Cramér-Rao Bounds for Wavelet Frequency Estimates

R.A. Scheper
Tactical Electronic Warfare Division

A. Teolis
AIMS, Inc.

Introduction: The problem of information extraction from noisy frequency-modulated (FM) signals is of fundamental importance in a variety of Naval applications including communication, radar, and electronic surveillance. As modulation techniques become more complex and signal bandwidths move into the GHz range, extraction algorithms must become ever more sophisticated and yet remain computationally tractable so as to process incoming signals at or near real time. We consider the well-studied problem of instantaneous frequency (IF) estimation from a novel perspective in the wavelet transform domain.

A very first step in the development of algorithms is to identify the fundamental limits of their performance. In estimation applications, this usually translates into determining a fundamental limit on the variance of the estimator. For unbiased estimators, this limit is the Cramér-Rao bound (CRB). In establishing the minimum attainable variance for any estimate, the CRB provides a standard against which to measure estimation algorithm performance in the presence of noise.

Wavelet-Based IF Estimation: Wavelet transforms offer themselves as powerful tools for the analysis of FM signals. A wavelet transform takes a onedimensional signal of time into a two-dimensional signal of time and frequency. The result is a joint time-frequency (TF) distribution that describes the evolution of the signal frequency content over time. Figure 1 depicts a wavelet filter bank and its response to a linear chirp signal. Judicious choice of the transform parameters yields TF distributions that tend to concentrate the signal energy along the frequency axis at or near the IF while simultaneously dispersing the noise energy uniformly. As such, they are inherently noise-tolerant and provide a basis for robust parameter estimation techniques.

FIGURE 1
A wavelet filter bank and its response to an input signal with a linear FM (chirp).

Both signal and noise are modeled under the wavelet transform to provide an analytical relationship between the parameters of interest and the wavelet transform of the signal. Figure 2 shows the response of a wavelet filter bank to a noiseless fixed frequency sinusoid (left) and the same signal in noise with a 10 dB signal-to-noise ratio (right). An asymptotic approximation is developed for a wavelet transform of an FM signal to yield a transform model. This model explicitly depends on the key parameters of signal amplitude, signal phase, signal instantaneous frequency, and noise level. Fitting the model to the observed wavelet output provides the IF estimation mechanism. Figure 3 shows the model matching procedure for IF estimation; similar estimation procedures are possible for the other parameters.

FIGURE 2
Time-frequency distributions associated with a fixed-frequency sinusoidal pulse:(left)distribution for signal only,and (right)distribution for signal plus noise.

Fig3 Image

FIGURE 3
Estimation of signal IF via transform matching technique using a wavelet filter bank.

Discussion: Obtaining simple transform approximation formulas is the key for developing fast and conceptually simple algorithms for estimating signal IF (and other parameters). Moreover, a model approximation of a given wavelet transform can provide the means to assess the performance of a particular algorithm when applied to noisy data. Some feasible tests include: comparisons of obtainable parameter variances between candidate algorithms for a given transform; specific comparisons of algorithm performance to the parent transform CRBs; comparisons of CRBs between different transforms; sensitivity of a given transform CRB to input parameters; and comparison of a given transform CRB to other extraction methods (e.g., polynomial phase model). In all, asymptotic approximations to wavelet transforms provide a good framework for both the development and analysis of IF estimation algorithms.

Summary: We exploit the property of the wavelet transform to concentrate signal energy along its FM trajectory in the wavelet domain to simultaneously achieve several objectives. These are (a) to develop an integral approximation of the wavelet transform; (b) to use the approximation to develop computational algorithms for IF estimation; and (c) to use the approximation to obtain the CRB of the IF estimate.

[Sponsored by NRL]