# Condition for modelling noise as white noise

I am reading a book on Communication System by Simon Haykin. In the topic of white noise there is a paragraph

"We may state that as long as the bandwidth of noise process at input of a system is appreciably larger than the bandwidth of the system itself we may model the noise as white noise".

1) What is the reason for this statement?

Further I also read about Noise Equivalent Bandwidth, in which we replace an arbitrary low pass filter with transfer function H(f) with an ideal low pass filter of constant frequency response H(0) and bandwidth B.

2) Is the reason for doing so is to make sure that bandwidth of noise process at input of a system is appreciably larger than the bandwidth of the system itself we may model the noise as white noise ?

3) Is there any other conditions to satisfy before modelling the noise as white noise?

Thank you in advance for your help.

I will try to answer your three questions as a whole, with the following insights:

"Whiteness" means: flat power spectral density, thus zero autocorrelation for lag different than zero (uncorrelated samples), thus wide sense stationary (time-independent autocorrelation and mean).

White Gaussian noise passing through a linear filter always produces Gaussian noise, only changing its parameters (mean and variance), since any linear combination of uncorrelated Gaussian random variables produces a Gaussian random variable. However, the whiteness is lost in general, and thus samples become correlated after passing a linear filter.

If your system's transfer function has a bandwidth in $\omega$ significantly lower than the noise's bandwidth, you may approximate the input noise as white, since what the system "sees" is a flat spectrum.

Anyway, you must take into account that white noise is always an approximation for real-life systems. There is no such thing as a noise with infinite bandwidth in practice. But you may be in presence of noise with very large bandwidth (compared to your system), and thus successive samples of such a noise are (nearly) uncorrelated.

• Thanks for the insights. What about the 2nd question about Noise equivalent bandwidth in which we replace the LPF with a LPF with zero frequency response H(0) ? – user1825567 Apr 18 '18 at 14:02
• I don't understand what you mean by "an ideal low pass filter of zero frequency response H(0)". Literally, this means $H(\omega) = 0$ for every $\omega$, which makes no sense. Do you mean "an ideal low pass filter with a constant frequency response of H(0)"? – Luis M Gato Apr 18 '18 at 14:15
• Yes you are right. It means that the frequency response is constant with value H(0) for frequencies -B to B (B is the noise equivalent bandwidth) and 0 elsewhere. I am sorry for the confusion, I just used those confusing technical terms just as it is from the book. – user1825567 Apr 18 '18 at 20:35