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

Introduction Digital Pulse Amplitude Modulation (PAM) modulates digital information onto amplitude of pulse and may be later modulated by sinusoid Digital Quadrature Amplitude Modulation (QAM) is two-dimensional extension of digital PAM that requires sinusoidal modulation 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

Amplitude Modulation by Cosine Review Amplitude Modulation by Cosine Example: y(t) = f(t) cos(wc t) Assume f(t) is an ideal lowpass signal with bandwidth w1 Assume w1 < wc Y(w) is real-valued if F(w) is real-valued Demodulation: modulation then lowpass filtering Similar derivation for modulation with sin(w0 t) w 1 w1 -w1 F(w) w Y(w) ½ -wc - w1 -wc + w1 -wc wc - w1 wc + w1 wc ½F(w - wc) ½F(w + wc)

Amplitude Modulation by Sine Review Amplitude Modulation by Sine Example: y(t) = f(t) sin(wc t) Assume f(t) is an ideal lowpass signal with bandwidth w1 Assume w1 < wc Y(w) is imaginary-valued if F(w) is real-valued Demodulation: modulation then lowpass filtering w 1 w1 -w1 F(w) w Y(w) j ½ -wc - w1 -wc + w1 -wc wc - w1 wc + w1 wc -j ½F(w - wc) j ½F(w + wc) -j ½

Matched delay matches delay through 90o phase shifter Digital QAM Modulator Serial/Parallel Map to 2-D constellation Impulse modulator Pulse shaping gT(t) Local Oscillator + 90o d[n] an bn a*(t) b*(t) s(t) 1 J Matched Delay Matched delay matches delay through 90o phase shifter bit stream

Phase Shift by 90 Degrees 90o phase shift performed by Hilbert transformer cosine => sine sine => – cosine Frequency response of ideal Hilbert transformer:

Hilbert Transformer Magnitude response For fc > 0 Phase response All pass except at origin For fc > 0 Phase response Piecewise constant For fc < 0 f 90o f -90o

Even-indexed samples are zero 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 t h(t) n h[n] Even-indexed samples are zero

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 a delay of length (N-1)/2 (N-1)/2 is an integer when N is odd (here N = 2 L + 1) How would you make sure that delay from local oscillator to sine modulator is equal to delay from local oscillator to cosine modulator?

Performance Analysis of PAM If we sample matched filter output at correct time instances, nTsym, without any ISI, received signal where the signal component is 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 Matched filter has impulse response gr(t) v(nT) ~ N(0; 2/Tsym) 3 d for i = -M/2+1, …, M/2 d -d -3 d 4-PAM

Performance Analysis of PAM Filtered noise T = Tsym Noise power s2 d(t1–t2)

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

Performance Analysis of QAM Received QAM signal Information signal s(nT) where i,k  { -1, 0, 1, 2 } for 16-QAM Noise, vI(nT) and vQ(nT) are independent Gaussian random variables ~ N(0; 2/T)

Performance Analysis of QAM 3 2 1 I Q 16-QAM Type 1 correct detection

Performance Analysis of QAM Type 2 correct detection Type 3 correct detection 3 2 1 I Q 16-QAM

Performance Analysis of QAM Probability of correct detection Symbol error probability

Average Power Analysis PAM and QAM signals are deterministic For a deterministic signal p(t), instantaneous power is |p(t)|2 4-PAM constellation points: { -3 d, -d, d, 3 d } Total power 9 d2 + d2 + d2 + 9 d2 = 20 d2 Average power per symbol 5 d2 4-QAM constellation points: { d + j d, -d + j d, d – j d, -d – j d } Total power 2 d2 + 2 d2 + 2 d2 + 2 d2 = 8 d2 Average power per symbol 2 d2