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Fourier Series Example

Module by: Don Johnson. E-mail the author

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.

s⁢t={sin2⁢π⁢tT  if  0≤t<T20  if  T2≤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⁢∫0T2sin2⁢π⁢tT⁢sin2⁢π⁢k⁢tTdt 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.
sinα⁢sinβ=12⁢(cosα−β−cosα+β) α β 1 2 α β α β (3)
cosα⁢cosβ=12⁢(cosα+β+cosα−β) α β 1 2 α β α β (4)
sinα⁢cosβ=12⁢(sinα+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⁢∫0T2sin2⁢π⁢tT⁢sin2⁢π⁢k⁢tTdt=1T⁢∫0T2cos2⁢π⁢(k−1)⁢tT−cos2⁢π⁢(k+1)⁢tTdt={12  if  k=10  if  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 ={(−2π)⁢1k2−1  if  k∈24…0  if  k 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
(a)
Figure 1(a) (spectrum2.png)
(b)
Figure 1(b) (fourier1.png)

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