Quick Revision Notes — Signals & Systems

Introduction

These revision notes distill the entire Signals and Systems course into essential concepts and results for rapid review before examinations. Each topic is summarized in its most compact yet complete form, emphasizing the relationships between concepts and the most commonly tested material. Use these notes after thorough study — they are meant to refresh memory, not replace understanding.

The notes follow the typical course progression: signal classification, system properties, convolution, Fourier methods, Laplace and Z-transforms, and finally sampling theory. Pay special attention to the interconnections between topics — understanding how Fourier Transform properties relate to system behavior, or how pole locations determine both stability and frequency response, is what separates strong students from average ones.

Signal Classification Summary

By continuity: Continuous-time $x(t)$ vs Discrete-time $x[n]$

By duration: Finite vs Infinite duration

By energy/power: Energy signal ($0<E<\infty$, $P=0$) vs Power signal ($E=\infty$, $0<P<\infty$)

By symmetry: Even $x(-t)=x(t)$ vs Odd $x(-t)=-x(t)$. Any signal decomposes uniquely into even + odd parts.

By periodicity: Periodic $x(t+T)=x(t)$ vs Aperiodic. Fundamental period = smallest positive $T$.

By causality: Causal ($x(t)=0$ for $t<0$) vs Non-causal

Discrete periodicity condition: $x[n] = A\cos(\omega_0 n)$ is periodic iff $\omega_0/(2\pi)$ is rational. Period $N = 2\pi m/\omega_0$ for smallest integer $m$ making $N$ integer.

Signal Operations Checklist

Order of operations for $x(at-b)$: ALWAYS factor out the scaling coefficient first. The shift amount is $b/a$, not $b$. This is the most common exam error in signal transformations.

System Properties — Quick Tests

Linear: $T\{ax_1+bx_2\} = aT\{x_1\}+bT\{x_2\}$ for all inputs and scalars. Test both additivity and homogeneity. Note: $y = x+3$ is NOT linear (fails homogeneity).

Time-Invariant: $T\{x(t-t_0)\} = y(t-t_0)$ for all $t_0$. Systems with explicit time dependence (e.g., $y(t) = tx(t)$) are time-variant.

Causal: Output at $t$ depends only on input at $\tau \leq t$. For LTI: $h(t)=0, t<0$. All physically realizable real-time systems must be causal.

Stable (BIBO): Bounded input → bounded output. For LTI: $\int|h(t)|dt < \infty$. Equivalent to all poles in the left half-plane (continuous) or inside unit circle (discrete).

Memoryless: $y(t)$ depends only on $x(t)$ (same instant). For LTI: $h(t)=c\delta(t)$, so the system is just a gain.

Invertible: Distinct inputs → distinct outputs. For LTI: $H(j\omega)\neq 0$ for all $\omega$ (no frequency completely zeroed out).

LTI Systems Core Results

1. Impulse response $h(t)$ completely characterizes the system
2. Output = Input * Impulse Response (convolution): $y = x * h$
3. Transfer function: $H(j\omega) = Y(j\omega)/X(j\omega) = \mathcal{F}\{h(t)\}$
4. Sinusoids are eigenfunctions: input $e^{j\omega t}$ → output $H(j\omega)e^{j\omega t}$
5. Cascade: $H_{total} = H_1 \cdot H_2$ (multiply transfer functions)
6. Parallel: $H_{total} = H_1 + H_2$ (add transfer functions)
7. Feedback: $H_{CL} = \frac{H_{forward}}{1 + H_{forward}\cdot H_{feedback}}$

Eigenfunction property explained: When you input a pure complex exponential $e^{j\omega_0 t}$ to an LTI system, the output is the same exponential scaled by a complex number $H(j\omega_0)$. This is why sinusoidal steady-state analysis works — the system only changes the amplitude and phase of each frequency component, never creates new frequencies.

Convolution Key Facts

• $x * \delta = x$ (identity element)
• $x * \delta(t-t_0) = x(t-t_0)$ (shifting property)
• $x * u(t) = \int_{-\infty}^t x(\tau)d\tau$ (running integral)
• Commutative: $x*h = h*x$
• Associative: $(x*h_1)*h_2 = x*(h_1*h_2)$
• Distributive: $x*(h_1+h_2) = x*h_1 + x*h_2$
• Output duration (finite signals): $L_y = L_x + L_h - 1$

Fourier Series Key Points

• Represents periodic signals as sum of harmonics
• Fundamental frequency: $\omega_0 = 2\pi/T$, harmonics at $n\omega_0$
• Exponential form: $c_n = \frac{1}{T}\int_T x(t)e^{-jn\omega_0 t}dt$
• Even signal → only $a_n$ (cosines); Odd signal → only $b_n$ (sines)
• Half-wave symmetry $x(t+T/2) = -x(t)$ → only odd harmonics
• Quarter-wave symmetry → combine even/odd with half-wave for further simplification
• Parseval: $P = \sum|c_n|^2$
• Gibbs phenomenon at discontinuities (~9% overshoot, does NOT disappear with more terms)
• Convergence: Dirichlet conditions (finite discontinuities, finite extrema, absolutely integrable per period)

Transform Domain Quick Reference

Stability Summary

Laplace Transform Essentials

• ROC determines uniqueness — same $X(s)$ expression can correspond to different time functions depending on ROC
• Right-sided signal ($x(t)=0$ for $t<t_0$): ROC is right of rightmost pole
• Left-sided signal: ROC is left of leftmost pole
• Two-sided signal: ROC is a strip between poles
• Poles on $j\omega$-axis → marginally stable (undamped oscillations)
• Initial value theorem: $x(0^+) = \lim_{s\to\infty}sX(s)$
• Final value theorem: $x(\infty) = \lim_{s\to 0}sX(s)$ (only if all poles of $sX(s)$ in LHP)

Z-Transform Essentials

• Discrete analog of Laplace: $z = e^{sT_s}$
• Right-sided ($x[n]=0, n<n_0$): ROC outside a circle
• Left-sided: ROC inside a circle
• Causal + stable: all poles strictly inside unit circle
• Long division of $X(z)$ gives $x[n]$ directly (useful for first few samples)
• Relationship: $z = e^{j\omega}$ on unit circle gives DTFT

Sampling Essentials

• Sample at $f_s > 2f_{max}$ (Nyquist criterion)
• Violating → aliasing (irreversible frequency folding)
• Anti-aliasing filter BEFORE sampling (analog LPF, cutoff at $f_s/2$)
• Reconstruction: ideal LPF (sinc interpolation)
• Practical: ZOH + analog smoothing filter
• Discrete-time frequency $\omega_d = \omega_a T_s = 2\pi f/f_s$ (normalized to $[-\pi, \pi]$)
• Folding frequency = $f_s/2$ = maximum representable frequency

Common Exam Mistakes to Avoid

1. Forgetting ROC with Laplace/Z-transforms — the expression is ambiguous without ROC
2. For $x(at-b)$: shifting by $b$ instead of $b/a$ — always factor out the coefficient first
3. Confusing convolution with multiplication — $y = x*h$ is NOT $y[n] = x[n] \cdot h[n]$
4. Stating $y=x+3$ is linear — it is NOT (constant offset violates homogeneity: $T\{0\} = 3 \neq 0$)
5. Forgetting the $1/|a|$ factor in the scaling property of Fourier Transform
6. Not checking ALL poles for stability (missing one pole ruins the conclusion)
7. Using final value theorem when it does not apply (poles must be in stable region)
8. Confusing energy and power signals — periodic signals are ALWAYS power signals (infinite energy)
9. Discrete periodicity: assuming $\cos(n)$ is periodic — it is NOT unless $1/(2\pi)$ is rational
10. Circular vs linear convolution — DFT multiplication gives circular convolution, not linear

Key Takeaways

• Master the system property tests — they appear on every exam
• LTI is the key class: know convolution, transfer functions, and stability criteria
• Transform properties save enormous computation time — memorize them
• Always specify ROC for Laplace and Z-transforms
• Sampling theorem + aliasing prevention is a universal design requirement
• Pole locations determine everything: stability, frequency response shape, transient behavior
• Practice partial fractions — they are needed for every inverse transform problem