# linear modulation

Modulation of amplitude or phase.

Better spectral properties than nonlinear modulation, but less power efficient.

Typically, the basis set of the signal space for linear modulation techniques are the sine and cosine functions

$$\phi_1(t) = \sqrt{\frac{2}{T}} \cos(2\pi f_c t)$$
$$\phi_2(t) = \sqrt{\frac{2}{T}} \cos(2\pi f_c t)$$

We include the square root normalization factor so that $$\int_0^T \phi^2(t)dt = 1$$. This is the notion of orthonormality in the basis function.

What this means is that every signal modulated in amplitude and phase only can be decomposed into a linear combination of some sine and cosine, which is a powerful idea used in several linear modulation schemes.

It also means that linear modulation signal constellations have exactly two axes, sine and cosine, which poises them to be drawn on the complex plane

## 1. Modulation

signal space representation of a quadrature signal:

$$s(t) = s_I(t) cos(2\pi f_c t) + s_Q(t)sin(2\pi f_c t) = s_{i1}\phi_1(t) + s_{i2}\phi_2(t)$$

Where $$\phi_i$$ are the basis functions above multiplied by a pulse shaping function $$g(t)$$. So now to transmit the $i$th message, you encode $$s_I = s_{i1}g(t)$$ and $$s_Q = s_{i2}{g(t)}$$.

The characteristics of the signal are determined by the pulse shaper:

• bandwidth is the bandwidth of the pulse shaper
• center frequency is $$f_c$$
• passband bandwidth is $$2B$$

Now, there are three fundamental linear modulation schemes