Signals and Systems

Signal Classification

A signal is a function carrying information, typically varying over time or space.

Continuous vs Discrete: Continuous signals are defined for all time (analog). Discrete signals are defined at specific time indices (sampled).

Deterministic vs Random: Deterministic signals are fully specified by a formula. Random signals require statistical description.

Periodic vs Aperiodic: Periodic: x(t) = x(t + T) for some period T. Aperiodic: no such T exists.

Energy vs Power signals: Energy signal has finite total energy (E = Σ|x[n]|²). Power signal has finite average power (P = lim (1/N) Σ|x[n]|²).

Basic Signal Operations

Time shifting: x[n - k] delays by k samples. x[n + k] advances by k.

Time scaling: x[an] compresses (|a| > 1) or expands (|a| < 1).

Time reversal (folding): x[-n] reflects around n = 0.

Amplitude scaling: c·x[n] scales amplitude by factor c.

Convolution

The output of a linear time-invariant (LTI) system with impulse response h[n]:

y[n] = x[n] * h[n] = Σₖ x[k] · h[n - k]

Properties: Commutative (x * h = h * x), associative ((x * h₁) * h₂ = x * (h₁ * h₂)), distributive (x * (h₁ + h₂) = x * h₁ + x * h₂).

Computation: For each output sample n, slide h reversed across x, multiply and sum overlapping samples.

PROCEDURE CONVOLVE(x, h)
    y ← array of 0.0, size LENGTH(x) + LENGTH(h) - 1
    FOR i ← 0 TO LENGTH(x) - 1 DO
        FOR j ← 0 TO LENGTH(h) - 1 DO
            y[i + j] ← y[i + j] + x[i] * h[j]
    RETURN y

Time: O(N·M). With FFT: O((N+M) log(N+M)).

LTI System Properties

Linearity: T{ax₁ + bx₂} = aT{x₁} + bT{x₂}. Superposition holds.

Time-invariance: If x[n] → y[n], then x[n-k] → y[n-k]. System behavior doesn't change over time.

Causality: Output depends only on present and past inputs. h[n] = 0 for n < 0.

Stability (BIBO): Bounded input → bounded output. Condition: Σ|h[n]| < ∞ (impulse response is absolutely summable).

Memory: Memoryless if y[n] depends only on x[n]. h[n] = c·δ[n].

Impulse Response

The response h[n] of an LTI system to a unit impulse δ[n]. Completely characterizes the system.

δ[n] = { 1 if n = 0
        { 0 otherwise

x[n] = Σₖ x[k]·δ[n-k]  →  y[n] = Σₖ x[k]·h[n-k] = x[n] * h[n]

FIR (Finite Impulse Response): h[n] has finite duration. Always stable.

IIR (Infinite Impulse Response): h[n] has infinite duration. May be unstable. Requires feedback.

Applications in CS

• Audio processing: Convolution for reverb, echo, filtering.
• Image processing: 2D convolution for blur, sharpen, edge detection.
• Telecommunications: Signal modulation, channel equalization.
• Machine learning: Convolutional neural networks use discrete convolution.
• Control systems: System response analysis via impulse/step response.