# Signal Plus Weakly Stationary Noise

I was reading the book "Spectral Analysis of Time Series" By Herman Koopmans. On Page 55, he explains that a specific type of non-stationary signal which is the result of adding weakly stationary ergodic noise to a deterministic signal can be decomposed to Wiener spectra. I wonder why there is a need for signal to be ergodic and why weakly stationarity is not enough to derive spectral summation formula for noise and deterministic part?

More specifically based on those assumptions he shows that one has: $$C_X(\tau) = \lim\limits_{ \tau\to\infty}\frac{1}{2T}\int^{T}_{-T} X(t + \tau)X(t) dt = C_S(\tau) + C_N(\tau), \text{ almost surely}.$$

And then $$F_X(A)=F_S(A)+F_N(A)$$ where $F_Z$ is the spectral distribution of stochastic process $Z(t)$, where $$X(t)=N(t)+S(t)$$

## 1 Answer

For ergodic processes, time averages (defined by integrals over time) and ensemble averages (defined by expectations with respect to probability distributions) are identical. This means that the autocovariance is the same, no matter if defined by a time integral or by an expectation:

$$C_X(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}X(t+\tau)X(t)dt= E\{X(t+\tau)X(t)\}$$