The Wigner function and non-stationarity

The Wigner function serves as a pivotal tool for analyzing
non-stationary processes, especially when considering the evolution of
spectral properties over time. In the context of a process that is mildly
non-stationary---where the process exhibits translational invariance in terms
of waveform shape but demonstrates variation in scale---this function can be
instrumental in capturing and quantifying the dynamics of such changes.

Given a time series $x [t]$ with its non-stationary auto-covariance function
\begin{equation}
  C_x (t_1, t_2) = \langle (x [t_1] - \mu [t_1]) (x [t_2] - \mu [t_2])^{\ast}
  \rangle
\end{equation}
where $\mu (t)$ represents the time-dependent mean and \tmrsup{$\ast$} denotes
the complex conjugate, the process involves first identifying the time-lagged
auto-correlation by considering the average time $t = \frac{t_1 + t_2}{2}$ and
time lag $\tau = t_1 - t_2$.

To analyze the non-stationary correlation that slowly grows over time,
indicating changes in the scale of waveform based on interval distances, we
utilize the Wigner function defined as:
\begin{equation}
  W_x (t, f) = \int_{- \infty}^{\infty} C_x \left( t + \frac{\tau}{2}, t -
  \frac{\tau}{2} \right) e^{- 2 \pi i \tau f} d \tau
\end{equation}
This definition leverages the Fourier transform of the lagged auto-correlation
function, effectively transitioning from a time-domain representation to a
time-frequency domain representation, where $f$ denotes frequency. For a
mean-zero time series, this simplifies to:
\begin{equation}
  W_x (t, f) = \int_{- \infty}^{\infty} x \left( t + \frac{\tau}{2} \right)
  x^{\ast}  \left( t - \frac{\tau}{2} \right) e^{- 2 \pi i \tau f} d \tau
\end{equation}
In the scenario where the scale of waveform changes over time, the Wigner
function's time-frequency representation will reveal how the spectral density,
or energy distribution across frequencies, evolves. For mildly non-stationary
processes, the gradual change in waveform scale can be traced as a function of
time within the Wigner distribution, thus providing a comprehensive view of
the process's dynamic behavior in both time and frequency domains.

The key advantage here is that, unlike traditional Fourier analysis which
assumes stationarity, the Wigner function accommodates non-stationarity,
allowing for the examination of how specific frequencies' contributions to the
process change over time. This is particularly useful in identifying and
characterizing the non-stationary correlation's evolution, providing insights
into the underlying dynamics of the process.

By analyzing the Wigner function across different time intervals, one can
detect variations in the spectral density that correspond to the slowly
growing non-stationary correlation, thereby capturing the essence of how the
process's characteristics evolve. This method is not only pertinent for
theoretical analysis but also for practical applications where understanding
the temporal evolution of spectral properties is crucial.