A four-parameter quadratic distribution

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 applying operators to variables, general quadratic distributions of more than two variables can be developed. We use this technique 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 mathematical 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 distributions 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 applying 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 frequencyshift operator. The time-shift operator when applied to a signal in the time domain is defined as (Tt0s)(u) = s(u− t0) (1) The frequency-shift operator is defined as (Ff0s)(u) = e 0s(u) (2) Now examine the AF. (AFs)(τ, θ) = ∫ s(t)s∗(t− τ)e−j2πθtdt (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(u1, u2)s(u1)s∗(u2)du1du2 (4) Looking at a quadratic distribution as a cross-correlation often 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 itself. (AFs)(τ, θ) = 〈s,F θ 2 T τ 2 s〉 (5) = ∫ s(t)s∗(t− τ)e−j2πθtdt (6) From (6), we can easily get to the form of (4) as follows. (AFs)(τ, θ) = ∫ s(t)s∗(t− τ)e−j2πθtdt (7) = ∫ e−j2πθ(u2+τ)s(u2 + τ)s∗(u2)du2 (8) = ∫ ∫ e−j2πθ(u2+τ)δ(u2 − τ − u1)s(u1)s∗(u2)du1du2 Where the kernel K = e−j2πθ(u2+τ)δ(u2 − τ − u1). The Wigner distribution will also be an important aspect of the four-parameter distribution and we would like to also define it in terms of the time and frequency shift operators. The subtle difference with the WD is the inversion of the variable of integration. WD(t, f) = ∫ s(t+ τ 2 )s∗(t− τ 2 )edτ (9) The WD is a cross-correlation of the signal with a timereversed, time-shifted, and frequency-modulated version of itself. To see this more clearly, let’s make a change of variables τ2 = u− t. WD(t, f) = ∫ s(t+ τ 2 )s∗(t− τ 2 )edτ (10) = e ∫ s(u)s∗(−u+ 2t)e−j4πfudu (11) WD(t, f) = e ‘ ∫ s(u)s∗(−u+ t‘)e−j4πfudu We cross-correlate the signal with a version that is reversed in time, shifted in time by an amount t , and modulated by the complex sinusoid e−j4πfu. This makes intuitive sense if we think about the WD. By shifting the signal in time and reversing it, we are folding a section of the future back onto the past, which tells us how localized in time a signal is. Modulating by the sinusoid gives us information about the frequency content at that time. The WD can also be represented in the form of (4) ; (WDs)(τ, θ) = ∫ ∫ K(u1, u2)s(u1)s∗(u2)du1du2 (12) = ∫ ∫ ej2πf(t−2u1)δ(u2 + u1 − t)s(u1)s∗(u2)du1du2 = ∫ ej2πfte−j4πfu1s(u1)s∗(−u1 + t)du1 = WD(t, f) Notice the form of the kernel that we used to get to the WD. It was a δ function that essentially forced an integration along a line in the two dimensional u1 − u2 plane. Cohen’s class of time-frequency distributions is a concise way of describing many time-frequency distributions in terms of convolving a kernel with the WD. We can obtain Cohen’s class from the general quadratic integral form with a little bit of work as shown [3]. (Ps)(t, f) = 〈KPF−fT−ts,F−fT−ts〉 (13) = ∫ ∫ KP (u2, u1)s(u1 + t)s∗(u2 + t) (14) e−j2πf(u1−u2)du1du2 = ∫ ∫ Π(u, τ)s(t+ u+ τ 2 )s∗(t+ u− τ 2 ) e−j2πfτdudτ = ∫ ∫ Φ(u− t, v − f)(Ws)(u, v)dudv Where Ws represents the Wigner distributions of the signal s. The motivation for the four-parameter distribution is to combine the operator form of the AF (5) with the operator form of Cohen’s class of time-frequency distributions (13). We will retain the generality of the quadratic integral form by including a yet unspecified kernel in the description. The end goal is to develop a time-frequency distribution that has variables of time, frequency, time lag, and frequency lag. We start by simply combining the forms of (5) and (13); (Ps)(t, f, τ, θ) = 〈KpF−θ 2 F−τ 2 F−t 2 F−f 2 s,F θ 2 F τ 2 F−t 2 F−f 2 s〉 = ∫ ∫ Kp(u1, u2)s(u1 + t+ τ 2 )s∗(u2 + t− τ2 ) e−jπθ(u1+u2)e−j2πf(u1−u2+τ)du1du2 (15) We now have a very general description of a t, f, τ, θ representation. We need to impose some further structure on the form of the kernel in order to gain some insight from such a general description. If we think of the signal in twodimensional time-frequency space, we are trying to obtain the AF of a specific region that is bandlimited in both time and frequency. We could vary the kernel with time and frequency to excise a particular t− f region. A better alternative is to make the kernel independent of t, f by making it a have low-pass properties in time and frequency and shifting the desired t−f portion of the signal under the kernel using operators. Since (15) implements the desired t − f shift, we follow the latter approach. We next discuss a few basic properties of the distribution.