Fourier Sequence | Fundamentals, Equation, Derivation of Fourier Coefficients

Within the early 1800’s, a French Mathematician generally known as Baron Jean Baptiste Joseph Fourier studied the conduction of warmth in several objects and got here up with a mathematical evaluation on the topic, which later turned the well-known Fourier Sequence. There are quite a few engineering issues within the fields of Communication, RADAR, Imaging, Audio, and many others. that entails the notion of Fourier Seres (and Fourier Transforms, which is a subject for another time). As an example, all energy indicators within the discipline of communication have a detailed affiliation with Fourier Sequence whereas power indicators with Fourier Transforms.

However what precisely is the Fourier Sequence? What are Fourier Coefficients and derive them? What are the functions of Fourier Sequence? We’ll discover out solutions to all these questions on this information.

Introduction

When Joseph Fourier launched his mathematical mannequin on conduction of warmth to the Academy of Sciences in Paris in 1807, they criticized and rejected his work. However later, when he printed this work in his ‘Théorie analytique de la Chaleur’ (or The Analytical Concept of Warmth) in 1822, it turned an integral a part of engineering.

Despite the fact that the unique mathematical evaluation was a examine of conduction of warmth in metals, engineers discovered that they might apply the identical method for different mathematical, engineering and physics issues.

Right this moment, you may see Fourier Sequence is essential within the fields of Communication (RADAR, Satellite tv for pc, Wi-fi basically), Acoustics, Picture Processing (Sign Processing basically), Electrical, Optics and lots of extra.

What’s the Fourier Sequence?

Okay. Sufficient introduction. Allow us to now give attention to the principle query: What’s the Fourier Sequence? Fourier Sequence is an Infinite Sequence of a periodic operate when it comes to Sine and Cosine capabilities.

Fourier Sequence of a periodic waveform is an infinite sum sine and cosine waveforms, the place the frequency of every wave (sine and cosine) is an integral a number of of the periodic waveform’s elementary frequency.

With none mathematical evaluation, these statements may sound ridiculous: why are we turning a easy waveform into a posh mixture of trigonometric capabilities?

Really, the ensuing Fourier Sequence simplifies the method of sign evaluation of the principle operate (or waveform) at totally different frequencies. We will derive Fourier Sequence for any periodic or steady capabilities or capabilities with discontinuous values and its derivatives.

Earlier than continuing additional, we have to perceive the fundamental terminology we talked about earlier: the Periodic Operate. A operate f(x) is periodic if for a continuing T>0, f(x + T) = f(x). Right here, T is called the Interval of the operate.

Periodic Operate: f(x + T) = f(x), T>0

f(x) = f(x +2T) = f(x + 3T) = … T is the interval of the operate f(x). The operate f(x) repeats itself after equal intervals. Trigonometric capabilities corresponding to sin(x), cos(x), tan(x) and many others. are some easy examples of periodic capabilities.

Now, coming again to the Fourier Sequence, if f(x) is a periodic operate, then we are able to categorical it as an infinite sum of sine and cosine capabilities as follows:

Right here, a₀, a_n and b_n are generally known as Fourier Coefficients. The values of those coefficients are what outline the Fourier Sequence of a operate. Fixed a₀ is the common worth of the periodic operate whereas a_n and b_n are the amplitudes of varied sinusoidal capabilities.

We will calculate a₀, a_n and b_n utilizing the next expressions. For instance, if f(x) is a periodic operate, then Fourier Coefficients of its Fourier Sequence within the interval T ≤ x ≤ T+2π are as follows:

The equations of a₀, a_n and b_n are generally known as Euler’s Formulae.

Within the earlier Fourier Sequence equation, we used each sine and cosine capabilities. However we are able to additional modify the equation to provide an equation solely when it comes to sinusoids.

We’ve the time period a_ncos(nx) + b_nsin(nx) within the equation. We will re-write this as follows:

Utilizing these phrases, we are able to derive the sinusoid solely Fourier Sequence Expression of a operate as:

Within the above equation, discover that for n = 1, the sinusoidal amount has the identical frequency as the principle operate (which is ‘x’ on this case) and it’s the Elementary Frequency of the principle waveform. All the following frequencies (for n = 2, n = 3 and so forth) are integral multiples of this elementary frequency which we name as Harmonic Frequencies.

So, for n = 2, the frequency of the corresponding sinusoid is called Second Harmonic. Equally, for n = 3, it’s Third Harmonic and many others.

Derivation of Fourier Coefficients

From the above dialogue, it’s clear that the Fourier Coefficients a₀, a_n and b_n are the essential values that we have to calculate for any Fourier Sequence. We’ve already seen the expressions for these constants however allow us to attempt to derive them.

For this, allow us to assume that f(x) is a periodic operate and its Fourier Sequence for the interval [T, T+2π] i.e., T ≤ x ≤ T+2π is given by:

 

Expression for a₀

Within the above equation, allow us to combine each side from x=T to x=T+2π. We get:

From the above equation, we are able to get the expression for a₀ as:

Expression for a_n

Now, take into account the unique Fourier Sequence expression as soon as once more. Multiply each side by ‘cos(mx)’ and combine the ensuing equation from x=T to x=T+2π.

Within the above expression, should you observe intently, the integrals akin to a₀ and b_n­ (first and third) are all the time zero. Coming to the second integral akin to a_n, for all m ≠ n instances, it turns into zero and the one attainable end result is for worth m = n. Due to this fact,

From the above equation, we are able to get the expression for a_n as:

Expression for b_n

Now, take into account the unique Sequence expression as soon as once more. Multiply each side by ‘sin(mx)’ and combine the ensuing equation from x=T to x=T+2π.

Within the above expression, the integrals akin to a₀ and a_n­ (first and second) are all the time zero. Coming to the third integral akin to b_n, for all m ≠ n instances, it turns into zero and the one attainable end result is for worth m = n. Due to this fact,

From the above equation, we are able to get the expression for b_n as:

Easy methods to Receive Fourier Sequence of a Operate f(x)?

Peter Gustav Lejeune Dirichlet formulated ample situations {that a} periodic operate should fulfill so as to derive Fourier Sequence. These are popularly generally known as ‘Dirichlet’s Situations’.

We will categorical any operate f(x) in its Fourier Sequence kind as:

The place a₀, a_n and b_n are constants if the next “ample situations” are happy.

• The operate f(x) is a periodic, single-valued, well-defined and finite operate.
• It has a finite variety of discontinuities in any interval.
• f(x) has finite variety of maxima and minima in a bounded interval.

If these situations are met, then we are able to get hold of the Fourier Sequence of any operate utilizing the next steps:

Assume the generic type of Sequence with respect to the operate as follows:

Calculate the worth of a₀ utilizing the next formulation:

Compute the worth of a_n utilizing the next expression:

Calculate the worth of b_n utilizing the next formulation:

Now, substitute the values of a₀, a_n and b_n­ within the authentic expression (from Step 1) and you’ve got the ultimate Fourier Sequence of the operate f(x).

Fourier Sequence Functions

The idea of representing any waveform (or operate) purely when it comes to sinusoids is a robust instrument in arithmetic, engineering (electronics, communication, mechanical, and many others.), acoustics, picture and video processing.

Fourier Sequence and Fourier Transforms, which collectively kind a brand new set of mathematical modelling generally known as Fourier Evaluation, are essential in a number of scientific functions to resolve unusual and partial differential equations, sign processing, statistics, seismography, oceanography, sonar, cryptography and many others.

Conclusion

This was an introductory information on Fourier Sequence. Joseph Fourier developed a mathematical mannequin of conduction of warmth in metals, which later turned the Fourier Sequence. On this, we are able to symbolize any steady, periodic operate (waveform) purely within the type of sinusoids. We’ve seen the fundamental Fourier Sequence expression, Fourier Coefficients, Dirichlet’s Situations and lots of extra.