Help please: x(T)= -1, t<0 0, t=0 1, t>0 system out: y(t)= u(t)(1-2(e^-t)) -2u(t)e^(-t)-u(-t) a) Fourier transform x(t) b) Fourier transform y(t)

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Help please: x(T)= -1, t<0 0, t=0 1, t>0 system out: y(t)= u(t)(1-2(e^-t)) -2u(t)e^(-t)-u(-t) a) Fourier transform x(t) b) Fourier transform y(t)

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fourier transform of x(t) is \[\int\limits_{-\infty}^{\infty} x(t) e ^{j \omega t} \delta t\]
so, \[\int\limits_{-\infty}^{\infty}xdx = \int\limits_{-\infty}^{0-} xdx+\int\limits_{0-}^{0+}xdx+\int\limits_{o+}^{\infty}\]
oops the last term is\[\int\limits_{0+}^{\infty}xdx\]

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so, substituting, we get, \[\int\limits_{-\infty}^{+\infty}x(t)e ^{j \omega t} \delta t = \int\limits_{-\infty}^{0-}-1e^{j \omega t} \delta t +0+\int\limits_{0+}^{\infty}e^{j \omega t} \delta t \]
\[=\int\limits_{0+}^{\infty} -1/e^{j \omega t} \delta t + \int\limits_{0+}^{\infty} e^{j \omega t} \delta t \]
got it?
yes, thanks, I think that I understand. I'm looking for the impulse response can u help me with that?
the second one you mean
i'm looking at that. hold on.
remind me again, u(t) = 1 for t = 0 and u(t) = 0 for all other values. Am i right?
yes
so y(t) = u(t)(1-2(e^-t)) -2u(t)e^(-t)-u(-t) = u(t) - 2u(t)e^-t - 2u(t)e^-t-u(-t) = u(t)-u(-t)-4u(t)e^-t
is u(-t) = -u(t)?
no it isn't.
don't bother. it is not -u(t)
so, fourier transform of y(t) is \[\int\limits_{0-}^{0+}u(t)-u(-t)-4e ^{-t} e ^{j \omega t } \delta t\]
notice the limits
nice! agree
now, u(t) = 1 for t = 0+ or t = 0- so the u(t) terms cancel out. (i am not sure about this. ask some one else about this)
what did you get as the fourier transform of y?
yes
impulse response and input x output y , how relate
impulse response is the response of a function at t = 0
or rather, when a brief signal, or impulse is input, the output that you get from the system is called impulse response
ok directly from the previous limit
i am not sure, but I think the fourier transform of the impulse response is 0.
One more question please: If we enter another signal to the system and
Y{2t}=h(t)*X2{t}
\[\sum_{-\infty}^{+\infty} C1e ^{jl2t} * h(t)
\[\sum_{-\infty}^{+\infty} C1e ^{jl2t} * h(t) \]
i am sorry. Post it again tomorrow. I am off to bed :)
ok dont problem thanks 4 erything

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