Question

How do you represent a periodic waveform using Fourier series?

Answer 1

To represent a periodic waveform using Fourier series, you decompose the waveform into a sum of sinusoidal functions (sine and cosine waves) of different frequencies and amplitudes. This technique is named after the French mathematician Joseph Fourier, who showed that any periodic function can be represented as an infinite sum of sine and cosine functions.

The general form of the Fourier series representation of a periodic function f(x) with period T is given by:


(

)
=

0
+
∑

=
1
∞
(


cos
⁡
(



)
+


sin
⁡
(



)
)
f(x)=a
0
    ​

+∑
n=1
∞
    ​

(a
n
    ​

cos(nωx)+b
n
    ​

sin(nωx))

where:


=
2


ω=
T
2π
    ​

 is the fundamental angular frequency,

0
a
0
    ​

 is the average value of the function over one period,


a
n
    ​

 and


b
n
    ​

 are the Fourier coefficients, which represent the amplitude of the cosine and sine components at frequency


nω respectively.

The Fourier coefficients are calculated using the following formulas:


0
=
1

∫
−

2

2

(

)
 


a
0
    ​

=
T
1
    ​

∫
−
2
T
    ​

2
T
    ​

    ​

f(x)dx



=
2

∫
−

2

2

(

)
cos
⁡
(



)
 


a
n
    ​

=
T
2
    ​

∫
−
2
T
    ​

2
T
    ​

    ​

f(x)cos(nωx)dx



=
2

∫
−

2

2

(

)
sin
⁡
(



)
 


b
n
    ​

=
T
2
    ​

∫
−
2
T
    ​

2
T
    ​

    ​

f(x)sin(nωx)dx

These integrals represent the average value of the product of the function with the corresponding trigonometric function over one period.

In practice, the Fourier series representation is often truncated to a finite number of terms, as using an infinite number of terms is usually impractical. The more terms you include in the series, the closer the representation will approximate the original waveform.

By calculating the appropriate Fourier coefficients, you can express a periodic waveform as a sum of harmonically related sinusoidal functions, providing valuable insights into its frequency components and behavior. This representation is widely used in various fields, including signal processing, electrical engineering, and physics.