fourier decomposition example

7 mins read · March 10, 2024

Fourier decomposition example: Three terms – constant, cosine, sine.

fourier_decomposition_example.pdf
- Intro 3
- For f(x) 6
- For f(t) 10
- An example (f1 = 1 Hz) 14
- Same example with f1 = 2 Hz 22
- General f1 value 30
- Discussion and assignment 34

integral identities $x$

$$\tag{1} \int_0^L \sin \left( m \frac{2\pi}{L} x \right) \sin \left( n \frac{2\pi}{L} x \right) \ dx = \left\{ \begin{array}{rcl} 0, & m \ne n, \newline \tfrac12 L, & m = n. \end{array} \right. $$

$$\tag{2} \int_0^L \cos \left( m \frac{2\pi}{L} x \right) \cos \left( n \frac{2\pi}{L} x \right) \ dx = \left\{ \begin{array}{rcl} 0, & m \ne n, \newline \tfrac12 L, & m = n. \end{array} \right. $$

$$\tag{3} \int_0^L \cos \left( m \frac{2\pi}{L} x \right) \sin \left( n \frac{2\pi}{L} x \right) \ dt = 0. $$

$$\tag{3b} \int_0^L \cos \left( m \frac{2\pi}{L} x \right) \ dt = 0. $$

$$\tag{3c} \int_0^L \sin \left( n \frac{2\pi}{L} x \right) \ dt = 0. $$

coefficients $x$

$$\tag{4} f(x) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{L} \right) x \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{L} \right) x \right] + c_0 $$

$$\tag{5} a_m = \frac{2}{L}\int_0^L f(x) \ \cos \left[ m \left( \frac{2\pi}{L} \right) x \right] \ dx $$

$$\tag{6} b_m = \frac{2}{L}\int_0^L f(x) \ \sin \left[ m \left( \frac{2\pi}{L} \right) x \right] \ dx $$

$$\tag{7} c_0 = \frac{1}{L} \int_0^L f(x) \ dx. $$

$$\tag{8} k_m = m \frac{2\pi}{L}, \ \ \ \ m = 1, 2, 3, .. $$

integral identities $t$

$$\tag{9} \int_0^T \sin \left( m \frac{2\pi}{T} t \right) \sin \left( n \frac{2\pi}{T} t \right) \ dt = \left\{ \begin{array}{rcl} 0, & m \ne n, \newline \tfrac12 T, & m = n. \end{array} \right. $$

$$\tag{10} \int_0^T \cos \left( m \frac{2\pi}{T} t \right) \cos \left( n \frac{2\pi}{T} t \right) \ dt = \left\{ \begin{array}{rcl} 0, & m \ne n, \newline \tfrac12 T, & m = n. \end{array} \right. $$

$$\tag{11} \int_0^T \cos \left( m \frac{2\pi}{T} t \right) \sin \left( n \frac{2\pi}{T} t \right) \ dt = 0. $$

$$\tag{11b} \int_0^T \cos \left( m \frac{2\pi}{T} t \right) \ dt = 0. $$

$$\tag{11c} \int_0^T \sin \left( n \frac{2\pi}{T} t \right) \ dt = 0. $$

coefficients $t$

$$\tag{12} f(t) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{T} \right) t \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{T} \right) t \right] + c_0 $$

$$\tag{13} a_m = \frac{2}{T}\int_0^T f(t) \ \cos \left[ m \left( \frac{2\pi}{T} \right) t \right] \ dt $$

$$\tag{14} b_m = \frac{2}{T}\int_0^T f(t) \ \sin \left[ m \left( \frac{2\pi}{T} \right) t \right] \ dt $$

$$\tag{15} c_0 = \frac{1}{T} \int_0^T f(t) \ dt. $$

$$\tag{16} \omega_m = m \frac{2\pi}{T}, \ \ \ \ m = 1, 2, 3, .. $$

example

$$\tag{17} f(t) = 2 + 5 \cos 20 \pi t + 3 \sin 8 \pi t. $$

$$\tag{18} \begin{array}{rcl} f(t) & = & \displaystyle 2 + 5 \cos \left( \frac{2\pi}{0.1} t \right) + 3 \sin \left( \frac{2\pi}{0.25} t \right) \newline \newline & = & \displaystyle 2 + 5 \cos \left( 10 \frac{2\pi}{1} t \right) + 3 \sin \left( 4 \frac{2\pi}{1} t \right) \end{array} $$

consine terms coefficients

Use Eqns (13), (10), (11) on (18).

$$\tag{19} \begin{array}{rcl} a_m & = & \displaystyle \frac{2}{T}\int_0^T 2 \ \cos \left[ m \left( \frac{2\pi}{T} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 5 \cos \left( 10 \frac{2\pi}{1} t \right) \ \cos \left[ m \left( \frac{2\pi}{1} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 3 \sin \left( 4 \frac{2\pi}{1} t \right) \ \cos \left[ m \left( \frac{2\pi}{1} \right) t \right] \ dt \newline \newline & = & \displaystyle 0 + \frac{2}{T} \tfrac{1}{2} T 5 \delta_{10,m} + 0 \ \ = \ \ 5 \delta_{10,m}. \end{array} $$

$$\tag{19b} \begin{array}{rl} a_m = 0, & 0 < m < 10, \newline a_{10} = 5, & \newline a_m = 0, & 10 < m. \end{array} $$

sine terms coefficients

Use Eqn (14), (9), (11) on (18).

$$\tag{20} \begin{array}{rcl} b_m & = & \displaystyle \frac{2}{T}\int_0^T 2 \ \sin \left[ m \left( \frac{2\pi}{T} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 5 \cos \left( 10 \frac{2\pi}{1} t \right) \ \sin \left[ m \left( \frac{2\pi}{1} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 3 \sin \left( 4 \frac{2\pi}{1} t \right) \ \sin \left[ m \left( \frac{2\pi}{1} \right) t \right] \ dt \newline \newline & = & \displaystyle 0 + 0 + \frac{2}{T} \tfrac{1}{2} T 3 \delta_{4,m} \ \ = \ \ 3 \delta_{4,m}. \end{array} $$

$$\tag{20b} \begin{array}{rl} b_m = 0, & 0 < m < 4, \newline b_{4} = 3, & \newline b_m = 0, & 4 < m. \end{array} $$

constant term

Use Eqn (15), (11b), (11c) on (18).

$$\tag{21} \begin{array}{rcl} c_0 & = & \displaystyle \frac{1}{T}\int_0^T 2 \ dt \newline \newline & + & \displaystyle \frac{1}{T}\int_0^T 5 \cos \left( 10 \frac{2\pi}{1} t \right) \ dt \newline \newline & + & \displaystyle \frac{1}{T}\int_0^T 3 \sin \left( 4 \frac{2\pi}{1} t \right) \ dt \newline \newline & = & \displaystyle 2 + 0 + 0 \ \ = \ \ 2. \end{array} $$

function $f(t)$

$$\tag{12} f(t) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{1} \right) t \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{1} \right) t \right] + c_0 $$

$$\tag{22} \begin{array}{rcl} f(t) & = & \displaystyle a_{10} \cos \left[ 10 \left( \frac{2\pi}{1} \right) t \right] + b_4 \sin \left[ 4 \left( \frac{2\pi}{1} \right) t \right] + c_0 \newline \newline & = & \displaystyle 5 \cos \left[ 10 \left( \frac{2\pi}{1} \right) t \right] + 3 \sin \left[ 4 \left( \frac{2\pi}{1} \right) t \right] + 2. \end{array} $$

$$\tag{17} f(t) = 2 + 5 \cos 20 \pi t + 3 \sin 8 \pi t. $$

$f_1 = 2 \ \rm Hz$

$$\tag{17} f(t) = 2 + 5 \cos 20 \pi t + 3 \sin 8 \pi t. $$

$$\tag{23} \begin{array}{rcl} f(t) & = & \displaystyle 2 + 5 \cos \left( \frac{2\pi}{0.1} t \right) + 3 \sin \left( \frac{2\pi}{0.25} t \right) \newline \newline & = & \displaystyle 2 + 5 \cos \left( 5 \frac{2\pi}{0.5} t \right) + 3 \sin \left( 2 \frac{2\pi}{0.5} t \right) \end{array} $$

consine terms coefficients

Use Eqns (13), (10), (11) on (23).

$$\tag{24} \begin{array}{rcl} a_m & = & \displaystyle \frac{1}{T}\int_0^T 2 \ \cos \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 5 \cos \left( 5 \frac{2\pi}{0.5} t \right) \ \cos \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 3 \sin \left( 2 \frac{2\pi}{0.5} t \right) \ \cos \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & = & \displaystyle 0 + \frac{2}{T} \tfrac{1}{2} T 5 \delta_{10,m} + 0 \ \ = \ \ 5 \delta_{5,m}. \end{array} $$

$$\tag{24b} \begin{array}{rl} a_m = 0, & 0 < m < 5, \newline a_{5} = 5, & \newline a_m = 0, & 5 < m. \end{array} $$

sine terms coefficients

Use Eqn (14), (9), (11) on (23).

$$\tag{25} \begin{array}{rcl} b_m & = & \displaystyle \frac{1}{T}\int_0^T 2 \ \sin \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 5 \cos \left( 5 \frac{2\pi}{0.5} t \right) \ \sin \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 3 \sin \left( 2 \frac{2\pi}{0.5} t \right) \ \sin \left[ m \left( \frac{2\pi}{0.5} \right) t \right] \ dt \newline \newline & = & \displaystyle 0 + 0 + \frac{2}{T} \tfrac{1}{2} T 3 \delta_{4,m} \ \ = \ \ 3 \delta_{2,m}. \end{array} $$

$$\tag{25b} \begin{array}{rl} b_m = 0, & 0 < m < 2, \newline b_{2} = 3, & \newline b_m = 0, & 2 < m. \end{array} $$

constant term

Use Eqn (15), (11b), (11c) on (23).

$$\tag{26} \begin{array}{rcl} c_0 & = & \displaystyle \frac{1}{T}\int_0^T 2 \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 5 \cos \left( 5 \frac{2\pi}{0.5} t \right) \ dt \newline \newline & + & \displaystyle \frac{2}{T}\int_0^T 3 \sin \left( 2 \frac{2\pi}{0.5} t \right) \ dt \newline \newline & = & \displaystyle 2 + 0 + 0 \ \ = \ \ 2. \end{array} $$

function $f(t)$

$$\tag{12} f(t) = \sum_{m=1}^\infty a_m \cos \left[ m \left( \frac{2\pi}{0.5} \right) t \right] + \sum_{m=1}^\infty b_m \sin \left[ m \left( \frac{2\pi}{0.5} \right) t \right] + c_0 $$

$$\tag{27} \begin{array}{rcl} f(t) & = & \displaystyle a_{5} \cos \left[ 5 \left( \frac{2\pi}{0.5} \right) t \right] + b_2 \sin \left[ 2 \left( \frac{2\pi}{0.5} \right) t \right] + c_0 \newline \newline & = & \displaystyle 5 \cos \left[ 5 \left( \frac{2\pi}{0.5} \right) t \right] + 3 \sin \left[ 2 \left( \frac{2\pi}{0.5} \right) t \right] + 2. \end{array} $$

$$\tag{17} f(t) = 2 + 5 \cos 20 \pi t + 3 \sin 8 \pi t. $$

refs

Michael Richmond, “Examples of Fsinier decomposition”, Phys283 Vibrations and Waves, Rochester Institute of Technology, Spring 2024, url http://spiff.rit.edu/classes/phys283/lectures/fourier_2/fourier_2.html [20240310].

Tags:

fi6004