Laboratory Work 15 harmonic analysis and synthesis of the periodical functions
Purpose of work: to learn to determine the periodical function harmonic composition.
15.1 Theoretical Data
The time function f(t) is called periodical if for it the condition
(15.1)
Where Т - period
Is equitable.
The periodic function harmonic analysis consists in determination of the coefficients ak , bk of the Fourier series:
(15.2)
Where
the
first harmonic circular frequency;
k – the harmonic serial number
Being limited by some final number of harmonics m in the formula (15.2) the approximating harmonic polynomial Qm(t) is obtained :
(15.4)
The Fourier coefficients are determined by the expressions
(15.5)
Using the rectangle method for the numerical integration of the formulae (15.5) when the integration interval [0,T] is split into n equal sections we get::
(15.6)
k=1,2,…,m,
(15.7)
(15.8)
(15.9)
Under
n=2k (15.10)
the function Qm(t) becomes the trigonometric interpolator.
Getting of the periodical function by summing up its harmonic components according to the formula (15.4).is called the harmonic synthesis
15.2 Task
Compute the coefficients of the interpolating trigonometric polynomial that approximate the table function given in the points
When
Build the interpolating function graph and plot the output tabel function in the form of the grid.
15.3 Methodical Recommendations
For vizualisation the integrity of the points on the approximation function graph is to be 5-10 times as much of the integrity of the points which split the period into the sections for numerical integration.
Observe the influence of the number of the harmonics m under the given number of the layout sections on the approximation accuracy..
Таблиця 15.1 – Вихідні дані до лабораторної роботи №15
Variant number |
Function table values |
1 |
2 |
1 |
1.00,1.803, 3.085,4.776,6.434,7.347,7.027,5.652,3.897,2.381, 1.347, 7.422, 0.419, 0.256, 0.176, 0.142, 0.136, 0.155, 0.209, 0.324, 0.554 |
2 |
7.38, 6.76, 5.22, 3.47, 2.07, 1.16, 0.64, 0.36, 0.23, 0.16, 0.13, 0.13, 0.16, 0.23, 0.37, 0.64, 1.16, 2.08, 3.48, 5.22, 6.76 |
3 |
-1.24, -1.17, -1.08, -0.96, -0.84, -0.79, -0.8, -0.9, -1.1, -1.21, -1.02, -1.28, -1.32, -1.34, -1.36, -1.37, -1.37, -1.36, -1.35, -1.33, -1.30 |
4 |
-3.0, -3.58, -4.12, -4.56, -4.86, -4.99, -4.94, -4.73, -4.36, -3.86, -3.30, -2.7, -2.13, -1.64, -1.26, -1.05, -1.00, -1.13, -1.43, -1.87, -2.43 |
5 |
1.0,1.05, 90.6, 520.4, 1714.7, 2915.0, 2439.2, 1020.6,230.7, 32.17, 3.29, 0.3, 0.03, 0.004, 0.001, 0.0003,0.0006, 0.002, 0.01, 0.09, 0.9 |
6 |
2980.1, 2089.3, 742.4, 146.6, 18.6, 1.8, 0.16, 0.02, 0.003, 0.001, 0.001,0.001,0.002,0.003, 0.018, 0.9, 1.22, 18.6, 146.6, 742.5, 2089.7 |
7 |
1.0, 1.34, 1.75, 2.18, 2.53, 2.71, 2.65, 2.37, 1.97, 1.54, 1.16, 0.86, 0.64, 0.5, 0.42, 0.37, 0.36, 0.39, 0.45, 0.56, 0.74 |
8 |
2.71, 2.6, 2.28, 1.86, 1.44, 1.07, 0.8, 0.46, 0.42, 0.4, 0.37, 0.37, 0.4, 0.48, 0.6, 0.8, 1.07, 1.44, 1.86, 2.28, 2.6 |
9 |
-1.32,-1.28,-1.26,-1.24, -1.25, -1.25, -1.25, -1.26, -1.27, -1.29, -1.29, -1.33, -1.34, -1.37, -1.37, -1.37, -1.37, -1.36, -1.36, -1.35, -1.34
|
Continuation of the table 15.1 |
|
1 |
2 |
10 |
-4.0, -4.2, -4.5, -4.7, -4.9, -5.0, -4.9, -4.8, -4.6, -4.4, -4.1, -3.8, -3.5, -3.1, -3.0, -3.0, -3.0, -3.1, -3.2, -3.4, -3.7 |
11 |
1.0, 2.4, 5.4, 10.4, 16.3, 19.9, 18.6, 13.4, 7.7, 3.6, 1.6, 0.64, 0.27, 0.13, 0.07, 0.05, 0.05, 0.06, 0.09, 0.18, 0.4 |
12 |
20.0, 17.5, 11.9, 6.4, 2.9, 1.2, 0.5, 0.2, 0.1, 0.06, 0.05, 0.05, 0.06, 0.1, 0.5, 1.0, 1.2, 2.9, 6.4, 11.9, 17.5 |
13 |
-1.1, -0.8, -0.3, 0.3, 0.7, 0.8, 0.7, 0.5, 0.04, -0.6, -0.9, 1.1, -1.27, -1.32, -1.35,-1.37, -1.37, -1.36, -1.34, -1.3, -1.2 |
14 |
-2.0, -2.8, -3.7, -4.3, -4.7, -4.9, -4.9, -4.5, -4.1, -3.3, -2.4, -1.5, -0.6, -0.04, 0.6, 0.02, 0.99, 0.79, 0.34, 0.3, -1.1 |
15 |
1.1, 3.2, 9.5, 22.8, 41.4, 53.9, 49.4, 31.9, 15.2, 5.7, 1.8, 0.55, 0.17, 0.06, 0.03, 0.02, 0.01, 0.02, 0.04, 0.1, 0.3 |
16 |
-0.78, -1.22, -1.34, -1.39, -1.42, -1.43, -1.42, -1.41, -1.37, -1.3, -1.1, -0.1, 1.1, 1.2, 1.33, 1.36, 1.37, 1.35, 1.3, 1.17, 0.65 |
17 |
54.5, 45.7, 27.2, 12.1, 4.3, 1.3, 0.4, 0.13, 0.05, 0.03, 0.02, 0.02, 0.03, 0.05, 0.13, 0.41, 1.3, 4.3, 12.1, 21.2, 45.7 |
18 |
-0.78, 0.18, 0.89, 1.13, 1.21, 1.24, 1.23, 1.18, 1.04,0.63, -0.38, -1.01, -1.22, -1.3, -1.35, -1.36, -1.37, -1.36, -1.33, -1.27, -1.1 |
19 |
-1.0, -2.1, 3.2, -4.1, -4.7, -4.9, -4.8, -4.4, -3.7, -2.7, -1.6, -0.4, 0.7, 1.7, 2.4, 2.9, 3.0, 2.7, 2.1, 1.2, 0.2 |
20 |
1.0 , 4.36, 16.7, 49.8, 105. 0, 146. 3, 130. 9, 75.9, 30.0,8.75, 2.1, 0.47, 0.11, 0.03, 0.01, 0.007, 0.006, 0.009, 0.02, 0.05, 0.2 |
21 |
148.4, 118.8, 62.6, 25.5, 6.21, 1.45, 0.33, 0.08, 0.02,0.01, 0.007,0.007, 0.01, 0.02, 0.08, 0.32, 1.45, 6.2, 22.6, 62.2, 119.0 |
22 |
0.0, 0.97, 1.23, 1.32, 1.36, 1.37, 1.36, 1.34, 1.28, 1.130.64, -0.64, -1.13, -1.28, -1.34, -1.37, -1.36, -1.32, -1.23, -0.9, -0.2 |
23 |
-0.0001, -1.47, -2.8, -3.9, -4.65, -4.98, -4.87, -4.33, -3.4, -2.16, -0.74, 0.74, 2.17, 3.14, 4.33, 4.87, 4.98, 4.65, 3.9, 2.8, 1.4 |
24 |
1.0,5.8, 29.3, 108.9, 266.4, 396.7, 347.1, 180.5, 59.2, 13.5, 2.4, 0.4, 0.07, 0.01, 0.005, 0.003, 0.002, 0.004, 0.009, 0.03, 0.1 |
25 |
403.4, 309.0, 142.2, 42.1, 8.9, 1.56, 0.26, 0.05, 0.01, 0.0044, 0.0026, 0.0026, 0.0044,0.01, 0.05, 0.263, 1.56, 8.95, 42.1, 142.2, 309.9 |
26
|
0.78, 1.22, 1.34, 1.39, 1.42, 1.43, 1.42, 1.41, 1.37, 1.3, 1.1, 0.1, -1.1, -1.2, -1.33, -1.36, -1.37, -1.35, -1.3, 1.17, -0.65 |
27 |
1.0, -0.77, -2.3, -3.6, -4.6, -4.9, -4.8, -4.1, -3.1, -1.6, 0.1, 1.9, 3.6, 5.1, 6.2, 6.84, 6.98, 6.58, 5.69, 4.4 , 2.7 |
28 |
1.0 , 7.8, 51.5, 238.1, 675.9, 1075.4, 620.1, 429.3, 110.8, 20.8, 2.83, 0.35, 0.04, 0.01, 0.002, 0.001, 0.001, 0.001,0.004, 0.02, 0.12 |
Continuation of the table 15.1 |
|
1 |
2 |
29 |
1.10, 1.32, 1.40, 1.43, 1.45, 1.46, 1.46, 1.44, 1.42, 1.37, 1.25, 0.76, -0.8, -1.22, -1.33, -1.36, -1.37, -1.35, -1.29, -1.1, -0.1 |
30 |
2.0 , -0.06, -1.9, -3.4, -4.9, -4.8, 4.0, -2.7, -1.1, 0.95, 3.0, 5.0, 6.7, 8.1, 8.8, 8.9, 8.5, 7.47, 5.94, 4.06 |