b Finite energy signal

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(b) Finite energy signal
The area is simply
And the total energy
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Example A repetitive triangular waveform as
shown in the Figure below. (i) Find the area,
and total energy of one pulse of the waveform.
(ii) Find the mean, mean square and variance of
the repetitive waveform.
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(a)
(b)
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(ii) Frequency domain average
(a) Finite mean power signals
Assume that the filter passes a narrow band of
frequencies of bandwidth df , with a centre
frequency f , and the gain is unity.
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For a continuous random signal, the plot of
meter reading versus centre frequency f
would be similar to
0
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The power spectrum P(f), sometimes called power
spectral density, is the distribution of mean
power per unit bandwidth, and is obtained by
dividing the mean square power voltage by df.
The total mean power
is given by
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Example Repetitive pulses
The zero line is simply the squares of mean
value, i.e.
Total mean power is just the sum of the power Pn
in the lines
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Fourier series for the function f(t) can be
written in the following trigonometric form

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The following Matlab code rectangular.m
demonstrates that a rectangular wave can be
generated from a series of sine
wave clear clear work space clf clear
figure space ninput('how many sine
waves?(gt2)') number of sine wave to be used,
try changing n from small to large for better
approximation i1500 500 data
point ti/5 5 points per time
step a(1,i)2.510/pisin(pit/4) the first
sine wave plus a constant term (offset) for
j2n a(j,i)10/(2j-1)/pisin(pit(2j-1)/4) e
nd wave forms that you generated wave_recsum(a
) the rectangular wave is form by summing these
waves plot(t,wave_rec) xlabel('t') ylabel('wave
forms') hold on plot(t,a(1,),'r') the first
sine wave (shown as red) plot(t,a(2,),'g') the
second sine wave (shown as green)
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n corresponds to frequency. The amplitude versus
frequency is determined by sinc function. Power
spectrum takes square value of amplitude.
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(b) Finite energy signals
Finite energy signal (discrete) have no power
spectrum, but it can be represented by a
continuous energy spectrum.
Total energy
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A Summary of time and frequency domain average
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• Time and ensemble averages

(a) Time average, e. g.
(b) Ensemble average, e. g.
If applied to the same signal, both must give the
same value.
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Consider Johnson noise produced by a large number
of identical resistors
Ensemble average from complete set of resistors
at instant t.
Time average from any one resistor average over
a long time.
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Example Triangle wave as shown below. Find the
mean (shown here), mean square and variance using
time average and ensemble averages.
Time average
Ensemble average