1

Please add comments line for matlab and show all step in paper, ASAP. Thanks in advance....

Question

Please add comments line for matlab and show all step in paper, ASAP. Thanks in advance....

Please add comments line for matlab and show all step in paper, ASAP0.24 Hyperbolic sinusoids-In filter design you will be asked to use hyper bolic functions. In this problem we relate these fu. Thanks in advance.

0.24 Hyperbolic sinusoids-In filter design you will be asked to use hyper bolic functions. In this problem we relate these functions to sinusoids and obtain a definition of these functions so that we can actually plot them (a) Consider computing the cosine of an imaginary number, i.e., use cos(x) 2 let j and find cos(x). The resulting function is called the hyper- bolic cosine or cos(je) = cosh(0) b) Consider then the computation of the hyperbolic sine sinh(0), how would you do it? Carefully plot it as a function of e. (c) Show that the hyperbolic cosine is always positive and bigger than 1 for all values of 0 d) Show that sinh(e)-sinh(-0) e) Write a MATLAB script to compute and plot these functions between -10 and 10 Answers: sinh(0) -j sin(je) and odd.

Answers

Question (a)

We know that

ejx +e-jx cos(x) = ?

Let

x= je

So

=(!)so! !!-a + arxra

= ()s05 + - 0.

cos(je) = cosh(O)

So

e-e tee cosh(O) = 2

Question (b)

We know that

لازم - لازم sin(x) =

Let

x= je

So

ejxje-e-jxje sin(jo) = j2

e-o-e sin(je) = = 12

j sin(je) =

-1 sin(jo) – (- -j sin(je) =

-j sin(je) = sinh(0)

So

sinh(0) = eº - e-º

Question (c)

e-e tee cosh(O) = 2

+ cosh(O) = 2

cocco 1+ 2e cosh(0) = 2e

1 cosh(O) = 300+1

This means that cosh(O) will be always greater than 1.

Question (d)

We know

sinh(0) = eº - e-º

So

sinh(-) = 2

sinh(-) = -sinh(0)

sinh(0) = - sinh(-)

Hence proved.

Question (e)

MATLAB Code

clc;
clear all;
close all;


th = -10:0.00001:10; % The range of theta for which the function is to be plotted
y1 = (exp(th) + exp(-th))/2; % Evaluating hyperpolic cosine
y2 = (exp(th) - exp(-th))/2; % Evaluating hyperpolic sine


plot(th,y1,'linewidth',2); % Plotting hyperpolic cosine
grid
xlabel('\theta');
ylabel('cosh(\theta)');
title('The hyperbolic cosine function');

figure
plot(th,y2,'linewidth',2); % Plotting hyperpolic sine
grid
xlabel('\theta');
ylabel('sinh(\theta)');
title('The hyperbolic sine function');

After executing, we get

The hyperbolic cosine function 12000 10000 8000 cosho) 4000 2000 H -10 0

104 The hyperbolic sine function 15 sinh(0)


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