lec 1.pdf

Fourier Series

1.1. Introduction

It is quite common that engineering applications would be under the influence of a time varying function. The time varying function may be part of the design of the system or may be appearing as part of the behavior of the system. For example, an RLC circuit by design is a vibrating system. A highway bridge is meant to be a static system, but when an automobile passes over it, the bridge vibrates; hence becoming a dynamic system. One tool that is indispensable in handling time varying, periodic signal is Fourier series.

The central theme of Fourier series is to represent a periodic — albeit not sinusoidal — signal into combination of sines and cosines. The concept had been in use even before the time of Jean Baptiste Joseph Fourier. Mathematicians like Euler, Lagrange, d’Alembert, Daniel Bernoulli had used it. Yet, there was considerable debate regarding the class of functions that could be successfully represented in terms of Fourier series, and the conditions required to do so. Part of the problem was that the concept of a ‘function’ was still under development. Therefore, when Fourier submitted his work ‘La Théorie Analytique de la Chaleur’ on the problem of heat diffusion in a thin long rod, the paper was rejected by the reviewers — none other than Laplace, Lagrange, and Euler — on the ground that mathematical analysis lacked sufficient rigor. Although Fourier did not invent Fourier series, he used them very effectively to develop insight into the physical and mathematical nature of the problem. This led other mathematicians of the time to use this series also.

1.2. Fourier Series

For any given function f(x), Fourier Series is defined as

(cid:2998)

(cid:2868)

(cid:3041)

(cid:3041)(cid:2880)(cid:2869)

(cid:2998)

(cid:3041)

(cid:3041)(cid:2880)(cid:2869)

(1.1)

We are trying to ascertain two objectives. First, given a function f(x), (i) under what condition can this function be described as a Fourier Series; and (ii) second how do we find the series.

Before embarking upon answering these questions, we need to learn two concepts.

1.2.1 Periodicity

Figure 1 A periodic function

When a function f(x) repeats after an interval l, we can write f(x + l) = f(x). The quantity l is termed as the period, or wavelength. An example of a periodic function is shown in figure 1.

1.2.2 Orthogonality & Orthonormality

We extend the concept of orthogonality and orthonormality that we studied in vectors. Two functions p(x) and q(x) defined in the domain  < x <  are considered orthogonal if

(cid:3081)

(cid:3080)

Similarly, the function p(x) is considered orthonormal if

(cid:3081)

(cid:3080)

The functions orthonormal in the domain –l < x < l. Therefore

, and

(cid:3039)

(cid:3041)(cid:3095)

are both orthogonal and

(cid:3039)

(cid:2879)(cid:3039) (cid:3039)

(cid:2879)(cid:3039)

For all values of m and n (1.1)

(1.2)

(1.3)

With these tools in hand, let us go back to answer the two questions those were posed.

A function f(x) defined in the domain –l ≤ x ≤ l can be represented in Fourier Series if it satisfies the following conditions (i) if the function has a period 2l; or f(x + 2l) = f(l) (ii) if the function is continuous, or piecewise continuous in the domain

–l ≤ x ≤ l

Given that these conditions above are satisfied, now we find constants a0, an and bn in equation (2.1).

First, we integrate equation (2.1) from –l to l. Therefore

y=2x^2-5

(cid:3039)

(cid:3505) 𝑓(𝑥)𝑑𝑥 (cid:2879)(cid:3039)

(cid:3039) = (cid:3505) (cid:2879)(cid:3039)

𝑎(cid:2868) 2

(cid:3039) 𝑑𝑥 + (cid:3533) (cid:3505) 𝑎(cid:3041) cos (cid:4672) (cid:2879)(cid:3039)

𝑛𝜋 𝑙

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

(cid:4673) 𝑥 𝑑𝑥

(cid:2998)

(cid:3039)

(cid:3533) (cid:3505) 𝑏(cid:3041) sin (cid:4672) (cid:2879)(cid:3039)

(cid:3041)(cid:2880)(cid:2869)

𝑛𝜋 𝑙

(cid:4673) 𝑥 𝑑𝑥

Performing the integration, we will obtain that the second and the third integration on the right-hand side would be 0. Therefore, we obtain

(cid:3042)

(cid:3039)

(cid:2879)(cid:3039)

(1.4)

Next we multiply equation (2.1) with to l. Therefore

(cid:3040)(cid:3095)

(cid:3039)

and integrate from –l

(cid:3039)

(cid:3505) 𝑓(𝑥) cos (cid:4672) (cid:2879)(cid:3039)

(cid:4673) 𝑥 𝑑𝑥

𝑚𝜋 𝑙 (cid:3039) = (cid:3505) (cid:2879)(cid:3039)

𝑎(cid:2868) 2

cos (cid:4672)

𝑚𝜋 𝑙

(cid:4673) 𝑥 𝑑𝑥

(cid:3533) 𝑎(cid:3041) cos (cid:4672)

𝑛𝜋 𝑙

(cid:4673) 𝑥 cos (cid:4672)

𝑚𝜋 𝑙

(cid:4673) 𝑥 𝑑𝑥

]

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

(cid:2998)

(cid:3533) 𝑏(cid:3041) sin (cid:4672)

(cid:3041)(cid:2880)(cid:2869)

𝑛𝜋 𝑙

(cid:4673) 𝑥 cos (cid:4672)

𝑚𝜋 𝑙

(cid:4673) 𝑥 𝑑𝑥

The first integration on the right-hand side will be 0, and keeping in mind the orthogonality and orthonormality conditions (1.1) and (1.3) above, the second integral will be l only when m = n, and the third integral will be 0. Therefore, we obtain

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

Similarly, multiplying equation (3.30) with from –l to l, we will obtain for bn

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(1.5)

(cid:3040)(cid:3095)

(cid:3039)

, and integrating

(2.6)

To summarize, we have a function f(x) defined in the domain (–l, l); periodic outside the domain; piecewise continuous. We want to expand f(x) in a Fourier series

(cid:2998)

(cid:2868)

(cid:3041)

(1.1)

(cid:3041)(cid:2880)(cid:2869) The constants a0, an and bn are given by

(cid:3041)(cid:2880)(cid:2869)

(cid:3042)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:3039)

(cid:2879)(cid:3039)

(1.4)

(1.5)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(1.6)

Equations (1.4)–(1.6) are called Euler–Fourier coefficients.

With these tools up our sleeves, let us look into some examples

Example

Find the Fourier Series for

f(x) = –x = x f(x + 2l) = f(x)

–l ≤ x < 0 0 < x ≤ l

The function represents a triangular wave as depicted in figure below

–4l

–3l

–2l

–l

The period of the function is 2l. We will use equations (1.4)–(1.6) to compute the coefficients of the Fourier Series.

Using equation (1.4)

(cid:3042)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2868)

(cid:2879)(cid:3039) (cid:2868)

(cid:3039)

(cid:2868) (cid:3039)

Similarly, from equation (2.5), we can write for an

(cid:2879)(cid:3039)

(cid:2868)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(cid:3041)

(cid:2868)

(cid:2879)(cid:3039)

(cid:3039)

(cid:2868) (cid:2870)

(cid:2868)

(cid:2879)(cid:3039) (cid:2870)

(cid:2870)

n = 1, 2, 3, 4, …

(cid:2870)

Finally, from equation (2.6) we can write for bn

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:2879)(cid:3039)

(cid:2870)

Therefore, the Fourier Series is

(cid:2998)

(cid:2870)

(cid:3041)(cid:2880)(cid:2869),(cid:2871),(cid:2873),…

(cid:3039)

(cid:2868) (cid:2870)

(cid:2868)

(cid:2879)(cid:3039) (cid:2870)

n = 1, 2, 3, 4, …

(cid:3039)

(cid:2868)

(cid:3039)

(cid:2868)

Figure 2. The function and its Fourier series with 1, 4 and 7 components.

(cid:2998)

(cid:2870)

(cid:3041)(cid:2880)(cid:2869) The graphical representation of f(x) and its Fourier series are shown in figure 2.

Example

Find the Fourier series expansion for

f(x) = x + 1 –1 < x < 0 0 < x < 1

= 1 f(x + 2) = f(x)

As before

(cid:2868)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:2879)(cid:2869)

(cid:2868)

(cid:2879)(cid:2869)

(cid:2869)

(cid:2868)

(cid:2869)

(cid:2868)

(cid:2870)

(cid:2879)(cid:2869)

(cid:2869) (cid:2868)

(cid:3039)

(cid:2868)

(cid:2869)

(cid:2868)

(cid:2879)(cid:2869)

(cid:2869) (cid:2868)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:2879)(cid:2869)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:2879)(cid:2869)

(cid:2870)

(cid:3039)

(cid:2868)

(cid:2869)

(cid:2868)

(cid:2879)(cid:2869)

(cid:3041)(cid:2878)(cid:2869)

(cid:2869) (cid:2868)

(cid:3041)(cid:2878)(cid:2869)

Therefore, the Fourier series is

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

The Fourier series representation of the function are shown in figure 3.

Figure 3. Function and its Fourier series with 1, 4, and 7 components

We observe a remarkable difference between the Fourier series representations in figures 2 and 3. The function represented in figure 2 is a continuous function – its first derivative is not continuous. We observe that for this function, its Fourier series representation very

Figure 4. Gibbs phenomenon

easily simulates the original function, with very few components. On the other hand, the function represented in figure 3 has a jump discontinuity at x = 1.0. We observe that as we continue adding more and more components to its Fourier series, the series begins to oscillate at the point of discontinuity. Figure 4 shows the same function with 20 terms. The oscillation at the point of discontinuity is very prominent. This is a common behavior of Fourier series representation of any discontinuous function, and is referred to as Gibbs phenomenon.

We will have much more to discuss about this shortly.

1.3. Half range expansion, even & odd functions

Strictly speaking, we do not need this in our discussion of Fourier Transform, but we are doing this for sake of completion.

Before proceeding any further, it is worthwhile to discuss the concept of even and odd functions.

A function is defined as an even function if f(x) = f(–x). An even function is depicted in figure 5 on the left.

A function is defined as an odd function if f(x) = –f(x). An odd function is depicted in figure 5 on the right.

The following results can be derived for odd and even functions

The sum and product of two even functions is an even function;
The sum of two odd functions is odd; and the product of two odd

functions is even.

The product of an even and an odd function is odd. If it becomes necessary to expand and even or odd function as Fourier Series, then the expressions (2.4)−(2.6) can be substantially simplified. The odd or even characteristics of the function f(x) associated with the

even and odd characteristics of cosine and sine functions simplifies the expression to a some extent.

If f(x) is even, then (cid:3039)

(cid:2868)

(cid:3039)

(cid:2868)

(cid:2879)(cid:3039) Substituting x = −x in the first integral, we obtain

(cid:2879)(cid:3039)

(cid:2868)

(cid:3039)

(cid:2868)

(cid:3039)

(cid:2868)

(cid:2879)(cid:3039)

(cid:2868)

As f(x) is even, therefore, f(−x) = f(x). Finally, we reverse the limits of integration, we obtain that for an even function

(cid:3039)

(cid:2868)

The Fourier coefficient an can be simplified

(1.7)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2868)

As before, we replace x with –x in the first integral.

(cid:2868)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2868)

Recalling that for an even function f(−x) = f(x), and cos (−x) = cos x, and finally reversing the limits of integration, we obtain that for an even function

(cid:3041)

(cid:3039)

(cid:2868)

(1.8)

Similarly, the Fourier coefficient bn can be simplified

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:2868)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2868)

As before, we replace x with –x in the first integral.

(cid:2868)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2868)

Recalling that for an even function f(−x) = f(x), and sin (−x) = −sin x, and finally reversing the limits of integration, we obtain that for an even function

(cid:3041)

Similarly, we can obtain, if f(x) is odd, then

(cid:2868)

(cid:3041)

(cid:3039)

(cid:2868)

(1.9)

(1.10) (1.11)

(1.12)

While finding solutions of partial differential equation it is often necessary to expand a function f(x) in either as a sine or as a cosine series. In case of sine expansion, bn will be 0; and in case of cosine expansion a0 and an will be 0. To perform this expansion, we will use the expressions (1.7)–(1.9) or (1.10)–(1.12). In all these cases, the function f(x) will be defined in the domain x > 0. Therefore, these expansions are called half-wave expansions.

Example

Find the cosine and sine expansion of the function

f(x) = 1 0 < x < 1 = 0 1 < x < 2

–1

Cosine expansion The wave for cosine expansion appears in the figure below

(cid:2868)

(cid:3039)

(cid:2868)

(cid:2869)

(cid:2868)

(cid:2870)

(cid:2869)

(cid:3041)

(cid:2869)

(cid:2868)

(cid:2870)

(cid:2869)

(cid:2868)

Therefore, the cosine series is

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

Sine expansion The wave for sine expansion appears in the figure below

a0 = an = 0 (cid:2869)

(cid:3041)

(cid:2868)

–1

(cid:2870)

(cid:2869)

(cid:2868)

Therefore, the sine series is (cid:2998)

(cid:3041)(cid:2880)(cid:2869)

1.4. Engineering interpretation of Fourier Series

We were given a periodic function with a period 2l, defined in the domain (–l, l), and piecewise continuous function (signal) f(x) that we expanded in a Fourier Series

(cid:2868)

(cid:3041)

(1.1)

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

(cid:2998)

(cid:3041)(cid:2880)(cid:2869)

The constants a0, an and bn were obtained from

(cid:3042)

(cid:3041)

(cid:3039)

(cid:2879)(cid:3039) (cid:3039)

(cid:2879)(cid:3039)

From an engineering perspective, what did we achieve?

In more detail, the constant term a0/2 can be written as (cid:3039)

(1.4)

(1.5)

(1.6)

(1.13)

(cid:2879)(cid:3039)

We know from fundamental calculus that integration gives us the area under the curve within the limit. Recognizing that 2l is the width of one period of the function – the wavelength, the expression in (1.13) gives

us the average height of one period – the statistical mean of the function.

With that thought, if we consider f(x) to be a signal, the constant term can also be thought of as the dc-value of the signal. We will immediately recognize the summation as the ac-component of the signal.

To make more sense of the ac-component, we expand the summation.

(cid:2869)

(cid:2871)

(cid:2870)

We know from our basic understanding in physics that in expressions A sinx and B cosx,  is the frequency, and A and B are the amplitudes. We also learn from physics that squaring the amplitudes gives us the power.

With this knowledge, we can further understand the ac-component. We can understand that a1 is the amplitude of the cosine component of frequency 1/l and b1 is the amplitude of the sine component at this (cid:2870) as the power of the 1/l frequency of frequency. This gives us (cid:2869) the ac-component. This is the first component of the series expansion. In engineering, we refer to this component as the primary harmonics. (cid:2870) is the power of the second harmonics, Similarly, (cid:2870) the power of the third harmonics, etc.

(cid:2870) is (cid:2871)

(cid:2870) (cid:2869)

(cid:2870) (cid:2871)

(cid:2870) (cid:2870)

We have just obtained the discrete power-spectrum of the signal f(x)!

We are ready to take this discussion further. We are given a signal f(x). Using the integrals (1.4)-(1.6) we broke this signal into its dc components and ac components. In signal processing, this step is referred to as analysis. Next, we took these components and attempted to reconstruct the original signal by putting them in (1.1). This step is referred to as synthesis. Therefore, Fourier analysis involves two steps: analysis and synthesis.

A very relevant question that we have to answer: Do we get our original signal back?

The answer, unfortunately, is no! This was exactly the reason why mathematicians like Lagrange, D’Alembert and Euler rejected Fourier’s paper. Their argument was that ‘you are using procedures those are still not clear to us’. Moreover, these issues carry over to Fourier Transform, and are still not very clearly understood.

This requires further clarification. We have seen in the graphic simulation that as we add more and more components of the signal, the summation seems to be approaching the original signal. But take the case of our first example of the triangular signal. We have seen in simulation that the summation quickly begins to look like the original signal. But the original signal, even though the signal is continuous, the first derivative of the signal is not continuous. But the summations of the components are results of integration; and integration is a smoothing operation. Therefore, the summation will always be smooth and continuous. Therefore, there is no way we can add continuous signals to obtain a discontinuous signal. This brings us to the last section of this discussion.

1.5. Error and convergence of Fourier series

In the examples of Fourier series expansions we have seen in the last section, we have observed that if f(x) is a continuous function in the domain it is defined, its Fourier series very nicely approximates the function. On the other hand, if f(x) has discontinuity, we have observed that its Fourier series representation demonstrates an oscillation that we called Gibbs phenomenon. So, it is very natural to ask the question how well does the Fourier series representation of the function f(x) represent the function itself?

To set things in motion, we define two terms: Nth partial sum and mean square error. The Nth partial sum is defined as the sum of the first N terms of the series.

(cid:2868)

(cid:3015)

(cid:3041)

(cid:3041)(cid:2880)(cid:2869)

(cid:2998)

(cid:3041)

(cid:3041)(cid:2880)(cid:2869)

The mean square error is defined as (cid:3039)

(cid:3015)

(cid:2870)

(cid:3015)

(1.14)

(1.15)

(cid:2879)(cid:3039)

To continue further on our study of mean-square error, we introduce a class of square integrable function on [a, b] which consists of a function f defined on [a, b], and such that following theorem without its proof.

. We state the

(cid:3029) (cid:3028)

Theorem 1 Suppose that f is square integrable on [a, b], then SN, the Nth partial sum of the Fourier series of f converges to f in the mean with an error EN that decreases to zero as N → ∞. In symbols, we have

(cid:3015)→(cid:2998)

(cid:3015)

(cid:3015)→(cid:2998)

(cid:2879)(cid:3039)

(cid:3039)

(cid:2870)

(cid:3015)

(1.16)

Theorem 1 is important to answer that question that we posed about the convergence of Fourier series. It essentially tells us that if we continue to add infinite terms in the Fourier series representation of the function f, the series will converge to the mean value of f.

We will go ahead and derive some very interesting and powerful results from our discussion. If we expand SN(x) in (1.16), and perform the integration, and remembering the orthogonality properties of sine and cosine functions, we can derive

(cid:3039)

(cid:3015)

(cid:2879)(cid:3039)

(cid:3015)

(cid:2870)

(cid:2870) (cid:2868)

(cid:2870) (cid:3041)

(1.17)

(cid:3041)(cid:2880)(cid:2869) We notice from (2.15) that EN(x) is positive. This, along with (1.17) leads to

(cid:2870) (cid:2868)

(cid:3015)

(cid:3041)(cid:2880)(cid:2869)

(cid:2870) (cid:3041)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2870)

(1.18)

This is known as Bessel’s inequality. Applying Theorem 1 to (1.18), if we let N → ∞, we obtain

(cid:3015)

(cid:2870) (cid:2868)

(cid:2870) (cid:3041)

(cid:3041)(cid:2880)(cid:2869)

(cid:3039)

(cid:2879)(cid:3039)

(cid:2870)

(1.19)

This very important expression is known as Parseval’s identity. We will study the meaning of this expression later.

007_index

lec 1.pdf

Graph View