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The following article appeared in the December 2012 edition of NCSLI Measure: The
Journal of Measurement Science, with all rights under copyright of NCSL International.
Spectrum Analyzer CW Power Measurements
and the Effects of Noise
The frequency discrimination capability of spectrum analyzers makes them a key component in
the electronic test and measurement industry. They are used in various applications to measure
the power of an electrical signal.. Although they do not have the same inherent amplitude
accuracy as other measurement devices such as broadband power sensors [1], they have superior
dynamic range that can extend, in some cases, to the environmental thermal noise limit. Previous
work has described the theoretical model of spectrum analyzer power measurements in the
presence of noise, but practical guidelines for actual measurements are less forthcoming. This
paper examines how to configure a spectrum analyzer to measure a low-power continuous wave
(CW) signal so that the trade-off between measurement time and accuracy is optimized. It
presents equations describing both the bias and the variance of spectrum analyzer measurements
due to noise.
Introduction
The topic of spectrum analyzer power measurements and noise has been previously addressed by
articles and application notes such as items [2] and [3]in the list of references at the end of this
document. This paper builds on these works and examines in greater detail the practical
implications for actual measurements. Spectrum analyzers are capable of a wide range of
measurements, but this paper shall simply consider the case of measuring the power of a
1
continuous wave (CW) signal of known frequency in the presence of noise. It is important to
remember the assumption of a CW signal, as different results will be obtained for other signals
such as those with pulsed or spread-spectrum characteristics [4].
The model of a spectrum analyzer signal + noise measurement will be reviewed, along with the
statistics of noise and signal + noise measurements for various averaging algorithms. Next, a
basic block diagram of a spectrum analyzer will be presented and the impact of each component
on the measurement will be discussed. Finally, recommendations for configuring the spectrum
analyzer will be summarized along with equations describing both the measurement bias and
variance due to noise.
Noise Model
A spectrum analyzer is fundamentally a voltage detector. The sophistication of the circuitry
producing the detected voltage, as well as the post-processing performed upon it, can be quite
impressive, but a CW signal can simply be represented as scalar for our purposes. The detected
voltage has a phase component, but scalar spectrum analyzers are unable to detect this and any
phase information is lost after the signal passes through the spectrum analyzer's envelope
detector. Any noise present before the envelope detector will also have a phase component. The
statistics of this noise can be modeled as having two orthogonal components of equal amplitude
and Gaussian distribution (see [3] and Fig. 1). All of the Monte Carlo analyses referenced in this
paper were performed based on this model.
Figure 1. Signal + noise voltage model.
2
Noise Measurements
Given the noise voltage. = + , the distribution of the noise voltage will be described
() =
by the Rayleigh distribution which can be expressed as
/
. (1)
where e = 2.71828... (Euler's number).
Figure 2. The Rayleigh distribution.
To measure the true noise power we must average power over time. Spectrum analyzers can be
configured to average not only power but also voltage and logarithmic power. However,
averaging noise on scales other than power will produce results that differ from the true power.
Power averaging
Averaging the noise power on a power scale will, by definition, produce the true average power,
but finding the variance of power averaging requires a bit more work. First, we must find the
average power when the voltage follows the Rayleigh distribution. This is given by
= PDF() =
. (2)
where
= average power
R = characteristic impedance of the system
v = instantaneous voltage
v2/R = instantaneous power
PDF(v) = probability distribution function for the voltage.
In this case it is the Rayleigh distribution given by Equation (1).
3
The variance of power averaging is given by
var() = ( - ) PDF() = - ,
/
(3)
where p is the power, which equals
( - 4 + 4) = =
/
. (4)
The standard deviation of a noise measurement made on the power scale therefore equals2/ ,
which is the same as the average power. Converting a power ratio, rp, from a linear ratio to the
(dB) = 10 log[ (ratio)]
equivalent value in decibels (dB) uses the formula
(5)
The conversion ratio is given by
= 10 log
()
( )
(6)
1