DSP: What is the Fourier Transform and how do we use it?

When you think of say, an audio signal or waveform, you think of this, right?


 Figure 1: Audio waveform in the time domain. 

Figure 1 depicts an audio signal in the time domain; essentially a visualisation of the variation of the signal’s amplitude (plotted on the y-axis) over time (plotted on the x-axis). So, by observing the signal in this way, how much information can actually be extracted[1]?

Window Functions and How We Use Them in DSP

A window function is a mathematical function that applies a weighting (often between 0 and 1) to each discrete time series sample in a finite set[1]. It should be noted that window functions can be applied in the frequency-domain, though this is a somewhat convoluted process and beyond the scope of this article. As a simple example, assume the finite set x of length N. If each value within x is assigned a weighting of 1 then it can be said that a new set y has been created to which the rectangular window function w (also of length N) has been applied:

\[y(n) = x(n) . w(n)\qquad0 \leq n \leq N\]

The above equation essentially states that a new set should be created whereby each value at index n is the product of the values at the nth index of the sample set x and window function w. Figure 1 depicts both an unmodified sine wave and a sine wave to which a rectangular window has been applied – they are analogous to each other.