QAM modulation and demodulation

Quadrature Amplitude Modulation (QAM) is an M-ary scheme in which both the amplitude and the phase of the carrier change. Two independent amplitude-modulated carriers, $\cos 2\pi f_ct$ (in-phase, I) and $\sin 2\pi f_ct$ (quadrature, Q), are added so that each symbol carries $\log_2 M$ bits.

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

where $(a_k,b_k)$ are integer levels, e.g. $\pm1,\pm3$ for 16-QAM, and $E_0$ is the energy of the lowest-amplitude signal.

Modulator

          +-> [2-to-L level] -> a_k -->(X)<- cos wc t
 binary   |                               |
 data -->[S/P]                           (+)--> QAM
          |                               |
          +-> [2-to-L level] -> b_k -->(X)<- -sin wc t

1. The serial data is split into two streams (half the bits each).
2. Each stream is grouped into $\log_2\sqrt{M}$ bits and converted to one of $L=\sqrt{M}$ levels.
3. The levels multiply the I and Q carriers and the products are added.

Demodulator (coherent)

          +->(X)->[int 0..T]->[L-level decision]-+
          |   ^ cos wc t                         |
 r(t) ----+                                    [P/S]-> data
          |   v -sin wc t                        |
          +->(X)->[int 0..T]->[L-level decision]-+

1. Two correlators (multiply and integrate) recover the I and Q amplitudes; the orthogonality of cos and sin removes cross-talk.
2. Each output is compared with $L-1$ thresholds to decide the level.
3. Level-to-bits conversion and a parallel-to-serial converter give the data back.

A carrier recovery circuit is needed because detection is coherent.

16-QAM constellation

            Q
   *    *   |   *    *     +3
            |
   *    *   |   *    *     +1
 -----------+-----------> I
   *    *   |   *    *     -1
            |
   *    *   |   *    *     -3
  -3   -1      +1   +3

16 points, 4 bits per symbol; with Gray coding, neighbours differ by one bit. For the same $M$, QAM has the same bandwidth efficiency as M-ary PSK ($\log_2 M$ bits per symbol), but because its points are spread over the whole plane rather than on one circle, they are farther apart for the same average power, so QAM gives a lower error probability.