Signals and systems

Map of content for signals and systems — how to describe signals as functions of time, how to characterize the systems that process them, and how to switch between time and frequency representations. The path: signal vocabulary → system properties → LTI theory → convolution → Fourier series → Fourier transform → Laplace transform → sampling → filters.

Signal fundamentals

What a signal is, and the four kinds we classify them into.

• Electric circuit fundamentals — passive vs active components, the physical substrate signals live on.
• Signal — the general concept; continuous-time vs discrete-time, continuous-value vs discrete-value.
• Continuous-time system — the system side: rule mapping input to output.
• Communication system — transmitter, channel, receiver — the canonical motivating example.
• Noise (signal) — the unpredictable component that filters have to deal with.
• Quantization — continuous-value to discrete-value.
• Encoding (digital) — discrete-value to bits.

Standard signal zoo

The basic building blocks out of which more complicated signals are constructed.

• Sinusoid — amplitude, period, frequency, phase; the single most important signal.
• Real exponential signal — $A{e}^{-t/\tau }$.
• Time constant — $\tau$, the characteristic time of a first-order response.
• Complex sinusoid — generalizes sinusoid and exponential at once.
• Euler’s formula — $e^{jx}=\cos x+j\sin x$; the most-used identity in the course.
• Signum function — the sign of $t$.
• Heaviside step function — $u(t)$, the workhorse for “switch on at $t=0$.”
• Unit ramp function — integral of the step.
• Dirac delta function — the unit impulse, with sifting/equivalence/scaling properties.
• Equivalence property of the impulse — $g(t)\delta (t-t_0)=g(t_0)\delta (t-t_0)$.
• Scaling property of the impulse — $\delta (at)=\delta (t)/|a|$.
• Periodic impulse train — ${\delta }_T(t)$, the sampling comb.
• Unit rectangle — $\mathrm{rect}(t)$, width-1 pulse centered at the origin.
• Rectangular window function — the general gating function.
• Unit triangle function — $\mathrm{rect}\ast \mathrm{rect}$.
• Sinc function — $\sin (\pi x)/(\pi x)$; transform of the rectangle.

Signal manipulation and properties

Operations on signals and their structural features.

• Signal transformations — amplitude scaling, time shift, time scaling, standard form.
• Even and odd signals — parity decomposition.
• Periodic signal — fundamental period, period of a sum (LCM rule).
• Signal energy — $\int |x{|}^{2}dt$.
• Signal power — time-average of $|x{|}^2$.
• Energy signal vs power signal — the mutually exclusive classification.

System properties

The seven properties we use to characterize systems.

• Zero-state response — output with all initial state at zero.
• Homogeneity (system property) — scaling input scales output.
• Additivity (system property) — sum of inputs gives sum of outputs.
• Linearity (system property) — homogeneous + additive.
• Time-invariance — delaying input just delays output.
• LTI system — linear and time-invariant; the central object of the rest of the course.
• BIBO stability — bounded inputs give bounded outputs.
• Causality — output depends only on past and present inputs.
• Memory (system property) — whether output depends on past inputs.
• Superposition principle — the operational consequence of linearity + time-invariance.

Impulse response and convolution

The time-domain characterization of LTI systems.

• Impulse response — the system’s response to $\delta (t)$, completely characterizes an LTI system.
• Convolution integral — $y(t)=x(t)\ast h(t)$; the LTI input-output formula.
• Convolution properties — commutativity, associativity, distributivity, identity, etc.
• Graphical convolution — the flip-and-slide method.
• Convolution theorem — convolution in time = multiplication in frequency.
• System interconnections — cascade and parallel combinations.
• RC step response — the canonical first-order step response.

Fourier series

Decomposing periodic signals into discrete harmonics.

• Fourier series — synthesis and analysis equations.
• Eigenfunction of LTI system — why complex exponentials are the natural basis.
• Orthogonality of complex sinusoids — the algebraic foundation.
• Trigonometric form of Fourier series — real cosines and sines for real signals.
• Conjugate symmetry of Fourier coefficients — $c_x[-k]=c_x^{\ast }[k]$ for real $x$.
• Parseval’s theorem — energy/power preserved across the Fourier representation.
• Kronecker delta — the stride-$m$ delta ${\delta }_m[k]$.
• Gibbs phenomenon — the 9% overshoot near discontinuities.

Fourier transform

Extending Fourier series to aperiodic signals.

• Continuous-time Fourier transform — the CTFT, in both f-form and ω-form.
• Spectrum (signal) — amplitude and phase as functions of frequency.
• Frequency response — $H(f)$, the system’s spectrum-shaping rule.
• Time-bandwidth tradeoff — narrow in time, broad in frequency, and vice versa.

Laplace transform

Generalizing the Fourier transform to complex frequencies.

• Laplace transform — definition and core properties (general).
• Bilateral Laplace transform — two-sided form, with ROC tracking.
• Unilateral Laplace transform — one-sided form for causal signals and IVPs.
• Complex frequency — $s=\sigma +j\omega$, the variable.
• s-plane — the geometric setting for poles, zeros, ROCs.
• Region of convergence — what distinguishes signals with the same algebraic $X(s)$.
• Pole and zero — roots of $D(s)$ and $N(s)$, the handles on system dynamics.
• Transfer function — $H(s)=Y(s)/X(s)$, the s-domain system description.
• Partial fraction decomposition — the inverse-transform workhorse.
• Polynomial division for improper rational functions — handle improper $X(s)$ first.
• Initial value theorem — $x(0^{+})={\lim }_{s\to \infty }sX(s)$.
• Final value theorem — $x(\infty )={\lim }_{s\to 0}sX(s)$.
• Inverse Laplace transform — recovery procedure (table + partial fractions).
• Method of Laplace transform — full ODE solution procedure.

Sampling and reconstruction

Bridging continuous and discrete signals.

• Sampling — impulse sampling and the sampled spectrum.
• Aliasing — what happens when you sample too slowly.
• Sampling theorem — the Nyquist–Shannon result.
• Nyquist rate — twice the highest frequency.
• Sinc interpolation — exact reconstruction formula.
• Anti-aliasing filter — analog lowpass before sampling.

Frequency response and filters

LTI systems designed to shape the spectrum.

• Filter (signal processing) — the general concept; ideal vs practical; bandwidth definitions.
• Lowpass filter — passes near DC, blocks high frequencies.
• Highpass filter — passes high frequencies, blocks DC.
• Bandpass filter — passes a band, blocks elsewhere.
• Bandstop filter — blocks a band, passes elsewhere (notch).
• Cutoff frequency — the 3 dB corner.
• Decibel — the logarithmic ratio unit.
• dBm and dBW — absolute power levels referenced to 1 mW and 1 W.
• Bode plot — $20\log |H|$ and $\angle H$ on a log frequency axis.

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Most of the analytical machinery here comes from Differential equations — the Laplace transform, characteristic equation, RLC circuit dynamics, and convolution all live in both courses. The discrete-signal counterpart is the natural sequel: when these continuous-time tools are applied to sampled signals via the z-transform and digital filters. Digital logic provides the physical layer that actually implements digital signal processing once we’ve moved into discrete time.