We find the trigonometric Fourier series (TFS) and compact TFS (CTFS) for a periodic "pulse-train" waveform.

Since the signal is even, the sinusoid components of the TFS are zero and thus bn = 0 for all n. Only the a0 and an coefficients terms need to be computed.

After computing the TFS, we also find the CTFS by both using the regular equations and also using a trigonometric identity to turn each sine term of the TFS into the desired cosine term of the CTFS.

We find the trigonometric Fourier series (TFS) and compact TFS (CTFS) for a periodic "pulse-train" waveform.

Since the signal is even, the sinusoid components of the TFS are zero and thus bn = 0 for all n. Only the a0 and an coefficients terms need to be computed.

After computing the TFS, we also find the CTFS by both using the regular equations and also using a trigonometric identity to turn each sine term of the TFS into the desired cosine term of the CTFS.

The Fourier Series representation of a continuous-time signal has a variety of properties that are noted/investigated in this three-part video sequence. We investigate the following properties in this video:

1. The FS representation on the time-interval T0 is always a T0 periodic-function.

2. The FS representation of a T0-periodic signal holds for all time, while the representation of a non-periodic signal only holds on the interval where the FS representation was found.

3. The amplitude spectrum of a signal is a plot of its CTFS coefficients Cn versus frequency. This plot visually quantifies the amplitude of each sinusoidal frequency component present in the signal.

4. The phase spectrum of a signal is a plot of its CTFS coefficients Thetan versus frequency. This plot visually quantifies the phase of each sinusoidal frequency component present in the signal.

The Fourier Series representation of a continuous-time signal has a variety of properties that are noted/investigated in this three-part video sequence. We investigate the following properties in this video:
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1. The FS exists for all signals that meet the weak and strong Dirchlet conditions (i.e. must be absolutely integrable on the T0 time interval and have finite number of min/max and discontinuities.

2. When a continuous-time signal is an even signal, all bn = 0 and a simplified equation can be used to compute a0 and an.

3. When a continuous-time signal is an odd signal, all an = 0 and a simplified equation can be used to compute bn.

4. Continuous-time signals with half-wave symmetry have no even harmonics in their FS representation (i.e. a2 = a4 = ... = b2 = b4 = ... = 0).

In this video we compute the exponential Fourier (EFS) series of a fully rectified sine wave signal |sin(t)|. This computation involves computing the EFS coefficients Dn by projecting the signal onto the the nth exponential basis signal.