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## (b) Finite energy signal

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**(b) Finite energy signal**The area is simply And the total energy**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.**(a)**(b)**(ii) Frequency domain average**(a) Finite mean power signals Assume that the filter passes a narrow band of frequencies of bandwidth δf , with a centre frequency f , and the gain is unity. 0 For a continuous random signal, the plot of meter reading versus centre frequency f would be similar to 0**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 δf. The total mean power is given by**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**Fourier series for the function f(t) can be written in the**following trigonometric form:**%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; n=input('how many sine waves?(>2)'); %number of sine wave to be used, try changing n from small to large for better approximation; i=1:500; %500 data point; t=i/5; %5 points per time step; a(1,i)=2.5+10/pi*sin(pi*t/4); % the first sine wave plus a constant term (offset); for j=2:n; a(j,i)=10/(2*j-1)/pi*sin(pi*t*(2*j-1)/4); end; % wave forms that you generated wave_rec=sum(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);**n corresponds to frequency. The amplitude versus frequency**is determined by sinc function. Power spectrum takes square value of amplitude.**(b) Finite energy signals**Finite energy signal (discrete) have no power spectrum, but it can be represented by a continuous energy spectrum. Total energy**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.**Consider Johnson noise produced by a large number of**identical resistors Ensemble average from complete set of resistors at instant τ. Time average from any one resistor average over a long time.**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: