Unit 1: Introduction to Signals

A signal is any function that conveys information about the behaviour of a physical system by describing how one quantity varies with an independent variable, usually time. Signals-and-systems analysis begins by classifying signals so that later transform tools (Fourier, Laplace, z) apply to the right class. This unit fixes those classifications and the elementary signals from which all others are built.

I. Orientation: The Signal Framework

Every signal in this unit is written as x(t) for continuous time or x[n] for discrete time, and is judged against a fixed set of properties.

• Independent variable: time — continuous t ∈ ℝ (round brackets) or discrete n ∈ ℤ (square brackets). Discrete signals exist only at integer indices; values between samples are undefined, not zero.
• Amplitude: the dependent quantity x, which may be real or complex. Its physical unit (volts, amperes, pascals) sets the units of energy and power.
• Determinism vs randomness: deterministic signals have an exact mathematical rule; this unit treats only these.
• Classification axes used throughout: continuous/discrete, energy/power, periodic/aperiodic, even/odd. Each axis is independent, so one signal carries several labels at once.

II. Continuous-Time and Discrete-Time Signals

Two domains and the energy–power test that classifies them

A signal is continuous-time when defined for every instant and discrete-time when defined only on integers, typically by sampling x[n] = x(nT) where T is the sampling period.

A. Continuous-time and discrete-time signals

• Continuous-time (CT): x(t) defined ∀ t, e.g. a mains voltage x(t) = 230√2 cos(100πt).
• Discrete-time (DT): x[n] defined at integers, e.g. x[n] = (0.5)^n u[n], a sampled decaying exponential.
• Digital vs discrete: discrete refers to the time axis; digital adds amplitude quantisation. DT signals here keep continuous amplitude.

B. Energy and power signals

The point is to measure a signal's "size" and decide which of two mutually exclusive classes it belongs to.

• Total energy (CT / DT):

E = ∫_{-∞}^{∞} |x(t)|² dt        E = Σ_{n=-∞}^{∞} |x[n]|²

• Average power (CT / DT):

P = lim_{T→∞} (1/2T) ∫_{-T}^{T} |x(t)|² dt
P = lim_{N→∞} 1/(2N+1) Σ_{n=-N}^{N} |x[n]|²

• Symbols: E = energy (joules into 1 Ω), P = mean power (watts), |x|² = instantaneous power.
• Energy signal: 0 < E < ∞, hence P = 0. Finite-duration or decaying signals qualify, e.g. one rectangular pulse.
• Power signal: 0 < P < ∞, hence E = ∞. Periodic and everlasting signals qualify.
• Neither: both infinite, e.g. x(t) = t.
• Worked example: for x(t)=A cos(ω₀t), E=∞ but P = A²/2, so it is a power signal.

III. Transformations of the Independent Variable

Reshaping the time axis, and the symmetry classes it defines

Replacing t by an affine expression αt + β shifts, scales or reverses a signal without altering its amplitude rule.

• Time shift: x(t − t₀) delays by t₀ > 0 (moves right); x(t + t₀) advances.
• Time reversal: x(−t) reflects about t = 0.
• Time scaling: x(αt) compresses if |α|>1, expands if |α|<1. In DT, x[αn] needs integer αn; downsampling discards samples.
• Combined order: for x(αt+β), shift by β/α first, then scale/reverse, to avoid errors.

A. Periodic signals

A signal is periodic if it repeats after a fixed interval.

• CT condition: x(t) = x(t + T) for all t; the fundamental period T₀ is the smallest such T > 0.
• DT condition: x[n] = x[n + N], N a positive integer.
• Fundamental frequency: ω₀ = 2π/T₀ (rad/s) or Ω₀ = 2π/N.
• DT periodicity rule: cos(Ω₀n) is periodic only if Ω₀/2π is rational; cos(n) is aperiodic because 2π is irrational.
• Sum of periodics: periodic iff the ratio of periods is rational; then T₀ is their LCM.

B. Even and odd signals

Every signal splits uniquely into symmetric and antisymmetric parts.

• Even: x(t) = x(−t), symmetric about the vertical axis (e.g. cos t).
• Odd: x(t) = −x(−t), x(0)=0 (e.g. sin t).
• Decomposition:

xₑ(t) = ½[x(t) + x(−t)]     xₒ(t) = ½[x(t) − x(−t)]
x(t) = xₑ(t) + xₒ(t)

• Product rule: even×even = even, odd×odd = even, even×odd = odd.
• Integral property: ∫_{-a}^{a} xₒ(t) dt = 0, useful in Fourier coefficients.

IV. Exponential and Sinusoidal Signals

The building-block waveforms of system analysis

These are the eigenfunctions of linear time-invariant systems, which makes them the natural basis for every transform in later units.

A. Exponential and sinusoidal signals

1. Real exponential: x(t) = C e^{at}, C, a real.
   • a > 0 growing, a < 0 decaying, a = 0 constant. Example: e^{−t/τ} with time constant τ.
2. Sinusoidal: x(t) = A cos(ω₀t + φ), amplitude A, angular frequency ω₀ (rad/s), phase φ (rad), period T₀ = 2π/ω₀.

• Complex exponential: x(t) = e^{jω₀t}, a unit-magnitude phasor rotating at ω₀.
• Euler's relation: links the two —

e^{jω₀t} = cos ω₀t + j sin ω₀t
A cos(ω₀t+φ) = A/2 e^{jφ}e^{jω₀t} + A/2 e^{−jφ}e^{−jω₀t}

• General complex exponential: x(t) = C e^{st} with s = σ + jω₀; gives a sinusoid with an e^{σt} envelope — growing (σ>0) or damped (σ<0).
• DT complex exponential: x[n] = e^{jΩ₀n}; distinct only for Ω₀ over a 2π range because Ω₀ and Ω₀+2π give identical sequences.

V. The Unit Impulse and Unit Step Functions

The two singularity signals that generate all others

These idealised signals let us represent arbitrary signals and probe system response.

A. The unit impulse and unit step functions

1. Unit step:

u(t) = 1, t > 0 ;  0, t < 0        u[n] = 1, n ≥ 0 ; 0, n < 0

• Models a switch turned on at t = 0; multiplying by u(t) makes any signal causal.

2. Unit impulse (Dirac delta) δ(t): zero everywhere except t=0, with unit area:

∫_{-∞}^{∞} δ(t) dt = 1        δ[n] = 1, n = 0 ; 0, n ≠ 0

• Step–impulse link:

δ(t) = du(t)/dt      u(t) = ∫_{-∞}^{t} δ(τ) dτ
δ[n] = u[n] − u[n−1]  u[n] = Σ_{k=-∞}^{n} δ[k]

• Sifting property: ∫ x(t) δ(t − t₀) dt = x(t₀) — samples a signal at t₀; the basis of convolution.
• Scaling: δ(at) = (1/|a|) δ(t).

VI. Operations on Signals

Amplitude and time manipulations combining the elements above

Operations act either on the amplitude or on the independent variable.

A. Operations on signals

• Amplitude scaling: y(t) = a·x(t) — an amplifier gain a.
• Addition: y(t) = x₁(t) + x₂(t) — pointwise; in DT, add matching indices.
• Multiplication: y(t) = x₁(t)·x₂(t) — modulation, e.g. m(t)cos ω_c t.
• Time shift, reversal, scaling: as defined in Section III, now applied to elementary signals; e.g. u(t) − u(t−2) builds a width-2 pulse.
• Differentiation / integration (CT): dx/dt, ∫x dt; differencing / accumulation (DT): x[n]−x[n−1], Σx[k].
• Worked example: r(t) = t·u(t) (ramp); its derivative is u(t), its derivative in turn is δ(t), chaining the singularity family.

VII. Software Simulation of Basic Operations on Elementary Signals

Numerical generation and manipulation of signals on a computer

Simulation represents a signal as a value vector over a sampled index vector, then applies vector operations that mirror the analytical ones.

A. Software simulation of basic operations on elementary signals

• Index and value vectors: define a time base t = t_start : step : t_end; each signal is an array aligned to it.
• Generating elementary signals (MATLAB/Octave-style):

n  = -10:10;
imp = (n==0);           % unit impulse δ[n]
stp = (n>=0);           % unit step u[n]
ramp = n.*(n>=0);       % ramp
expo = 0.8.^n;          % exponential
sine = sin(0.2*pi*n);   % sinusoid

• Time shift: delay x[n−k] implemented by shifting the index vector n → n+k while keeping values, so the axis, not the data, moves.
• Time reversal: fliplr(x) with negated index array.
• Arithmetic on unequal supports: pad shorter vectors with zeros over the common index range before adding or multiplying, so samples align by index.
• Verification: plot with stem (DT) or plot (CT), and confirm computed energy sum(abs(x).^2) matches the analytical E.
• Purpose: lets a large signal be built, transformed and measured numerically before hardware realisation, and confirms hand derivations of shifts, energy and periodicity.