Cone-shape distribution function

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution, (acronymized as the ZAM distribution or ZAMD ), is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the t, \tau plane. The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel.

The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution, (acronymized as the ZAM distribution or ZAMD ), is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the t, \tau plane. The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel. C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_x(\eta,\tau)\Phi(\eta,\tau)\exp (j2\pi(\eta t-\tau f))\, d\eta\, d\tau, where A_x(\eta,\tau)=\int_{-\infty}^{\infty}x(t+\tau /2)x^*(t-\tau /2)e^{-j2\pi t\eta}\, dt, and the kernel function is \Phi \left(\eta,\tau \right) = \frac{\sin \left(\pi \eta \tau \right)}{ \pi \eta \tau }\exp \left(-2\pi \alpha \tau^2 \right). The kernel function in t, \tau domain is defined as: \phi \left(t,\tau \right) = \begin{cases} \frac{1}{\tau} \exp \left(-2\pi \alpha \tau^2 \right), & |\tau | \ge 2|t|, \\ 0, & \mbox{otherwise}. \end{cases} Following are the magnitude distribution of the kernel function in t, \tau domain. Following are the magnitude distribution of the kernel function in \eta, \tau domain with different \alpha values. As is seen in the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the \tau axis in the \eta, \tau domain, or the ambiguity domain. Therefore, unlike the Choi-Williams distribution function, the cone-shape distribution function can effectively reduce the cross-term results form two component with same center frequency. However, the cross-terms on the \eta axis are still preserved. The cone-shape distribution function is in the MATLAB Time-Frequency Toolbox and National Instruments' LabVIEW Tools for Time-Frequency, Time-Series, and Wavelet Analysis \int_{-\infty}^{\infty} C_x(t,f) df = |x(t)|^2 (Time Marginal) \int_{-\infty}^{\infty} C_x(t,f) dt = |X(f)|^2 (Frequency Marginal) In the case of the Cone-Shape Distribution: Time Marginal: The kernel in the ambiguity domain \Phi(\eta, \tau) must satisfy \Phi(\eta, 0) = 1 . For the ZAM kernel, \Phi(\eta, 0) = \lim_{\tau \to 0} \frac{\sin(\pi \eta \tau)}{\pi \eta \tau} e^{-2\pi \alpha \tau^2} = 1 . Thus, the ZAM distribution preserves the time marginal, meaning the summation over frequency at a given time t yields the instantaneous power of the signal. Frequency Marginal: The condition is \Phi(0, \tau) = 1 . For the ZAM kernel, substituting \eta=0 gives \Phi(0, \tau) = e^{-2\pi \alpha \tau^2} . Since this expression depends on \tau and \alpha and is not strictly 1 (unless \alpha=0 , which trivializes the kernel), the ZAM distribution does not preserve the frequency marginal. This implies that integrating the distribution over time does not perfectly recover the energy spectrum |X(f)|^2 . If y(t) = x(t-t_0) , then C_y(t,f) = C_x(t-t_0, f) . If y(t) = x(t)e^{j2\pi f_0 t} , then C_y(t,f) = C_x(t, f-f_0) . These properties are crucial for analyzing signals where events may occur at arbitrary times or frequencies without distorting the representation shape. Since the kernel function in the time-lag domain \phi(t, \tau) satisfies the conjugate symmetry \phi(t, -\tau) = \phi^*(t, \tau) , the resulting distribution C_x(t,f) is always real-valued. This allows for a straightforward physical interpretation of signal energy, although negative values may still appear (a common trait in Cohen's class excluding the Spectrogram). The Wigner Distribution has an all-pass kernel \Phi_{WDF}(\eta, \tau) = 1 , preserving all cross-terms. The Choi-Williams Distribution uses an exponential kernel that covers both axes but decays away from them. The Cone-Shape Distribution has a unique kernel support region. In the t, \tau domain, the kernel is non-zero only within the cone defined by |\tau| \ge 2|t| . This geometric constraint in the t-\tau domain translates to a low-pass filtering effect on the cross-terms in the Time-Frequency domain. Specifically, the ZAM kernel strongly attenuates interference components that result from signals separated in frequency (which appear on the \tau axis in the ambiguity domain) while preserving the resolution of components separated in time. This makes ZAM particularly effective for signals composed of short-duration events occurring at different frequencies. ==References==

Text derived from https://en.wikipedia.org/w/index.php?oldid=1329225506 under CC BY-SA 4.0; modifications have been made.