Complex Exponentials and “Orthogonality”

• Recall Euler's Formula, which relates sinusoidal functions and a complex exponential by:
    $$e^{j \omega t} = \cos(\omega t) + j\sin(\omega t)$$ (7)

• From the previous discussion, it should be clear that a complex exponential function can represent the sinusoidal terms needed when computing the inner products used to extract sinusoidal information from a signal.

• The only difference between Eq. (6) and that on the right-hand side of Eq. (7) is the presence of the complex variable $j$ in front of the sine term. The Euler formula distinguishes the two terms using this notation, which effectively maps the result onto a two-dimensional complex space for each evaluated frequency.

• The ability to represent these sinusoidal terms as a complex exponential function allows us to more easily manipulate and evaluate mathematical expressions involving the procedures we have discussed so far in this section.

• For example, it is easy to calculate some of the properties found above using complex exponentials.

• The definition for the dot-product given earlier must be slightly modified for use with complex variables. The new definition is
    $$\langle x_n, y_n \rangle = \sum_{n=0}^{N-1} x[n] y^{\ast}[n],$$ (8)
  where $y^{\ast}[n]$ denotes the complex conjugate of $y[n]$.

• With this definition, the result of an inner product of a complex signal $x[n]$ with itself produces a real value:
    $$\langle x_n, x_n \rangle = \sum_{n=0}^{N-1} x[n] x^{\ast}[n] = \sum_{n=0}^{N-1}\left\vert x[n] \right\vert^2.$$ (9)

• For a set of discrete-time complex exponential signals given by $e^{j \hat{\omega}_{k}n}$, we can easily show that
    $$\langle e^{j \hat{\omega}_{k}n}, e^{j \hat{\omega}_{m}n} \rangle ... ...e{0.1in} k \neq m \\ N, & \mbox{for} \hspace{0.1in} k = m \end{array} \right.$$ (10)

• Because our dot-product is now computed for two sinusoids (embedded in the complex exponential), the dot-product of complex exponentials of the same frequency is $N$ (instead of $N/2$).

• This property of the inner product of sinusoids is referred to as “orthogonality”. It implies that the various “basis functions” (complex exponential signals of different frequencies) are mutually independent from one another.