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EE445S Real-Time Digital Signal Processing Lab Spring 2014 Lecture 15 Quadrature Amplitude Modulation (QAM) Transmitter Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin

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Introduction Digital Pulse Amplitude Modulation (PAM) Modulates digital information onto amplitude of pulse May be later upconverted (e.g. to radio frequency) Digital Quadrature Amplitude Modulation (QAM) Two-dimensional extension of digital PAM Baseband signal requires sinusoidal amplitude modulation May be later upconverted (e.g. to radio frequency) Digital QAM modulates digital information onto pulses that are modulated onto Amplitudes of a sine and a cosine, or equivalently Amplitude and phase of single sinusoid

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Amplitude Modulation by Cosine y 1 (t) = x 1 (t) cos( c t) Assume x 1 (t) is an ideal lowpass signal with bandwidth 1 Assume 1 << c Y 1 ( ) is real-valued if X 1 ( ) is real-valued Demodulation: modulation then lowpass filtering 0 1 -- X1()X1() 0 Y1()Y1() ½ - c - - c + c c - c + cc ½X 1 c ½X 1 c Review Baseband signalUpconverted signal

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Amplitude Modulation by Sine y 2 (t) = x 2 (t) sin( c t) Assume x 2 (t) is an ideal lowpass signal with bandwidth 2 Assume 2 << c Y 2 ( ) is imaginary-valued if X 2 ( ) is real-valued Demodulation: modulation then lowpass filtering Y2()Y2() j ½ - c – - c + cc c – c + cc -j ½X 2 c j ½X 2 c -j ½ 0 1 -- X2()X2() Review Baseband signalUpconverted signal

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Baseband Digital QAM Transmitter Continuous-time filtering and upconversion i[n]i[n] gT(t)gT(t) + q[n]q[n] gT(t)gT(t) Serial/ parallel converter 1 Bits Map to 2-D constellation J Pulse shapers (FIR filters) Index Impulse modulator s(t)s(t) Local Oscillator 90 o Delay Delay matches delay through 90 o phase shifter Delay required but often omitted in diagrams 4-level QAM Constellation I Q d d -d

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Phase Shift by 90 Degrees 90 o phase shift performed by Hilbert transformer cosine => sine sine => – cosine Frequency response f -90 o 90 o f Magnitude ResponsePhase Response All-pass except at origin

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Hilbert Transformer Continuous-time ideal Hilbert transformer Discrete-time ideal Hilbert transformer h(t) = 1/( t) if t 0 0 if t = 0 h[n] = if n 0 0 if n=0 Even-indexed samples are zero t h(t)h(t) n h[n]h[n]

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Discrete-Time Hilbert Transformer Approximate by odd-length linear phase FIR filter Truncate response to 2 L + 1 samples: L samples left of origin, L samples right of origin, and origin Shift truncated impulse response by L samples to right to make it causal L 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 samples

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Baseband Digital QAM Transmitter i[n]i[n] gT(t)gT(t) + q[n]q[n] gT(t)gT(t) Serial/ parallel converter 1 Bits Map to 2-D constellation J Pulse shapers (FIR filters) Index Impulse modulator s(t)s(t) Local Oscillator 90 o Delay i[n]i[n] gT[m]gT[m] L + cos( 0 m) q[n]q[n] gT[m]gT[m] L sin( 0 m) Serial/ parallel converter 1 Bits Map to 2-D constellation J L samples/symbol (upsampling factor) Pulse shapers (FIR filters) Index s[m]s[m] D/A s(t)s(t) 100% discrete time

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Performance Analysis of PAM If we sample matched filter output at correct time instances, nT sym, without any ISI, received signal where transmitted signal is v(t) output of matched filter G r ( ) for input of channel additive white Gaussian noise N(0; 2 ) G r ( ) passes frequencies from - sym /2 to sym /2, where sym = 2 f sym = 2 / T sym Matched filter has impulse response g r (t) for i = -M/2+1, …, M/2 v(nT) ~ N(0; 2 /T sym ) 4-level PAM Constellation d dd 3 d 3 d

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Performance Analysis of PAM Decision error for inner points Decision error for outer points Symbol error probability -7d-5d-3d-dd3d5d7d O-O- IIIIIIO+O+ 8-level PAM Constellation

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Performance Analysis of QAM If we sample matched filter outputs at correct time instances, nT sym, without any ISI, received signal Transmitted signal where i,k { -1, 0, 1, 2 } for 16-QAM Noise For error probability analysis, assume noise terms independent and each term is Gaussian random variable ~ N(0; 2 /T sym ) In reality, noise terms have common source of additive noise in channel 4-level QAM Constellation I Q d d -d

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Performance Analysis of 16-QAM Type 1 correct detection I Q 16-QAM 1 - interior decision region 2 - edge region but not corner 3 - corner region

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Performance Analysis of 16-QAM Type 2 correct detection Type 3 correct detection I Q 16-QAM 1 - interior decision region 2 - edge region but not corner 3 - corner region

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Performance Analysis of 16-QAM Probability of correct detection Symbol error probability (lower bound) What about other QAM constellations?

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Average Power Analysis Assume each symbol is equally likely Assume energy in pulse shape is 1 4-PAM constellation Amplitudes are in set { -3d, -d, d, 3d } Total power 9 d 2 + d 2 + d d 2 = 20 d 2 Average power per symbol 5 d 2 4-QAM constellation points Points are in set { -d – jd, -d + jd, d + jd, d – jd } Total power 2d 2 + 2d 2 + 2d 2 + 2d 2 = 8d 2 Average power per symbol 2d 2 4-level PAM Constellation d -d -3 d 3 d 4-level QAM Constellation I Q d d -d

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