Biosignal Processing - Introduction to Biomedical Engineering - page 685

Combining results

1

5
2 þ

10
m p cos

x ð t Þ¼ ð m o o t þ f m Þ

m ¼ 1

where f m is as just defined. An interesting point regards the similarity of standard and compact
versions of the Fourier series for this square wave example. In the standard form, the coefficient
a m alternates between positive and negative values, while for the compact form the Fourier coef-
ficient, or -) of

the standard Fourier coefficient is now consumed in the phase term, which alternates between
0 and p . This forces the cosine to alternate in its external sign because

A m , is identical in magnitude to a m , but it is always a positive quantity. The sign ( þ

.
The two equations are therefore mathematically identical, differing only in the way that the trigo-
nometric functions are written out.

cos ð x Þ¼ cos ð x þ pÞ

MATLAB implementation:

%Plotting Fourier Series Approximation
time

¼

-2:0.01:2; %Time Axis

¼

x 5/2; %Initializing Signal

¼

m 1:10;

¼

A (10*sin(m*pi/2)./m/pi); %Fourier Coefficients

¼

P angle(A); %Phase Angle

¼

A abs(A); %Fourier Magnitude

for m

¼

1:10

¼ þ þ

x x A(m)*cos(m*pi*time P(m));

end
subplot(211)
plot(time,x,'k') %Plotting and Labels
xlabel('Time (sec)')
ylabel('Amplitude')
set(gca,'Xtick',[-2:2])
set(gca,'Ytick',[0 5])
set(gca,'Box','off')

%Plotting Fourier Magnitudes
subplot(212)
m

¼

1:10;

¼

A [5/2 A]; %Fourier Magnitudes

Faxis

¼

(0:10)*.5; %Frequency Axis
plot(Faxis,A,'k.') %Plotting
axis([0 5 -2 4])
set(gca,'Box','off')
xlabel('Frequency (Hz)')
ylabe l ('Fourier Amplitudes')

The results are identical to those shown in Figures 11.8a and b.