0.2 Filtering of noise

We will see the effects on the power spectrum when noise is passed through a filter.

Filtered output:

If $h(t)$ is the impulse response, filtered output $n_o(t)$ obtained by convolution.

• Approximate continuous time by discrete interval.

• In the interval impulse response is deterministic and known, but $n_i$ depends on sample chosen,
• For white Gaussian noise $n_i$ in interval k is different from and independent of that in interval l.
• Thus output at any time $t_0$ is a linear superposition of independent Gaussian variables: hence it is Gaussian.
• If we split filter $H(f)$ into $H_1(f)$ and $H_2(f)$ , output of $H_1(f)$ is Gaussian but NOT white (due to filtering)

• Hence output $n_o$ of $H_2(f)$ is also non-white.
• output $n_o^{\text{'}}$ of $H_1(f)$ is non-white, thus $n_o^{\text{'}}$ in intervals k and l are dependent, but despite of this $n_o$ is Gaussian.
• Hence superposition of non-white dependent Gaussian variables also gives Gaussian output.

 Spectral components:

• Let spectral component k be

• Then though coefficients are random over ensemble, they are deterministic for a specific case and depend on the particular sample chosen.
• Hence spectral component is stationary but not ergodic.
• Its normalized power is

• As $n_k(t)$ is stationary, t can be any time. Choose t as $t_1$ when cosine is 1, sine is 0. Then

and similarly

Since

• Then $p_k$ can be written as showing a constant term and a sin-cos time dependent term.

thus

• But $p_k$ is stationary and independent of time, so

Hence the coefficients are seen to be uncorrelated.

• Also at $t_1$ , we have $n_k(t_1)=a_k$ . But $n_k$ is Gaussian as it can be considered output of a narrowband filter whose input is Gaussian.

Thus $a_k$ and $b_k$ are Gaussian.

• Also $a_k$ is output of filter in a non zero frequency interval k, it is not DC, thus mean value of $a_k=0$ . Hence coefficients are Zero-mean Gaussian.
• Taking product of two samples $n_k(t)$ and $n_l(t)$ where

The ensemble averages of the result must be independent of time as each component is stationary, hence

• Hence coefficients at a given frequency are uncorrelated, and also coefficients at different frequencies are uncorrelated.

Summary of noise characteristics:

Random, Gaussian, Ergodic, Linear superposition of random spectral components with coefficients which are Gaussian, zero-mean, with variances related to power spectral density and uncorrelated with each other and with other coefficients at different frequencies.