Dtft examples

Let's compute the discrete-time Fourier transform of the exponentially decaying sequence $s(n)=a^{n}u(n)$ , where $u(n)$ is the unit-step sequence. Simply plugging the signal's expression into the Fourier transform formula,

This sum is a special case of the geometric series .

Using Euler's relation, we can express the magnitude and phase of this spectrum.

No matter what value of $a$ we choose, the above formulae clearly demonstrate the periodicnature of the spectra of discrete-time signals. [link] shows indeed that the spectrum is a periodic function. We need only consider the spectrumbetween $-\left(\frac{1}{2}\right)$ and $\frac{1}{2}$ to unambiguously define it. When $a> 0$ , we have a lowpass spectrum — the spectrum diminishes as frequency increases from $0$ to $\frac{1}{2}$ — with increasing $a$ leading to a greater low frequency content; for $a< 0$ , we have a highpass spectrum ( [link] ).

Analogous to the analog pulse signal, let's find the spectrum of the length- $N$ pulse sequence.

The Fourier transform of this sequence has the form of a truncated geometric series.

For the so-called finite geometric series, we know that