Quadrature Amplitude Modulation (QAM) Transmitter

1. Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Quadrature Amplitude Modulation (QAM) Transmitter

2. 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

3. Review Y1(w) ½X1(w + wc) ½X1(w - wc) X1(w) ½ 1 w -wc - w1 -wc + w1 wc - w1 wc + w1 0 -wc wc w -w1 w1 0 Amplitude Modulation by Cosine • y1(t) = x1(t) cos(wct) Assume x1(t) is an ideal lowpass signal with bandwidth w1 Assume w1 << wc Y1(w) is real-valued if X1(w) is real-valued • Demodulation: modulation then lowpass filtering Baseband signal Upconverted signal

4. Review Y2(w) j ½X2(w + wc) -j ½X2(w - wc) X2(w) j ½ 1 wc wc – w2 wc + w2 w -wc – w2 -wc + w2 -wc w -j ½ -w2 w2 0 Amplitude Modulation by Sine • y2(t) = x2(t) sin(wct) Assume x2(t) is an ideal lowpass signal with bandwidth w2 Assume w2 << wc Y2(w) is imaginary-valued if X2(w) is real-valued • Demodulation: modulation then lowpass filtering Baseband signal Upconverted signal

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

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

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

8. 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

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

10. 3 d d d 3 d 4-level PAM Constellation Performance Analysis of PAM • If we sample matched filter output at correct time instances, nTsym, without any ISI, received signal where transmitted signal is v(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) v(nT) ~ N(0; 2/Tsym) for i = -M/2+1, …, M/2

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

12. Q d -d d I -d 4-level QAM Constellation Performance Analysis of QAM • If we sample matched filter outputs at correct time instances, nTsym, 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/Tsym) In reality, noise terms have common source of additive noise in channel

13. Q 2 2 3 3 1 1 2 2 I 2 2 1 1 3 3 2 2 16-QAM Performance Analysis of 16-QAM • Type 1 correct detection 1 - interior decision region2 - edge region but not corner3 - corner region

14. Q 2 2 3 3 1 1 2 2 I 2 2 1 1 3 3 2 2 16-QAM Performance Analysis of 16-QAM • Type 2 correct detection • Type 3 correct detection 1 - interior decision region2 - edge region but not corner3 - corner region

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

16. 3 d d -d -3 d Q d -d d I -d 4-level QAM Constellation 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 d2 + d2 + d2 + 9 d2 = 20 d2 Average power per symbol 5 d2 • 4-QAM constellation points Points are in set { -d – jd, -d + jd, d + jd, d – jd } Total power 2d2 + 2d2 + 2d2 + 2d2 = 8d2 Average power per symbol 2d2 4-level PAM Constellation