Robust spectral analyzer for one-dimensional and multi-dimensional data analysis
View Patent ↗A method of analyzing a spectrum of one-dimensional or multi-dimensional signal X(t) requires a number of steps including deriving coefficients [A N (ω), B N (ω)] of an Lp-norm harmonic regression of tie signal with 0<p≦∞ and pγ2, squaring the coefficients, summing the squared coefficients, and scaling the summed, squared coefficients with a constant c to realize a periodogram of X(t) as L N (ω)=c{[A N (ω)]+[B N (ω)] 2 }. The method may include receiving the signal X(t), storing the received signal X(t), and outputting the periodogram L N (ω). The method may still further include scanning to maximize the periodogram L N (ω) by identifying its largest peak(s) and comparing the amplitude of the identified largest peak(s) with a threshold to determine if the largest peak(s) is(are) attributable to a presence of a periodic signal. The coefficients are preferable derived from a time series signal X(t), t=1, 2, . . . , N, but may include receiving a continuous-time signal and converting it to the time series signal.
1. A method of analyzing a spectrum of a real-valued time series signal [X(t), t=1, 2, . . . , N], comprising the steps of,
deriving coefficients [A N (ω), B N (ω)] of an Lp-norm harmonic regression of the real-valued time series signal with 0<p≦∞ and pγ2;
squaring the coefficients;
summing the squared coefficients; and
scaling the summed, squared coefficients by a positive constant c to realize an Lp-norm periodogram of X(t) as L N (ω)=c{[A N (ω)] 2 +[B N (ω)] 2 }.
2. The method as set forth in claim 1 , further comprising steps of:
receiving the real-valued time series signal [X(t), t=1, 2, . . . , N]; and
storing the received real-valued time series signal [X(t), t=1, 2, . . . , N].
3. The method as set forth in claim 1 , further comprising a step of outputting the Lp-norm periodogram, L N (ω)=c{[A N (ω)] 2 +[B N (ω)] 2 }.
4. The method as set forth in claim 3 , further comprising a step of storing the Lp-norm periodogram, L N (ω)=c{[A N (ω)] 2 +[B N (ω)] 2 }.
5. The method as set forth in claim 1 , wherein the coefficients of the Lp-norm harmonic regression representation of the real-valued time series signal are obtained using a simplex algorithm.
6. The method as set forth in claim 1 , wherein the coefficients of the Lp-norm harmonic regression representation of the real-valued time series signal are obtained using an interior-point algorithm.
7. The method as set forth in claim 1 , wherein the Lp-norm harmonic regression includes the L1-norm harmonic regression that produces the Laplace periodogram.
8. The method as set forth in claim 1 , wherein the L1-norm harmonic regression may comprise any one of the following: least absolute deviations (LAD), least absolute errors (LAE), least absolute values (LAV) and median regression.
9. The method as set forth in claim 1 , further comprising steps of:
scanning to maximize the Lp-norm periodogram by identifying its largest peak(s); and
comparing the amplitude of the identified largest peak(s) with a threshold to determine if the largest peak(s) is(are) attributable to a presence of a periodic signal within the real-valued time series signal.
10. The method as set forth in claim 9 , wherein the largest peak(s) of the Lp-norm periodogram provides an accurate estimate of the frequency of the identified periodic signal within the time series.
11. The method as set forth in claim 1 , further comprising:
receiving a real-valued, continuous-time signal, Y(t); and
converting the real-valued, continuous-time signal to the real-valued time series signal [X(n), n=1, 2, . . . , N].