Method and apparatus for sinusoid detection
A system and method for detection of complex sinusoidal signals. The method can include obtaining a set of complex-valued samples of one or more signals, in the presence of unknown noise, using a plurality of sensors. The presence of κ candidate complex sinusoids can be assumed. Then, κ candidate complex sinusoids in the set of complex-valued samples can be estimated. The method can include comparing a measurement of the energy of the κ estimated candidate complex sinusoids with a threshold value. Based on the comparison, a determination can be made whether to assume a different number of candidate complex sinusoids and repeat the estimation and the comparison steps, or to specify the estimated κ candidate complex sinusoids as detected complex sinusoids. The threshold value can be determined using a penalty value which reduces the probability of overestimating the number of complex sinusoids.
1. A method for detection of complex sinusoidal signal components, the method comprising:
obtaining a set of complex-valued samples of one or more signals, in the presence of unknown noise, using a plurality of sensors;
assuming the presence of κ candidate complex sinusoidal signal components in the set of complex-valued samples;
estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples;
comparing a measurement of the energy of the κ estimated candidate complex sinusoidal signal components with a threshold value; and
based on the comparison, determining whether to assume a different number of candidate complex sinusoidal signal components and repeat the estimation and the comparison steps, or to specify the estimated κ candidate complex sinusoidal signal components as detected complex sinusoidal signal components,
wherein the threshold value is determined using a penalty value which reduces the probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples.
2. The method of claim 1 , wherein the threshold value comprises a threshold signal-to-noise ratio (SNR) value.
3. The method of claim 1 , wherein comparing the measurement of the energy of the κ estimated candidate complex sinusoidal signal components with the threshold value comprises determining whether each of the candidate complex sinusoidal signal components has a signal-to-noise ratio that exceeds the threshold value.
4. The method of claim 3 , further comprising repeating the estimation and the comparison steps until a SNR value for a candidate complex sinusoidal signal component falls below the threshold value.
5. The method of claim 1 , wherein the threshold is determined using extreme value theory (EVT).
6. The method of claim 1 , wherein the probability of underestimating the number of complex sinusoidal signal components in the set of complex-valued samples is asymptotically vanishing with an increasing number of samples.
7. The method of claim 1 , wherein the probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples is asymptotically vanishing with an increasing number of samples.
8. The method of claim 1 , wherein estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples comprises using maximum likelihood estimation.
9. The method of claim 1 , wherein estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples comprises determining frequency and amplitude values of the κ candidate complex sinusoidal signal components.
10. The method of claim 1 , further comprising extracting the detected complex sinusoidal signal components from the set of complex-valued samples of the one or more signals.
11. The method of claim 1 , wherein obtaining the set of complex-valued samples comprises identifying a signal path in a multiple-input and multiple-output (MIMO) channel.
12. The method of claim 11 , wherein each candidate complex sinusoidal signal component comprises a postulated specular ray for the signal path.
13. The method of claim 1 , wherein the set of complex-valued samples comprises a time-varying channel impulse response of a multiple-input and multiple-output (MIMO) channel.
14. The method of claim 1 , wherein determining the number of complex sinusoidal signal components present in the set of complex-valued samples comprises determining a model order of a multiple-input and multiple-output (MIMO) channel.
15. The method of claim 14 , wherein the penalty value comprises a model order selection penalty value.
16. The method of claim 1 , wherein the complex-valued samples comprise measurements of radio waves, and wherein the sensors comprise radio antennas.
17. The method of claim 1 , wherein the noise has an unknown correlation matrix.
18. The method of claim 1 , wherein the unknown noise comprises colored noise.
19. The method of claim 1 , wherein the unknown noise comprises correlated Gaussian noise.
20. A device for detection of complex sinusoidal signal components, the device comprising:
a plurality of sensors; and
a processor configured to perform a method comprising
obtaining a set of complex-valued samples of one or more signals, in the presence of unknown noise, using the plurality of sensors;
assuming the presence of κ candidate complex sinusoidal signal components in the set of complex-valued samples;
estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples;
comparing a measurement of the energy of the κ estimated candidate complex sinusoidal signal components with a threshold value; and
based on the comparison, determining whether to assume a different number of candidate complex sinusoidal signal components and repeat the estimation and the comparison steps, or to specify the estimated κ candidate complex sinusoidal signal components as detected complex sinusoidal signal components,
wherein the threshold value is determined using a penalty value which reduces the probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples.
21. A method for detection of complex sinusoidal signal components, the method comprising:
obtaining a set of complex-valued samples of one or more signals, in the presence of unknown noise, using a plurality of sensors;
assuming the presence of κ candidate complex sinusoidal signal components in the set of complex-valued samples;
estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples;
comparing a function defined by a scaled ratio of likelihood functions with a threshold value to determine if at least κ complex sinusoidal signals are present; and
based on the comparison, determining whether to increment the number of candidate complex sinusoidal signal components and repeat the estimation and the comparison steps, or to specify the estimated κ candidate complex sinusoidal signal components as detected complex sinusoidal signal components,
wherein the threshold value is determined using a penalty value which leads to a probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples that is about or below a constant false alarm rate.
22. The method of claim 21 , wherein SNR values of detected sinusoidal signal components are above a minimum SNR threshold required to achieve a predetermined probability of model order underestimation.
23. The method of claim 21 , wherein the threshold is determined using extreme value theory (EVT).
24. The method of claim 21 , wherein the probability of underestimating the number of complex sinusoidal signal components in the set of complex-valued samples is asymptotically vanishing with an increasing number of samples.
25. The method of claim 21 , wherein the probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples is asymptotically vanishing with an increasing number of samples.
26. The method of claim 21 , wherein estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples comprises using maximum likelihood estimation.
27. The method of claim 21 , wherein estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples comprises determining frequency and amplitude values of the κ candidate complex sinusoidal signal components.
28. The method of claim 21 , wherein obtaining the set of complex-valued samples comprises identifying a signal path in a multiple-input and multiple-output (MIMO) channel.
29. The method of claim 28 , wherein each candidate complex sinusoidal signal component comprises a postulated specular ray for the signal path.
30. The method of claim 21 , wherein the set of complex-valued samples comprises a time-varying channel impulse response of a multiple-input and multiple-output (MIMO) channel.
31. The method of claim 21 , wherein determining the number of complex sinusoidal signal components present in the set of complex-valued samples comprises determining a model order of a multiple-input and multiple-output (MIMO) channel.
32. The method of claim 21 , wherein the complex-valued samples comprise measurements of radio waves, and wherein the sensors comprise radio antennas.
33. The method of claim 21 , wherein the noise has an unknown correlation matrix.
34. The method of claim 21 , wherein the unknown noise comprises noise that is at least partially correlated between sensors.
35. The method of claim 21 , wherein the unknown noise comprises correlated Gaussian noise.
36. A device for detection of complex sinusoidal signal components, the device comprising:
a plurality of sensors; and
a processor configured to perform a method comprising
obtaining a set of complex-valued samples of one or more signals, in the presence of unknown noise, using a plurality of sensors;
assuming the presence of κ candidate complex sinusoidal signal components in the set of complex-valued samples;
estimating κ candidate complex sinusoidal signal components in the set of complex-valued samples;
comparing a function defined by a scaled ratio of likelihood functions with a threshold value to determine if at least κ complex sinusoidal signals are present; and
based on the comparison, determining whether to increment the number of candidate complex sinusoidal signal components and repeat the estimation and the comparison steps, or to specify the estimated κ candidate complex sinusoidal signal components as detected complex sinusoidal signal components,
wherein the threshold value is determined using a penalty value which leads to a probability of overestimating the number of complex sinusoidal signal components in the set of complex-valued samples that is about or below a constant false alarm rate.
37. The method of claim 1 , wherein κ and associated sinusoid signal component parameters, as well as a noise covariance matrix, are estimated through joint detection, with a model order overestimation probability, alpha, and a model order underestimation probability, epsilon, where alpha can be arbitrarily set between 0 and 1, and where epsilon is related to the threshold value.