Cohen's class distribution function

For many years, signal analysis in both time and frequency domain has enjoyed high degree of popularity. The time-frequency analysis techniques are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time, for example the acoustic signals. Among the time-frequency analysis techniques, Cohen's class distribution function is one of the most powerful techniques.

Cohen's class distribution function was first proposed by Leon Cohen in 1966. This distribution function is a generalized time-frequency representation which utilizes bilinear transformations. Comparing with other time-frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation based Cohen’s class distribution function has obviously higher clarity. But on the other hand, it suffers from the inherent cross-term when analyzing multi-component signals. Fortunately, by a properly chosen mask kernel function, the interference can be significantly mitigated.

Mathematical definition

The definition of the Cohen’s class distribution function is as follows:

:$C\_x(t,\; f)=int\_\{-infty\}^\{infty\}int\_\{-infty\}^\{infty\}A\_x(eta,\; au)Phi(eta,\; au)exp\; (j2pi(eta\; t-\; au\; f)),\; deta,\; d\; au,$

where $A\_xleft(eta,\; au\; ight)$ is the ambiguity function (AF) which will be further discussed later, and $Phi\; left(eta,\; au\; ight)$ is the kernel function which is usually a low-pass function and is used to mask out the interference.

Ambiguity function

Since the Cohen’s class distribution function can be easily illustrated through the ambiguity function, we will first explain the meaning of the ambiguity function.

Let’s get started from the well known power spectral density $P\_xleft(f\; ight)$ and the signal auto-correlation function $R\_xleft(\; au\; ight)$. The relationship between the power spectral density and auto-correlation function is as follows:

:$P\_x(f)=\; int\_\{-infty\}^\{infty\}R\_x(\; au)e^\{-j2pi\; f\; au\},\; d\; au,$

:$R\_x(\; au)\; =\; Eleft\; [x(t+\; au\; /2)*x(t-\; au\; /2)\; ight]$.

For a non-stationary signal $xleft(t\; ight)$, we can generalize the time-dependent power spectral density or equivalently the famous Wigner distribution function of $xleft(t\; ight)$ as follows:

:$W\_x(t,\; f)=\; int\_\{-infty\}^\{infty\}R\_x(t,\; au)e^\{-j2pi\; f\; au\},\; d\; au,$

:$R\_xleft(t\; ,\; au\; ight)\; =\; x(t+\; au\; /2)*x(t-\; au\; /2).$

On the other hand, if we take the Fourier transform of the auto-correlation function with respect to $t$ instead of $au$, we get the ambiguity function as follows:

:$A\_x(eta,\; au)=int\_\{-infty\}^\{infty\}x(t+\; au\; /2)*x(t-\; au\; /2)e^\{-j2pi\; teta\},\; dt.$

The relationship between the Wigner distribution function, the auto-correlation function and the ambiguity function can then be illustrated by the following figure.

By comparing the definition of Cohen’s class distribution function and of the Wigner distribution function, we can easily find that the latter is a special case of the former with $Phi\; left(eta,\; au\; ight)\; =\; 1$. Alternatively, Cohen’s class distribution function can be regarded as a masked version of the Wigner distribution function if a kernel function $Phi\; left(eta,\; au\; ight)\; eq\; 1$ is chosen. A properly chosen kernel function can significantly reduce the undesirable cross-term of the Wigner distribution function.

How can we get benefit from the additional kernel function? Let’s examine the property of the ambiguity function and Wigner distribution function a little more closely. The following figure shows the distribution of the auto-term and the cross-term of a multi-component signal in both the ambiguity and the Wigner distribution function.

For multi-component signals in general, the distribution of its auto-term and cross-term within its Wigner distribution function is generally not predictable, and hence the cross-term cannot be removed easily. However, as shown in the figure, for the ambiguity function, the auto-term of the multi-component signal will inherently tend to close the origin in the $eta,\; au$ plane, and the cross-term will tend to be away from the origin. With this property, the cross-term in can be filtered out effortlessly if a proper low-pass kernel function is applied in $eta,\; au$ domain. Following is an example that demonstrates how the cross-term is filtered out.

Some members of Cohen’s class distribution function

Wigner distribution function

Aforementioned, the Wigner distribution function is a member of Cohen's class with the kernel function $Phi\; left(eta,\; au\; ight)\; =\; 1$. The definition of Wigner distribution function is as follows:

:$W\_x(t,\; f)=\; int\_\{-infty\}^\{infty\}x(t+\; au\; /2)*x(t-\; au\; /2)e^\{-j2pi\; f\; au\},\; d\; au.$

Choi-Williams distribution function

The kernel of Choi-William distribution function is defined as follows:

:$Phi\; left(eta,\; au\; ight)\; =\; exp\; left\; [-alpha\; left(eta\; au\; ight)^2\; ight]\; ,$

where $alpha$ is an adjustable parameter.

Cone-shape distribution function

The kernel of Cone-shape distribution function is defined as follows:

:$Phi\; left(eta,\; au\; ight)\; =\; frac\{sin\; left(pi\; eta\; au\; ight)\}\{\; pi\; eta\; au\; \}exp\; left(-2pi\; alpha\; au^2\; ight),$

where $alpha$ is an adjustable parameter.

References

*Jian-Jiun Ding, Time frequency analysis and wavelet transform class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
*S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996.
*L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
*H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE. Trans. Acoustics, Speech, Signal Processing, vol. 37, no. 6, pp. 862-871, June 1989.
*Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 38, no. 7, pp. 1084-1091, July 1990.