DISTRIBUTED SENSING OF SIGNALS LINKED BY SPARSE FILTERING
First Claim
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1. A method for reconstructing a pair of signals, wherein the first signal is linked to the second signal by an unknown filter, comprising:
- observing with a first sensor first samples of the first signal;
observing with a second sensor second samples of the second signal;
exploiting the knowledge that the first and the second signal are linked by a filter for almost surely reconstructing said first signal and said second signal from said first and second samples.
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Abstract
Certain aspects of the present disclosure provide methods for distributed sensing and centralized reconstruction of two correlated signals, modeled as the input and output of an unknown sparse filtering operation.
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Citations
59 Claims
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1. A method for reconstructing a pair of signals, wherein the first signal is linked to the second signal by an unknown filter, comprising:
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observing with a first sensor first samples of the first signal; observing with a second sensor second samples of the second signal; exploiting the knowledge that the first and the second signal are linked by a filter for almost surely reconstructing said first signal and said second signal from said first and second samples. - View Dependent Claims (2, 3, 4, 5)
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6. An apparatus for reconstructing a pair of signals, wherein the first signal is linked to the second signal by an unknown filter, comprising:
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a first receiver configured to observe first samples of the first signal a second receiver configured to receive second samples of the second signal; a reconstructing circuit configured to almost surely reconstruct said first signal and said second signal from said first and second samples, wherein said reconstructing circuit exploits the knowledge that the first and the second signal are linked by a filter. - View Dependent Claims (7, 8, 9, 10)
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11. A method for signal processing, comprising:
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receiving discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; computing 2K consecutive DFT coefficients of a filter matrix using the received DFT coefficients of the first and second signals; obtaining an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; reconstructing the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and reconstructing the second signal using the impulse response of the filter and the received DFT coefficients of the first signal. - View Dependent Claims (12, 13, 14, 15, 16)
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17. An apparatus for signal processing, comprising:
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a receiver configured to receive discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; a computer configured to compute 2K consecutive DFT coefficients of a filter matrix using the received DFT coefficients of the first and second signals; a calculator configured to obtain an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal. - View Dependent Claims (18, 19, 20, 21, 22, 23, 24)
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25. An apparatus for signal processing, comprising:
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means for receiving discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; means for computing 2K consecutive DFT coefficients of a filter using the received DFT coefficients of the first and second signals; means for obtaining an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; means for reconstructing the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and means for reconstructing the second signal using the impulse response of the filter and the received DFT coefficients of the first signal. - View Dependent Claims (26, 27)
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28. A computer-program product for signal processing, comprising a computer readable medium comprising instructions executable to:
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receive discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; compute 2K consecutive DFT coefficients of a filter using the received DFT coefficients of the first and second signals; obtain an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal.
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29. A headset, comprising:
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a receiver configured to receive discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; a computer configured to compute 2K consecutive DFT coefficients of a filter using the received DFT coefficients of the first and second signals; a calculator configured to obtain an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal; and a transducer configured to provide an audio output based on the reconstructed first and second signals.
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30. A monitor for monitoring patient vital signs, comprising:
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a receiver configured to receive discrete Fourier transform (DFT) coefficients of first and second signals, the DFT coefficients including K+1 coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; a computer configured to compute 2K consecutive DFT coefficients of a filter using the received DFT coefficients of the first and second signals; a calculator configured to obtain an impulse response of the filter using the computed 2K DFT coefficients, wherein K is a number of non-zero elements of the impulse response of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal; and a user interface for displaying parameters related to the patient vital signs derived from the reconstructed first and second signals.
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31. A method for signal processing, comprising:
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sampling of first and second signals to obtain samples of the first and second signals; performing a discrete Fourier transform (DFT) on the samples of the first and second signals to obtain DFT coefficients of the first and second signals; sending first L+1 DFT coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; and wherein the first signal is an input in a sparse filter, the second signal is an output of the sparse filter, L≧
K, and K is a number of non-zero elements of an impulse response of the sparse filter. - View Dependent Claims (32, 33, 34, 35)
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36. An apparatus for signal processing, comprising:
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a sampler configured to sample first and second signals to obtain samples of the first and second signals; a first circuit configured to performing a discrete Fourier transform (DFT) on the samples of the first and second signals to obtain DFT coefficients of the first and second signals; a transmitter configured to send first L+1 DFT coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; and wherein the first signal is an input in a sparse filter, the second signal is an output of the sparse filter, L≧
K, and K is a number of non-zero elements of an impulse response of the sparse filter. - View Dependent Claims (37, 38, 39, 40)
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41. An apparatus for signal processing, comprising:
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means for sampling first and second signals to obtain samples of the first and second signals; means for performing a discrete Fourier transform (DFT) on the samples of the first and second signals to obtain DFT coefficients of the first and second signals; means for sending first L+1 DFT coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; and wherein the first signal is an input in a sparse filter, the second signal is an output of the sparse filter, L≧
K, and K is a number of non-zero elements of an impulse response of the sparse filter. - View Dependent Claims (42, 43, 44, 45)
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46. A computer-program product for signal processing, comprising a computer readable medium comprising instructions executable to:
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sample first and second signals to obtain samples of the first and second signals; perform a discrete Fourier transform (DFT) on the samples of the first and second signals to obtain DFT coefficients of the first and second signals; send first L+1 DFT coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; and wherein the first signal is an input in a sparse filter, the second signal is an output of the sparse filter, L≧
K, and K is a number of non-zero elements of an impulse response of the sparse filter.
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47. A sensing device, comprising:
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a sampler configured to sample first and second signals to obtain samples of the first and second signals; a first circuit configured to performing a discrete Fourier transform (DFT) on the samples of the first and second signals to obtain DFT coefficients of the first and second signals; a transmitter configured to transmit first L+1 DFT coefficients for each of the first and second signals and complementary subsets of remaining DFT coefficients for each of the first and second signals; a sensor configured to provide data to be transmitted via the transmitter; and wherein the first signal is an input in a sparse filter, the second signal is an output of the sparse filter, L≧
K, and K is a number of non-zero elements of an impulse response of the sparse filter.
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48. A method for signal processing, comprising:
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receiving first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; generating a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;generating a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;generating a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; obtaining DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; computing the impulse response of the filter using the obtained DFT coefficients of the filter; reconstructing the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and reconstructing the second signal using the impulse response of the filter and the received DFT coefficients of the first signal. - View Dependent Claims (49, 50)
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51. An apparatus for signal processing, comprising:
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a receiver configured to receive first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; a first generator configured to generate a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;a second generator configured to generate a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;a third generator configured to generate a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; a calculator configured to obtain DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; a computer configured to compute the impulse response of the filter using the obtained DFT coefficients of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal. - View Dependent Claims (52, 53, 55, 56)
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54. An apparatus for signal processing, comprising:
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means for receiving first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; means for generating a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;means for generating a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;means for generating a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; means for obtaining DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; means for computing the impulse response of the filter using the obtained DFT coefficients of the filter; means for reconstructing the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and means for reconstructing the second signal using the impulse response of the filter and the received DFT coefficients of the first signal.
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57. A computer-program product for signal processing, comprising a computer readable medium comprising instructions executable to:
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receive first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; generate a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;generate a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;generate a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; obtain DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; compute the impulse response of the filter using the obtained DFT coefficients of the filter; reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; and reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal.
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58. A headset, comprising:
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a receiver configured to receive first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; a first generator configured to generate a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;a second generator configured to generate a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;a third generator configured to generate a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; a calculator configured to obtain DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; a computer configured to compute the impulse response of the filter using the obtained DFT coefficients of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal; and a transducer configured to provide an audio output based on the reconstructed first and second signals.
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59. A monitor for monitoring patient vital signs, comprising:
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a receiver configured to receive first L+1 discrete Fourier transform (DFT) coefficients for each of first and second signals; a first generator configured to generate a filter matrix of dimension L×
(L+1) using the received DFT coefficients of the first and second signals;a second generator configured to generate a rank-K filter matrix by setting L−
K+1 smallest singular values of the filter matrix to zero, if a ratio of (K+1)th singular value of the filter matrix to Kth singular value of the filter matrix is not smaller than a defined threshold value, wherein K is a number of non-zero elements of an impulse response of the filter, and L≧
K;a third generator configured to generate a Toeplitz rank-K filter matrix by averaging coefficients along diagonals of the rank-K filter matrix; a calculator configured to obtain DFT coefficients of the filter based on elements of the first row and the first column of the Toeplitz rank-K filter matrix, if a ratio of (K+1)th singular value of the Toeplitz rank-K filter matrix to Kth singular value of the Toeplitz rank-K filter matrix is smaller than the defined threshold value; a computer configured to compute the impulse response of the filter using the obtained DFT coefficients of the filter; a first reconstructing circuit configured to reconstruct the first signal using the impulse response of the filter and the received DFT coefficients of the second signal; a second reconstructing circuit configured to reconstruct the second signal using the impulse response of the filter and the received DFT coefficients of the first signal; and a user interface for displaying parameters related to the patient vital signs derived from the reconstructed first and second signals.
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Specification