The discrete Fourier series representation for the following sequence:

\(x\left( n \right) = \cos \frac{\pi }{4}n\) is

Concept:

The Fourier series representation of the discrete-time periodic sequence is given by:

\(x\left( n \right) = \mathop \sum \limits_{k = 0}^{N - 1} {a_k}{e^{jk{\omega _0}n}}\)

\(x\left( n \right) = \ldots + {a_{ - 1}}{e^{ - j{\omega _0}n}} + {a_1}{e^{j{\omega _0}n}} + \ldots \)

ak = Fourier series coefficient periodic by N.

ω0 = Fundamental frequency.

Application:

Given sequence is: \(x\left( n \right) = \cos \frac{\pi }{4}n\)

The fundamental frequency of the sequence \({\omega _0} = \frac{\pi }{4}\)

We know that, \(\cos \theta = \frac{{{e^{j\theta }} + {e^{ - j\theta }}}}{2}\)

Now, we can rewrite the given sequence as

\(x\left( n \right) = \frac{{{e^{\frac{{j\pi }}{4}n}} + {e^{ - \frac{{j\pi }}{4}n}}}}{2}\)

\( = \frac{1}{2}{e^{\frac{{j\pi }}{4}n}} + \frac{1}{2}{e^{\frac{{ - j\pi }}{4}n}}\)

We can write \({e^{\frac{{ - j\pi }}{4}n}} = {e^{\frac{{j7\pi }}{4}n}}\)

Now, x(n) becomes

\(x\left( n \right) = \frac{1}{2}{e^{\frac{{j\pi }}{4}n}} + \frac{1}{2}{e^{\frac{{j7\pi }}{4}n}}\)