HARMONIC ANALYSIS T MANIKANDAN AP MATHEMATICS MSEC HARMONIC
- Slides: 12
HARMONIC ANALYSIS T. MANIKANDAN AP / MATHEMATICS MSEC
HARMONIC ANALYSIS If f(x) is defined analytically in (0, 2π) we know how to find the Fourier coefficients a 0, an and bn. But in practical Engineering problems, the function f(x) to be expanded in Fourier series is not defined by analytical expression. Instead, we know a few values of the function or its graph. In such cases, the Fourier co-efficients a 0, and bn in Fourier series are calculated by means of approximate integration.
HARMONIC ANALYSIS The process of finding the Fourier series for a function given by numerical value is known as harmonic analysis.
The term a 1 cosx+b 1 sinx is called the fundamental or first harmonic, the term a 2 cos 2 x+b 2 sin 2 x is called the second harmonic and so on.
Example: 1 Find the first harmonic of Fourier series of y=f(x) from the given data x (in degree) y = f(x) 0 60 120 180 240 300 360 1. 98 1. 3 1. 05 1. 3 -0. 88 -0. 25 1. 98 Solution: Since the function is periodic with period 2π, We exclude the last point x = 360. The Fourier series for the first harmonic is
x y Cos x Sin x y cos x 0 1. 98 1 0 0 60 1. 3 1. 98 0. 5 0. 866 1. 1258 120 1. 05 0. 65 -0. 5 0. 866 0. 9093 180 1. 3 -0. 525 -1 0 -1. 3 0 240 -0. 88 -0. 5 -0. 866 0. 44 0. 762 300 -0. 25 4. 5 0. 5 -0. 886 -0. 125 1. 12 Total y sin x 0. 2165 3. 013 f(x)=0. 75 + 0. 373 cos x + 1. 004 sin x
Example: 2 Find the first two harmonics of Fourier series of y=f(x) from the given data x 0 y = f(x) 1 2π 1. 4 1. 9 1. 7 1. 5 1. 2 1 Solution: Since the function is periodic with period 2π, We exclude the last point x = 2π. The Fourier series for the first two harmonic is
x 0 y Cos x 1 1 1 0 0 0. 5 -0. 5 0. 866 0. 7 -0. 7 1. 2124 -0. 5 0. 866 -0. 95 1. 6454 -1 1 0 0 -1. 7 0 0 -0. 5 -0. 866 -0. 75 -1. 299 -0. 5 -0. 866 0. 6 -1. 0392 1. 4 1. 9 1. 7 1. 5 1. 2 Total 8. 7 0. 5 Cos 2 x Sin 2 x y cos x -1. 1 y cos 2 x y sin x -0. 3 y sin 2 x 0. 5196 -0. 1732
f(x)=1. 45 – 0. 37 cos x – 0. 1 cos 2 x + 0. 173 sin x – 0. 577 sin 2 x
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