EE 445 S RealTime Digital Signal Processing Lab

EE 445 S Real-Time Digital Signal Processing Lab Spring 2014 Quadrature Amplitude Modulation (QAM) Transmitter Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Lecture 15

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 15 - 2

Review Amplitude Modulation by Cosine • y 1(t) = x 1(t) cos(wc t) Assume x 1(t) is an ideal lowpass signal with bandwidth w 1 Assume w 1 << wc Y 1(w) is real-valued if X 1(w) is real-valued X 1(w) Y 1(w) ½X 1(w wc) ½X 1(w + wc) 1 -w 1 0 ½ w 1 Baseband signal w -w c - w 1 wc -w c + w 1 0 wc - w 1 Upconverted signal wc wc + w 1 • Demodulation: modulation then lowpass filtering 15 - 3 w

Review Amplitude Modulation by Sine • y 2(t) = x 2(t) sin(wc t) Assume x 2(t) is an ideal lowpass signal with bandwidth w 2 Assume w 2 << wc Y 2(w) is imaginary-valued if X 2(w) is real-valued X 2(w) Y 2(w) -j ½X 2(w wc) j ½X 2(w + wc) 1 j -w 2 0 w 2 Baseband signal w ½ wc – w 2 -w c – w 2 - wc wc wc + w 2 -w c + w 2 -j ½ Upconverted signal • Demodulation: modulation then lowpass filtering 15 - 4 w

Baseband Digital QAM Transmitter • Continuous-time filtering and upconversion Impulse modulator i[n] Index Bits 1 Serial/ parallel converter Pulse shapers (FIR filters) Map to 2 -D constellation J q[n] Q g. T(t) Impulse modulator Local Oscillator s(t) Delay + 90 o g. T(t) d -d 4 -level QAM Constellation I Delay matches delay through 90 o phase shifter Delay required but often omitted in diagrams 15 - 5

Phase Shift by 90 Degrees • 90 o phase shift performed by Hilbert transformer cosine => – cosine • Frequency response Magnitude Response Phase Response 90 o f All-pass except at origin f -90 o 15 - 6

Hilbert Transformer • Continuous-time ideal Hilbert transformer • Discrete-time ideal Hilbert transformer 1/( t) if t 0 if n 0 h(t) = h[n] = 0 if t = 0 0 h(t) if n=0 h[n] t n Even-indexed samples are zero 15 - 7

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 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 15 - 8

Baseband Digital QAM Transmitter i[n] Impulse modulator Index Bits 1 Serial/ parallel converter J 100% discrete time i[n] Index Bits 1 Serial/ parallel converter J L samples/symbol (upsampling factor) Pulse shapers (FIR filters) Map to 2 -D constellation q[n] g. T(t) Impulse modulator L Local Oscillator s(t) Delay + 90 o g. T(t) g. T[m] s[m] Pulse shapers (FIR filters) Map to 2 -D constellation q[n] L g. T[m] cos( 0 m) sin( 0 m) + s(t) D/A 15 - 9

Performance Analysis of PAM • If we sample matched filter output at correct time instances, n. Tsym, without any ISI, received signal where transmitted signal is v(n. T) ~ N(0; 2/Tsym) 3 d for i = -M/2+1, …, M/2 v(t) output of matched filter Gr( ) for input of channel additive white Gaussian noise N(0; 2) Gr( ) passes frequencies from - sym/2 to sym/2 , where sym = 2 fsym = 2 / Tsym d d 3 d 4 -level PAM • Matched filter has impulse response gr(t) Constellation 15 - 10

Performance Analysis of PAM • Decision error for inner points • Decision error for outer points • Symbol error probability 8 -level PAM Constellation O- I I I -7 d -5 d -3 d -d d 3 d 5 d O+ 7 d 15 - 11

Performance Analysis of QAM • If we sample matched filter outputs at correct time instances, n. Tsym, without any ISI, received signal Q • Transmitted signal where i, k { -1, 0, 1, 2 } for 16 -QAM d -d d I -d 4 -level QAM • Noise Constellation For 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 15 - 12

Performance Analysis of 16 -QAM • Type 1 correct detection Q 3 2 2 1 1 2 2 3 3 2 1 1 2 2 I 2 3 16 -QAM 1 - interior decision region 2 - edge region but not corner 3 - corner 15 -region 13

Performance Analysis of 16 -QAM • Type 2 correct detection Q 3 2 2 1 1 2 2 3 • Type 3 correct detection 3 2 1 1 2 2 I 2 3 16 -QAM 1 - interior decision region 2 - edge region but not corner 3 - corner 15 -region 14

Performance Analysis of 16 -QAM • Probability of correct detection • Symbol error probability (lower bound) • What about other QAM constellations? 15 - 15

Average Power Analysis 3 d • Assume each symbol is equally likely • Assume energy in pulse shape is 1 • 4 -PAM constellation Amplitudes are in set { -3 d, -d, d, 3 d } Total power 9 d 2 + 9 d 2 = 20 d 2 Average power per symbol 5 d 2 d -d -3 d 4 -level PAM Constellation • 4 -QAM constellation points Points are in set { -d – jd, -d + jd, d – jd } Total power 2 d 2 + 2 d 2 = 8 d 2 Average power per symbol 2 d 2 Q d -d 4 -level QAM 15 - 16 Constellation I