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Quadrature Amplitude Modulation (QAM) Transmitter

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

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Quadrature Amplitude Modulation (QAM) Transmitter

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  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 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

  3. Y(w) ½F(w + wc) ½F(w - wc) F(w) ½ 1 w -wc - w1 -wc + w1 wc - w1 wc + w1 0 -wc wc w -w1 w1 0 Review Amplitude Modulation by Cosine • Example:y(t) = f(t) cos(wct) 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)

  4. Y(w) j ½F(w + wc) -j ½F(w - wc) F(w) j ½ 1 wc wc - w1 wc + w1 w -wc - w1 -wc + w1 -wc w -j ½ -w1 w1 0 Review Amplitude Modulation by Sine • Example: y(t) = f(t) sin(wct) 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

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

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

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

  8. 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

  9. 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?

  10. 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 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) 3 d for i = -M/2+1, …, M/2 d -d -3 d 4-PAM

  11. Performance Analysis of PAM Filtered noise T = Tsym Noise power s2d(t1–t2)

  12. 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-PAM Constellation

  13. 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)

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

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

  16. Performance Analysis of QAM • Probability of correct detection • Symbol error probability

  17. 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

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