0.3 Filtering of noise - 2

Psd after filtering:

The relation between a ,b and c and $\phi$ which describe the noise components can be seen to be identical with that between X,Y and R and $\theta$ .

Hence pdf of c is Rayleigh and that of $\theta$ is uniform.

$f\left(c_k\right)=\frac{c_k}{P_k}{e}^{-c_k^2/{\mathrm{2P}}_k}{c}_{k}0$ ,

Let a spectral component of noise be the input to a filter whose transfer function at frequency $\mathrm{k\Delta f}$ is

The output spectral component of noise is

The power associated with the input component is

As $\mid H\left(\mathrm{k\Delta f}\right)\mid$ is a deterministic function, $\overline{{\left[\mid H\left(\mathrm{k\Delta f}\right)\mid a_k\right]}^2}={\mid H\left(\mathrm{k\Delta f}\right)\mid }^2\overline{a_k^2}$

Similarly for $b_k$ , and thus the power associated with noise output is

And the power spectral densities are related by

Where the $\mathrm{k\Delta f}$ has been replaced by $f$ as a continuous variable as $\mathrm{\Delta f}$ tends to 0.

Superposition of noises:

 Noise can be represented as superposition of (orthogonal) harmonics of $\mathrm{\Delta f}$ therefore total power is the result of superposition of component powers.

Consider Two processes $n_1$ and $n_2$ with overlapping spectral components.

Power of the sum of $n_1$ and $n_2$ will be $p_1+p_2+2E\left[n_1{n}_2\right]$ and since $n_1$ and $n_2$ are uncorrelated, the last term = 0.

Then these noises also obey the superposition of powers rule.

Mixing of noise with a sinusoid

 If $k^{\text{th}}$ component of noise is mixed with a sinusoid

Sum and difference frequency noise spectral components with 1/2 amplitude are generated and

Considering power spectral components at $\mathrm{k\Delta f}$ and $\mathrm{l\Delta f}$ , let the mixing frequency be $f_0=\left(k+l\right)\mathrm{\Delta f}$ . This will generate 2 difference frequency components at the same frequency: $\mathrm{p\Delta f}=f_0-\mathrm{k\Delta f}=\mathrm{l\Delta f}-f_0$

Then difference frequency components are

But as $\overline{a_k{a}_l}=\overline{a_k{b}_l}=\overline{b_k{a}_l}=\overline{b_k{b}_l}=0$ , We find $E\left[n_{\mathrm{p1}}\left(t\right)n_{\mathrm{p2}}\left(t\right)\right]=0$

and $E\left\{{\left[n_{\mathrm{p1}}\left(t\right)+n_{\mathrm{p2}}\left(t\right)\right]}^2\right\}=E\left\{{\left[n_{\mathrm{p1}}\left(t\right)\right]}^2\right\}+E\left\{{\left[n_{\mathrm{p2}}\left(t\right)\right]}^2\right\}$

Thus superposition of power applies even after shifting due to mixing.