Baseband vs Passband M-QAM Signals

 

For M-QAM (M-ary Quadrature Amplitude Modulation), the baseband symbols are complex numbers with different amplitudes and phases, unlike M-PSK, where the amplitude is constant.

The general baseband symbol is

$$\boxed{s_k=I_k+jQ_k}$$

where $I_k$ is the in-phase component and $Q_k$ is the quadrature component.

1. 4-QAM

4-QAM has 4 symbols, with 2 possible values for each component:

$$I,Q\in\{-1,+1\}$$

The baseband symbols are

$$\boxed{S=\{-1-j,-1+j,1-j,1+j\}}$$

Symbol	$I$	$Q$
$-1-j$	-1	-1
$-1+j$	-1	+1
$1-j$	+1	-1
$1+j$	+1	+1

Each symbol represents $\log_2 4=2$ bits.

2. 16-QAM

16-QAM has 16 symbols. The possible values of each component are

$$I,Q\in\{-3,-1,+1,+3\}$$

Therefore,

$$\boxed{s_k=I+jQ}$$

for all 16 combinations of $I$ and $Q$.

16-QAM constellation

Each point represents one symbol $I+jQ$. There are 4 × 4 = 16 points.

Examples of symbols:

$$\begin{aligned} s_1&=-3-j3\\ s_2&=-3-j1\\ s_3&=-1+j3\\ s_4&=1+j1\\ s_5&=3+j3 \end{aligned}$$

There are 16 possible combinations in total. Each symbol represents $\log_2 16=4$ bits.

3. 32-QAM

32-QAM has 32 symbols. Unlike 16-QAM, it does not have one universally fixed constellation layout.

One common rectangular arrangement uses 4 levels on the $I$-axis and 8 levels on the $Q$-axis:

$$I\in\{-3,-1,+1,+3\}$$

$$Q\in\{-7,-5,-3,-1,+1,+3,+5,+7\}$$

The baseband symbols are

$$\boxed{s_k=I+jQ}$$

for all combinations of these values, giving $4\times8=32$ symbols.

Rectangular 32-QAM constellation

This is one possible 32-QAM layout: 4 × 8 = 32 constellation points.

Examples of symbols:

$$\begin{aligned} s_1&=-3-j7\\ s_2&=-1-j5\\ s_3&=1+j3\\ s_4&=3+j7 \end{aligned}$$

Each symbol represents $\log_2 32=5$ bits.

Final comparison

Modulation	Possible $I$ levels	Possible $Q$ levels	Bits/symbol
4-QAM	$\pm1$	$\pm1$	2
16-QAM	$\pm1,\pm3$	$\pm1,\pm3$	4
32-QAM (rectangular)	$\pm1,\pm3$	$\pm1,\pm3,\pm5,\pm7$	5

Passband M-QAM

The passband signal at the transmitter is given by

$$s(t)=I(t)\cos(2\pi f_ct)-Q(t)\sin(2\pi f_ct)$$

where $I(t)$ and $Q(t)$ denote the in-phase and quadrature components of the baseband signal, respectively, and $f_c$ represents the carrier frequency. The cosine and negative sine terms form two orthogonal carrier signals.

In a typical transmission system, digital data symbols are mapped into I/Q sample sequences and passed through pulse-shaping filters. The resulting signals modulate the corresponding carrier components, which are combined to produce the final passband signal for transmission.