Chapter Summary

Chapter Summary

Key Points

• 1.

  From LRT to matched filter. For the binary AWGN problem $\mathcal{H}_1: \mathbf{Y} = \mathbf{s} + \mathbf{W}$ versus $\mathcal{H}_0: \mathbf{Y} = \mathbf{W}$ with $\mathbf{W} \sim \mathcal{N}(\mathbf{0}, \sigma^2\mathbf{I})$, the LRT reduces to a threshold test on $T(\mathbf{y}) = \mathbf{s}^{\mathsf T}\mathbf{y}$. This scalar is simultaneously the correlator output and the matched filter output sampled at the optimal time.

• 2.

  Matched filter maximises SNR. Among all linear filters with fixed norm, $h(t) = s(T-t)$ maximises the output SNR at $t=T$; the maximum output SNR equals $2E_s/N_0$ and is achieved uniquely up to scaling.

• 3.

  Exact performance. For the matched-filter receiver $P_d = Q(Q^{-1}(P_f) - d)$ with $d = \sqrt{2E_s/N_0}$ the deflection coefficient. For equal priors the minimum error probability is $P_e = Q(d/2)$ --- the formula underlying every BPSK/orthogonal-signalling curve.

• 4.

  GLRT for composite hypotheses. When signal parameters (amplitude, phase, timing) are unknown, replace them by their MLEs under each hypothesis: $L_G = \max_{\theta_1} f_1(y;\theta_1)/\max_{\theta_0} f_0(y;\theta_0)$. The GLRT is not necessarily optimal but is the standard practical construction.

• 5.

  Colored noise via prewhitening. For $\mathbf{W} \sim \mathcal{N}(\mathbf{0}, \mathbf{C})$, apply $\mathbf{C}^{-1/2}$ to observation and signal; the resulting problem is standard AWGN detection, and the test statistic becomes the Mahalanobis inner product $\mathbf{s}^{\mathsf T} \mathbf{C}^{-1} \mathbf{y}$. Equivalently, the detection SNR is the Mahalanobis norm $\mathbf{s}^{\mathsf T}\mathbf{C}^{-1}\mathbf{s}$.

• 6.

  Continuous-time view. In $L^2[0,T]$ the sufficient statistic is $\langle y, s\rangle = \int_0^T y(t)s(t)\,dt$. Gram--Schmidt orthogonalisation reduces $M$-ary signalling to detection of a vector in $\mathbb{R}^N$ with $N\leq M$ --- the signal-space representation used throughout digital modulation.

• 7.

  Sufficiency as structural dimension reduction. The matched-filter output is a sufficient statistic in the Fisher--Neyman sense: the likelihood depends on $\mathbf{y}$ only through this scalar (or through the basis projections, in $M$-ary detection). No optimal receiver needs to keep anything outside the signal subspace.