Determination of statistical error bounds and uncertainty measures for estimates 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 error bound for an estimate of a power spectral density of a 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 error bound includes computing the statistical error bound based on an expression of the form √{square root over (A 2 +T 2 )}, wherein A depends on the number n, wherein T depends on the number n and a coherence function associated with the complex-valued averaged cross spectral density;
storing the statistical error bound in a memory.
2. 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 graphical indication of the statistical error bound in association with the graph.
3. The method of claim 1 , wherein said determining the statistical error 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 A depends on the effective number, wherein T depends on the effective number and the coherence function.
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 error 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 A depends on the effective number, wherein T depends on the effective number and on a coherence function associated with the spectrally smoothed cross spectral density.
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. 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:
determining a statistical error bound for an estimate of a power spectral density of a 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 on n respective two-channel data sets, wherein each two-channel data set includes a set of samples of a signal a(t) acquired from a first channel and a corresponding set of samples of a signal b(t) acquired from a second channel, wherein each set of samples of the signal a(t) and the corresponding set of samples of the second signal b(t) are acquired over the same interval of time, wherein the signal a(t) is a sum of a first interfering noise signal and the target noise signal y(t), wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal y(t), wherein said determining the statistical error bound includes computing the statistical error bound based on an expression of the form √{square root over (A 2 +T 2 )}, wherein A depends on the number n, wherein T depends on the number n and a coherence function associated with the complex-valued averaged cross spectral density;
storing the statistical error bound in a memory medium.
8. The system of claim 7 , 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 graphical indication of the statistical error bound in association with the graph.
9. The system of claim 7 , wherein said determining the statistical error 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 A depends on the effective number, wherein T depends on the effective number and the coherence function.
10. The system of claim 7 , 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 error 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 A depends on the effective number, wherein T depends on the effective number and on a coherence function associated with the spectrally smoothed cross spectral density.
11. The system of claim 10 , wherein the filter used to perform the spectrally smoothing varies with frequency.
12. The system of claim 11 , wherein the smoothing redistributes the complex-valued averaged cross spectral density from a linear frequency scale to a logarithmic frequency scale.