- Recall Euler's Formula, which relates sinusoidal functions and a complex exponential by:

- 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

(8) *y*[*n*]. - With this definition, the result of an inner product of a complex signal
*x*[*n*] with itself produces a real value:

(9) - For a set of discrete-time complex exponential signals given by
, we can easily show that

(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.

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