1. Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

2. Overview Signal + Noise Model
Bias & Variance of Noise and Signal + Noise Measurements
Variance of Averaged Measurements
Spectrum Analyzer Block Diagram and Measurement Recommendations
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

3. Signal + Noise Model
A bandpassed in the presence of noise can be modeled as a signal having two orthogonal noise components with Gaussian distribution.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

4. Noise Only Measurements
Noise voltage follows a Rayleigh distribution
The absolute voltage of the noise will follow a Rayleigh distribution.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

5. Spectrum Analyzer Averaging Methods
Averaging Types:
Voltage
Power
Log Power
Although any given measurement can be expressed in terms of voltage, power, or log power, averaging will produce different results based on the quantities averaged.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

6. Noise – Measurement Bias
Averaging
Bias (dB)
Notes
Power
by definition
Voltage
-1.05
-20 log 𝜋 4
Log Power
-2.51
-10𝛾 log 10 e
 = Euler’s constant: …
Power averaging produces, by definition, the true average power of a pure noise signal. Voltage and log power averaging under-report the power by amounts determined by the underlying mathematics of the noise distribution.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

7. Noise – Standard Deviation of Discrete Measurements
Averaging
Std Dev (dB)
Notes
Power
4.34
10 log 10 e
Voltage
4.54
−𝜋 𝜋 log 10 𝑒
Log Power
5.77
Monte Carlo
Similarly, the standard deviation of discrete noise measurements for various averaging types can be determined from the underlying mathematics or by using Monte Carlo analysis.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

8. Signal + Noise – Measurement Bias
Averaging
Bias (dB)
Power
10 log 𝑚
Voltage
20 log 𝜋 4𝑚 e −𝑚 𝑘=0 ∞ 𝑚 ×3×5… 2𝑘 𝑘! 2
Log Power
10 log 10 e − ln 𝑚−𝛾+ 𝑒 −𝑚 𝑘=1 ∞ 𝑚 𝑘 𝑘! …+ 1 𝑘
The bias of signal measurements in the presence of noise varies based on the averaging type as well. A close-form solution is available for power averaging, while infinite series have been derived that produce the bias when using voltage or log power averaging.
m = signal-to-noise ratio (W/W)
 = Euler’s constant: …
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

9. Signal + Noise – Measurement Bias
Averaging using log powers has the benefit of the bias becoming vanishingly small for higher signal to noise ratios.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

10. Signal + Noise – Standard Deviation of Discrete Measurements
Averaging
Std Dev (dB)
Power
10 log 10 e 𝑚 1+𝑚
Voltage
10 log 10 e 𝑚 − 𝑚 1+𝑚 𝑚 − (model based on Monte Carlo)
Log Power
10log 10 e 𝑚 − 𝑚 1+𝑚 𝑚 − (model based on Monte Carlo)
A closed-form solution for the standard deviation for signal+noise measurements can be derived for power averaging, but Monte Carlo analysis must be used to derive the standard deviation when using voltage or log power averaging. The mathematical models listed above match the Monte Carlo results to within 1%.
m = signal-to-noise ratio (W/W)
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

11. Signal + Noise – Standard Deviation of Discrete Measurements
The results listed in the previous table are charted in the graph above. Note that for larger signal-to-noise ratios, the standard deviation is independent of averaging type. Two approximate formulas for larger signal-to-noise ratios are also included on the graph. For smaller signal-to-noise ratios, the results approach the values for noise only measurements.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

12. Standard Deviation of Averaged Measurements
Averaging Discrete Measurements – reduced by 𝑁
N = number of samples.
Time Averaged Measurements – reduced by 𝑡 int NBW
tint = integration (measurement) time
NBW = noise bandwidth
Filter Type
Application
NBW/RBW
4-pole sync
Most SAs analog
1.128
5-pole sync
Some SAs analog
1.111
FFT/digital
FFT/digital IF swept SAs
1.056
The values obtained so far for standard uncertainty due to noise have assumed discrete measurements. The standard deviation for the average obtained by combining several discrete measurements is reduced by the square root of the number of independent measurements. Values obtained by averaging results over time (using the spectrum analyzer’s average detector) are reduced in a similar manner by the square root of the product of the overall measurement time and the noise bandwidth. The relationship of the noise bandwidth to the resolution bandwidth varies based on the spectrum analyzer’s architecture, but sample values are given in the table above.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

13. Spectrum Analyzer Block Diagram
A typical spectrum analyzer block diagram is given in the diagram above.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

14. Spectrum Analyzer Block Diagram – Input Attenuator and Preamplifier
Attenuation = 0 dB
Preamplifier On
but… don’t overdrive the mixer!
For the best sensitivity, the input attenuation should be set to 0 and the preamplifier, if available, turned on. The only caution is that the input mixer must not be overdriven by the input signal, but this is presumably not a concern when dealing with low amplitude signals.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

15. Spectrum Analyzer Block Diagram – Preselector
set based on impact to noise floor
The preselector is used to filter out spurious images in swept mode. It is recommended that the spectrum analyzer be tuned to the signal of interest and set to a frequency span of zero, so spurious images will not be a concern. The preselector will have an impact on the noise floor, but depending on the frequency it may either increase of decrease background noise, so the decision to use or bypass the preselector should be based on the actual impact to the noise floor.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

16. Spectrum Analyzer Block Diagram – Frequency Span & Sweep Time
By using zero span, all measurement time is spent at the frequency of interest. The sweep time should be set based on the desired uncertainty due to noise and your patience. Increasing sweep time by a factor or four will decrease the uncertainty due to noise by a factor of two.
span = zero
sweep time = based on noise uncertainty
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

17. Spectrum Analyzer Block Diagram – RBW
Signal + Noise:
set RBW to give at least 8 dB SNR. Reducing RBW beyond this does not give any benefit (assumes log power averaging)
Noise: set RBW wide
𝜎= dB 𝑡 int NBW (power averaging)
For noise only measurements, the uncertainty due to noise is inversely proportional to the square root of the noise bandwidth (which is proportional to the resolution bandwidth), so measuring in wider bandwidth will produce lower uncertainty. For signal + noise measurements, it turns out that uncertainty due to noise is not impacted by the resolution bandwidth.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

18. Spectrum Analyzer Block Diagram – RBW (Signal + Noise)
SNR > 8 dB produces negligible bias when using log power averaging
The resolution bandwidth chosen for signal+noise measurements must still be narrow enough that the signal-to-noise ratio is at least 8 dB to eliminate bias. (This assumes that log power averaging is used.)
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

19. Spectrum Analyzer Block Diagram – RBW (Signal + Noise)
SNR approximated by
10 log 10 e 𝑚
In this region, the uncertainty due to noise is relatively independent of the averaging type and approximated by the formula listed above.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

20. Spectrum Analyzer Block Diagram – RBW (S+N)
Standard Uncertainty of discrete measurements approximated by:
σ=10 log 10 e 𝑚 dB
m = SNR (W/W) = 10 SNR(dB) 10
σ= − SNR(dB) 20
For time-averaged measurements:
σ= − SNR(dB) 𝑡 int NBW = − SNR(dB) 𝑡 int NBW RBW RBW
But SNR dB = SNR RBW=1 Hz −10 log 10 RBW
Therefore: σ= − SNR 1Hz (dB) 𝑡 int NBW RBW dB (time-averaged, SNR > 8 dB)
For time-averaged measurements, changing RBW does not change uncertainty due to noise
Developing this formula, it can be shown that the uncertainty due to noise is independent of resolution bandwidth for assumptions of low power averaging and a signal-to-noise ratio of at least 8 dB.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

21. Spectrum Analyzer Block Diagram – Envelope Detector
detector = average
Use the average detector as opposed to the peak or sample detector. Using the average detector maximizes the amount of information you collect in a given time period.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

22. Spectrum Analyzer Block Diagram – Video Filter
video filter = “wide open”
Given the use of time averaging, video filtering produces no additional benefit. It should be set as least as wide as the resolution bandwidth.
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

23. Spectrum Analyzer Block Diagram – Averaging Type
When measuring pure noise, noise averaging produces lower variance. When measuring a signal in the presence of noise, log power averaging produces negligible bias if the signal-to-noise ratio is at least 8 dB.
averaging type
noise: power
signal + noise: log power
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

24. Spectrum Analyzer Block Diagram – Trace Averaging & Trace Points
Because time averaging is use, trace averaging does not provide any additional benefit. Likewise the number of trace points does not affect the result since the measured value should be averaged over the entire trace.
trace averages = 1
trace points = any
function: average entire trace
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium

25. Spectrum Analyzer Block Diagram – Summary of Recommendations
Function
Noise
Signal + Noise
Input Attenuation
Preamplifier On
Preselector
based on impact to noise floor
Frequency Span
Zero
Resolution Bandwidth
wide
narrow enough for SNR > 8 dB
Detector
average
Video Bandwidth
Averaging Type
power
log power
Trace Averages 1
Trace Points
n/a
average entire trace
Spectrum Analyzer CW Power Measurements and Noise
2012 NCSL International Workshop and Symposium