Page 245 - Organic Electronics in Sensors and Biotechnology

222 Cha pte r S i x

ΔP = ℘Δf

℘(f )

Δf f
FIGURE 6.16 The noise spectral density for a white noise source. The power
is distributed evenly over all frequencies up to some (extremely high)
maximum cutoff frequency; if the amplitude of the power spectrum (known
as the power spectral density) is ℘, then the power ΔP contained within a
frequency range Δf is given by ΔP = ℘Δf.

for I . The electrons inside any resistor undergo random motion
Vphoto
as they are constantly buffeted backward and forward by atomic col-
lisions, leading (in short circuit) to an overall current of zero. This
motion results in a constantly changing charge distribution and so
induces a randomly fluctuating voltage across the terminals of the
resistor, known as thermal or Johnson noise. A process that is random
in time is evenly distributed in the frequency domain. It follows that
†
thermal noise is “white” in its characteristics; i.e., the time-averaged
power per unit frequency, known as the power spectral density ℘( f ),
‡
is the same at all frequencies (Fig. 6.16). The power P dissipated by a
2
resistor is related to its voltage V by P = V /R. Hence, we can write for
the mean squared voltage
V 2 = R P = R℘() Δ f = σ 2 f Δ (6.32)
f
V
2
where σ = R℘ ( f ) is the voltage variance per unit frequency and Δf is
V
the frequency range of the measurement. Since ℘ ( f ) is constant for a
2
white noise source, so too is σ , and the mean squared voltage is
V
therefore directly proportional to the bandwidth Δf. Note, in this chap-
ter, we will use tilde (~) notation to indicate bandwidth-corrected
2
parameters. The voltage variance σ = V 2 has units V / Hz (volts
2
V
squared per hertz) whereas the corresponding standard deviation σ
V
has units V/ Hz (volts per square-root hertz).
† Or, more precisely, one whose autocorrelation is a delta function.
‡ In reality, this can be true only up to some maximum cutoff frequency, or else the
integrated power would be infinite. In practice, though, the cutoff frequency is
sufficiently high as to be of no practical relevance to our discussion.