Quadrature Amplitude Modulation QAM

1
Quadrature Amplitude Modulation (QAM)

• Problem 2.16

2
General Problem

• Two i.i.d (independent/identically distributed)
  sequences put through a filter
• h(n)
• (1/2)n, ngt 0
• 0 else
• Fourier Transform of h(n) infinite, bad for
  computation
• Need another form that is exact but finite

3
Z-transform

• Thus W(z)H(z) X(z)
• W(z) X(z)/H(z)
• W(z) X(z)/(1/(1-.5z-1)) X(z)-.5X(z)z-1
• W(z).5X(z)z-1 X(z)
• x(n) 1/2xn-1(n) w(n)
• Now have an exact finite form
• Can easily obtain xcn and xsn

h(n)
w(n)
x(n)
4
xcn and xsn
5
QAM

• Each message signal multiplied by a carrier wave,
  then summed
• cos(np/2) for primary
• sin(np/2) for quadrature

6
QAM, Continued
7
Autocorrelation

• Since signals are i.i.d, their autocorrelation
  functions should be flat except near x0.

8
Autocorrelation, Cont.
9
Power Spectra

• Easy to calculate using autocorrelation
• i.e. Sc FRc

10
PSD
11
Power Spectral Density

• Alternate method of calculation
• Sc Fxc2

12
PSD, Continued
13
x(n) recovery

• Normally, need to multiply by cosine or sin
  (depending on which signal you want) and then
  lowpass the result.
• Signal m1(t)cos(2pf0t) m2(t)sin(2pf0t)
• Multiplying by cos gives (reduced via identities)
  
• .5m1(t) .5m1(t)cos(4pf0t) m2(t)sin(4pf0t)
• Lowpass filter removes influence of m2(t)
• Repeat for sin(2pf0t)
• However, special case here need only multiply
  by carrier signals

14
Recovery