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Updated January 26, 2011.  See Document History at end for details.

## Simple Noise Calculations

#### Some Theory

Contrary to adding signal voltage or current, noise adds by power.  I would attempt to explain why by the following line of reasoning.  Consider that noise is random.  It would be random as to its magnitude and phase.  There is no trouble adding random voltage magnitudes directly.  The random phase is what adds the additional difficulty.  Consider adding a signal varying -180° through 180° with respect to the signal to which it is added.  According to the rules of trignonometry the result will change with the angle.  Symmetry between positive and negative angles would simplify the argument by allowing consideration of only the positive range 0 through 180° (Positive and negative angles produce the same magnitude result in this vector addition).  Now we can see that the average vector angle for noise voltage or current addtion is 90°.  Thus noise voltage or current adds by root-sum-square.  In the case of only two operands, this is the familiar Pythagorean theorem.  Thus
(1)
 vn =
 vn12 + vn22 + vn32 + ...
 and, in =
 in12 + in22 + in32 + ...
Noise power is easily seen to add directly by substituting the power formulas below in the above equations and reducing (you can verify this yourself):
 (2) P = V2 R = I2R
The root-sum-square operation is associative and commutative.  That is:
This allows you to build on your calculations without excessive concern about order.

#### Resistor Noise

White resistor noise is the most common source of noise.  It is caused by the random motion of atoms associated with heat.1
(4)
 vnr =

 4kTRB

 and, inr =

 4kTB R

 Where k = 1.3806504 x 10-23 (Boltzmann constant) T = Kelvin temperature Conversions:  K° = C° + 273.15, C° = (F°-32) x 5/9 R = Resistance (Ohms) B = Bandwidth of system (Hz)

#### Noise Bandwidth

In some cases, frequency response alters the effective noise bandwidth of a noise calculation.
In the case that noise is filtered by a first order RC lowpass response the noise bandwidth is higher than the 3dB point (fo).2
 (5) Bnoise-lowpass = 1 4RC = π 2 fo
This is because noise in the stopband still contributes to the overall noise up to the useful upper limit of the system.

Substituting 1/(4RC) for BW in the resistor noise equation (4) and reducing gives the capacitor noise equation.3
(6)
 vn =

 kT C
The capacitor is band limiting the resistor noise here in a manner independent of the resistance.  The change in bandwidth due to the change in resistance is exactly canceled by an opposite change in its effect on noise power per unit bandwidth.  If the pole related to this calculation is above the useful upper frequency limit of the system (as the 20kHz upper limit of an audio system) it would be better to calculate using that limit rather than use the capacitor noise equation.

Bandwidth is often separated out of the noise calculation to be added in later to simplify them or to specify noise where the bandwidth is not yet known.
(7)
 vnr =
 4kTRB
 =
 4kTR
 B
 The noise specification separated from bandwidth is in units of V/√Hz or A/√Hz.

#### RIAA Noise Bandwidth by Numerical Integral

RIAA networks alter the bandwidth of their own resistance and the noise coming into them in a somewhat more complicated way.  First, I tried to solve for RIAA noise bandwidth by utilizing the capacitor noise equation and obtained an implausable result of 313.3Hz.  (This would be the result if calculated from a 212.2Hz first-order lowpass response by equation (5) and subtracted the subsonic 20Hz.)  I could not rest until I resolved the issue.  As a result, I used a textbook numerical integral4 to obtain the correct RIAA noise bandwidth.  With near infinite system bandwidth (0-1MHz), I obtained 109.325Hz.  Within a system bandwidth of 20-20kHz, I obtained a noise bandwidth of 88.0903Hz.  I verified the integration algorithm first by calculating the noise bandwidth of a first order lowpass response and comparing to the result obtained from equation (5) above.  The correct result is very close to an earlier calculation based on my mathematical intuition, which I discarded in favor of one based on a textbook equation.  :-(

#### C Code Listing

This program verified to work with the Linux gcc and Microsoft Visual C++ 6.0 compilers.

/* nbwcons.c
Calculate Noise Bandwidth by Numerical Integral
(Trapezoidal Approximation)
*/

#include <stdio.h>
#include <math.h>

/* define integral parameters in Hz */
#define INCREMENT 0.1
#define LOWERBND 20.0
#define UPPERBND 20000.0
#define twopi 6.283185307
/* time constants */
#define pole1  3180.0e-6
#define pole2  75.0e-6
#define zero1  318.0e-6

int main()
{
int i,count;
double increment,lowerbnd,upperbnd;
double omega,interval,value,bw;

/* convert to radian frequency */
increment = INCREMENT * twopi;
lowerbnd = LOWERBND * twopi;
upperbnd = UPPERBND * twopi;
/* calculate integral */
bw = 0.0;
count = (int)(((upperbnd-lowerbnd)/increment)+0.5);
for(omega = lowerbnd,i = 0; i <= count; omega += increment, i++)
{
/* set up trapezoidal intervals */
if(i > 0 && i < count)
interval = increment;
else
interval = increment*0.5;
/* execute squared normalized magnitude function (RIAA, no gain) */
value = (1+(omega*omega*zero1*zero1))
/((1+(omega*omega*pole1*pole1))*(1+(omega*omega*pole2*pole2)));
/* multiply by interval and accumulate bandwidth */
bw += (value * interval);
}
/* convert radian frequency to Hz */
bw /= twopi;
/* print result */
printf("Bandwidth = %gHz\n",bw);
return 0;
}

#### Gain Considerations

Stage gain is often determined by the value of a load resistor in the case of a small signal class A circuit.  Since the signal gain is proportional to the value of this resistor but its own noise is proportional to the square root of the same value, signal to noise ratio in just that resistor increases with the square root of that resistor's value.  This is why I believe signal to noise ratio is benefited by maximizing the gain of the first stage of a multi-stage amplifier.
(16)
S/N =
vsignal
vnoise
=

proportional to R

 proportional to

 R

 = proportional to

 R

Noise sources are placed in schematics where they have their effect and then circuit analysis determines the order that they are added.  Resistor noise adds in series with the affected resistor for voltage and in parallel for current.5  Resistor voltage and current noise are aspects of the same noise phenomena, use one or the other but not both in a calculation depending on the context.  Input referred transistor noise is probably best reflected to the gate or emitter to be added to the noise of the local feedback resistor.

Figure 5 of the datasheet of TI's OPA227 op amp gives a fair illustration of how op amp noise is calculated.

The next article provided on noise is Flicker Noise Calculations.

1Ferrel G. Stremler, "Band-limited White Noise", Introduction to Communication Systems (Reading, Massachusetts, 1982), p. 179, equations (4.58) and (4.59)
2Ibid., p. 184.
3Ibid., p. 182, example 4.7.2.
4Ibid., p. 184, numerical integral calcution based on noise bandwidth integral of equation (4.67)
5Ibid., p. 179, figure 4.11 and related text.

Document history
February 24, 2010  Created
February 26, 2010  Added supporting footnotes.
February 27, 2010  Recalculated RIAA noise bandwidth by numerical integral.
February 28, 2010  Made minor change to C code listing to ensure broad compiler compatibility.  Now will compile and run with Linux gcc compiler as well as Microsoft Visual C++.
August 7, 2010  Add missing word in sentence after equation (3).
October 13, 2010  Improved some wording for better clarity.
January 17, 2011  Made minor symbol improvements
.
January 26, 2011  Added link to new noise article "Flicker Noise Calculations".