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Fourier Series Example

Module by: Don Johnson

Summary: This module provides an example of the Fourier Series representation of a half-wave rectified sinusoid.

Example 1

Let's find the Fourier series representation for the half-wave rectified sinusoid.

st=sin2πtTif0≤t<T20ifT2≤t<T s t 2 t T 0 t T 2 0 T 2 t T (1)
Begin with the sine terms in the series; to find b k b k we must calculate the integral
b k =2T∫0T2sin2πtTsin2πktTdt b k 2 T t 0 T 2 2 t T 2 k t T (2)
The key to evaluating such integrals is the classic trigonometric identities.
sinαsinβ=12cosα-β-cosα+β α β 1 2 α β α β (3)
cosαcosβ=12cosα+β+cosα-β α β 1 2 α β α β (4)
sinαcosβ=12sinα+β+sinα-β α β 1 2 α β α β (5)
Using these identities turns our integral of a product of sinusoids into a sum of integrals of individual sinusoids, which are much easier to evaluate.
2T∫0T2sin2πtTsin2πktTdt=1T∫0T2cos2πk-1tT-cos2πk+1tTdt=12ifk=10if{k|k≥2∧k∈ℕ} 2 T t 0 T 2 2 t T 2 k t T 1 T t 0 T 2 2 k 1 t T 2 k 1 t T 1 2 k 1 0 k k 2 k (6)
Thus,
b 1 =12 b 1 1 2 (7)
b 2 = b 3 =…=0 b 2 b 3 … 0 (8)

On to the cosine terms. The average value, which corresponds to a 0 a 0 , equals 1π 1 The remainder of the cosine coefficients are easy to find, but yield the complicated result

a k =-2π1k2-1ifk∈24…0ifk is odd a k 2 1 k 2 1 k 2 4 … 0 k is odd (9)

Thus, the Fourier series for the half-wave rectified sinusoid has non-zero terms for the average, the fundamental, and the even harmonics. Plotting the Fourier coefficients reveals at what component frequencies the half-wave rectified sinusoid has energy ( Figure 1 ). Furthermore, this figure shows what the Fourier series sum looks like with these coefficients as we add more and more terms. Presumably, you now believe more in the Fourier series.

Figure 1: The Fourier series spectrum of a half-wave rectified sinusoid is shown in the upper portion. The index indicates the multiple of the fundamental frequency at which the signal has energy. The cumulative effect of adding terms to the Fourier series for the half-wave rectified sine wave is shown in the bottom portion. The dashed line is the actual signal, with the solid line showing the finite series approximation to the indicated number of terms k k
Fourier Series Spectrum of a Half-Wave Rectified Sine Wave
Subfigure 1.1
Fourier Series Spectrum of a Half-Wave Rectified Sine Wave, Subfigure 1.1 (spectrum2.png)
Subfigure 1.2
Fourier Series Spectrum of a Half-Wave Rectified Sine Wave, Subfigure 1.2 (fourier1.png)

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