The Idea of Orthogonality

To understand the idea of orthogonality, we consider both real and complex signals.

Real sinusoids

The figure below shows the continuous-time versions of two sinusoids containing $N$ samples, one with frequency $f_s\cdot 1/N$ and the other with frequency $f_s\cdot 2/N$. The discrete frequencies are $1/N$ and $2/N$, but it’s easier to understand this idea with a continuous curve than with a discrete sinusoid.

Observe the “Q” part (the vertical plane at the back of the screen) of the two waves and look at the number of samples in each.

• The wave with frequency $1/N$ has one cycle in $N$ samples.

• The wave with frequency $2/N$ has one cycle in $N$ samples.

Clearly, the wave with frequency $2/N$ completes one cycle for the positive half of the first sinusoid and another cycle for the negative half of the first sinusoid. Due to these opposite signs, a summation of their sample-by-sample product is zero. Orthogonality between two signals means that a summation of their sample-by-sample product (i.e., correlation at time $0$) is zero.

Mathematically, we write for the “Q” part in the figure above:

$$\begin{array}{c}{\displaystyle }\end{array}$$ ...