Use the symbolic toolbox of Matlab to calculate the energy of the signals x(t) and f(t). Verify theoretically. How is the energy of these two signals related?

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Aaron Connell 2016-10-16

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/307494-use-the-symbolic-toolbox-of-matlab-to-calculate-the-energy-of-the-signals-x-t-and-f-t-verify-the

评论： Karan Gill 2016-10-20

I have two piecewise defined functions, x(t) and a reversed and shifted version of x(t) as shown below...I am supposed to use the symbolic toolbox in Matlab to calculate the energy of these two signals and explain how they are related. Not sure what the symbolic toolbox is...is it using syms function? Code for both signals is below.

 % code
x = @(t) zeros(size(t)) +(t >= 0 & t < 2).*((1/2)*exp(t)) + (t >= 2 & t < 4).*(2*t-8);
t = linspace(0,6);
plot(t,x(t))
axis([-1 8 -4 4])
xlabel('Time(t)')
ylabel('x(t)')
title('Function x(t)')
% code
x = @(t) zeros(size(t)) +(t >= 0 & t < 2).*((1/2)*exp(t)) + (t >= 2 & t < 4).*(2*t-8);
f = @(t) x(-t-1);
t = linspace(-7, -1);
plot(t,f(t))
axis([-8 1 -4 4])
xlabel('Time(t)')
ylabel('f(t)');
title('Left shift and time reversal of x(t)')

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Answer 1

Walter Roberson 2016-10-16

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/307494-use-the-symbolic-toolbox-of-matlab-to-calculate-the-energy-of-the-signals-x-t-and-f-t-verify-the#answer_239198

在 MATLAB Online 中打开

That code I showed before,

syms x(t) shifted_x(t)
x(t) = evalin(symengine, 'piecewise([0 <= t and t < 2, exp(t)/2], [2 <= t and t < 4, 2*t-8], [t < 0 or t >= 4, 0])');
shifted_x(t) = x(-t-1);

is the Symbolic Toolbox implementation of a piecewise function. This is something that I would not expect very many people to know.

Possibly you are expected to implement in terms of heaviside functions and dirac delta

syms F(t)
F(t) = -exp(t)/2 * heaviside(-2 + t) + exp(t)/2 * heaviside(t) - (2 * t - 8) * heaviside(-4 + t) + (2 * t  -8 ) * heaviside(-2 + t) + dirac(t)*(1/2) + dirac(t-2)*(-4)

The value of heaviside at the point where the parameter is 0 is something that can be adjusted with sympref() from R2015a onwards. The issue here is that heaviside(t) is 0 for t < 0 and is 1 for t > 0, but what is it at exactly t = 0 ? The dirac() is there to give explicit values at the boundary conditions

According to https://www.mathworks.com/matlabcentral/newsreader/view_thread/320985

the energy of a signal F is var(F)

In turn, var(F(t)) is E(F(t)^2)-E(F(t))^2 where E standards for expected value. The expected value of F(t) is int(t*F(t), t, -inf, inf)

You can now put those formulas together with heaviside version of your piecewise functions to come up with a value for their energies. F(t) and F(-1-t) are the formula...