Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data. Such methods are used where one needs to deal with a situation where the frequency composition of a signal may be changing over time;[1] this sub-field used to be called time–frequency signal analysis, and is now more often called time–frequency signal processing due to the progress in using these methods to a wide range of signal-processing problems.
Methods for analysing time series, in both signal analysis and time series analysis, have been developed as essentially separate methodologies applicable to, and based in, either the time or the frequency domain. A mixed approach is required in time–frequency analysis techniques which are especially effective in analyzing non-stationary signals, whose frequency distribution and magnitude vary with time. Examples of these are acoustic signals. Classes of "quadratic time-frequency distributions" (or bilinear time–frequency distributions") are used for time–frequency signal analysis. This class is similar in formulation to Cohen's class distribution function that was used in 1966 in the context of quantum mechanics. This distribution function is mathematically similar to a generalized time–frequency representation which utilizes bilinear transformations. Compared with other time–frequency analysis techniques, such as short-time Fourier transform (STFT), the bilinear-transformation (or quadratic time–frequency distributions) may not have higher clarity for most practical signals, but it provides an alternative framework to investigate new definitions and new methods. While it does suffer from an inherent cross-term contamination when analyzing multi-component signals, by using a carefully chosen window function(s), the interference can be significantly mitigated, at the expense of resolution. All these bilinear distributions are inter-convertible to each other, cf. transformation between distributions in time–frequency analysis.
See main article: Wigner distribution function. The Wigner–Ville distribution is a quadratic form that measures a local time-frequency energy given by:
PVf(u,\xi
infty | |
)=\int | |
-infty |
f\left(u+\tfrac{\tau}{2}\right)f*\left(u-\tfrac{\tau}{2}\right)e-i\taud\tau
The Wigner–Ville distribution remains real as it is the fourier transform of f(u + τ/2)·f*(u - τ/2), which has Hermitian symmetry in τ. It can also be written as a frequency integration by applying the Parseval formula:
PVf(u,\xi)=
1 | |
2\pi |
infty | |
\int | |
-infty |
\hat{f}\left(\xi+\tfrac{\gamma}{2}\right)\hat{f}*\left(\xi-\tfrac{\gamma}{2}\right)ei\gammad\gamma
Proposition 1. for any f in L2(R)
infty | |
\int | |
-infty |
PVf(u,\xi)du=|\hat{f}(\xi)|2
infty | |
\int | |
-infty |
PVf(u,\xi)d\xi=2\pi|f(u)|2
Moyal Theorem. For f and g in L2(R),
2\pi\left|
infty | |
\int | |
-infty |
f(t)g*(t)dt
2=\iint{P | |
\right| | |
V |
f(u,\xi)}PVg(u,\xi)dud\xi
Proposition 2 (time-frequency support). If f has a compact support, then for all ξ the support of
PVf(u,\xi)
\hat{f}
PVf(u,\xi)
\hat{f}
Proposition 3 (instantaneous frequency). If
i\phi(t) | |
f | |
a(t)=a(t)e |
\phi'(u)=
| ||||||||||
|
Let
f=f1+f2
PVf=PVf1+PVf2+PV\left[f1,f2\right]+PV\left[f2,f1\right]
PV[h,g](u,\xi
infty | |
)=\int | |
-infty |
h\left(u+\tfrac{\tau}{2}\right)g*\left(u-\tfrac{\tau}{2}\right)e-i\taud\tau
I[f1,f2]=PV[f1,f2]+PV[f2,f1]
(u,\xi)
fa(t)
The interference terms are oscillatory since the marginal integrals vanish and can be partially removed by smoothing
PVf
P\thetaf(u,\xi
infty | |
)=\int | |
-infty |
infty | |
{\int | |
-infty |
{PVf(u',\xi')}}\theta(u,u',\xi,\xi')du'd\xi'
The time-frequency resolution of this distribution depends on the spread of kernel θ in the neighborhood of
(u,\xi)
P\thetaf(u,\xi)\ge0, \forall(u,\xi)\in{{R
The spectrogram and scalogram are examples of positive time-frequency energy distributions. Let a linear transform
Tf(\gamma)=\left\langlef,\phi\gamma\right\rangle
\left\{\phi\gamma\right\}\gamma
(u,\xi)
\phi\gamma
(u,\xi)
PTf(u,\xi)=\left|\left\langlef,\phi\gamma\right\rangle\right|2
From the Moyal formula,
PTf(u,\xi)=
1 | |
2\pi |
infty | |
\int | |
-infty |
infty | |
\int | |
-infty |
PVf(u',\xi')PV\phi\gamma(u',\xi')du'd\xi'
which is the time frequency averaging of a Wigner–Ville distribution. The smoothing kernel thus can be written as
\theta(u,u',\xi,\xi')=
1 | |
2\pi |
PV\phi\gamma(u',\xi')
The loss of time-frequency resolution depends on the spread of the distribution
PV\phi\gamma(u',\xi')
(u,\xi)
A spectrogram computed with windowed fourier atoms,
\phi\gamma(t)=g(t-u)ei\xi
\theta(u,u',\xi,\xi')=
1 | |
2\pi |
PV\phi\gamma(u',\xi')=
1 | |
2\pi |
PVg(u'-u,\xi'-\xi)
For a spectrogram, the Wigner–Ville averaging is therefore a 2-dimensional convolution with
PVg
PVg
PVf
PVf
Wigner Theorem. There is no positive quadratic energy distribution Pf that satisfies the following time and frequency marginal integrals:
infty | |
\int | |
-infty |
Pf(u,\xi)d\xi=2\pi|f(u)|2
infty | |
\int | |
-infty |
Pf(u,\xi)du=|\hat{f}(\xi)|2
The definition of Cohen's class of bilinear (or quadratic) time–frequency distributions is as follows:
Cx(t,
infty | |
f)=\int | |
-infty |
infty | |
\int | |
-infty |
Ax(η,\tau)\Phi(η,\tau)\exp(j2\pi(ηt-\tauf))dηd\tau,
where
Ax(η,\tau)
\Phi(η,\tau)
\Phi\equiv1
An equivalent definition relies on a convolution of the Wigner distribution function (WD) instead of the AF :
Cx(t,
infty | |
f)=\int | |
-infty |
infty | |
\int | |
-infty |
Wx(\theta,\nu)\Pi(t-\theta,f-\nu)d\thetad\nu=[Wx\ast\Pi](t,f)
where the kernel function
\Pi(t,f)
\Pi=\delta(0,0)
\Phi=l{F}t
-1 | |
l{F} | |
f |
\Pi
i.e.
\Phi(η,\tau)=
infty | |
\int | |
-infty |
infty | |
\int | |
-infty |
\Pi(t,f)\exp(-j2\pi(tη-f\tau))dtdf,
or equivalently
\Pi(t,f)=
infty | |
\int | |
-infty |
infty | |
\int | |
-infty |
\Phi(η,\tau)\exp(j2\pi(ηt-\tauf))dηd\tau.
See main article: Ambiguity function. The class of bilinear (or quadratic) time–frequency distributions can be most easily understood in terms of the ambiguity function, an explanation of which follows.
Px(f)
Rx(\tau)
Px(f)=
infty | |
\int | |
-infty |
-j2\pif\tau | |
R | |
x(\tau)e |
d\tau,
Rx(\tau)=
infty | |
\int | |
-infty |
x\left(t+\tfrac{\tau}{2}\right)x*\left(t-\tfrac{\tau}{2}\right)dt.
For a non-stationary signal
x(t)
x(t)
Wx(t,f)=
infty | |
\int | |
-infty |
Rx(t,\tau)e-j2\pid\tau,
Rx(t,\tau)=x\left(t+\tfrac{\tau}{2}\right)x*\left(t-\tfrac{\tau}{2}\right).
If the Fourier transform of the auto-correlation function is taken with respect to t instead of τ, we get the ambiguity function as follows:
Ax(η,\tau)=\int
infty | |
-infty |
x\left(t+\tfrac{\tau}{2}\right)x*\left(t-\tfrac{\tau}{2}\right)ej2\pidt.
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 bilinear (or quadratic) time–frequency distributions with that of the Wigner distribution function, it is easily found that the latter is a special case of the former with
\Phi(η,\tau)=1
\Phi(η,\tau) ≠ 1
What is the benefit of the additional kernel function? 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 ητ-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 ητ-domain. The following is an example that demonstrates how the cross-term is filtered out.
The Fourier transform of
\theta(u,\xi)
\hat{\theta}(\tau,\gamma
infty | |
)=\int | |
-infty |
infty | |
\int | |
-infty |
\theta(u,\xi)e-i(u\gammadud\xi
The following proposition gives necessary and sufficient conditions to ensure that
P\theta
Proposition: The marginal energy properties
infty | |
\int | |
-infty |
P\thetaf(u,\xi)d\xi=2\pi|f(u)|2,
infty | |
\int | |
-infty |
P\thetaf(u,\xi)du=|\hat{f}(\xi)|2
are satisfied for all
f\inL2(R)
\forall(\tau,\gamma)\inR2: \hat{\theta}(\tau,0)=\hat{\theta}(0,\gamma)=1
Aforementioned, the Wigner distribution function is a member of the class of quadratic time-frequency distributions (QTFDs) with the kernel function
\Phi(η,\tau)=1
Wx(t,f)=
infty | |
\int | |
-infty |
x\left(t+\tfrac{\tau}{2}\right)x*\left(t-\tfrac{\tau}{2}\right)e-j2\pid\tau.
See main article: Modified Wigner distribution function.
We can design time-frequency energy distributions that satisfy the scaling property
1 | |
\sqrt{s |
as does the Wigner–Ville distribution. If
g(t)= | 1 |
\sqrt{s |
then
P\thetag(u,\xi)=P\thetaf\left(\tfrac{u}{s},s\xi\right).
This is equivalent to imposing that
\foralls\inR+: \theta\left(su,\tfrac{\xi}{s}\right)=\theta(u,\xi),
and hence
\theta(u,\xi)=\theta(u\xi,1)=\beta(u\xi)
The Rihaczek and Choi–Williams distributions are examples of affine invariant Cohen's class distributions.
The kernel of Choi–Williams distribution is defined as follows:
\Phi(η,\tau)=\exp(-\alpha(η\tau)2),
where α is an adjustable parameter.
The kernel of Rihaczek distribution is defined as follows:
\Phi(η,\tau)=\exp\left(-i2\pi
η\tau | |
2 |
\right),
With this particular kernel a simple calculation proves that
Cx(t,f)=x(t)\hat{x}*(f)ei
See main article: Cone-shape distribution function. The kernel of cone-shape distribution function is defined as follows:
\Phi(η,\tau)=
\sin(\piη\tau) | |
\piη\tau |
\exp\left(-2\pi\alpha\tau2\right),
where α is an adjustable parameter. See Transformation between distributions in time-frequency analysis. More such QTFDs and a full list can be found in, e.g., Cohen's text cited.
A time-varying spectrum for non-stationary processes is defined from the expected Wigner–Ville distribution. Locally stationary processes appear in many physical systems where random fluctuations are produced by a mechanism that changes slowly in time. Such processes can be approximated locally by a stationary process. Let
X(t)
R(t,s)=E[X(t)X(s)]
The covariance operator K is defined for any deterministic signal
f\inL2(R)
infty | |
Kf(t)=\int | |
-infty |
R(t,s)f(s)ds
For locally stationary processes, the eigenvectors of K are well approximated by the Wigner–Ville spectrum.
The properties of the covariance
R(t,s)
\tau=t-s
u= | t+s |
2 |
R(t,s)=R\left(u+\tfrac{\tau}{2},u-\tfrac{\tau}{2}\right)=C(u,\tau)
The process is wide-sense stationary if the covariance depends only on
\tau=t-s
infty | |
Kf(t)=\int | |
-infty |
C(t-s)f(s)ds=C*f(t)
The eigenvectors are the complex exponentials
ei\omega
PX
infty | |
(\omega)=\int | |
-infty |
C(\tau)e-i\omegad\tau
For non-stationary processes, Martin and Flandrin have introduced a time-varying spectrum
PX(
infty | |
u,\xi)=\int | |
-infty |
C(u,\tau)e-i\xid\tau
infty | |
=\int | |
-infty |
E\left[X\left(u+\tfrac{\tau}{2}\right)X\left(u-\tfrac{\tau}{2}\right)\right]e-i\xid\tau
To avoid convergence issues we suppose that X has compact support so that
C(u,\tau)
\tau
PX(u,\xi)=E[PVX(u,\xi)]
which proves that the time varying spectrum is the expected value of the Wigner–Ville transform of the process X. Here, the Wigner–Ville stochastic integral is interpreted as a mean-square integral:[2]
PV(
infty | |
u,\xi)=\int | |
-infty |
\left\{X\left(u+\tfrac{\tau}{2}\right)X\left(u-\tfrac{\tau}{2}\right)\right\}e-i\xid\tau