Physics and Engineering of Radiation Detection

512 Chapter 8. Signal Processing

average value (see Fig.8.9.1(a)). These fluctuations are due to the thermal noise of
the conductor. The question is, if the average voltage is zero, then why we should
care about these fluctuations. The answer lies in the fact that it is actually the
signal power that determines the information content of the signal. Since power is
proportional to the square of the voltage (V^2 ), therefore in this case it will have a
value greater than zero whenever the actual voltage fluctuates from its average value
of zero (see Fig.8.9.1(b)). Let us now see how this noise can be quantified.

0

V

or

I

Time

Time

Power

0

(a)

(b)

Figure 8.9.1: (a) Typical variation of volt-

age or current in a conductor not connected

to any current source. The average voltage

or current is zero. (b) Power, which is pro-

portional to square of voltage or current is

non-zero even in the absence of a current

source.

Johnson noise is usually represented by spectral noise power densityp^2 j,whichis
simply the noise power per unit frequency bandwidth. This quantity has been found
to be proportional to the absolute temperatureT.

p^2 j≡

dPnoise

df

=4kBT (8.9.2)

HerekB is the Boltzmann’s constant. This can also be represented in terms of
current and voltage by noting that the power carried by a currentIin the presence
of a voltageV is given by

P=VI=I^2 R=

V^2

R

, (8.9.3)

where we have used the Ohm’s lawV =IRfor a conductor having resistanceR.
Hence the spectral noise current density can be obtained by substituting this in the
above definition ofpjas

i^2 j≡

dInoise^2

df

=

4 kBT

R

. (8.9.4)

Similarly the spectral noise voltage density is given by

v^2 j≡

dVnoise^2

df

=4kBTR. (8.9.5)