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

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. 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)
-wc - w1
-wc + w1
-wc
wc - w1
wc + w1 wc
½F(w - wc)
½F(w + wc)

4. 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

5. 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

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

7. 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

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

11. 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

12. 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)

13. Performance Analysis of QAM3 2 1 I Q
16-QAM
Type 1 correct detection

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

15. Performance Analysis of QAM
Probability of correct detection
Symbol error probability

16. 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