Information Systems Reference

Control systems, communication theory, information theory, and coding. Mathematical transforms and probability content has been consolidated in Mathematical Foundations.

Source: Information Data Book (2017 Edition, revised 2019 & 2021), Cambridge University Engineering Department

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1. Control Systems

1.1 Closed-Loop System

Return ratio: $L(s) = H(s)G(s)K(s)$

Closed-loop transfer function: $\frac{G(s)K(s)}{1+L(s)}$

1.2 Stability

Stable iff roots of $1 + L(s) = 0$ have negative real parts.

1.3 Routh-Hurwitz Criteria

For polynomial: $a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0$

Second order: All $a_i > 0$

Third order: $a_3 a_2 > a_1 a_0$

Fourth order: $a_3 a_2 a_1 > a_4 a_1^2 + a_3^2 a_0$

1.4 Nyquist Criterion

Encirclement of $-1/k$ equals number of RHP poles of $g(s)$.

1.5 Root Locus

Roots of $1 + k g(s) = 0$.

Angle condition: $\angle g(s) = (2m+1)\pi$

Magnitude condition: $|g(s)| = \frac{1}{k}$

1.6 Bode Diagrams

Standard first- and second-order forms (see plots in original databook).

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2. Communication

2.1 Analogue Modulation

AM (Amplitude Modulation): $s(t) = [a_0 + x(t)]\cos(2\pi f_c t)$

FM (Frequency Modulation): $s(t) = a_0 \cos\left(2\pi f_c t + 2\pi k_F \int_0^t x(u) \, du\right)$

Carson’s rule (FM bandwidth): $B \approx 2W + 2\Delta f$

2.2 Digital Communication

Quantisation SNR: $\mathrm{SNR} = 1.76 + 6.02n \text{ dB}$

where $n$ is the number of bits.

PAM (Pulse Amplitude Modulation): $x(t) = \sum_k X_k p(t - kT)$

QAM (Quadrature Amplitude Modulation): $x(t) = \sum_k \Re\{X_k e^{j2\pi f_c t}\} p(t - kT)$

2.3 Wireless Channel

If $h \sim \mathcal{CN}(0, \sigma^2)$, then: $|h|^2 \sim \text{Exponential}\left(\frac{1}{\sigma^2}\right)$

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3. Information Theory

3.1 Entropy

$H(X) = \sum_x P(x) \log \frac{1}{P(x)}$

3.2 Mutual Information

$I(X;Y) = H(X) - H(X|Y) = D(P_{XY} \| P_X P_Y)$

3.3 Differential Entropy

$h(X) = \int p(x) \log \frac{1}{p(x)} \, dx$

3.4 Key Inequalities

Data-processing inequality: $I(X;Y) \ge I(X;Z)$

Fano’s inequality: $1 + P_e \log|\mathcal{X}| \ge H(X|Y)$

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4. Coding Theory

4.1 Linear Block Codes

Rate: $R = \frac{k}{n}$

Singleton bound: $d_{\min} \le n - k + 1$

4.2 LDPC Codes

Density evolution (BEC): $p_t = \varepsilon \lambda\big(1 - \rho(1-p_{t-1})\big)$

LLR for AWGN: $L(y) = \frac{2y}{\sigma^2}$

4.3 Finite Fields and Reed-Solomon Codes

DFT over GF($q$): $X_k = \sum_{m=0}^{n-1} x_m \alpha^{mk}$

Inverse: $x_m = \frac{1}{n^*} \sum_{k=0}^{n-1} X_k \alpha^{-mk}$

Reed-Solomon codes are MDS with: $d_{\min} = n - k + 1$

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Cross-References

• For Fourier transforms, Z-transforms, and Laplace transforms, see Mathematical Foundations
• For probability and statistics fundamentals, see Mathematical Foundations

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Source

Information Data Book (2017 Edition, revised 2019 & 2021), Cambridge University Engineering Department