Editing Bandwidth (signal processing) (section)

== Noise equivalent bandwidth ==
[[File:Enbw.svg|right|300px|thumb|Setup for the measurement of the noise equivalent bandwidth <math>B_n</math> of the system with frequency response <math>H(f)</math>.]] 
{{further|Spectral leakage#Noise bandwidth}}

The '''noise equivalent bandwidth''' (or '''equivalent noise bandwidth (enbw)''') of a system of [[frequency response]] <math>H(f)</math> is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing <math>H(f)</math> when both systems are excited with a [[white noise]] source.
The value of the noise equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals <math>|H(f)|</math> at its center frequency,<ref name="Jeruchim">{{cite book |last1=Jeruchim |first1=M. C.|last2=Balaban |first2=P.|last3=Shanmugan |first3=K. S. |title=Simulation of Communication Systems. Modeling, Methodology, and Techniques. |publisher=Kluwer Academic |isbn=0-306-46267-2 |edition=2nd|date=2000}}</ref> but it can also equal the peak value of <math>|H(f)|</math>. 

The noise equivalent bandwidth <math>B_n</math> can be calculated in the frequency domain using <math>H(f)</math> or in the time domain by exploiting the [[Parseval's theorem]] with the system [[impulse response]] <math>h(t)</math>.  If <math>H(f)</math> is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then: 

<math display="block"> B_n = \frac{\int_{-\infty}^{\infty} |H(f)|^2 df}{2|H(0)|^2} =  \frac{\int_{-\infty}^{\infty} |h(t)|^2 dt}{2\left|\int_{-\infty}^{\infty} h(t)dt\right|^2} \, . </math>

The same expression can be applied to bandpass systems by substituting the [[Baseband#Equivalent baseband signal|equivalent baseband]] frequency response for <math>H(f)</math>. 

The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.