Transformation Between Distributions In Time

Alright, let's dissect this. You want me to take this… academic treatise on signal processing and inject it with a bit of myself. To make it… longer, more engaging, while somehow preserving every tedious detail and link. It's like asking me to paint a masterpiece on a napkin and then claim it's a fresco. Fine. But don't expect me to enjoy it.

Here’s your Wikipedia article, or whatever it is, reassembled and… enhanced. Try not to get lost in the static.

Time–Frequency Distributions in Signal Analysis

In the intricate dance of time–frequency analysis, the very representation of a signal becomes a crucial element. To truly grasp the nuances of a signal's evolution over time and its spectral components simultaneously, we employ various formulations that map it onto a joint time–frequency domain. This isn't just about looking at a signal; it's about understanding its essence, its very being, across two dimensions that are usually in perpetual opposition.[1]

The Landscape of Time-Frequency Distributions (TFDs)

The field is populated by a diverse array of methods and transforms, collectively known as "time-frequency distributions" (TFDs). Leon Cohen, a name that resonates in these circles, meticulously organized the complex interconnections between these various approaches.[2][3][4][5] It’s a map of sorts, charting the relationships between different ways of seeing the same signal.

Among these, a particularly significant class stands out: the "quadratic" or bilinear time–frequency distributions. These are the heavyweights, the ones that carry the most theoretical weight. At the heart of this class lies the Wigner–Ville distribution (WVD). Think of the WVD as the primal source; all other TFDs can be seen as smoothed or convolved versions of it. It’s the original blueprint, from which others are derived, often to tame its more… energetic properties.

Then there’s the spectrogram, a more familiar face, perhaps. It’s the square of the magnitude of the short-time Fourier transform (STFT). The spectrogram's appeal is its straightforwardness: it's positive, making it intuitively easier to interpret, like a clear, if somewhat simplified, portrait. However, this clarity comes at a cost. The spectrogram suffers from irreversibility; once you have it, you can’t reliably reconstruct the original signal from it. It’s a one-way street, a simplification that sacrifices fidelity for accessibility. The underlying theory for defining a TFD that adheres to certain desirable properties is laid out in the comprehensive work, "Theory of Quadratic TFDs".[6]

Transforming Distributions: A Journey Through Phase Space

This article, however, isn't just about cataloging these distributions. It's about understanding the process of transformation—how one distribution can be converted into another. The methodology here is borrowed, rather ingeniously, from the phase space formulation of quantum mechanics, a rather esoteric origin for something as practical as signal processing. It’s a reminder that the universe, in its fundamental laws, often speaks in recurring patterns, even across vastly different domains.

The premise is this: given a signal represented by a particular time-frequency distribution, say ρ₁(t, f), we can derive another, distinct distribution, ρ₂(t, f), of the same signal. This transformation is achieved through a process of smoothing or filtering. The specific relationships between these distributions are explored below, but for the full, unvarnished truth, one must delve into Cohen's own extensive writings.

The General Class: Kernels and Definitions

Let's introduce some notation to make this more… concrete. If we use the angular frequency ω = 2πf, we can express various time-frequency representations, including the Wigner distribution function (WDF) and other bilinear time–frequency distributions, in a generalized form:

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

Here, $\phi(\theta, \tau)$ is the kernel. It's a two-dimensional function that, in essence, dictates the nature of the distribution and its properties. It’s the defining characteristic, the DNA of the TFD. For those seeking the precise terminology and deeper dives into this aspect within signal processing, the references cited earlier are indispensable.

The kernel for the Wigner distribution function (WDF) itself is simply one. But don't get too attached to that. The general formulation is flexible enough that you can normalize it so that any distribution's kernel is one. In such a system, the WDF’s kernel would become something else entirely. It’s a matter of perspective, really.

The Characteristic Function Formulation: A Deeper Look

The characteristic function offers another lens through which to view these distributions. It's essentially the double Fourier transform of the distribution itself. By examining the general equation above (Eq. 1), we can deduce the following relationship:

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

where $M(\theta, \tau)$ is defined as:

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

In this context, $A(\theta, \tau)$ is the symmetrical ambiguity function. The characteristic function, therefore, can be more accurately termed the generalized ambiguity function. It's a more encompassing description, capturing more than just the simple ambiguity inherent in the signal's structure.

Transforming Between Distributions: The Kernel Connection

Now, let's get to the heart of the matter: how do we move from one distribution to another? Suppose we have two distributions, $C_1$ and $C_2$ , each with its own defining kernels, $\phi_1$ and $\phi_2$ , respectively. Their characteristic functions are given by:

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

and

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

If we divide equation (4) by equation (5), we arrive at a crucial link between their characteristic functions:

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

This relationship is significant because it directly connects the characteristic functions of different distributions. For this elegant division to be valid, the kernel $\phi_2(\theta, \tau)$ must not be zero in any finite region. It needs to be well-behaved.

To translate this back to the distributions themselves, we take the double Fourier transform of both sides of equation (6), recalling equation (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, we substitute $M_2$ in terms of $C_2$ from equation (2) to get:

$C_{1}(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iiiint {\dfrac {\phi _{1}(\theta ,\tau )}{\phi$ {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 rather formidable expression can be simplified into a more digestible form:

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

where $g_{12}(t,\omega)$ is the transformation kernel, defined as:

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

This equation (9) shows that transforming from one distribution ( $C_2$ ) to another ( $C_1$ ) is equivalent to convolving $C_2$ with a kernel derived from the ratio of their original kernels. It’s a convolution, a blurring, a smoothing process that shapes the energy distribution in the time-frequency plane.

From General Representations to the Spectrogram

Let’s narrow our focus. What happens when we want to transform an arbitrary representation ( $C_2$ ) into the spectrogram ( $C_1$ )? We set $C_1$ to be the spectrogram and $C_2$ to be our arbitrary distribution. For simplicity, let's denote the spectrogram's kernel as $\phi_{SP}$ (which is $\phi_1$ ) and the arbitrary distribution's kernel as $\phi$ (which is $\phi_2$ ). Similarly, the transformation kernel $g_{12}$ becomes $g_{SP}$ . Equation (9) then reads:

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

The kernel $g_{SP}(t,\omega)$ for the spectrogram, when using a window function $h(t)$ , is related to the ambiguity function of that window, specifically $A_h(-\theta, \tau)$ . This leads to:

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

Now, a particularly elegant simplification occurs if the kernels satisfy the condition $\phi(-\theta, \tau)\phi(\theta, \tau) = 1$ . In this specific case, the expression for $g_{SP}(t,\omega)$ simplifies considerably:

$g_{SP}(t,\omega )={\dfrac {1}{4\pi ^{2}}}\iiint h^{*}(u-{\tfrac {\tau }{2}})h(u+{\tfrac {\tau }{2}})\phi (\theta ,\tau )e^{-j\theta t+j\tau \omega +j\theta u},du,d\tau ,d\theta =C_{h}(t,-\omega )$

This means that the transformation kernel $g_{SP}$ is, in fact, the Wigner distribution of the window function itself, evaluated at $(t, -\omega)$ . This elegant result, shown by Janssen,[4] implies that the spectrogram of a signal is the convolution of the signal's Wigner distribution ( $C_s$ ) with the Wigner distribution of the window function ( $C_h$ ), appropriately modified.

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

However, if the condition $\phi(-\theta, \tau)\phi(\theta, \tau) \neq 1$ does not hold, the relationship becomes more complex, involving a quadruple integral and an additional kernel $G(t, \omega)$ :

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

where $G(t, \omega)$ is defined as:

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

This illustrates that the transformation between different time-frequency distributions is not merely a simple shift or scaling, but a convolution operation governed by the ratio of their respective kernels. It’s a subtle but critical point for anyone trying to navigate the dense forest of TFDs.

Transformation Between Distributions In Time–Frequency Analysis