QAM modulation, demodulation and constellation

Quadrature Amplitude Modulation (QAM) is a digital modulation in which each symbol is sent by changing both the amplitude and phase of the carrier. It is built from two amplitude-modulated carriers of the same frequency in quadrature, so an $M$-point QAM carries $\log_2 M$ bits per symbol.

$$s_k(t)=\sqrt{\frac{2E_0}{T}}\,a_k\cos 2\pi f_ct-\sqrt{\frac{2E_0}{T}}\,b_k\sin 2\pi f_ct,\quad 0\le t\le T$$

For 16-QAM, $a_k,b_k\in\{-3,-1,+1,+3\}$.

Modulator

              +->[2-bit to 4-level]-a_k->(X)<-phi1
 bits -->[S/P]                             |
  4 bits/sym  |                           (+)--> s(t)
              +->[2-bit to 4-level]-b_k->(X)<-phi2

$\phi_1(t)=\sqrt{2/T}\cos 2\pi f_ct$, $\phi_2(t)=-\sqrt{2/T}\sin 2\pi f_ct$.

1. Serial-to-parallel converter sends 2 bits to the I branch and 2 bits to the Q branch.
2. Each 2-bit group is mapped (Gray code) to a level: 00 to -3, 01 to -1, 11 to +1, 10 to +3.
3. The levels amplitude-modulate the quadrature carriers and the outputs are summed.

Demodulator

          +->(X)->[int 0..T]->[4-level slicer]--+
          |   ^ phi1                            |
 r(t) ----+                                   [P/S]-> bits
          |   v phi2                            |
          +->(X)->[int 0..T]->[4-level slicer]--+

Each correlator recovers one coordinate ($\approx a_k\sqrt{E_0}$ and $b_k\sqrt{E_0}$ plus noise). The slicer compares it with thresholds $0,\pm2\sqrt{E_0}$, picks the nearest level, and the level-to-bits converter plus P/S give the data. Carrier and symbol timing recovery are needed (coherent detection).

16-QAM constellation (Gray coded)

               Q (phi2)
  0010  0110   |   1110  1010   +3
               |
  0011  0111   |   1111  1011   +1
 --------------+--------------> I (phi1)
  0001  0101   |   1101  1001   -1
               |
  0000  0100   |   1100  1000   -3
   -3    -1        +1    +3

First two bits select the I level, last two the Q level; neighbouring points differ by one bit, so most symbol errors cause only one bit error. The minimum distance between points is $2\sqrt{E_0}$.

Merits

• 4 bits/symbol: bandwidth is 1/4 of BPSK for the same bit rate.
• Better error performance than 16-PSK at the same average power.
• Not constant envelope, so linear amplifiers are required.