SIGNALS SYSTEMS ENT 281 Chapter 3 Fourier Series

SIGNALS & SYSTEMS (ENT 281) Chapter 3: Fourier Series DR. HASIMAH ALI SCHOOL OF MECHATRONIC ENGINEERING UNIVERSITY MALAYSIA PERLIS

3. 1 INTRODUCTION 1. The representation of a periodic signal in terms of complex exponentials, or equivalently in terms of sine and cosine waveforms, leads to the Fourier series that are used extensively used in all fields of science and engineering. 2. The Fourier series is named after the French physicist Jean Baptiste Fourier (1768 -1830) –who was the first to suggest – periodic signals could be represented by a sum of sinusoids. 3. Introduction of representation. the concept of frequency-domain 4. We learn how to decompose periodic signals into their frequency components.

3. 1 INTRODUCTION

3. 1 INTRODUCTION 1. Cosine Function The function f(x) = cos x is an even function. That is, it is symmetrical about the vertical axis. We have: cos (-x) = cos x

3. 1 INTRODUCTION 2. Sine Function The function f(x) = sin x is an odd function. That is, it is symmetrical about the origin. We have: sin (-x) = -sin x

3. 1 INTRODUCTION 3. Multiples of for sine and cosine curve. Consider the function y = sin x From the graph, for n = 0, 1, 2, 3, …

3. 1 INTRODUCTION y = cos x for n = 0, 1, 2, 3, …

3. 2 ORTHOGONAL SET OF FUNCTIONS A set of functions {φ1(x), φ2(x), … } is an orthogonal set of functions on the interval [a, b] if any two functions in the set are orthogonal to each other.

3. 2 ORTHOGONAL SET OF FUNCTIONS Show that: form an orthogonal set on [-π, π].

3. 2 ORTHOGONAL SET OF FUNCTIONS Show that: Solution: is proven!

3. 2 ORTHOGONAL SET OF FUNCTIONS 2 nd Case: Solution:

3. 2 ORTHOGONAL SET OF FUNCTIONS 3 rd Case: Solution:

3. 3 TRIGONOMETRIC FORM OF FOURIER SERIES • A sinusoidal signal, is a periodic signal with period. • The sum of two sinusoids is periodic provided that their frequencies are integral multiples of a fundamental frequency. • We can show that a signal x(t), a sum of sine and cosine functions whose frequencies are integral multiples of, is periodic signal. • Let the signal be

PERIODIC FUNCTIONS A function is periodic if it is defined for all real and if there is some positive number, such that .

FOURIER SERIES Let the signal x(t) be: i. e.

FOURIER SERIES For the signal x(t) to be periodic, it must satisfy the condition x(t) = x(t + T) for all t, i. e.

FOURIER SERIES This proves that the signal x(t), which is summation of sine and cosine functions of frequencies 0, ω0, 2ω0, …, kω0 is a periodic signal x(t) with period T.

FOURIER SERIES The infinite series of sine and cosine terms of frequencies 0, ω0, 2ω0, …, kω0 is known as trigonometric form of Fourier series and can be written as: or (1) The constants a 0, a 1, …, an, b 0, b 1, …, bn, are called Fourier coefficients.

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine a 0 Integrate both sides of (1) over one period (t 0 to t 0 + T)

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine a 0 We know that and since the net areas of sinusoids over complete periods are zero for any nonzero integer n and any time t 0.

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine a 0 Thus, we obtain or

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine an, bn We can use the following results: 1. 3. for all m and n 2.

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine an, bn To find an, multiply the equation (1) by cos mω0 t and integrate over one period. That is,

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Determine an, bn To first and third integrals are equal zero and the second integral is equal to T/2 when m = n. 0 0

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Therefore, or

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES To find bn, multiply both sides of equation (1) by sin mω0 t and integrate over one period. Then,

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES The first and second integrals are zero, and the third integral is equal to T/2 when m = n. Thus, 0 0

EVALUATION OF FOURIER COEFFICIENTS OF THE TRIGONOMETRY FOURIER SERIES Thus, or *a 0, an and bn are called trigonometric Fourier series coefficients.

EXAMPLE: Find the Fourier series expansion of the half wave rectified sine wave as shown in Figure 1. x(t) ---- A -----2π -π 0 π 2π 3π

SOLUTION: The periodic waveform shown in Figure with period 2 π is half of a sine wave with period 2π. Now the fundamental period: T = 2π. Fundamental frequency:

SOLUTION: Let: Thus,

For odd n: For even n: Therefore: (for even n)

This is zero for all values of n except for n = 1. For n = 1:

Therefore, the trigonometric Fourier series is:

EXAMPLE: Obtain the trigonometric Fourier series for the waveform shown in Figure 2. x(t) A ----- -4π -3π -2π -π 0 π 2π 3π

SOLUTION: The waveform shown in Figure is periodic with a period T = 2π. Let: Then, Fundamental frequency: The waveform is described by:

SOLUTION: Let: Thus,

Thus, for odd n for even n

For odd n For even n

The trigonometric Fourier series is:

EXPONENTIAL FOURIER SERIES COMPLEX EXPONENTIALS

EXPONENTIAL FOURIER SERIES The set of complex exponential functions: Forms a closed orthogonal set over an interval (t 0, t 0 + T) where T=(2π/ω0) for any value of t 0, and therefore it can be used as a Fourier series. Using Euler’ identity, we can write:

Substituting this in the definition of cosine Fourier representation, we obtain:

Letting n = -k in the second summation, we have On comparing the above equations for x(t), we get n>0 k<0

Let us define Thus i. e This is known as exponential form of Fourier series. The above equation is called the synthesis equation.

So the exponential series from cosine series is:

DETERMINATION OF THE COEFFICIENTS OF EXPONENTIAL FOURIER SERIES We have: Multiplying both sides by and integrating over one period,

We know Thus, or or Where Cn are the Fourier coefficients of the exponential series.

q The Fourier coefficients of x(t) have only a discrete spectrum because values of Cn exist only for discrete values of n. q As it represents a complex spectrum, it has both magnitude and phase spectra. q Noted: a. The magnitude line spectrum is always an even function of n b. The phase line spectrum is always an odd function of n.

TRIGONOMETRIC FOURIES SERIES FROM EXPONENTIAL FOURIER SERIES The complex exponential Fourier series is given by:

TRIGONOMETRIC FOURIES SERIES FROM EXPONENTIAL FOURIER SERIES Comparing with standard trigonometric Fourier series: We get conversion of exponential to Trigonometric

EXPONENTIAL FOURIER SERIES FROM TRIGONOMETRIC FOURIES SERIES From exponential series, we know:

EXPONENTIAL FOURIER SERIES FROM TRIGONOMETRIC FOURIES SERIES We get conversion of trigonometric to exponential FS:

COSINE FOURIER SERIES FROM EXPONENTIAL FOURIER SERIES We know that:

Example 3: Obtain the exponential Fourier series for the waveform shown in Figure 3. Also draw the frequency spectrum. Figure 3

Solution: The periodic waveform shown in Figure 3 with a period T = 2π can be expressed as: Let And fundamental frequency:

Solution: Exponential Fourier Series:

Cont. …

Cont. … Cn -5 -4 -3 -2 -1 0 1 2 3 4 5

Example 4: Find the exponential Fourier series and plot the frequency spectrum for full wave rectified sine wave shown in Figure 4. Figure 3

Solution: The waveform shown in Figure 4 can be expressed over one period (0 to π) as: where Because it is part of a sine wave with period = 2 π. The full wave rectified sine wave is periodic with T = π.

Exponential Fourier Series:

Cont. …

Cont. … Cn -5 -4 -3 -2 -1 0 1 2 3 The frequency spectrum 4 5

GIBBS PHENOMENON Gibbs discovered that for a periodic signal with discontinuities, if the signal is reconstructed by adding the Fourier series, overshoots appear around the edges.

GIBBS PHENOMENON These overshoots decay outwards in a damped oscillatory manner away from edges. This is called Gibbs phenomenon. The overshoots at the discontinuity according to the Gibbs are found to be around 9 percent of the height of discontinuity irrespective of the number of terms in the Fourier series.

GIBBS PHENOMENON q It has also been observed that as more number of terms in the series are added, the frequency increases, overshoots, get sharper, but the adjoining oscillation amplitude reduces. q That is error between the original signal x(t) and the truncated signal xn(t) reduced except at the edges as n increases. q Thus, the truncated Fourier series approaches x(t) as the number of terms in approximation increases.

Gibbs Phenomenon – Cont’d

Gibbs Phenomenon – Cont’d overshoot: overshoot about 9 % of the signal magnitude (present even if )

PARSEVAL’S THEOREM If, and, Then, the Parseval’s relation states that, [for complex x 1(t) and x 2(t)] And, Parseval’s identity states that,

PARSEVAL’S THEOREM Proof: Parseval’s relation, Interchanging the order of integration and summation in the RHS,

PARSEVAL’S THEOREM Therefore, Proved. Parseval’s Identity: If x 1(t)=x 2(t)=x(t), then the relation changes to

PARSEVAL’S THEOREM Since, Substitute this value, Proved.

REFERENCES § A. Anand Kumar, “Signals & Systems”, PHI Learning Private Limited. 2 nd edition. New Delhi. § Simon Haykin and Barry Van Veen, “Signals and Systems”, Wiley, 2 nd Edition, 2002 § M. J. Roberts, “Signals and Systems”, International Edition, Mc. Graw Hill, 2 nd Edition 2012