...

/

The Idea of Orthogonality

The Idea of Orthogonality

Explore why many DSP algorithms rely on orthogonality.

We'll cover the following...

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 NN samples, one with frequency fs1/Nf_s\cdot 1/N and the other with frequency fs2/Nf_s\cdot 2/N. The discrete frequencies are 1/N1/N and 2/N2/N, but it’s easier to understand this idea with a continuous curve than with a discrete sinusoid.

Press + to interact
Two complex sinusoids with frequencies 1/N and 2/N
Two complex sinusoids with frequencies 1/N and 2/N

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/N1/N has one cycle in NN samples.

  • The wave with frequency 2/N2/N has one cycle in NN samples.

Clearly, the wave with frequency 2/N2/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 00) is zero.

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

...