c – -c + cc – c + c-j ½X2cj ½X2c-j ½01-X2()Review     ccXjXjY22222Baseband signal Upconverted signal15 - 4Baseband Digital QAM Transmitter•Continuous-time filtering and upconversion15 - 5i[n]gT(t)+q[n]gT(t)Serial/parallelconverter1BitsMap to 2-D constellationJPulse shapers(FIR filters)IndexImpulsemodulatorImpulsemodulators(t)Local Oscillator90oDelayDelay matches delay through 90o phase shifterDelay required but often omitted in diagrams4-level QAM ConstellationIQdd-d-d15 - 6Phase Shift by 90 Degrees•90o phase shift performed by Hilbert transformercosine => sinesine => – cosine•Frequency response)(21)(21) 2cos(000fffftf )(2)(2) 2sin(000ffjffjtf )sgn()( fjfH f)( fH-90o90of|)(| fHMagnitude Response Phase ResponseAll-pass except at origin15 - 7Hilbert Transformer•Continuous-time ideal Hilbert transformer•Discrete-time ideal Hilbert transformerh(t) = 1/( t) if t  00 if t = 0h[n] =if n00 if n=0nn )2/(sin22Even-indexed samples are zeroth(t))sgn()( fjfH  )sgn()(jH nh[n]15 - 8Discrete-Time Hilbert Transformer•Approximate by odd-length linear phase FIR filterTruncate response to 2 L + 1 samples: L samples left of origin, L samples right of origin, and originShift truncated impulse response by L samples to right to make it causalL is odd because every other sample of impulse response is 0•Linear phase FIR filter of length N has same phase response as an ideal delay of length (N-1)/2(N-1)/2 is an integer when N is odd (here N = 2 L + 1)•Matched delay block on slide 15-5 would be an ideal delay of L samples15 - 9Baseband Digital QAM Transmitteri[n]gT(t)+q[n]gT(t)Serial/parallelconverter1BitsMap to 2-D constellationJPulse shapers(FIR filters)IndexImpulsemodulatorImpulsemodulators(t)Local Oscillator90oDelayi[n]gT[m] L+cos(0 m)q[n]gT[m] Lsin(0 m)Serial/parallelconverter1BitsMap to 2-D constellationJL samples/symbol (upsampling factor)Pulse shapers(FIR filters)Indexs[m]D/As(t)100% discrete time15 - 10Performance Analysis of PAM•If we sample matched filter output at correct time instances, nTsym, without any ISI, received signalwhere transmitted signal isv(t) output of matched filter Gr() for input ofchannel additive white Gaussian noise N(0; 2)Gr() passes frequencies from -sym/2 to sym/2 ,where sym = 2  fsym = 2 / Tsym•Matched filter has impulse response gr(t))()()(symsymsymnTvnTsnTx dianTsnsym)12()( for i = -M/2+1, …, M/2v(nT) ~ N(0; 2/Tsym)4-level PAM Constellationdd3 d 3 d15 - 11Performance Analysis of PAM•Decision errorfor inner points•Decision errorfor outer points •Symbol error probability-7d -5d -3d -d d 3d 5d 7dO-I I I I I I O+symsymITdQdnTvPeP 2))(()(symsymOTdQdnTvPeP))(()(symsymsymOTdQdnTvPdnTvPeP))(())(()(symOOITdQMMePMePMePMMeP)1(2)(1)(1)(2)(8-level PAM Constellation15 - 12Performance Analysis of QAM•If we sample matched filter outputs at correct time instances, nTsym, without any ISI, received signal•Transmitted signalwhere i,k  { -1, 0, 1, 2 } for 16-QAM•NoiseFor error probability analysis, assume noise terms independent and each term is Gaussian random variable ~ N(0; 2/Tsym) In reality, noise terms have common source of additive noise in channel)()()(symsymsymnTvnTsnTx dkjdibjanTsnnsym)12( )12( )( )( )()(symQsymIsymnTvjnTvnTv 4-level QAM ConstellationIQdd-d-d15 - 13Performance Analysis of 16-QAM•Type 1 correct detection))(&)(()(1dnTvdnTvPcPsymQsymI   dnTvPdnTvPsymQsymI )( )(     dnTvPdnTvPsymQsymI )(1 )(1221symTdQ)(2 TdQ)(2 TdQ33332 2222 222111 1IQ16-QAM1 - interior decision region2 - edge region but not corner3 - corner region15 - 14Performance Analysis of 16-QAM•Type 2 correct detection•Type 3 correct detection))(&)(()(2dnTvdnTvPcPsymQsymI))(())(( dnTvPdnTvPsymQsymIsymsymTdQTdQ211))(&)(()(3dnTvdnTvPcPsymQsymI))(())(( dnTvPdnTvPsymQsymI21symTdQ33332 2222