Question

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

Answer 1

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 will be always greater than 1.

cosh(O)

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)