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

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

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

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

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

Noise – Measurement Bias Averaging Bias (dB) Notes Power by definition Voltage -1.05 -20 log 10 𝜋 4 Log Power -2.51 -10𝛾 log 10 e  = Euler’s constant: 0.577216… 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

Noise – Standard Deviation of Discrete Measurements Averaging Std Dev (dB) Notes Power 4.34 10 log 10 e Voltage 4.54 20 4−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

Signal + Noise – Measurement Bias Averaging Bias (dB) Power 10 log 10 1+ 1 𝑚 Voltage 20 log 10 𝜋 4𝑚 e −𝑚 𝑘=0 ∞ 𝑚 2 2 1×3×5… 2𝑘+1 𝑘! 2 Log Power 10 log 10 e − ln 𝑚−𝛾+ 𝑒 −𝑚 𝑘=1 ∞ 𝑚 𝑘 𝑘! 1+ 1 2 + 1 3 +…+ 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: 0.577216… Spectrum Analyzer CW Power Measurements and Noise 2012 NCSL International Workshop and Symposium

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

Signal + Noise – Standard Deviation of Discrete Measurements Averaging Std Dev (dB) Power 10 log 10 e 1+2𝑚 1+𝑚 Voltage 10 log 10 e 3.99𝑚 −1.733 + 1+2𝑚 1+𝑚 3.11 𝑚 −1.733 +1 (model based on Monte Carlo) Log Power 10log 10 e 0.648𝑚 −1.4 + 1+2𝑚 1+𝑚 0.62 𝑚 −1.4 +1 (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

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

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

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

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

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

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

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= 4.34 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

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

Spectrum Analyzer Block Diagram – RBW (Signal + Noise) SNR approximated by 10 log 10 e 2 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

Spectrum Analyzer Block Diagram – RBW (S+N) Standard Uncertainty of discrete measurements approximated by: σ=10 log 10 e 2 𝑚 dB m = SNR (W/W) = 10 SNR(dB) 10 σ=6.14 10 − SNR(dB) 20 For time-averaged measurements: σ= 6.14 10 − SNR(dB) 20 𝑡 int NBW = 6.14 10 − SNR(dB) 20 𝑡 int NBW RBW RBW But SNR dB = SNR RBW=1 Hz −10 log 10 RBW Therefore: σ= 6.14 10 − SNR 1Hz (dB) 20 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

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

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

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

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

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