Fourier Analysis |

페이지 정보

작성자 cemtool 작성일13-12-31 18:10 조회9,598회 댓글19건

본문

Fourier Analysis

Fourier Analysis

overview

This tutorial is part of the CEMWARE series. Each tutorial in this series will teach you a specific topic of common applications by explaining theoretical concepts and providing practical examples.

This tutorial is to demonstrate the use of CEMTool for solving electronics problems. In this tutorial, Fourier analysis will be discussed. Topics covered are Fourier series expansion, Fourier transform, discrete Fourier transform, and fast Fourier transform. Some applications of Fourier analysis, using CEMTool, will also be discussed.

Table of Contents

1. Fourier series

2. Fourier transforms

3. Discrete and fast Fourier transform

1. Fourier series

If a function g(t) is periodic with period Tp, i.e., (1)

and in any finite interval g(t) has at most a finite number of discontinuities and a finite number of maxima and minima (Dirichlets conditions), and in addition,

(2)

then g(t) can be expressed with series of sinusoids. That is,

(3)

where (4)

and the Fourier coefficients an and bn are determined by the following equations.

(5)

(6)

Equation (8.3) is called the trigonometric Fourier series. The term in Equation (3) is the dc component of the series and is the average value of g(t) over a period. The term is called the n-th harmonic. The first harmonic is obtained when n = 1. The latter is also called the fundamental with the fundamental frequency of ωo. When n = 2,

we have the second harmonic and so on.

Equation (3) can be rewritten as (7)

where and (8,9)

The total power in g(t) is given by the Parseval’s equation:

(10)

where (11)

The following example shows the synthesis of a square wave using Fourier series expansion.

Example 1: Using Fourier series expansion, a square wave with a period of 2 ms, peak-to-peak value of 2 volts and average value of zero volt can be expressed as

(12)

where . if a(t) is given as (13)

Write a CEMTool program to plot a(t) from 0 to 4 ms at intervals of 0.05 ms and to show that a(t) is a good approximation of g(t).

Solution

CEMTool script

% fourier series expansion

f = 500; c = 4/pi; dt = 5.0e-05;

tpts = (4.0e-3/5.0e-5) + 1;

for (n = 1;n<= 12;n++ )

{

for (m = 1;m<= tpts;m++ )

s1(n,m) = (4/pi)*(1/(2*n - 1))*sin((2*n - 1)*2*pi*f*dt*(m-1));

}

for (m = 1;m<=tpts;m++)

{ a1 = s1(:,m);

a2(m) = sum(a1); }

f1 = a2';

t = 0.0:4.0e-3:5.0e-5;

plot(t,f1)

xlabel("Time, s")

ylabel("Amplitude, V")

title("Fourier series expansion")

Figure 1 shows the plot of a(t)

설명: chap8_1.jpg

Figure 1: Approximation to square wave

By using the Euler’s identity, the cosine and sine functions of Equation (3) can be replaced by exponential equivalents, yielding the expression

(14)

where (15)

Equation (14) is termed the exponential Fourier series expansion. The coefficient cn is related to the coefficients an and bn of Equations (5) and (6) by the expression

(16)

In addition, cn relates to An and φn of Equations (8) and (8.9) by the relation

(17)

The plot of versus frequency is termed the discrete amplitude spectrum or the line spectrum. It provides information on the amplitude spectral components of g(t). A similar plot of ∠cn versus frequency is called the discrete phase spectrum and the latter gives information on the phase components with respect to the frequency of g(t) .

If an input signal xn(t): (18)

passes through a system with transfer function Hw(), then the output of the system yn(t) is

(19)

The block diagram of the input/output relation is shown in Figure 2.

설명: chap8_2.jpg

Figure 2: Input/Output relationship

However, with an input x(t) consisting of a linear combination of complex excitations,

(20)

the response at the output of the system is (21)

The following two examples show how to use CEMTool to obtain the coefficients of Fourier series expansion.

Example 2: For the full-wave rectifier waveform shown in Figure 3, the period is 0.0333s and the amplitude is 169.71 Volts.

(a) Write a CEMTool program to obtain the exponential Fourier series coefficients cn for n = 0,1, 2, .. , 19

(b) Find the dc value.

설명: chap8_3.jpg

Figure 3:Full-wave rectifier waveform

Solution

% generate the full-wave rectifier waveform

f1 = 60;

inv = 1/f1; inc = 1/(80*f1); tnum = 3*inv;

t = 0:tnum:inc;

g1 = 120*sqrt(2)*sin(2*pi*f1*t);

g = abs(g1);

N = length(g);

% obtain the exponential Fourier series coefficients

num = 20;

for( i = 1;i<=num;i++ )

{ for (m = 1;m<=N;m++)

cint(m) = exp(-j*2*pi*(i-1)*m/N)*g(m);

c(i) = sum(cint)/N;

}

cmag = abs(c);

cphase = angle(c);

%print dc value

disp("dc value of g(t)"); cmag(1)

% plot the magnitude and phase spectrum

f = (0:num-1)*60;

subplot(121), stem(f(1:5),cmag(1:5))

title("Amplitude spectrum")

xlabel("Frequency, Hz")

subplot(122), stem(f(1:5),cphase(1:5))

title("Phase spectrum")

xlabel("Frequency, Hz")

dc value of g(t)

ans =

107.5344

Figure 4 shows the magnitude and phase spectra of Figure 3.

설명: chap8_4.jpg

Figure 4: Magnitude and Phase spectra of a Full-wave rectification waveform

Example 3: The periodic signal shown in Figure 5 can be expressed as

(i) Show that its exponential Fourier series expansion can be expressed as

(22)

(ii) Using a CEMTool program, synthesize g(t) using 20 terms, i.e.,

설명: chap8_5.jpg

Figure 5: Periodic exponential signal

Solution

(i)

where and ,

thus

(ii) CEMTool script

% synthesis of g(t) using exponential Fourier series expansion

dt = 0.05;

tpts = 8.0/dt +1;

cst = exp(2) - exp(-2);

for (n = -10;n<=10;n++)

{ for( m = 1;m<=tpts;m++ )

g1(n+11,m) = ((0.5*cst*((-1)^n))/(2+j*n*pi))*(exp(j*n*pi*dt*(m-1)));

}

for( m = 1;m<= tpts;m++)

{ g2 = g1(:,m);

g3(m) = sum(g2);

}

g = g3';

t = -4:4.0:0.05;

plot(t,g)

xlabel("Time, s")

ylabel("Amplitude")

title("Approximation of g(t)")

Figure 6 shows the approximation of g(t)

설명: chap8_6.jpg

Figure 6: An approximation of g(t)

2. Fourier transforms

If g(t) is a non-periodic deterministic signal expressed as a function of time t, then the Fourier transform of g(t) is given by the integral expression:

(23)

where and f denotes frequency

g(t) can be obtained from the Fourier transform G(f) by the inverse Fourier transform formular:

(24)

For a signal g(t) to be Fourier transformable, it should satisfy the Dirichlet’s. If g(t) is continuous and non-periodic, then G(f) will be continuous and periodic. However, if g(t) is continuous and periodic, then G(f) will discrete and non-periodic; that is (25)