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The derivative of the Dirac delta impulse is written as $\delta'(\tau)$. This helps with notation because the mistake you made is to write $h(-\tau)=\frac{d}{d\tau}\delta(-\tau)$, which is not the case because $\delta(\tau)$ is an even (generalized) function, whereas the derivative operator $\delta'(\tau)$ is an odd (generalized) function: $$\delta'(\tau)=-\...


2

If the $x[n]$ are i.i.d. random variables with mean $\mu$ and variance $\sigma^2$, then the power spectral density of this discrete-time random process (which is effectively white noise plus a possibly nonzero mean $\mu$) is what you have calculated. The shape of the common pdf of the random variables is irrelevant in all this except insofar as the shape ...


1

The autocorrelation of $x(t)$ is $$r_x(t)=x(t)\star x(-t)\tag{1}$$ where $\star$ denotes convolution. Taking the Fourier transform of $(1)$ gives $$S_x(\omega)=X(\omega)X^*(\omega)=|X(\omega)|^2\tag{2}$$ $S_x(\omega)$ is the energy density of $x(t)$, and according to $(2)$ it equals the squared magnitude of the Fourier transform of $x(t)$. So if $x(t)$ is ...


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