Chapter Summary

Chapter 16 Summary

Key Points

• 1.

  A WSS process is mean-square continuous if and only if its autocorrelation $r_{xx}(\tau)$ is continuous at $\tau = 0$, equivalently if the process has finite average power.

• 2.

  The m.s. derivative $X'(t)$ exists if and only if $\int (2\pi f)^2 P_x(f)\, df < \infty$; the PSD of the derivative is $(2\pi f)^2 P_x(f)$, showing that differentiation amplifies high frequencies.

• 3.

  Higher-order m.s. derivatives require faster PSD decay: the $n$-th derivative exists iff $\int (2\pi f)^{2n} P_x(f)\, df < \infty$. Bandlimited processes are infinitely differentiable.

• 4.

  The m.s. integral $\int_a^b X(t)\,dt$ exists whenever $X(t)$ is m.s.-continuous — integration is a smoothing operation that does not require differentiability.

• 5.

  A bandlimited WSS process (PSD vanishing for $|f| > W$) can be perfectly reconstructed from Nyquist-rate samples $X(nT_s)$ with $T_s = 1/(2W)$ via sinc interpolation, in the mean-square sense.

• 6.

  The sampling theorem for random processes justifies the discrete-time models used throughout digital communications, including the OFDM transceiver chain.

• 7.

  The Karhunen-Loève expansion $X(t) = \sum_n Z_n \phi_n(t)$ diagonalizes the autocorrelation operator, with eigenfunctions $\phi_n$ and uncorrelated coefficients $Z_n$ of variance $\lambda_n$.

• 8.

  The KL expansion minimizes the $N$-term mean-square truncation error among all orthonormal expansions — it is the continuous-time analogue of PCA.