Method and apparatus of spectral estimation

• US 20070067167A1
• Filed: 09/16/2005
• Published: 03/22/2007
• Est. Priority Date: 09/16/2005
• Status: Active Grant

• Current Assignee: Keysight Technologies, Inc.
• Original Assignee: Agilent Technologies, Inc.
• Inventors: Barford, Lee
• Granted Patent: US 7,720,644 B2
• Time in Patent Office: Days
• US Class Current: 704/226
• CPC Class Codes: G06F 17/141 Discrete Fourier transforms
• CPC Class Codes: H04L 43/50 Testing arrangements

Abstract

A spectrum of a set of samples from a data stream of sampled data is estimated until a targeted signal to noise ratio is achieved.

Claims

1. A method of spectral estimation for a data stream D of sampled data, the method comprising:

selecting a first set of samples from the data stream D; and

selecting other sets of samples to successively combine with the first set to create a combined set of samples until a measure of a signal to noise ratio for an estimated spectrum of the combined set of samples indicates a target signal to noise ratio is achieved.

2. The method of spectral estimation of claim 1, wherein the target signal to noise ratio is achieved when the measure of the signal to noise ratio is less than a predetermined limit λ

.

3. The method of spectral estimation of claim 1, wherein the measure of the signal to noise ratio is proportional to a condition number of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to the estimated spectrum.

4. The method of spectral estimation of claim 1, wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of a least squares matrix (L-S) matrix A of a least squares fit of the combined sample set to an estimated spectrum meets a condition given by

min(abs(diag(R)))>

0ˆ

1/min(abs(diag(R)))<

λ

where λ

is a predetermined limit equal to or greater than approximately 1.1 and less than approximately 1,000.

5. The method of spectral estimation of claim 1, wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of a least squares matrix (L-S) matrix A of a least squares fit of the combined sample set to an estimated spectrum meets a condition given by

min(abs(diag(R)))>

0ˆ

max(abs(diag(R)))/min(abs(diag(R)))<

λ

where λ

is a predetermined limit equal to or greater than approximately 1.1 and less than approximately 1,000

6. The method of spectral estimation of claim 1, further comprising:

determining an upper right triangular factor matrix R of a QR factorization of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to an estimated spectrum,

wherein the measure of the signal to noise ratio is a ratio of selected diagonal elements of the QR factorization of the L-S matrix A when the factor matrix R is nonsingular.

7. The method of spectral estimation of claim 6, wherein the ratio of selected diagonal elements is one or both of a maximum value ratio of an element selected from an element-wise absolute value of a main diagonal of the factor matrix R to a minimum value element selected from the element-wise absolute value of the main diagonal and a ratio that is an inverse of a minimum value element selected from an element-wise absolute value of a main diagonal of the factor matrix R.

8. The method of spectral estimation of claim 7, wherein the predetermined limit λ

is equal to or greater than approximately 1.1 and less than approximately 1,000.

9. The method of spectral estimation of claim 1, wherein selecting other sets of samples uses a minimum time step Δ

t_min between corresponding samples in the selected sets, and wherein selecting other sets of samples comprises taking a quantity of samples equaling at least two times a number of frequencies in a set of frequencies of interest F plus 1 from at least two periods of a lowest frequency of interest f₁ the lowest frequency of interest f₁ being greater than zero Hertz.

10. The method of spectral estimation of claim 1, wherein a minimum time step Δ

t_min between selected sets is given by

Δ

t_min=2/f₁/(2|F|+1)

where |F| is a number of frequencies in a set of frequencies of interest F and f₁ is a lowest frequency greater than zero Hertz in the set of frequencies of interest F.

11. The method of spectral estimation of claim 1, further comprising:

computing an estimated spectrum by solving a least-squares fitting problem for the combined set; and

extracting Fourier coefficients of the estimated spectrum from a solution to the least-squares fitting problem.

12. The method of spectral estimation of claim 1, the data stream D comprises irregularly spaced, time-sampled data.

13. The method of spectral estimation of claim 1, wherein the samples of the selected sets from the data stream D comprise pairs of values, each pair comprising an explicit timestamp and a sample value.

14. A method of estimating a spectrum of sampled data at a set of frequencies of interest F, the method comprising:

selecting timestamped samples from the sampled data to produce a set of samples;

evaluating a measure of a signal to noise ratio of an estimated spectrum for the set of samples;

successively selecting additional timestamped samples to combine with the set of samples; and

re-evaluating the measure of the signal to noise ratio of the estimated spectrum for the combined set of samples,

wherein successively selecting and re-evaluating are repeated until the measure is less than a predetermined limit λ

, the measure being less than the predetermined limit λ

indicating a target signal to noise ratio is achieved.

15. The method of estimating a spectrum of claim 14, wherein both evaluating and re-evaluating comprise:

computing an estimated spectrum by solving a least-squares fitting problem for the set of samples, the targeted signal to noise ratio being achieved when both (a) an upper right triangular factor matrix R of a QR factorization of an L-S matrix A of the least-squares fitting problem is nonsingular and (b) a ratio of selected diagonal elements of the factor matrix R is less than the predetermined limit λ

.

16. The method of estimating a spectrum of claim 15, wherein the ratio of selected diagonal elements is one or both of a maximum value ratio of an element selected from an element-wise absolute value of a main diagonal of the factor matrix R to a minimum value element selected from the element-wise absolute value of the main diagonal and a ratio that is an inverse of a minimum value element selected from an element-wise absolute value of a main diagonal of the factor matrix R.

17. The method of estimating a spectrum of claim 14, wherein the predetermined limit λ

is equal to or greater than approximately 1.1 and less than approximately 1000.

18. The method of estimating a spectrum of claim 14, wherein selecting timestamped samples produces a minimum time step Δ

t_min between the selected timestamped samples, and wherein selecting timestamped samples comprises taking a quantity of samples equaling at least two times a number of frequencies in a set of frequencies of interest F plus 1 from at least two periods of a lowest frequency of interest f₁ the lowest frequency of interest f₁ being greater than zero Hertz.

19. The method of estimating a spectrum of claim 14, wherein selecting timestamped samples comprises selecting a plurality of adjacent samples, a first sample of each selected plurality being separated by a minimum time step Δ

t_min from a first sample of another selected plurality, the minimum time step Δ

t_min being given by

Δ

t_min=2/f₁/(2|F|+1)

where |F| is a number of frequencies in the set of frequencies of interest F and where f₁ is a lowest frequency of interest that is greater than zero Hertz.

20. A method of spectral estimation for a data stream D of sampled data comprising:

constructing a least-squares (L-S) matrix A, the L-S matrix A comprising a basis set of a Fourier Transform, and timestamps of samples selected from the data stream D;

solving a least-squares fitting problem for a solution vector x representing Fourier coefficients of an estimated spectrum using a vector b comprising values of the selected samples; and

extracting the Fourier coefficients of the estimated spectrum from the solution vector x, the Fourier coefficients representing the estimated spectrum,

wherein the samples are successively selected from the data stream D, and wherein the least-squares fitting problem is solved until a targeted signal to noise ratio of the estimated spectrum is achieved.

21. The method of spectral estimation of claim 20, wherein the targeted signal to noise ratio is achieved when both an upper right triangular factor matrix R of a QR factorization of the L-S matrix A is nonsingular and a maximum value ratio from a set of ratios of selected diagonal elements of the factor matrix R is less than a predetermined limit λ

.

22. The method of spectral estimation of claim 21, wherein a ratio of the set of ratios is a diagonal element selected from an element-wise absolute value of a main diagonal of the factor matrix R to a minimum value element selected from the element-wise absolute value of the main diagonal.

23. The method of spectral estimation of claim 21, wherein the targeted signal to noise ratio is further achieved when an inverse of a minimum value element selected from an element-wise absolute value of a main diagonal of the factor matrix R is less than the predetermined limit λ

.

24. The method of spectral estimation of claim 20, wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of the L-S matrix A meets a condition given by

min(abs(diag(R)))>

0ˆ

max(abs(diag(R)))/min(abs(diag(R)))<

λ

.

25. The method of spectral estimation of claim 20, wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of the L-S matrix A is nonsingular and an inverse of a minimum value element of an absolute value of main diagonal elements of the factor matrix R is less than a predetermined limit λ

.

26. The method of spectral estimation of claim 20, wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of the L-S matrix A meets a condition given by

min(abs(diag(R)))>

0ˆ

1/min(abs(diag(R)))<

λ

27. The method of spectral estimation of claim 20, wherein the successively selected samples have a minimum time step Δ

t_min between the selected samples, such that a quantity of samples are taken from at least two periods of a lowest frequency of interest f₁ the quantity equaling at least two times a number of frequencies in a set of frequencies of interest F plus 1, the lowest frequency of interest f₁ being greater than zero Hertz.

28. The method of spectral estimation of claim 20, wherein there is a minimum time step Δ

t_min between the successively selected samples that is given by

Δ

t_min=2/f₁/(2|F|+1)

where F is a set of frequencies of interest in the estimated spectrum, and where f₁ is a lowest frequency in the set of frequencies that is greater than zero Hertz.

29. The method of spectral estimation of claim 20, wherein the samples are successively selected in sets comprising a plurality of adjacent samples, the sets being separated by a minimum time step Δ

t_min.

30. The method of spectral estimation of claim 20, wherein the represented estimated spectrum is specified at predetermined frequencies of interest.

31. The method of spectral estimation of claim 30, wherein the L-S matrix A is given by

$A=[\begin{array}{cc}\frac{1}{\sqrt{2}}& \sin \end{array}$

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 1 ) cos ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 1 ) 1 2 sin ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 2 ) cos ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 2 ) 1 2 sin ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 3 ) cos ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 3 ) ⋮

⋮

⋮

1 2 sin ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 

F 

) cos ⁡

( 2 ⁢

 

⁢

π

⁢

 

⁢

tf 

F 

) ]

where f_m represents an m-th one of an ordered set of frequencies of interest F indexed on m, the index m ranging from 1 to |F|, and where the functions sin(2π

tf_m) and cos(2π

tf_m) are each row vectors indexed on a set of selected samples of the data given by

sin(2π

tf_m)=[(2π

t₁f_m)sin(2π

t₂f_m) . . . sin(2π

t_Nf_m)]
cos(2π

tf_m)=[(2π

t₁f_m)cos(2π

t₂f_m) . . . cos(2π

t_Nf_m)] 

 

(7)

where t_n are timestamps associated with samples selected from the data stream D indexed on n, such that t represents a vector comprising the timestamps t_n.

32. A spectral estimator comprising:

a computer processor;

memory; and

a computer program stored in the memory and executed by the computer processor,

wherein the computer program comprises instructions that, when executed by the processor, implement selecting a first set of samples from a data stream of timestamped samples, and further implement subsequently selecting other sets of timestamped samples to successively combine with the first set to create a combined set of samples until a measure of a signal to noise ratio for an estimated spectrum of the combined set of samples indicates a target signal to noise ratio is achieved.

33. The spectral estimator of claim 32, wherein the target signal to noise ratio is achieved when the measure of the signal to noise ratio is less than a predetermined limit λ

, the measure of the signal to noise ratio being proportional to a condition number of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to the estimated spectrum.

34. The spectral estimator of claim 32, wherein in the executed instructions further implement determining an upper right triangular factor matrix R of a QR factorization of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to the estimated spectrum, the measure of the signal to noise ratio being a ratio of selected diagonal elements of the QR factorization of the L-S matrix A when the factor matrix R is nonsingular.

35. The spectral estimator of claim 34, wherein the ratio of selected diagonal elements is one or both of a maximum value ratio of an element selected from an element-wise absolute value of a main diagonal of the factor matrix R to a minimum value element selected from the element-wise absolute value of the main diagonal and a ratio that is an inverse of a minimum value element selected from an element-wise absolute value of a main diagonal of the factor matrix R, and wherein the predetermined limit λ

is equal to or greater than approximately 1.1 and less than approximately 1,000.

36. The spectral estimator of claim 32, wherein the data stream comprises one or both of randomly sampled data and irregularly sampled data.

37. The spectral estimator of claim 32, wherein the instructions that implement subsequently selecting other sets of samples select timestamped sample amplitude samples from the data stream such that the successively combined sample sets are separated by a minimum time step Δ

t_min given by

Δ

t_min=2/f₁/(2|F|+1)

where F is a set of frequencies of interest in the estimated spectrum, and where f₁ is a lowest frequency in the set of frequencies that is greater than zero Hertz.

38. A spectral estimator that uses timestamped samples from sampled data, the spectral estimator comprising:

a computer processor;

memory; and

a computer program stored in the memory and executed by the computer processor,

wherein the computer program comprises instructions that, when executed by the processor, implement selecting timestamped samples from the sampled data to produce a set of samples, implement evaluating a measure of a signal to noise ratio of an estimated spectrum for the set of samples, implement successively selecting additional timestamped samples to combine with the set of samples, and implement re-evaluating the measure of the signal to noise ratio of the estimated spectrum for the combined set of samples, successively selecting and re-evaluating being repeated until the measure is less than a predetermined limit λ

, wherein a target signal to noise ratio is achieved when the measure is less than the predetermined limit λ

.

39. The spectral estimator of claim 38, wherein the executed instructions that implement evaluating a measure and re-evaluating the measure implement computing the estimated spectrum by solving a least-squares fitting problem, the targeted signal to noise ratio being further achieved when both (a) an upper right triangular factor matrix R of a QR factorization of an L-S matrix A of the least-squares fitting problem is nonsingular and (b) a ratio of selected diagonal elements of the factor matrix R is less than the predetermined limit λ

.

40. The spectral estimator of claim 38, wherein the predetermined limit λ

is equal to or greater than approximately 1.1 and less than approximately 1000.

41. The spectral estimator of claim 38 used in a system employing spectral estimation, the system comprising:

an excitation source that connects to an input of a device under test; and

a sampler that connects to an output of the device under test, the sampler having an output connected to the spectral estimator, the sampler providing the sampled data from the device under test to the spectral estimator.

42. The spectral estimator of claim 41, wherein the system further comprises a test results evaluator connected to an output of the spectral estimator, the test results evaluator determining whether the device under test meets predetermined specifications.

43. A test system employing spectral estimation comprising:

an excitation source that connects to an input of a device under test;

a sampler that connects to an output of the device under test; and

a spectral estimator connected to an output of the sampler, the sampler providing sampled data from the device under test to the spectral estimator,

wherein the spectral estimator implements selecting timestamped samples from the sampled data to produce a set of samples, implements evaluating a measure of a signal to noise ratio of an estimated spectrum for the set of samples, implements successively selecting additional timestamped samples to combine with the set of samples, and implements re-evaluating the measure of the signal to noise ratio of the estimated spectrum for the combined set of samples, successively selecting and re-evaluating being repeated until the measure is less than a predetermined limit λ

, wherein a target signal to noise ratio is achieved when the measure is less than the predetermined limit λ

.

44. The test system of claim 43, further comprising a test results evaluator connected to an output of the spectral estimator, the test results evaluator determining whether the device under test meets predetermined specifications.