Problem 1.6 Determine an expression for an exponentially decaying sinusoid that

Question

Problem 1.6 Determine an expression for an exponentially decaying sinusoid that oscillates three times per second and whose amplitude decreases by 50% every 2 seconds. Use MATLAb to plot the signal over-2 a) t 2 b) When a bel is struck with a mallet, t produces a ringing sound. Write an equation that approximates the sound produced by a small i.e., light) bell. Your equation should include the amplitude of the sound wave. Does amplitude remain constant over time? Justify your answer. In addition, you should be able to hear the sound from this bell: sound is a wave in 20 Hz - 20 kHz frequency range c) You probably know that smaller bell will produce higher frequency sound compared to the larger and heavier bell. Also, larger/heavier bell will produce a louder sound (i.e., higher amplitude). Now let write an equation for the larger/heavier bell.

Explanation / Answer

Here, Starting from the First Part we have,

a. here, 3 cycles per second so f=3 Hz

w=2*pi*3=6pi

So the Sinusoid expression is sin(wt) = sin(6t)

Now the decaying function is given as:-

exp(-at) where a is calculated as

at t=0 the amplitude is exp(-0) = 1

and t=2 it is 0.5 so

0.5=exp(-a*2)

-0.693=-2a

a=0.346

So the total signal expression is

sin(6t)*exp(-0.346t) (Put the time between -2 to 2 for MATLAB)

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B.  

Let the equation be 1.0 x sin(100t) x cos(100t) x exp(-0.5t)

Use the following matlab code:

clear all;
clc;
t=linspace(1,10,10000);

n = size(t,2);

for(i = 1:n)
      y(i) = 1.0*sin(100*t(i))*cos(100*t(i))*exp(-0.5*t(i));
end

sound(y)

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