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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 ...