Designing a low-pass digital filter

Joseph Fourier discovered that a complex signal could be described by a sum of sinusoids that is known as a Fourier series, and applying this idea enables us to visualize a signal frequency spectrum. A spectrum analyzer is a device that allows the frequency content of a signal to be displayed and measurements to be made. Two parameters, known as magnitude (amplitude) and phase, describe a sinusoidal signal. The magnitude spectrum describes the amplitude of each sinusoidal component that is summed, and the phase spectrum describes its associated phase. Often, we ignore the phase information and focus on the magnitude spectrum, but for some applications, particularly those that involve feedback, the phase of the signal is very important.

The magnitude spectrum of a pure 10 kHz sinusoidal signal is illustrated in the following diagram (the left panel) and that of a sampled version of the signal (the right panel):

When we sample a signal, the steps in the digitized...