(New page: Correction: (A) We first change the form of y[n] that we are given by separating the exponentials and converting the complex exponential using Euler's method. <math> \begin{align} y[n...) |
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y(t) &= \int_{-\infty}^{t} \delta(t + 2) - \delta(t - 2) dt = u(t + 2) - u(t - 2) | y(t) &= \int_{-\infty}^{t} \delta(t + 2) - \delta(t - 2) dt = u(t + 2) - u(t - 2) | ||
\\ so | \\ so | ||
| − | \\ E\infty &= \int_{-\infty}^{\infty} [u(t + 2) - u(t - 2)] dt\\ | + | \\ E\infty &= \int_{-\infty}^{\infty} |[u(t + 2) - u(t - 2)]|^2 dt\\ |
&= \int_{-2}^{2} |1|^2 dt \\ | &= \int_{-2}^{2} |1|^2 dt \\ | ||
&= 2 - (-2) = 4 | &= 2 - (-2) = 4 | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 18:53, 2 July 2008
(A)
We first change the form of y[n] that we are given by separating the exponentials and converting the complex exponential using Euler's method.
$ \begin{align} y[n] &= e^{2 + j4.72135n} = e^{2}e^{j4.72135n}\\ &= e^{2}[cos(4.72135n) + jsin(4.72135n)]\\ \end{align} $
Next, we use the equation for energy in discrete time. Finding the magnitude of y[n], we see that cos^2 + sin^2 = 1 Therefore the energy for infinite interval is the infinite sum of $ e^{4} $ which = $ \infty $
$ \begin{align} \\ E\infty &= \Sigma_{n=-\infty}^{\infty}|y[n]|^2\\ &= \Sigma_{n=-\infty}^{\infty}\sqrt{(e^{2})^2[cos^2(4.72135n) + jsin^2(4.72135n)]}^2\\ &= \Sigma_{n=-\infty}^{\infty}\sqrt{e^4}^2 = \Sigma_{n=-\infty}^{\infty} e^{4} = \infty \end{align} $
(B)
We start with part B by noticing that the integral of the delta function is a step function. So the energy over an infinite interval is just the integral of the step function $ u(t + 2) - u(t - 2) $
$ \begin{align} y(t) &= \int_{-\infty}^{t} \delta(t + 2) - \delta(t - 2) dt = u(t + 2) - u(t - 2) \\ so \\ E\infty &= \int_{-\infty}^{\infty} |[u(t + 2) - u(t - 2)]|^2 dt\\ &= \int_{-2}^{2} |1|^2 dt \\ &= 2 - (-2) = 4 \end{align} $
