A FOUR-PARAMETER QUADRATIC DISTRIBUTION David S. Roberts and Douglas L. Jones University of Illinois Coordinated Science Laboratory Urbana, IL ABSTRACT Quadratic distributions such as time-frequency distributions and ambiguity functions have many useful applications. In some cases it is desirable to have a quadratic distribution of more than two variables. Using the technique of apply- ing operators to variables, general quadratic distributions of more than two variables can be developed. We use this tech- nique to develop a four-parameter quadratic distribution that includes variables of time, frequency, lag, and doppler. A general distribution is first developed and some of the math- ematical properties are discussed. The distribution is then applied to the improvement of an adaptive time-frequency distribution. An example signal is shown to evaluate the performance of the technique. 1. FOUR PARAMETER DISTRIBUTION Quadratic distributions have a variety of applications to fields such as time-frequency analysis, RADAR, and analysis of biological signals. Most two-variable quadratic distribu- tions involve the variables of time and frequency, or as in the case of the ambiguity function, lag and doppler. In some cases it would be useful to have a joint distribution of all four of these variables. O’Neil and Williams [1] have developed a quartic version of a time, frequency, lag, and doppler distribution. In order to circumvent the problem of excessive cross-terms inherent to a quartic distribution, we develop here a quadratic version of such a distribution. In order to develop the four-parameter distribution, we will first show how the ambiguity function (AF) and the Wigner distribution (WD) [2] can be cast into a general quadratic form using operator notation. The concept of ap- plying operators to the signal is central to development of the four-parameter distribution. We utilize two operators in this development, the time-shift operator and the frequency- shift operator. The time-shift operator when applied to a signal in the time domain is defined as (T t0 s)(u)= s(u - t 0 ) (1) The frequency-shift operator is defined as (F f0 s)(u)= e j2πf0u s(u) (2) Now examine the AF. (AFs)(τ,θ)=  s(t)s ∗ (t - τ )e -j2πθt dt (3) The AF takes a signal and cross-correlates it with a time and frequency shifted version of the signal. The general quadratic integral representation easily accommodates such a cross-correlation interpretation.  K(u 1 ,u 2 )s(u 1 )s ∗ (u 2 )du 1 du 2 (4) Looking at a quadratic distribution as a cross-correlation of- ten adds valuable insight. The AF can be cast into the form of (4). We can start with the conceptual cross-correlation, which is the signal correlated with a time and frequency shifted version of it- self. (AFs)(τ,θ)= 〈s, F θ 2 T τ 2 s〉 (5) =  s(t)s ∗ (t - τ )e -j2πθt dt (6) From (6), we can easily get to the form of (4) as follows. (AFs)(τ,θ)=  s(t)s ∗ (t - τ )e -j2πθt dt (7) =  e -j2πθ(u2+τ ) s(u 2 + τ )s ∗ (u 2 )du 2 (8) =  e -j2πθ(u2+τ ) δ(u 2 - τ - u 1 )s(u 1 )s ∗ (u 2 )du 1 du 2 Where the kernel K = e -j2πθ(u2+τ ) δ(u 2 - τ - u 1 ). The Wigner distribution will also be an important aspect of