Define Fourier series.
1 Answer
Fourier series is a mathematical representation of a periodic function as an infinite sum of sine and cosine functions (or complex exponentials) with different frequencies and amplitudes. It was introduced by the French mathematician Joseph Fourier in the early 19th century and has profound applications in various fields, including signal processing, engineering, physics, and mathematics.
The general form of a Fourier series for a periodic function f(x) with period 2π is given as:
(
)
=
0
+
∑
=
1
∞
[
cos
(
)
+
sin
(
)
]
f(x)=a
0
+∑
n=1
∞
[a
n
cos(nx)+b
n
sin(nx)]
Here, the terms a_0, a_n, and b_n are known as Fourier coefficients, and they are calculated using the following formulas:
0
=
1
2
∫
−
(
)
a
0
=
2π
1
∫
−π
π
f(x)dx
=
1
∫
−
(
)
cos
(
)
a
n
=
π
1
∫
−π
π
f(x)cos(nx)dx
=
1
∫
−
(
)
sin
(
)
b
n
=
π
1
∫
−π
π
f(x)sin(nx)dx
The coefficients a_0, a_n, and b_n represent the DC component and the amplitudes of the harmonics (sine and cosine waves) at different frequencies n. The higher the value of n, the higher the frequency of the corresponding sine and cosine functions in the series.
By using Fourier series, a periodic function can be accurately approximated by truncating the infinite sum at a finite number of terms. This makes it possible to analyze and synthesize complex periodic signals and study their frequency components, which is especially useful in analyzing periodic phenomena in various fields of science and engineering.
The general form of a Fourier series for a periodic function f(x) with period 2π is given as:
(
)
=
0
+
∑
=
1
∞
[
cos
(
)
+
sin
(
)
]
f(x)=a
0
+∑
n=1
∞
[a
n
cos(nx)+b
n
sin(nx)]
Here, the terms a_0, a_n, and b_n are known as Fourier coefficients, and they are calculated using the following formulas:
0
=
1
2
∫
−
(
)
a
0
=
2π
1
∫
−π
π
f(x)dx
=
1
∫
−
(
)
cos
(
)
a
n
=
π
1
∫
−π
π
f(x)cos(nx)dx
=
1
∫
−
(
)
sin
(
)
b
n
=
π
1
∫
−π
π
f(x)sin(nx)dx
The coefficients a_0, a_n, and b_n represent the DC component and the amplitudes of the harmonics (sine and cosine waves) at different frequencies n. The higher the value of n, the higher the frequency of the corresponding sine and cosine functions in the series.
By using Fourier series, a periodic function can be accurately approximated by truncating the infinite sum at a finite number of terms. This makes it possible to analyze and synthesize complex periodic signals and study their frequency components, which is especially useful in analyzing periodic phenomena in various fields of science and engineering.