Concept:

Convolution of a signal x(t) with unit impulse δ(t) is the signal itself. i.e. x(t) ⊕ δ(t) = x(t) Fourier transform of auto-correlation function of a power signal x(t) is power spectral density Sx(f). i.e. ({R_X}left( tau right)mathop leftrightarrow limits^{FT} {S_X}left( f right))

And E(x2 (t)) = RX (0)

The variance of the signal x(t) is defined as:

(varleft( {xleft( t right)} right) = Eleft( {{x^2}left( t right)} right) - (E{left( {xleft( t right)} right)^2})

Fourier transform of unit impulse is 1.

(delta left( t right)mathop leftrightarrow limits^{FT} 1)

Calculation:

Let n(t) be the input white noise with zero mean and (frac{{{N_0}}}{2}) power spectral density.

Mean of the white noise = E(n(t)) = 0

Power spectral density is:

({S_n}left( f right) = frac{{{N_0}}}{2}) ;

And the auto-correlation function is:

({R_n}left( tau right)mathop leftrightarrow limits^{FT} {S_n}left( f right))

(frac{{{N_0}}}{2}mathop to limits^{IFT} frac{{{N_0}}}{2}delta left( t right))

({R_n}left( tau right) = frac{{{N_0}}}{2}delta left( t right))

Let yn(t) is the output noise.

## Match Filter Theory

Mean of the output noise:

( = Eleft( {{y_n}left( t right)} right) = Eleft( {nleft( t right) times mathop smallint nolimits_{ - infty }^infty hleft( t right)dt} right))

( = Eleft( {nleft( t right)} right) times mathop smallint nolimits_{ - infty }^infty hleft( t right)dt)

( = 0 times mathop smallint nolimits_{ - infty }^infty hleft( t right)dt = 0)

The variance of the output noise is:

(Varleft( {{y_n}left( t right)} right) = Eleft( {y_n^2left( t right)} right) - (E{left( {{y_n}left( t right)} right)^2})

( = Eleft( {y_n^2left( t right)} right))

(Eleft( {y_n^2left( t right)} right) = {R_{{y_n}}}left( 0 right)) ({R_{{y_n}}}left( tau right) = hleft( tau right)*{h^*}left( { - tau } right)*{R_n}left( tau right))