Determination of statistical upper bound for estimate of noise power spectral density
View Patent ↗Systems/methods for computing a power spectral density estimate for a noise signal. Where the noise signal appears in two channels (a single channel), n successive data acquisitions from the two channels (the single channel) are used to compute n respective cross (power) spectral densities, which are then averaged. The averaged cross (power) spectral density may then be smoothed in the spectral domain. The magnitude of the smoothed cross (power) spectral density comprises an estimate for the noise power spectral density. An effective number of independent averages may be computed based on the number n, the time-domain window applied to the acquired sample sets, the amount of overlap between successive sample sets, and the shape of the frequency-domain smoothing function. A statistical error bound (or uncertainty measure) may be determined for the power spectral density estimate based on the effective number of averages and the averaged single-channel and cross-channel spectral estimates.
1. A method comprising:
acquiring n two-channel data sets and storing the n two-channel data sets in a memory, wherein for each two-channel data set said acquiring comprises:
acquiring a set of samples of a signal a(t) from a first channel, wherein the signal a(t) is a sum of a first interfering noise signal and a target noise signal y(t);
acquiring a set of samples of a signal b(t) from a second channel, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal y(t), wherein the sample set of the signal a(t) and the sample set of the signal b(t) are acquired over the same interval of time;
utilizing a computational device to implement:
determining a statistical upper bound for an estimate of a power spectral density of the target noise signal y(t), wherein the estimate is determined by averaging n complex-valued cross spectral densities to obtain a complex-valued averaged cross spectral density, wherein the n complex-valued cross spectral densities are computed based respectively on the n two-channel data sets, wherein said determining the statistical upper bound includes computing the statistical upper bound based on the number n, the complex-valued averaged cross spectral density, a first spectral density and a second spectral density, wherein the first spectral density is an averaged power spectral density for the signal a(t), wherein the second spectral density is an averaged power spectral density for the signal b(t); and
storing the statistical upper bound in a memory.
2. The method of claim 1 , wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including: the number n and a relative amount of time overlap between successive ones of the sample sets of the signal a(t) and between successive ones of the sample sets of the signal b(t), wherein the statistical upper bound is computed based on the effective number of independent averages, the complex-valued averaged cross spectral density, the first spectral density and the second spectral density.
3. The method of claim 1 , further comprising:
displaying a graph of the estimate of the power spectral density of the target noise signal y(t); and
displaying a graph of the statistical upper bound in association with the graph of the power spectral density estimate.
4. The method of claim 1 , wherein said determining the power spectral density estimate includes spectrally smoothing the complex-valued averaged cross spectral density to obtain a spectrally smoothed cross spectral density, wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including:
the number n;
information specifying a time-domain window that is applied to the sample sets of the signal a(t) and to the sample sets of the signal b(t); and
information specifying the filter used to perform said spectral smoothing;
wherein the statistical upper bound is computed based on the effective number of independent averages, the spectrally smoothed cross spectral density, and spectrally smoothed versions of the first and second spectra.
5. The method of claim 4 , wherein the filter used to perform the spectral smoothing varies with frequency.
6. The method of claim 5 , wherein the spectral smoothing redistributes the complex-valued averaged cross spectral density from a linear frequency scale to a logarithmic frequency scale.
7. The method of claim 1 , wherein the target noise signal y(t) is a phase noise signal or a voltage noise signal or a current noise signal or an optical noise signal or a displacement noise signal.
8. A system comprising:
a memory configured to store program instructions; and
a processor configured to access the program instructions from the memory and execute the program instructions, wherein the program instructions, when executed by the processor, cause the processor to implement:
acquiring n two-channel data sets and storing the n two-channel data sets in a memory, wherein for each two-channel data set said acquiring comprises:
acquiring a set of samples of a signal a(t) from a first channel, wherein the signal a(t) is a sum of a first interfering noise signal and a target noise signal y(t); and
acquiring a set of samples of a signal b(t) from a second channel, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal y(t), wherein the sample set of the signal a(t) and the sample set of the signal b(t) are acquired over the same interval of time;
determining a statistical upper bound for an estimate of a power spectral density of the target noise signal y(t), wherein the estimate is determined by averaging n complex-valued cross spectral densities to obtain a complex-valued averaged cross spectral density, wherein the n complex-valued cross spectral densities are computed based respectively on the n two-channel data sets, wherein said determining the statistical upper bound includes computing the statistical upper bound based on the number n, the complex-valued averaged cross spectral density, a first spectral density and a second spectral density, wherein first spectral density is an averaged power spectral density for the signal a(t), wherein the second spectral density is an averaged power spectral density for the signal b(t); and
storing the statistical upper bound in a memory medium.
9. The system of claim 8 , wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including: the number n and a relative amount of time overlap between successive ones of the sample sets of the signal a(t) and between successive ones of the sample sets of the signal b(t), wherein the statistical upper bound is computed based on the effective number of independent averages, the complex-valued averaged cross spectral density, the first spectral density and the second spectral density.
10. The system of claim 8 , wherein the program instructions, when executed by the processor, further cause the processor to implement:
displaying a graph of the estimate of the power spectral density of the target noise signal y(t); and
displaying a graph of the statistical upper bound in association with the graph of the power spectral density estimate.
11. The system of claim 8 , wherein said determining the power spectral density estimate includes spectrally smoothing the complex-valued averaged cross spectral density to obtain a spectrally smoothed cross spectral density, wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including:
the number n;
information specifying a time-domain window that is applied to the sample sets of the signal a(t) and to the sample sets of the signal b(t); and
information specifying the filter used to perform said spectral smoothing;
wherein the statistical upper bound is computed based on the effective number of independent averages, the spectrally smoothed cross spectral density, and spectrally smoothed versions of the first and second spectra.
12. The system of claim 11 , wherein the filter used to perform the spectral smoothing varies with frequency.
13. The system of claim 12 , wherein the spectral smoothing redistributes the complex-valued averaged cross spectral density from a linear frequency scale to a logarithmic frequency scale.
14. The system of claim 8 , wherein the target noise signal y(t) is a phase noise signal or a voltage noise signal or a current noise signal or an optical noise signal or a displacement noise signal.
15. A non-transitory computer-readable memory medium storing program instructions that are executable by a computer system, wherein the program instructions, when executed by the computer system, cause the computer system to implement:
acquiring n two-channel data sets and storing the n two-channel data sets in a memory, wherein for each two-channel data set said acquiring comprises:
acquiring a set of samples of a signal a(t) from a first channel, wherein the signal a(t) is a sum of a first interfering noise signal and a target noise signal y(t); and
acquiring a set of samples of a signal b(t) from a second channel, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal y(t), wherein the sample set of the signal a(t) and the sample set of the signal b(t) are acquired over the same interval of time;
determining a statistical upper bound for an estimate of a power spectral density of the target noise signal y(t), wherein the estimate is determined by averaging n complex-valued cross spectral densities to obtain a complex-valued averaged cross spectral density, wherein the n complex-valued cross spectral densities are computed based respectively on the n two-channel data sets, wherein said determining the statistical upper bound includes computing the statistical upper bound based on the number n, the complex-valued averaged cross spectral density, a first spectral density and a second spectral density, wherein the first spectral density is an averaged power spectral density for the signal a(t), wherein the second spectral density is an averaged power spectral density for the signal b(t); and
storing the statistical upper bound in a memory.
16. The memory medium of claim 15 , wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including: the number n and a relative amount of time overlap between successive ones of the sample sets of the signal a(t) and between successive ones of the sample sets of the signal b(t), wherein the statistical upper bound is computed based on the effective number of independent averages, the complex-valued averaged cross spectral density, the first spectral density and the second spectral density.
17. The memory medium of claim 15 , wherein the program instructions, when executed by the computer system, further cause the computer system to implement:
displaying a graph of the estimate of the power spectral density of the target noise signal y(t); and
displaying a graph of the statistical upper bound in association with the graph of the power spectral density estimate.
18. The memory medium of claim 15 , wherein said determining the power spectral density estimate includes spectrally smoothing the complex-valued averaged cross spectral density to obtain a spectrally smoothed cross spectral density, wherein said determining the statistical upper bound includes computing an effective number of independent averages corresponding to said power spectral density estimate based on data including:
the number n;
information specifying a time-domain window that is applied to the sample sets of the signal a(t) and to the sample sets of the signal b(t); and
information specifying the filter used to perform said spectral smoothing;
wherein the statistical upper bound is computed based on the effective number of independent averages, the spectrally smoothed cross spectral density, and spectrally smoothed versions of the first and second spectra.
19. The memory medium of claim 18 , wherein the filter used to perform the spectral smoothing varies with frequency.
20. The memory medium of claim 19 , wherein the spectral smoothing redistributes the complex-valued averaged cross spectral density from a linear frequency scale to a logarithmic frequency scale.