Fourier series coefficients
Consider an audio signal given by
This is a major triad in a non-well-tempered scale. The first tone is A-440. The third is approximately E, with a frequency 3/2 that of A-440. The middle term is approximately C sharp, with a frequency 5/4 that of A-440. It is these simple frequency relationships that result in a pleasant sound. We choose the non-well-tempered scale because it makes it much easier to construct a Fourier series expansion for this waveform.
To construct the Fourier series expansion, we can follow these steps:
- Find p, the period. The period is the smallest number p such that s(t) = s(t − p). To do this, note that
- Find A_{0}, the constant term. By inspection, there is no constant component in s(t), only sinusoidal components, so A_{0} = 0.
- Find A_{1}, the fundamental term. By inspection, there is no component at 110 Hz, so A_{1} = 0.
- Find A_{2}, the first harmonic. By inspection, there is no component at 220 Hz, so A_{2 }= 0.
- Find A_{3}. By inspection, there is no component at 330 Hz, so A_{3 }= 0.
- Find A_{4}. There is a component at 440 Hz, sin(440× 2π t). We need to find A_{4} and φ _{4} such that A_{4}cos(440× 2π t +φ _{4}) = sin(440× 2π t). By inspection, φ _{4} = - π /2 and A_{4} = 1.
- Similarly determine that A_{5 }= A_{6} = 1, φ _{5} = φ _{6} = - π /2, and all other terms are zero.
where ω _{0} = 220π.
Clearly this method for determining the Fourier series coefficients is tedious and error prone, and will only work for simple signals. We will see much better techniques.
Exercise
Determine the fundamental frequency and the Fourier series coefficients for the well-tempered major triad,