Why We Use Three Different, Equivalent Forms of the Fourier Series

We’ve now developed three different, equivalent forms of the Fourier series: Trigonometric form One-sided form Two-sided or complex exponential form

Each form has advantages and disadvantages. For some applications, one form may be more suitable than others, and yet for another application, another form may be more suitable. The next few slides contain a partial list of the advantages and disadvantages of each form.

Advantages: This is the most basic of the three forms and is very intuitive (we see x(t) as a summation of sinusoids). Disadvantages: There are two terms for each frequency component (a n and b n ). This means that we cannot easily generate the magnitude, phase, and power spectra, and so we lose a very powerful graphic tool. Trigonometric Form

Advantages: 1) This form has only one term at each frequency component, so it is very easy to generate magnitude, phase, and power spectra. 2) Form is still intuitive (we still see x(t) as a summation of sinusoids). Disadvantages: The dc component of the magnitude and power spectra is calculated in a different manner than the ac components. One-Sided Form

Advantages: 1) The dc component is calculated in the same way as all other components, which is not the case for either the one- sided or trigonometric form. 2) Mathematics is simpler using exponentials than using sines and cosines (exponentials are simpler to integrate, no need to memorize trig. identities, etc.). 3) The complex exponential form can be conceptually extended to include nonperiodic waveforms (the Fourier transform) and is the basis for the discrete Fourier transform (DFT and FFT). (continued) Two-Sided (Complex Exponential) Form

Disadvantages: This form is non-intuitive for three reasons: a) The form uses complex exponentials—it is now harder to see x(t) as a summation of sinusoids. b) The form uses complex coefficients. c) The form introduces the abstract concept of “negative frequency.” As we’ve discussed, negative frequency does not physically exist, it's just an abstract mathematical concept needed to correlate the two-sided Fourier series with physical parameters from the one-sided Fourier series. Two-Sided (Complex Exponential) Form