Transformation between distributions in time–frequency analysis

In time-frequency analysis, there should be a procedure to transform one distribution into another. It has be shown that a signal can recovered from a particular distribution if the kernel is not zero in a finite region. Given a distribution for which the signal can be recovered, the recovered signal can be taken to calculate any other distribution, so in these cases a relationship to expected to exist between them.

General Class

All time-frequency representations can be obtained from

   $C(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iiint s^{*}(u-{\dfrac {1}{2}}\tau )s(\mu +{\dfrac {1}{2}}\tau )\phi (\theta ,\phi )e^{-j\theta t-j\tau \omega +j\theta u}\,du\,d\tau \,d\theta$     (1)

where $\phi (\theta ,\phi )$ is a two dimensional function called the kernel, which determines the distribution and its properties.

Characteristic Function Formulation

The characteristic function is the double Fourier transform of the distribution. By inspection of Eq. (1),

   $C(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iint M(\theta ,\tau )e^{-j\theta t-j\tau \omega }\,d\theta \,d\tau$     (2)

where

   ${\begin{alignedat}{2}M(\theta ,\tau )&=\phi (\theta ,\tau )\int s^{*}(u-{\dfrac {1}{2}}\tau )s(u+{\dfrac {1}{2}}\tau )e^{j\theta u}\,du\\&=\phi (\theta ,\tau )A(\theta ,\tau )\\\end{alignedat}}$     (3)

and where $A(\theta ,\tau )$ is the symmetrical ambiguity function. The characteristic function may be appropriately called the generalized ambiguity function.

Transformation between Distribution

To obtain that relationship suppose that there are two distributions, $C_{1}$ and $C_{2}$, with corresponding kernels, $\phi _{1}$ and $\phi _{2}$. Their characteristic functions are

   $M_{1}(\phi ,\tau )=\phi _{1}(\theta ,\tau )\int s^{*}(u-{\dfrac {1}{2}}\tau )s(u+{\dfrac {1}{2}}\tau )e^{j\theta u}\,du$     (4)

   $M_{2}(\phi ,\tau )=\phi _{2}(\theta ,\tau )\int s^{*}(u-{\dfrac {1}{2}}\tau )s(u+{\dfrac {1}{2}}\tau )e^{j\theta u}\,du$     (5)

Divide one equation by the other to obtain

   $M_{1}(\phi ,\tau )={\dfrac {\phi _{1}(\theta ,\tau )}{\phi _{2}(\theta ,\tau )}}M_{2}(\phi ,\tau )$     (6)

This is an important relationship because it connects the characteristic functions. For the division to be proper the kernel cannot to be zero in a finite region.

To obtain the relationship between the distributions take the double Fourier transform of both sides and use Eq. (2)

   $C_{1}(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iint {\dfrac {\phi _{1}(\theta ,\tau )}{\phi _{2}(\theta ,\tau )}}M_{2}(\theta ,\tau )e^{-j\theta t-j\tau \omega }\,d\theta \,d\tau$     (7)

Now express $M_{2}$ in terms of $C_{2}$ to obtain

   $C_{1}(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iiiint {\dfrac {\phi _{1}(\theta ,\tau )}{\phi _{2}(\theta ,\tau )}}C_{2}(t,\omega ^{'})e^{j\theta (t^{'}-t)+j\tau (\omega ^{'}-\omega )}\,d\theta \,d\tau \,dt^{'}\,d\omega ^{'}$     (8)

This relationship can be written as

   $C_{1}(t,\omega )=\iint g_{12}(t^{'}-t,\omega ^{'}-\omega )C_{2}(t,\omega ^{'})\,dt^{'}\,d\omega ^{'}$     (9)

with

   $g_{12}(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iint {\dfrac {\phi _{1}(\theta ,\tau )}{\phi _{2}(\theta ,\tau )}}e^{j\theta t+j\tau \omega }\,d\theta \,d\tau$      (10)

Relation of the Spectrogram to Other Bilinear Representations

Now specialize to the case where one transform from an arbitrary representation to the spectrogram. In Eq. (9), set $C_{1}$ to be the spectrogram and $C_{2}$ to be arbitrary. To simplify notation, set $\phi _{SP}=\phi _{1}$ and $\phi _{=}\phi _{2}$ and $g_{SP}=g_{12}$ and write

   $C_{SP}(t,\omega )=\iint g_{SP}(t^{'}-t,\omega ^{'}-\omega )C(t,\omega ^{'})\,dt^{'}\,d\omega ^{'}$     (11)

The kernel for the spectrogram with window, $h(t)$, is $A_{h}(-\theta ,\tau )$ and therefore

   ${\begin{alignedat}{3}g_{SP}(t,\omega )&={\dfrac {1}{4\pi ^{2}}}\iint {\dfrac {A_{h}(-\theta ,\tau )}{\phi (\theta ,\tau )}}e^{j\theta t+j\tau \omega }\,d\theta \,d\tau \\&={\dfrac {1}{4\pi ^{2}}}\iiint {\dfrac {1}{\phi (\theta ,\tau )}}h^{*}(u-{\dfrac {1}{2}}\tau )h(u+{\dfrac {1}{2}}\tau )\phi (\theta ,\phi )e^{j\theta t+j\tau \omega -j\theta u}\,du\,d\tau \,d\theta \\&={\dfrac {1}{4\pi ^{2}}}\iiint h^{*}(u-{\dfrac {1}{2}}\tau )h(u+{\dfrac {1}{2}}\tau ){\dfrac {\phi (\theta ,\tau )}{\phi (\theta ,\tau \phi (-\theta ,\tau )}}e^{j\theta t+j\tau \omega -j\theta u}\,du\,d\tau \,d\theta \\\end{alignedat}}$      (12)

If taking the kernels for which $\phi (-\theta ,\tau )\phi (\theta ,\tau )=1$, $g_{SP}(t,\omega )$ is just the distribution of the window function, except that it is evaluated at $-\omega$. Therefore,

   $g_{SP}(t,\omega )=C_{h}(t,-\omega )$     (13)

for kernels that satisfy $\phi (-\theta ,\tau )\phi (\theta ,\tau )=1$

and

   $C_{SP}(t,\omega )=\iint C_{s}(t^{'},\omega ^{'})C_{h}(t^{'}-t,\omega ^{'}-\omega )\,dt^{'}\,d\omega ^{'}$     (14)

for kernels that satisfy $\phi (-\theta ,\tau )\phi (\theta ,\tau )=1$

This was shown by Janssen[4]. For the case where $\phi (-\theta ,\tau )\phi (\theta ,\tau )$ does not equal one, then

   $C_{SP}(t,\omega )=\iiiint G(t^{''},\omega ^{''})C_{s}(t^{'},\omega ^{'})C_{h}(t^{''}+t^{'}-t,-\omega ^{''}+\omega -\omega ^{'})\,dt^{'}\,dt^{''}\,d\omega ^{\,}d\omega ^{''}$     (15)

where

   $G(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iint {\dfrac {e^{-j\theta t-j\tau \omega }}{\phi (\theta ,\tau )\phi (-\theta ,\tau )}}\,d\theta \,d\tau$     (16)

References

[1] L. Cohen, TIME-FREQUENCY ANALYSIS, Prentice-Hall, New York, 1995.

[2] L. Cohen, "Generalized phase-space distribution functions," Jour. Math. Phys., vol.7, pp.781-786, 1966.

[3] L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," Jour. Math. Phys., vol.7, pp. 1863-1866, 1976.

[4] A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time frequency plane," Philips Journal of Research, vol. 37, pp. 79-110, 1982.