Chapter 3: Relationship Between Frequency and Time Domains
1. Correspondence
The frequency-domain and time-domain Wiener filters are related through the Fourier transform. The correspondence is summarized below.
| Frequency Domain | Time Domain | Meaning |
|---|---|---|
| $P_S(\omega)$ | $\boldsymbol{R}_{ss}$ | Signal autocorrelation |
| $P_N(\omega)$ | $\boldsymbol{R}_{nn}$ | Noise autocorrelation |
| $H^*(\omega)P_S(\omega)$ | $\boldsymbol{p} = E[\boldsymbol{x}d^*]$ | Cross-correlation |
| $|H(\omega)|^2 P_S(\omega) + P_N(\omega)$ | $\boldsymbol{R} = E[\boldsymbol{x}\boldsymbol{x}^H]$ | Observation autocorrelation |
| $G(\omega)$ | $\boldsymbol{w}_{\rm opt}$ | Optimal filter |
By the Wiener-Khinchin theorem, the power spectral density $P(\omega)$ and the autocorrelation function $R(\tau)$ form a Fourier transform pair:
\begin{equation} P(\omega) = \int_{-\infty}^{\infty} R(\tau) e^{-j\omega\tau} d\tau \end{equation}2. Proof of Essential Equivalence
2.1 Both Starting Points
Frequency-domain optimal solution (Chapter 1):
\begin{equation} G(\omega) = \frac{H^*(\omega) P_S(\omega)}{|H(\omega)|^2 P_S(\omega) + P_N(\omega)} \label{eq:wiener_freq} \end{equation}Time-domain optimal solution (Chapter 2, Wiener-Hopf equation):
\begin{equation} \boldsymbol{w}_{\rm opt} = \boldsymbol{R}^{-1}\boldsymbol{p} \end{equation}2.2 Derivation of Equivalence
For the observation $y(t) = (h * s)(t) + n(t)$ ($h$ is the degradation filter's impulse response, $s$ and $n$ are uncorrelated: $E[s(t)n^*(t')] = 0$), we relate each time-domain quantity to its frequency-domain counterpart via the Fourier transform. We write the signal autocorrelation as $R_s(\tau) = E[s(t)s^*(t-\tau)]$ and the noise autocorrelation as $R_n(\tau) = E[n(t)n^*(t-\tau)]$.
Step 1: Fourier transform of the observation autocorrelation $R_y$
Write out $y(t)$ and $y^*(t-\tau)$ using the convolution definition:
\begin{align} y(t) &= \int h(u) \, s(t-u) \, du + n(t) \nonumber \\ y^*(t-\tau) &= \int h^*(v) \, s^*(t-\tau-v) \, dv + n^*(t-\tau) \end{align}Multiplying and taking the expectation. Since $s$ and $n$ are uncorrelated, the cross-terms vanish:
\begin{align} R_y(\tau) &= E[y(t)y^*(t-\tau)] \nonumber \\ &= \iint h(u) h^*(v) \, E[s(t-u) s^*(t-\tau-v)] \, du\,dv + R_n(\tau) \nonumber \\ &= \iint h(u) h^*(v) \, R_s(\tau + v - u) \, du\,dv + R_n(\tau) \end{align}The last equality uses stationarity: $E[s(t-u)s^*(t-\tau-v)] = R_s((t-u)-(t-\tau-v)) = R_s(\tau+v-u)$.
This double integral equals a convolution of $R_s$ with $h$ and then with $h^*(-t)$. By the convolution theorem, convolutions become products under the Fourier transform:
$$\mathcal{F}\{R_y\} = \underbrace{\mathcal{F}\{h\}}_{H(\omega)} \cdot \underbrace{\mathcal{F}\{h^*(-t)\}}_{H^*(\omega)} \cdot \underbrace{\mathcal{F}\{R_s\}}_{P_S(\omega)} + \underbrace{\mathcal{F}\{R_n\}}_{P_N(\omega)}$$Here $\mathcal{F}\{h^*(-t)\} = H^*(\omega)$ (time reversal and complex conjugation correspond to conjugation in the Fourier domain). Therefore:
\begin{equation} P_Y(\omega) = |H(\omega)|^2 P_S(\omega) + P_N(\omega) \end{equation}Step 2: Fourier transform of the cross-correlation
The cross-correlation $R_{sy}(\tau) = E[s(t) y^*(t-\tau)]$ between the target $d = s$ and the observation $y$ is:
\begin{align} R_{sy}(\tau) &= E\!\left[s(t) \left(\int h^*(v) s^*(t-\tau-v) dv + n^*(t-\tau)\right)\right] \nonumber \\ &= \int h^*(v) \, E[s(t) s^*(t-\tau-v)] \, dv + \underbrace{E[s(t) n^*(t-\tau)]}_{=\,0} \nonumber \\ &= \int h^*(v) \, R_s(\tau + v) \, dv \end{align}The last equality uses $E[s(t)s^*(t-\tau-v)] = R_s(\tau+v)$ (the same stationarity argument as Step 1 with $u=0$). This integral is a convolution of $R_s(\tau)$ with $h^*(-v)$, so the Fourier transform gives:
\begin{equation} P_{SY}(\omega) = \underbrace{\mathcal{F}\{h^*(-t)\}}_{H^*(\omega)} \cdot \underbrace{\mathcal{F}\{R_s\}}_{P_S(\omega)} = H^*(\omega) \, P_S(\omega) \end{equation}Step 3: Fourier transform of the Wiener-Hopf equation
Let $g(\tau)$ be the optimal filter's impulse response, so that $\hat{s}(t) = \int g(\tau') y(t-\tau') d\tau'$. The Chapter 2 Wiener-Hopf equation $\boldsymbol{R}\boldsymbol{w} = \boldsymbol{p}$ becomes, for stationary processes, a convolution equation — because each matrix element $R_{ij}$ depends only on the time lag $i-j$ (Toeplitz structure), so the matrix-vector product reduces to convolution:
\begin{equation} \int_{-\infty}^{\infty} R_y(\tau - \tau') \, g(\tau') \, d\tau' = R_{sy}(\tau) \label{eq:convolution} \end{equation}The left-hand side is a convolution of $R_y$ and $g$. By the convolution theorem, the Fourier transform turns it into a product, yielding a per-frequency scalar equation:
\begin{equation} P_Y(\omega) \cdot G(\omega) = P_{SY}(\omega) \end{equation}Substituting the results from Steps 1 and 2 and solving for $G(\omega)$:
\begin{equation} G(\omega) = \frac{P_{SY}(\omega)}{P_Y(\omega)} = \frac{H^*(\omega) P_S(\omega)}{|H(\omega)|^2 P_S(\omega) + P_N(\omega)} \end{equation}This is identical to \eqref{eq:wiener_freq}.
Essence of the Equivalence
The time-domain Wiener-Hopf equation is a convolution equation for stationary processes. The Fourier transform decomposes it into per-frequency scalar equations $P_Y(\omega) G(\omega) = P_{SY}(\omega)$: the value of $G(\omega)$ at each frequency is determined independently of all other frequencies. This is precisely the frequency-domain Wiener filter.
Conclusion
The frequency-domain and time-domain Wiener filters are fully equivalent via the Fourier transform. They are simply different representations of the same optimal solution.
3. Interpretation
3.1 Frequency-Domain Perspective
- Each frequency component is processed independently
- High-SNR frequencies pass through; low-SNR frequencies are attenuated
- Intuitively easy to understand
3.2 Time-Domain Perspective
- Directly computable from finite-length data
- Natural extension to adaptive filters
- Easier to handle causality constraints
3.3 When to Use Which
Frequency domain is suited when:
- Stationary process with long data available
- Spectral characteristics are clearly known
- Non-causal filtering is acceptable
Time domain is suited when:
- Short data or nonstationary signals
- Real-time processing is required
- Causal filter is mandatory
- Adaptive coefficient updates are needed
4. Summary
Key Points
- The frequency-domain and time-domain Wiener filters are equivalent representations connected by the Fourier transform
- Both are derived from the orthogonality principle (uncorrelation of error and observations)
- Choose the appropriate domain based on the problem's nature and constraints