Signals over a fixed observation interval can be compared using an inner product.

For complex continuous-time signals on $[t_0,t_1]$,

$$\langle x,y\rangle=\int_{t_0}^{t_1}x(t)y^*(t),dt.$$

The corresponding squared norm is

$$|x|^2=\langle x,x\rangle,$$

which equals signal energy over that interval.

Two signals are orthogonal when

$$\langle x,y\rangle=0.$$

Orthogonal waveforms can carry independent coefficients because projecting onto one does not pick up a contribution from the other.

For example, sine and cosine at the same frequency are orthogonal over any integer number of periods. This is why in-phase and quadrature carriers can share the same frequency while remaining separable by coherent projection.

Treating waveforms as vectors turns modulation and detection into geometry: transmitted signals become points or directions, and a receiver estimates their coefficients by inner products.