What is the relationship between bandwidth and noise bandwidth

What is the mathematical formulae for noise bandwidth for a 1st order low pass system (and a second order LP systems analysis if possible). What is the importance of this concept, how and when is it applied? A nice graph showing the noise BW and the BW and mathematical derivations would be excellent (for both 1st and 2nd order systems). Extensions to bandpass filters would be helpful too.

• Are you interested in the thermal noise stuff and that for a given bandwidth, thermal noise is x and that with a tighter filter, the thermal noise will be some fraction of x? – Andy aka May 22 '13 at 14:24
• @Andyaka question edited for clarity, there is no mention on the site so far concerning this fundamental noise concept. This is more than about the fact that if you decrease the BW the noise decreases, it's about how to calculate that noise properly. There are necessary constraints on the characteristics of the noise but it need not be thermal. – placeholder May 22 '13 at 14:41
• Other than thermal, are you considering that shot noise is what you want in the answer? – Andy aka May 22 '13 at 16:31
• @Andyaka That's entirely up to you. Although a good answer will allow the reader to understand when the concept is applicable. I will answer the question if there isn't a good one forth coming but I'll give it a couple of days so see what happens. Provide your answer based upon your knowledge/research as you see fit I don't want to lead the witness ... – placeholder May 22 '13 at 16:54

The noise bandwidth $B_N$ of a (linear time-invariant) system is defined as the bandwidth which an ideal filter with a rectangular frequency response would need to have to get the same noise power at the output, given that the input noise to both systems is identical and white. The ideal filter is usually assumed to have the same maximum gain as the system under consideration. From this definition, it follows that the noise bandwidth is given by

$$B_N=\frac{1}{H_{max}^2}\int_{0}^{\infty}|H(f)|^2\;df\tag{1}$$

where $H(f)$ is the system's frequency response and

$$H_{max}^2=\max_{f}|H(f)|^2$$

Definition (1) is valid irrespective of the specific characteristics of $H(f)$. So it is valid for lowpass systems as well as for bandpass systems, or other types of filters.

For a simple first order lowpass system (e.g. an $RC$ lowpass), we have

$$H(f)=\frac{1}{1+jf/f_c}$$ with $f_c$ the -3dB cut-off frequency. From (1) we get for the noise bandwidth

$$B_N=\int_{0}^{\infty}\frac{1}{1+(f/f_c)^2}df= f_c\int_{0}^{\infty}\frac{1}{1+x^2}dx= f_c\arctan(x)|_{0}^{\infty}=f_c\frac{\pi}{2}$$

In this case we see that the noise bandwidth is larger than the -3dB frequency by a factor of $\pi/2$.

For a second order system everything is similar, but slightly more complex. The integral gets a bit more involved and the value of $H^2_{max}$ needs to be determined, because there can be overshoot in the frequency response, depending on the damping. If I have more time later on I might add details about second order systems. For the time being I hope that the answer is clear enough so that everybody can derive the noise bandwidth of any system they are interested in.