EECS20N: Signals and Systems

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Week 8

• Frequencies
• Musical scale
• Phase
• Timbre
• Images
• Periodicity
• Fourier series
• Examples
• Aperiodicity
• Images
• Coefficients
• Discrete signals
• DFS
• Examples

Fourier series coefficients

Consider an audio signal given by

s(t) = sin(440× 2π t) + sin(550 × 2π t) + sin(660× 2π t).

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

sin(2πft) = sin(2πf (t − p)) if f p is an integer. Thus, we want to find the smallest p such that 440p, 550p, and 660p are all integers. Equivalently, we want to find the largest fundamental frequency f₀ such that 440/ f₀, 550/ f₀, and 660/ f₀ are all integers. Such an f₀ is called the greatest common divisor of 440, 550, and 660. This can be computed using the gcd function in Matlab. In this case, however, we can do it in our heads, observing that f₀ = 110.

• Find A₀, the constant term. By inspection, there is no constant component in s(t), only sinusoidal components, so A₀ = 0.
• Find A₁, the fundamental term. By inspection, there is no component at 110 Hz, so A₁ = 0.
• Find A₂, the first harmonic. By inspection, there is no component at 220 Hz, so A₂ = 0.
• Find A₃. By inspection, there is no component at 330 Hz, so A₃ = 0.
• Find A₄. There is a component at 440 Hz, sin(440× 2π t). We need to find A₄ and φ ₄ such that A₄cos(440× 2π t +φ ₄) = sin(440× 2π t). By inspection, φ ₄ = - π /2 and A₄ = 1.
• Similarly determine that A₅ = A₆ = 1, φ ₅ = φ ₆ = - π /2, and all other terms are zero.

Putting this all together, the Fourier series expansion can be written

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,

s(t) = sin(440 × 2π t) + sin(554 × 2π t) + sin(659 × 2π t).