Fourier3

Agung Budi Santoso, ST

Sintesis Bentuk Gelombang
& Spektrum Garis

SINTESIS BENTUK GELOMBANG

Sintesis (penyatuan) adalah gabungan dari bagian-bagian sehingga
membentuk satu kesatuan. Sintesis Fourier adalah penggabungan
kembali suku-suku deret trigonometri, lazimnya empat atau lima
suku pertama agar menghasilkan gelombang mula-mula.

SPEKTRUM GARIS

Spektrum linier adalah sebuah kurva yang memperlihatkan masing-
masing amplitudo harmonik di dalam gelombang. Garis-garis
bertambah secara cepat pada gelombang-gelombang dengan deret
yang mengecil (konvergen) secara cepat.
Kandungan harmonik dan spektrum garis dari sebuah gelombang
adalah bagian yang sangat alamiah dari gelombang tersebut dan
tidak pernah berubah, tanpa memperhatikan metode analisis.

%
% Filename: example8.m
%
% Description: This M-file plots the truncated Fourier Series
% for a square wave as well as its amplitude
% spectrum.

clear; % clear all variables
clf; % clear all figures

N = 11; % summation limit (use N odd)
wo = pi; % fundamental frequency (rad/s)
c0 = 0; % dc bias
t = -3:0.01:3; % declare time values

figure(1) % put first two plots on figure 1

% Compute yce, the Fourier Series in complex exponential form

yce = c0*ones(size(t)); % initialize yce to c0

for n = -N:2:N, % loop over series index n
cn = 2/(j*n*wo); % Fourier Series Coefficient
yce = yce + cn*exp(j*n*wo*t); % Fourier Series computation
end

subplot(2,1,1)
plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t)
[-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':');
hold;

plot(t,yce);
grid;
xlabel('t (seconds)')
ylabel('y(t)');
ttle = ['EE341.01: Truncated Exponential Fourier Series with N = ',...
num2str(N)];
title(ttle);
hold

% Compute yt, the Fourier Series in trigonometric form

yt = c0*ones(size(t)); % initialize yce to c0

for n = 1:2:N, % loop over series index n
yt = yt + 2*abs(cn)*cos(n*wo*t+angle(cn)); % Fourier Series computation
end

subplot(2,1,2)
plot([-3 -2 -2 -1 -1 0 0 1 1 2 2 3],... % plot original y(t)
[-1 -1 1 1 -1 -1 1 1 -1 -1 1 1], ':');
hold;
plot(t,yt);
grid;
xlabel('t (seconds)')
ylabel('y(t)');
ttle = ['EE341.01: Truncated Trigonometric Fourier Series with N = ',...
num2str(N)];
title(ttle);
hold

% Draw the amplitude spectrum from exponential Fourier Series

figure(2) % put next plots on figure 2

subplot(2,1,1)
stem(0,c0); % plot c0 at nwo = 0

hold
for n = -N:2:N, % loop over series index n
stem(n*wo,abs(cn)) % plot |cn| vs nwo
end
for n = -N+1:2:N-1, % loop over even series index n
cn = 0; % Fourier Series Coefficient
stem(n*wo,abs(cn)*180/pi); % plot |cn| vs nwo
end

xlabel('w (rad/s)')
ylabel('|cn|')
ttle = ['EE341.01: Amplitude Spectrum with N = ',num2str(N)];
title(ttle);
hold

% Draw the phase spectrum from exponential Fourier Series

subplot(2,1,2)
stem(0,angle(c0)*180/pi); % plot angle of c0 at nwo = 0

hold
for n = -N:2:N, % loop over odd series index n
stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo
end
for n = -N+1:2:N-1, % loop over even series index n
cn = 0; % Fourier Series Coefficient
stem(n*wo,angle(cn)*180/pi); % plot |cn| vs nwo
end

xlabel('w (rad/s)')
ylabel('angle(cn) (degrees)')
ttle = ['EE341.01: Phase Spectrum with N = ',num2str(N)];
title(ttle);
hold

EE341.01: Truncated Exponential Fourier Series with N = 11
2

1
y(t)
0

-1

-2
-3 -2 -1 0 1 2 3
t (seconds)
EE341.01: Truncated Trigonometric Fourier Series with N = 11
2

1
y(t)

0

-1

-2
-3 -2 -1 0 1 2 3
t (seconds)

EE341.01: Amplitude Spectrum with N = 11
0.8

0.6
|cn|
0.4

0.2

0
-40 -30 -20 -10 0 10 20 30 40
w (rad/s)
EE341.01: Phase Spectrum with N = 11
100
angle(cn) (degrees)

50

0

-50

-100
-40 -30 -20 -10 0 10 20 30 40
w (rad/s)

KESIMETRISAN DALAM GELOMBANG
PERIODIK

1. SIMETRIS GENAP
2. SIMETRIS GANJIL
3. SIMETRIS GELOMBANG SETENGAH

SIMETRI GELOMBANG
SETENGAH

Fourier3

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