Projection, Inner Product and the DFT

• The coefficient of projection of a signal $x$ onto another signal $y$:
  • ``a measure of how much $y$ is present in $x$''
  • is computed using the inner product $\langle x,y \rangle$:
    $$\langle x,y \rangle \triangleq \sum_{n=0}^{N-1}x(n)\overline{y(n)}.$$

• The vectors (signals) $x$ and $y$ are said to be orthogonal if $\langle x,y \rangle$ = 0:
  $$x \perp y \Leftrightarrow \langle x,y\rangle = 0$$

• Consider the projection of

  Figure 7: Two orthogonal vectors for $N=2$

  
  $$\langle x,y \rangle = 1\cdot\overline{1} + 1\cdot\overline{(-1)} = 0.$$