Joseph D. Warner and Onoufrios Theofylaktos

Glenn Research Center, Cleveland, Ohio

Correlation Between Analog Noise Measurements 

and the Expected Bit Error Rate of a Digital Signal 

Propagating Through Passive Components

NASA/TM—2012-217238

October 2012
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Joseph D. Warner and Onoufrios Theofylaktos

Glenn Research Center, Cleveland, Ohio

Correlation Between Analog Noise Measurements 

and the Expected Bit Error Rate of a Digital Signal 

Propagating Through Passive Components

NASA/TM—2012-217238

October 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217238 1

Correlation Between Analog Noise Measurements and the 

Expected Bit Error Rate of a Digital Signal Propagating 

Through Passive Components

Joseph D. Warner and Onoufrios Theofylaktos

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

A method of determining the bit error rate (BER) of a digital circuit from the measurement of the 

analog S-parameters of the circuit has been developed. The method is based on the measurement of the 

noise and the standard deviation of the noise in the S-parameters. Once the standard deviation and the 

mean of the S-parameters are known, the BER of the circuit can be calculated using the normal Gaussian 

function.

Introduction

It is possible to estimate the contribution to the bit error rate (BER) of passive components of a digital 

circuit by measuring the S-parameters of those components. This involves measuring the S-parameters 

many times so that an estimate of the standard deviation and average of the S-parameters can be 

determined. It is this noise in the S-parameters of the passive components that contributes to the BER of 

the circuit. 

This effort was motivated by the need to relate the noise generated in very long Quadrax cables for 

use on NASA’s Orion project. The cables were to connect communication components in the service 

module with those in the command module—a length of 25 m. It became necessary to estimate the 

amount of noise generated through the cable during launch. To make this estimate, the cables would be 

shaken at 32g (root-mean-square). Accordingly, performance network analyzers (PNAs) were chosen to 

measure the noise in the S-parameters of the cables before, during, and after shaking. However, the noise 

data alone were not very useful in determining the expected BER from the cables being shaken. 

Consequently, the development of a methodology to relate the noise measured by the PNA to the BER 

was necessary. This report explains how to relate analog noise measurements to the expected BER of a 

digital signal going through a passive component.

Noise

The type of random noise in a communication system that determines the BER of a circuit is thermal 

noise. Thermal noise can be described by an average white noise (AWN) across a narrow frequency band 

of interest. Although it is possible to have other types of noise from interfering signals combine with 

thermal noise in the final BER, the type of noise addressed in this report is purely Gaussian noise.

Because of the assumption of AWN, each noise source can be described by a single temperature. These 

temperatures are additive so that the total noise energy in the system is given by kBTtotal,, where kB is 

Boltzmann’s constant and Ttotal is the system noise temperature in Kelvin. Note Ttotal is the sum of all the 

individual noise temperatures in the system that is normalized by the previous gains in the system (Ref. 1).

Because the measurements are assumed to be on the long wavelength side of the peak of the 

blackbody radiation, the appropriate distribution function to be used for the measurement is the Gaussian 

distribution. The Gaussian distribution gives the probability of detecting an observed value, given the 

average of the measurement (i.e., its expected value) and the standard deviation of the measurement.
NASA/TM—2012-217238 2

In Figure 1, the y-axis is the value of the Gaussian function, and the x-axis is the number of standard 

deviations. The average of the Gaussian function μ has been set to 0.

For binary digital circuits there is a measurement threshold. Below that threshold, the value of the 

measurement is usually referred to as a digital 0, and above that threshold it is a digital 1. In a real circuit 

a signal is sent out with an amplitude or phase of one value to represent a 0 and a different value to 

represent a 1. For the discussion of noise in this report, amplitude variation is to be used.

For noise analysis purposes assume that the signal at the output of a transmitter has no noise, similar 

to the signal depicted in Figure 2; it is then reasonable to ask what the signal at the input of the receiver 

looks like with the Gaussian noise it has picked up between the transmitter and the receiver. What is then 

the probability of the receiver has misinterpreted a 1 for a 0 or a 0 for a 1. This misinterpretation will only 

occur when the noise increases the signal level for a 0 above the threshold or reduces the signal level 

below the threshold for a 1.

The probability of detecting the wrong value is dependent on the amplitude of the signal, how close 

the signal is to the threshold value, and the standard deviation in the signal amplitude due to noise. As can 

be seen in Figure 3, the larger the standard deviation, the smaller the number of standard deviations that 

need to be added to the average of the signal for the signal to cross the threshold. 

To know the probability of a signal being misinterpreted by the system, one only needs to know the 

number of standard deviations that is added or subtracted from the average signal level so that the signal 

will cross the threshold value. Then this number of standard deviations is related to the probability of an 

error occurring, using the Gaussian function.
NASA/TM—2012-217238 3

Relationship Between Noise and BER

The relationship of the probability of an error occurring and the number of standard deviations can be 

found in many books on digital communications. One useful book on this topic is Digital Modulation 

Techniques, Second Edition, by Fuqin Xiong (Ref. 2). Chapter 2 and Appendix B in this book discuss 

how to determine the BER for various modulation and coding techniques.

One can calculate the probability (P) of an error occurring:

P = A∗Q(x) (1)

where A is a constant dependent on the modulation and coding techniques used, x is the number of 

standard deviation of the detection level away from the mean signal level, and Q(x) is called the Qfunction, which is simply related to the error function (erf) by





 











 





 ≡ − 2

1 erf

2

1 ( ) x Q x

(2)

Q(x) is only the tail of the error function on either the low or high side. The definition of the error 

function is

x e t

x

t d 2 erf( )

0

2

∫ −

π ≡

(3)

From Fuqin Xiong’s book (Ref. 2), the signal energy per bit divided by Eb/N0, the average noise power 

within the frequency band of measurement time (the time period of the bit), is a function of x. Here, Eb is 
NASA/TM—2012-217238 4

the energy in one bit and N0 is the noise energy derived from the noise power and the bit period. The 

quantity Eb/N0 can be expressed as

2

0

B x

N

Eb = ∗

(4)

where B is a constant that depends on the modulation and coding schemes used.

For the simplest modulation scheme and no coding (A = B = 1), therefore





















 



 = 2

1

N0

E P Q b

(5)

It is this equation that is used in analog measurements of the S-parameters to determine the effect of 

vibration on the BER of the system.

Relating P to the S-Parameters

Forces (voltages) are what is normally measured in electronic systems, and then those measurements 

are related to current or power through calibration. Looking at the argument of the Q-function, Eb and N0

are both energies; therefore, the square root of the ratio of the two energies is a ratio of voltages that will 

be used later to determine the BER. Also, note the S-parameters from the PNA are ratios of energies. 

For a digital circuit, the link budget equation of interest can be written as

RNP IP

TransmitPower TransmissionCoefficient

N

Eb

+

∗ = 0 (6)

where RNP is the random noise power and IP is the power associated with any interfering signal like 

reflections or outside signals.

For an internal digital circuit, the outside interfering signals can be neglected. Also, power due to 

reflections will be ignored for clarity. For most well-designed passive circuits the reflected power is a

fraction of the noise power and usually can be ignored unless the communications system is operating at 

cryogenic temperatures.

By multiplying the numerator and the denominator by the time period of 1 bit, Equation (6) becomes 

0 N0

TransmitEnergy TransmissionCoefficient

N

Eb ∗ = (7)

It is now clear that the transmission coefficient is the measured S-parameter of the circuit, S21, because Eb

is the received power.

0 0 0

21 21

N

E S

N

TransmitEnergy S

N

Eb t ∗ = ∗ = (8)

where Et is the transmitted energy. The next procedure relates N0 to the standard deviations of S21.

The signal energy at the detector within the bit period with noise can be written as

SE = SE0 + N0 (9)

where N0 is the noise energy in 1 bit, SE is the instantaneous signal energy, and SE0 is the signal energy 

without noise.
NASA/TM—2012-217238 5

The transmission coefficient S21 can be written as 

21 = 21 + σ21 S S (10)

where σ21 is the standard deviation of S21 and 21 S is the average of S21.

Now multiplying Equation (10) by the initial transmit energy Et results in

21 21 σ21 ∗ = ∗ + ∗ t t Et E S E S (11)

It is easy to see that SE = Et ∗ S21 and 0 21 SE E S t = ∗

∴ N0 = Et ∗σ21 (12)

Substituting Equation (12) into Equation (8) one obtains

21

21

0 σ = SN

Eb

(13)

Therefore, by measuring the S-parameters and their standard deviations of the link one can obtain the 

Eb/N0 of the link.

Conclusion

It has been shown that one can use analog measurements to determine the energy per bit divided by 

the noise energy density Eb/N0 of a circuit if one can measure the analog S-parameters of the circuit and 

their standard deviations. For this technique to be useful, the power input in the analog circuit must be the 

same as in the digital circuit. When this condition exists, Eb N0 = S21 σ21 is the correct value to be 

considered for the link in the circuit, where S21 is the transmission coefficient and T21 is its standard 

deviation. This is because the noise in the circuit is independent of the input power into the circuit.

References

1. Sklar, Bernard: Digital Communications, Fundamentals and Applications. Prentice-Hall, Englewood 

Cliffs, NJ, 1988, p. 219.

2. Xiong, Fuqin: Digital Modulation Techniques. Second ed., Artech House, Inc., Boston, MA, 2006.
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Correlation Between Analog Noise Measurements and the Expected Bit Error Rate of a 

Digital Signal Propagating Through Passive Components 

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Warner, Joseph, D.; Theofylaktos, Onoufrios 

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14. ABSTRACT

A method of determining the bit error rate (BER) of a digital circuit from the measurement of the analog S-parameters of the circuit has 

been developed. The method is based on the measurement of the noise and the standard deviation of the noise in the S-parameters. Once the 

standard deviation and the mean of the S-parameters are known, the BER of the circuit can be calculated using the normal Gaussian 

function. 

15. SUBJECT TERMS

Network analyzer; Passive components; BER measurements 

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