0.4 Noise bandwidth

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

And

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

Minimizing noise in systems by filtering:

Assume white noise with $G_n\left(f\right)=\frac{\eta }{2}$

To minimize noise entering the demodulator, a filter of bandwidth B can be placed, with B just wide enough to pass signal of interest. Output noise depends on the filter used.

Ideal LPF with white noise has $N_o=\mathrm{\eta B}$

rectangular BPF with white noise has $N_o=2\frac{\eta }{2}\left(f_2-f_1\right)=\eta \left(f_2-f_1\right)$

Rc low pass filter:

The filter transfer function is

Using $G_{\text{no}}\left(f\right)={\mid H\left(f\right)\mid }^2{G}_{\text{ni}}\left(f\right)$

we have

and noise power at filter o/p is

using $x=\frac{f}{f_c}$ and noting that $\underset{-\infty }{\overset{\infty }{\int }}\frac{\text{dx}}{1+x^2}=\pi$ , we have

Differentiating filter:

 The transfer function is

white noise creates output psd

and following this by a rectangular lpf of bandwidth B, noise at o/p is

$N_o=\underset{-B}{\overset{B}{\int }}{G}_{\text{no}}\left(f\right)\text{df}=\frac{{\mathrm{4\pi }}^2}{3}{\text{ητ}}^2{B}^3$  

Integrating filter:

 An integrator integrating over an interval T has transfer function

and thus

The noise power o/p with white noise input

(The integral has a value =π)

Noise bandwidth:

If a real filter with transfer fn $H\left(f\right)$ centered at $f_o$ is used, we can consider an equivalent rectangular filter centered at $f_o$ with a bandwidth $B_N$ passing the same noise power.

• $B_N$ is called the noise bandwidth of the real filter

• For the RC Filtered curves, the area can be shown to be

• For a rectangular filter we have

• Setting $N_0(\text{RC})=N_0(\text{rect})$ , $B_N=\frac{\pi }{2}{f}_c$

Hence noise BW of RC filter is 1.57 times its 3 dB BW.