A High Resolution Approach to Estimating Time-Frequency Spectra and Their Amplitudes

A high resolution approach to estimating time-frequency spectra (TFS) and associated amplitudes via the use of variable frequency complex demodulation (VFCDM) is presented. This is a two-step procedure in which the previously developed time-varying optimal parameter search (TVOPS) technique is used to obtain TFS, followed by using the VFCDM to obtain even greater TFS resolution and instantaneous amplitudes associated with only the specific frequencies of interest. This combinational use of the TVOPS and the VFCDM is termed the TVOPS-VFCDM. Simulation examples are provided to demonstrate the performance of the TVOPS-VFCDM for high resolution TFS as well as instantaneous amplitude estimation. The simulation results show that the TVOPS-VFCDM approach provides the highest resolution and most accurate amplitude estimates when compared to the smoothed pseudo Wigner–Ville, continuous wavelet transform and Hilbert–Huang transform methods. Application of the TVOPS-VFCDM to renal blood flow data indicates some promise of a quantitative approach to understanding the dynamics of renal autoregulatory mechanisms as well as a possible approach to quantitatively discriminating between different strains of rats.

Appendices

APPENDIX A: HILBERT TRANSFORM

For an arbitrary time series, X(t) (we assume that X(t) contains center frequency oscillations, i.e., f ₀), its Hilbert transform, Y(t), is as follows:

$$Y(t) = H[X(t)] = \frac{1}{\pi }\int {\frac{{X(t')}}{{t - t'}}dt'}$$

(A.1)

$${\rm The}\,{\rm instantaneous}\,{\rm amplitude}\;A(t) = [X^2 (t) + Y^2 (t)]^{1/2}$$

(17)

$${\rm The}\,{\rm instantaneous}\,{\rm phase}\;\varphi (t) = \arctan \left( {\frac{{Y(t)}}{{X(t)}}} \right)$$

(A.2)

$${\rm The}\,{\rm instantaneous}\,{\rm frequency}\;f(t) = \frac{1}{{2\pi }}\frac{{d\varphi (t)}}{{dt}}$$

(A.3)

APPENDIX B: TIME-VARYING OPTIMAL PARAMETER SEARCH (TVOPS)

The TVOPS method is based on estimating only a few time-varying coefficients that best characterize the dynamics of the system to within a specified degree of accuracy. Thus, the resulting time-varying spectra are not data length dependent, meaning that they are high-resolution time-frequency spectra which are immune from cross-term spectral artifacts when there are multiple components in the signal. To obtain TV spectra, the data are formulated into a time-varying autoregressive (TVAR) model of the form:

$$y(n) = \sum\limits_{i = 1}^P {a(i,n)y(n - i) + e(n)}$$

(B.1)

where $a(i,n)$ are the TVAR coefficients to be determined, and are functions of time. Index P is the maximum order of the AR model. The term e(n) is the residual error. The TVAR coefficients are expanded onto a set of basis functions and then the optimal parameter search algorithm is used to select only the significant TV terms among the chosen P terms. The next step is then to use either the least squares or total least squares methods to estimate TV coefficients. For full details of the TVOPS algorithm, the reader is referred to Reference.18

After obtaining the TV coefficients a(i, n), we can calculate the TV spectrum, S(n, ω) as follows:

$$S(n,\omega ) = \frac{T}{{\left| {1 + \sum\nolimits_{k_1 = 1}^p {a(k_1 ,n)e^{ {-} jwTk_1 } } } \right|^2 }}$$

(B.2)