Write this on the A4

Typed or handwritten, both sides allowed. Copy it by hand once. Skip anything you can re-derive in 30 seconds. The 2026–27 guide does not promise a provided formula sheet.

Do not waste space on: a 2-tap Wiener layout (legacy 2019), a full BT/WOSA derivation, Pascal’s PMF, or MAP algebra for a posterior the paper will hand you.

Probability

$$P(B\mid A)=\frac{P(A\mid B)P(B)}{P(A\mid B)P(B)+P(A\mid B^c)P(B^c)}$$

Binomial \(\binom{n}{x}p^x(1-p)^{n-x}\). Geometric (first success at \(x\)): \(p(1-p)^{x-1}\).

$$P(Y>y_{\mathrm{th}})=\tfrac12\bigl[1-\mathrm{erf}\bigl((y_{\mathrm{th}}-\mu)/(\sigma\sqrt{2})\bigr)\bigr],\quad Q(z)=\int_z^\infty\frac{e^{-u^2/2}}{\sqrt{2\pi}}\,du$$

\(\mathrm{Var}=E[X^2]-\mu^2\). \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\). \(\mathrm{Cov}=E[XY]-\mu_X\mu_Y\), \(\rho=\mathrm{Cov}/(\sigma_X\sigma_Y)\in[-1,1]\).

Processes

WSS: constant mean, \(r[n_1,n_2]=r[n_1-n_2]\). Valid ACF: even, \(r[0]\ge|r[\ell]|\), \(P(e^{j\theta})\ge 0\).

$$P(e^{j\theta})=r[0]+2\sum_{k\ge 1}r[k]\cos(k\theta)$$

White: \(r[\ell]=\sigma^2\delta[\ell]\leftrightarrow P=\sigma^2\). Filtered: \(P_y=|H|^2 P_x\).

Factorization: \(P(z)=\sigma_i^2 L(z)L(z^{-1})\), keep poles and zeros of \(L\) inside the unit circle, \(l[0]=1\). \(\cos\theta=(z+z^{-1})/2\). \(|1-\alpha e^{-j\theta}|^2=1+\alpha^2-2\alpha\cos\theta\).

Ergodic in the mean: time average \(=\) ensemble, and variance of that estimator \(\to 0\). Random constant \(X(t)=A\): WSS, not ergodic.

Estimation

MSE \(=\mathrm{Var}+b(\theta)^2\). IID: \(p(\mathbf{x};\theta)=\prod p(x[n];\theta)\). MLE: \(\partial\ln p/\partial\theta=0\) if interior, else check the boundary. Invariance: \(\widehat{h(\theta)}=h(\hat\theta_{\mathrm{ML}})\).

$$\mathcal{I}(\theta)=-E\Bigl[\frac{\partial^2\ln p}{\partial\theta^2}\Bigr],\qquad \mathrm{Var}(\hat\theta)\ge\frac1{\mathcal{I}},\qquad N\text{ iid}\Rightarrow N\,i(\theta)$$

Transform: \([h'(\theta)]^2/\mathcal{I}\). WGN: \(\sigma^2/\sum(\partial s/\partial\theta)^2\). Efficient iff \(\partial\ln p/\partial\theta=\mathcal{I}(\theta)(g-\theta)\).

If the support of \(p(x;\theta)\) depends on \(\theta\), the CRLB does not exist. Do not grind Fisher.

LS: \(\hat\theta=(H^\top H)^{-1}H^\top x\). BLUE: \((H^\top C^{-1}H)^{-1}H^\top C^{-1}x\). Gauss–Newton: \(\theta_{k+1}=\theta_k+(H_k^\top H_k)^{-1}H_k^\top(x-s(\theta_k))\).

MMSE \(=E[\theta\mid x]\) (posterior mean). MAP \(=\arg\max p(\theta\mid x)\) (mode). BPSK \(X=\pm1\) eq.prob in \(\mathcal N(0,\sigma^2)\): MMSE \(=\tanh(y/\sigma^2)\), MAP \(=\mathrm{sign}(y)\). Gamma(2, \(\beta\)): mean \(2/\beta\), mode \(1/\beta\).

1-step predictor / YW: \(Rw=r\), \(J_{\min}=r[0]-w^\top r\), \(w=-a_1\).

Spectra

Periodogram \(\frac1N\bigl|\sum x[n]e^{-jn\theta}\bigr|^2\): asymptotically unbiased, var \(\propto P^2\), inconsistent. Correlogram uses the biased ACF or the PSD can go negative.

Bartlett, \(K\) non-overlap blocks: var \(\propto 1/K\), bias up. Welch/WOSA: overlap, var not \(1/K\). Blackman–Tukey: lag window, consistent. Main lobe = resolution. Side lobes = leakage. Zero-padding = interpolation, not resolution.

$$P_{\mathrm{AR}}=\frac{\sigma_i^2}{\bigl|1+\sum a_k e^{-jk\theta}\bigr|^2},\qquad a_1=-\frac{r[1]}{r[0]},\quad\sigma_i^2=r[0]+a_1 r[1]$$

AR(2): Toeplitz in \(r[0],r[1]\). MA(\(q\)): \(P=\) correlogram of lags \(|\ell|\le q\). Two real roots \(\Rightarrow\) keep min-phase \(|b|<1\).

Detection

$$L=\frac{p(x\mid\mathcal H_1)}{p(x\mid\mathcal H_0)}\gtrless\gamma$$

NP: fix \(P_{\mathrm{FA}}=\alpha\) from the \(\mathcal H_0\) tail only, then max \(P_D\). Bayes 0-1: \(\gamma=P_0/P_1\).

\(P_{\mathrm{FA}}=P(\text{decide }H_1\mid H_0)\). \(P_D=P(\text{decide }H_1\mid H_1)\). \(P_M=1-P_D\). Shade: FA is \(\mathcal H_0\) density on the \(\mathcal H_1\) side; miss is \(\mathcal H_1\) density on the \(\mathcal H_0\) side.

Mean-shift Gaussian: one threshold, right tail. Unequal-variance Gaussians / Laplace vs Gauss: two thresholds, outer tails if \(\mathcal H_1\) is heavier. Faster exponential / narrower Laplace: inner region — the inequality flips. Energy detector: \(\sum x_n^2\).

Matched filter maximises output SNR. \(h[n]=s[N-1-n]\) (or \(s[-n]\)), sample at the end of the pulse. ROC: \(P_D\) vs \(P_{\mathrm{FA}}\), above the diagonal; more separable \(\Rightarrow\) steeper.

How to copy it

1. One block per side-column. Tiny writing is fine; you will look at it.
2. Write the Yule–Walker 2×2 and the energy log-LRT once each. Those two eat time if you rebuild them.
3. Write \(\tanh(y/\sigma^2)\) and “support of \(p\) depends on \(\theta\) \(\Rightarrow\) no CRLB” in a corner. They are free points.
4. The night before: sit Nov 2022 or the 2022 resit in 3 hours with only this sheet.

5CTA0 Q1 2026–27 · from the course reader, lectures and official keys.