Summary: How to compute discrete-time Fourier transforms for decaying sequences.

Let's compute the discrete-time Fourier transform of the
exponentially decaying sequence

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

Analogous to the analog pulse signal, let's find the spectrum
of the length-

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

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

Derive this formula for the finite geometric series sum.
The "trick" is to consider the difference between the
series'; sum and the sum of the series multiplied by

Applying this result yields (Figure 3.)

The ratio of sine functions has the generic form of