Vector smoothing of complex-valued cross spectra to estimate power spectral density of a noise signal
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 for estimating a power spectral density of a target noise signal, the method comprising:
performing n two-channel acquisitions, wherein each of the two-channel acquisitions includes:
acquiring a set of samples of a signal a(t) from a first channel,
simultaneously acquiring a set of samples of a signal b(t) from a second channel, and
storing the sample set of the signal a(t) and the sample set of the signal b(t) in a memory, wherein the signal a(t) is a sum of a first interfering noise signal and the target noise signal, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal;
utilizing a computational device to implement:
smoothing a complex-valued averaged cross spectral density to obtain a complex-valued smoothed cross spectral density, wherein the complex-valued averaged cross spectral density is an average of n complex-valued cross spectral densities, wherein n is an integer greater than one, wherein each of the n complex-valued cross spectral densities is computed based on a corresponding one of the n two-channel acquisitions; and
determining an estimate of the power spectral density of the target noise signal based on the complex-valued smoothed cross spectral density, wherein the estimate of the power spectral density of the target noise signal is stored in a memory.
2. The method of claim 1 , wherein a filter used to perform the smoothing varies with frequency.
3. The method of claim 1 , further comprising utilizing the computational device to:
display the estimate of the power spectral density of the target noise signal as a graph.
4. 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.
5. The method of claim 1 , wherein successive sets of the samples of the signal a(t) are overlapping in time, wherein successive sets of the samples of the signal b(t) are overlapping in time.
6. The method of claim 1 , wherein, for each of the n two-channel acquisitions, the sample set of the signal a(t) is windowed with a time domain window to obtain a first windowed sample set, and the sample set of the signal b(t) is windowed with the time domain window to obtain a second windowed sample set, wherein each of the n complex-valued cross spectral densities is based on the first windowed sample set and the second windowed sample set of the corresponding two-channel acquisition.
7. The method of claim 1 , further comprising:
averaging the n complex-valued cross spectral densities to obtain said complex-valued averaged cross spectral density, wherein said averaging comprises computing an updated average after each of the n two-channel acquisitions.
8. A system for estimating a power spectral density of a target noise signal, the system comprising:
a memory configured to store program instructions; and
a processor configured to access the memory and execute the program instructions accessed from the memory, wherein the program instructions, when executed by the processor, cause the processor to implement:
smoothing a complex-valued averaged cross spectral density to obtain a complex-valued smoothed cross spectral density, wherein the complex-valued averaged cross spectral density is an average of n complex-valued cross spectral densities, wherein n is an integer greater than one, wherein each of the n complex-valued cross spectral densities is computed based on a corresponding one of n two-channel acquisitions, wherein each of the two-channel acquisitions includes an acquisition of a set of samples of a signal a(t) from a first channel and a simultaneous acquisition of a set of samples of a signal b(t) from a second channel, wherein the signal a(t) is a sum of a first interfering noise signal and the target noise signal, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal; and
determining an estimate of the power spectral density of the target noise signal based on the complex-valued smoothed cross spectral density, wherein the estimate of the power spectral density of the target noise signal is stored in the memory.
9. The system of claim 8 , wherein a filter used to perform the smoothing varies with frequency.
10. The system of claim 8 , wherein the program instructions, when executed by the processor, further cause the processor to implement:
displaying the estimate of the power spectral density of the target noise signal as a graph.
11. 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.
12. The system of claim 8 , wherein successive sets of the samples of the signal a(t) are overlapping in time, wherein successive sets of the samples of the signal b(t) are overlapping in time.
13. The system of claim 8 , wherein, for each of the n two-channel acquisitions, the sample set of the signal a(t) is windowed with a time domain window to obtain a first windowed sample set, and the sample set of the signal b(t) is windowed with the time domain window to obtain a second windowed sample set, wherein each of the n complex-valued cross spectral densities is based on the first windowed sample set and the second windowed sample set of the corresponding two-channel acquisition.
14. The system of claim 8 , wherein the program instructions, when executed by the processor, further cause the processor to implement:
averaging the n complex-valued cross spectral densities to obtain said complex-valued averaged cross spectral density, wherein said averaging comprises computing an updated average after each of the n two-channel acquisitions.
15. A non-transitory computer-readable memory medium for estimating a power spectral density of a target noise signal, wherein the memory medium stores 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:
smoothing a complex-valued averaged cross spectral density to obtain a complex-valued smoothed cross spectral density, wherein the complex-valued averaged cross spectral density is an average of n complex-valued cross spectral densities, wherein n is an integer greater than one, wherein each of the n complex-valued cross spectral densities is computed based on a corresponding one of n two-channel acquisitions, wherein each of the two-channel acquisitions includes an acquisition of a set of samples of a signal a(t) from a first channel and a simultaneous acquisition of a set of samples of a signal b(t) from a second channel, wherein the signal a(t) is a sum of a first interfering noise signal and the target noise signal, wherein the signal b(t) is a sum of a second interfering noise signal and the target noise signal; and
determining an estimate of the power spectral density of the target noise signal based on the complex-valued smoothed cross spectral density, wherein the estimate of the power spectral density of the target noise signal is stored in a memory.
16. The memory medium of claim 15 , wherein a filter used to perform the smoothing varies with frequency.
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 the estimate of the power spectral density of the target noise signal as a graph.
18. The memory medium of claim 15 , wherein successive sets of the samples of the signal a(t) are overlapping in time, wherein successive sets of the samples of the signal b(t) are overlapping in time.
19. The memory medium of claim 15 , wherein, for each of the n two-channel acquisitions, the sample set of the signal a(t) is windowed with a time domain window to obtain a first windowed sample set, and the sample set of the signal b(t) is windowed with the time domain window to obtain a second windowed sample set, wherein each of the n complex-valued cross spectral densities is based on the first windowed sample set and the second windowed sample set of the corresponding two-channel acquisition.
20. The memory medium of claim 15 , wherein the program instructions, when executed by the computer system, further cause the computer system to implement:
averaging the n complex-valued cross spectral densities to obtain said complex-valued averaged cross spectral density, wherein said averaging comprises computing an updated average after each of the n two-channel acquisitions.