How to derive the transmitted signal from a constellation diagram

Question

Let us assume we have $16$ symbols to transmit. We can represent these $16$ symbols by $16$ unique signals and transmit. If at the receiver we can identify these $16$ signals correctly, we have identified the symbols transmitted correctly. I have used qammod([0:15],16,0) in MATLAB and got the output that is attached here.

>> a = qammod([0:15],16,0); a'

ans =

  -3.0000 - 3.0000i
  -3.0000 - 1.0000i
  -3.0000 + 1.0000i
  -3.0000 + 3.0000i
  -1.0000 - 3.0000i
  -1.0000 - 1.0000i
  -1.0000 + 1.0000i
  -1.0000 + 3.0000i
   1.0000 - 3.0000i
   1.0000 - 1.0000i
   1.0000 + 1.0000i
   1.0000 + 3.0000i
   3.0000 - 3.0000i
   3.0000 - 1.0000i
   3.0000 + 1.0000i
   3.0000 + 3.0000i

$0,1,\ldots,15$ represent the $16$ symbols. $16$ represents the number of symbols or the number of unique signals - sinusoids with different amplitudes and phases - to be transmitted. $0$ represents the offset phase.

Now I want to interpret the output. I actually got $16$ complex numbers.

• What do they represent?
• What are the magnitude, phase, real part and imaginary part of these complex numbers that they represent? I guess they represent the parameters of sinusoidal signals.
• In both the real and imaginary parts we see $-1, -3 , 1$ and $3$. We do not see $-2, 0$ and $2$. Why?

Answer 1

This is a basic question about how passband pulse-amplitude-modulation (PAM) works. It has nothing to do specifically with QAM, but it applies to any type of passband PAM (such as PSK, or any other choice of a constellation).

The question is how to obtain a transmit signal from a given point in a constellation diagram. Let's say that at time $t=kT$ we want to transmit a signal corresponding to the point $a_k$ in the constellation diagram, where $a_k$ is a complex number:

$$a_k=\text{Re}(a_k)+j\text{Im}(a_k)=|a_k|e^{j\phi_k}$$

The complex number is multiplied by a (real-valued) transmit pulse $g(t)$ and by a complex carrier, and the transmitted signal is the real part of that complex-valued signal:

$$\begin{align}s_k(t)&=\text{Re}\left(a_kg(t-kT)e^{j\omega_0t}\right)\\&=\text{Re}(a_k)g(t-kT)\cos(\omega_0t)-\text{Im}(a_k)g(t-kT)\sin(\omega_0t)\\&=|a_k|g(t-kT)\cos(\omega_0t+\phi_k)\end{align}\tag{1}$$

In practice you transmit a sequence of symbols, and the corresponding transmitted signal is given by

$$\begin{align}s(t)&=\sum_ks_k(t)\\&=\cos(\omega_0t)\sum_k\text{Re}(a_k)g(t-kT)-\sin(\omega_0t)\sum_k\text{Im}(a_k)g(t-kT)\\&=\sum_k|a_k|g(t-kT)\cos(\omega_ot+\phi_k)\end{align}\tag{2}$$

As you can see from $(1)$ and $(2)$, the transmitted signal can be represented either as two amplitude-modulated orthogonal carriers (sine and cosine), or as an amplitude and phase modulated carrier. In the first case the real and imaginary parts of $a_k$ determine the amplitudes of the two carriers, and in the second case the magnitude $|a_k|$ determines the amplitude of the carrier, and $\phi_k=\arg\{a_k\}$ is the phase of the carrier.