Choi–Williams distribution function

The Choi–Williams distribution function is a quadratic time-frequency representation belonging to Cohen's class. Introduced by H.-I. Choi and W. J. Williams in 1989, it uses an exponential kernel to reduce interference between signal components.^[1] The amount of attenuation depends on the positions of the interference terms in the ambiguity domain, and stronger smoothing can also change the concentration of the signal's auto-terms.^[2]

Mathematical definition

Let $x (t)$ be a complex signal of finite energy, and use frequency measured in cycles per unit time. With the Fourier-transform convention $X (f) = \int_{ℝ} x (t) e^{- i 2 π f t} d t,$ the centered correlation and ambiguity function are $R_{x} (u, τ) = x (u + τ / 2) x^{*} (u - τ / 2),$ $A_{x} (ν, τ) = \int_{ℝ} R_{x} (u, τ) e^{- i 2 π ν u} d u .$ Here $τ$ is delay, $ν$ is Doppler frequency, and the asterisk denotes complex conjugation. The Choi–Williams distribution is $C_{x}^{(α)} (t, f) = \iint_{ℝ^{2}} A_{x} (ν, τ) Φ_{α} (ν, τ) e^{i 2 π (ν t - f τ)} d ν d τ,$ where $Φ_{α} (ν, τ) = \exp [- α (ν τ)^{2}], α > 0.$ This exponential-kernel convention is used in Ding's lecture notes on Cohen-class distributions.^[3] The multiplier acts on the ambiguity function before the inverse transform to the time-frequency plane. The constant multiplier $Φ = 1$ gives the Wigner distribution.^[4]

Equivalent time-lag representation

An equivalent representation is obtained by transforming the multiplier in its Doppler coordinate. For nonzero delay, $k_{α} (v, τ) = \int_{ℝ} Φ_{α} (ν, τ) e^{i 2 π ν v} d ν = \frac{\sqrt{π}}{\sqrt{α} | τ |} \exp [- \frac{π^{2} v^{2}}{α τ^{2}}] .$ The distribution can therefore be written as $C_{x}^{(α)} (t, f) = \int_{ℝ} [\int_{ℝ} k_{α} (t - u, τ) R_{x} (u, τ) d u] e^{- i 2 π f τ} d τ .$ The inner integral smooths the correlation in time using a Gaussian whose width depends on the delay.^[2] For each nonzero delay, the Gaussian kernel has unit integral in $v$ . Its width tends to zero with the delay, and its zero-delay interpretation is the distributional limit $k_{α} (v, 0) = δ (v)$ . Thus the apparent division by $| τ |$ is handled by a limiting kernel rather than by evaluating that quotient at zero.^[2]

Kernel geometry and cross-term suppression

For a multicomponent signal, the quadratic correlation contains auto-terms from the individual components and cross-terms between different components. The Choi–Williams multiplier weights both types of terms in the ambiguity domain.^[2] On either coordinate axis it satisfies $Φ_{α} (0, τ) = Φ_{α} (ν, 0) = 1.$ Away from the axes, its attenuation depends on the squared product $(ν τ)^{2}$ . Consequently its pass region is cross-shaped rather than circular.^[4] Interference near either axis can remain prominent even when interference farther from the axes is strongly attenuated. The orientation of the signal components therefore affects the reduction of cross-terms. The Time-Frequency Toolbox tutorial illustrates this effect with Gaussian atoms arranged at different orientations in the time-frequency plane.^[4] In particular, components with equal time centers or equal frequency centers place their cross-ambiguity centers on an axis. Unit gain at such a center does not mean that every point of a finite-width cross-ambiguity lobe is unchanged: points away from the axis can still be attenuated.^[4] Increasing the attenuation may also change the auto-terms. Cross-term suppression and preservation of component concentration are therefore separate considerations.^[2]

Properties

Energy marginals

For sufficiently regular signals, the time marginal is $\int_{ℝ} C_{x}^{(α)} (t, f) d f = | x (t) |^{2},$ and the frequency marginal is $\int_{ℝ} C_{x}^{(α)} (t, f) d t = | X (f) |^{2} .$ The corresponding conditions on a Cohen-class multiplier are $Φ (ν, 0) = 1$ and $Φ (0, τ) = 1$ , respectively. The Choi–Williams multiplier satisfies both.^[2] These conditions follow by integrating the defining expression over one output coordinate. Integration over frequency selects zero delay, giving $\int_{ℝ} C_{x}^{(α)} (t, f) d f = \int_{ℝ} A_{x} (ν, 0) e^{i 2 π ν t} d ν = R_{x} (t, 0) .$ Integration over time selects zero Doppler, giving $\int_{ℝ} C_{x}^{(α)} (t, f) d t = \int_{ℝ} A_{x} (0, τ) e^{- i 2 π f τ} d τ .$ The latter expression is the Fourier transform of the signal's autocorrelation and equals its energy spectrum.^[2] In particular, the signed integral over the whole time-frequency plane equals the signal energy. Marginal preservation does not imply pointwise nonnegativity.^[2]

Reality and covariance

The ambiguity function satisfies the conjugate symmetry $A_{x} (- ν, - τ) = A_{x} (ν, τ)^{*} .$ The exponential multiplier is real and unchanged by simultaneous reversal of its coordinates. The product $A_{x} Φ_{α}$ therefore has the conjugate symmetry required for a real-valued time-frequency distribution.^[2] For a translated and modulated signal $y (t) = e^{i 2 π f_{0} t} x (t - t_{0}),$ the representation obeys $C_{y}^{(α)} (t, f) = C_{x}^{(α)} (t - t_{0}, f - f_{0}) .$ This is covariance under time and frequency shifts: the distribution shifts with the signal.^[2]

Parameter conventions

Numerical parameter values must be interpreted using the kernel formula in which they appear. In the convention $\exp [- α (ν τ)^{2}]$ , larger $α$ gives stronger attenuation away from the axes. As $α \to 0^{+}$ , the multiplier tends to one, giving the Wigner limit.^[4] Hlawatsch and Boudreaux-Bartels use the convention $Φ (ν, τ) = \exp [- \frac{(2 π ν τ)^{2}}{σ_{H}}] .$ Comparison of the exponents gives $α = \frac{4 π^{2}}{σ_{H}} .$ Their Gaussian time-lag kernel is the same kernel after this change of parameter.^[2] The Time-Frequency Toolbox tutorial uses $Φ (ν, τ) = \exp [- \frac{(π ν τ)^{2}}{2 σ_{T}^{2}}],$ which corresponds to $α = \frac{π^{2}}{2 σ_{T}^{2}} .$ In that convention, increasing $σ_{T}$ reduces the attenuation.^[4] Parameters bearing the same name in different references thus need not represent the same degree of smoothing. The balance between interference reduction and auto-term distortion depends on the signal as well as on the chosen convention.^[2]^[4]

References

↑ Choi, H.-I.; Williams, W. J. (1989). "Improved time-frequency representation of multicomponent signals using exponential kernels". IEEE Transactions on Acoustics, Speech, and Signal Processing. 37 (6): 862–871. doi:10.1109/ASSP.1989.28057.
1 2 3 4 5 6 7 8 9 10 11 12 Hlawatsch, Franz; Boudreaux-Bartels, G. Faye (1992). "Linear and quadratic time-frequency signal representations" (PDF). IEEE Signal Processing Magazine. 9 (2): 21–67. doi:10.1109/79.127284. See Tables IV–V, pp. 40–41.
↑ Ding, Jian-Jiun. "Time-Frequency Analysis and Wavelet Transform". Department of Electrical Engineering. National Taiwan University. Lecture 6, section VI-A-3, printed p. 186. Retrieved 2026-10-12. Lecture 6 PDF, PDF p. 10.
1 2 3 4 5 6 7 Auger, François; Flandrin, Patrick; Gonçalvès, Paulo; Lemoine, Olivier (2005-10-26). Time-Frequency Toolbox Tutorial (PDF). pp. 78–82.