Signals and Systems — Complete Notes for B.Tech ECE

Unit 1: Signals — Classification

A signal is a function that conveys information about a physical phenomenon.

Types of Signals:

| Classification | Types | |---|---| | Time domain | Continuous-time (CT) / Discrete-time (DT) | | Value domain | Analog (continuous values) / Digital (discrete values) | | Periodicity | Periodic (repeats) / Aperiodic | | Determinism | Deterministic (fully predictable) / Random | | Energy | Energy signal (finite energy) / Power signal (finite power) |

Energy and Power:

• Energy: E = ∫|x(t)|² dt (CT) or Σ|x[n]|² (DT)
• Power: P = lim(T→∞) (1/2T)∫|x(t)|² dt
• Energy signal: E < ∞, P = 0
• Power signal: P < ∞, E = ∞

Standard Signals:

• Unit step: u(t) = 1 for t≥0, 0 for t<0
• Unit impulse (Dirac delta): δ(t) = ∞ at t=0, 0 elsewhere; ∫δ(t)dt = 1
• Ramp: r(t) = t·u(t)
• Exponential: x(t) = Ae^(αt); decaying if α<0
• Sinusoidal: x(t) = A·sin(ωt + φ)

Operations on Signals:

• Time shifting: x(t-t₀) → shift right by t₀
• Time reversal: x(-t) → flip about vertical axis
• Time scaling: x(at) → compress if a>1, expand if a<1

---

Unit 2: Systems — Properties

A system transforms an input signal to an output signal.

Properties of LTI Systems:

| Property | Definition | Test | |---|---|---| | Linearity | Superposition holds: af(x₁)+bf(x₂)=f(ax₁+bx₂) | Check homogeneity + additivity | | Time-Invariance | Shift in input → same shift in output | x(t-t₀) → y(t-t₀) | | Causality | Output at t depends only on present/past inputs | h(t)=0 for t<0 | | Stability (BIBO) | Bounded input → bounded output | ∫|h(t)|dt < ∞ | | Memory | Memoryless: output at t depends only on x(t) | | | Invertibility | Unique input for each output | |

---

Unit 3: Convolution

Continuous-time convolution: y(t) = x(t) * h(t) = ∫ x(τ)h(t-τ)dτ

Discrete-time convolution: y[n] = x[n] * h[n] = Σ x[k]h[n-k]

Properties of Convolution:

• Commutative: xh = hx
• Associative: (xh₁)h₂ = x(h₁h₂)
• Distributive: x*(h₁+h₂) = xh₁ + xh₂
• Identity: x(t)*δ(t) = x(t) (impulse is identity)
• Shift: x(t)*δ(t-t₀) = x(t-t₀)

Steps for Convolution (Graphical):

1. Keep x(τ) as is
2. Flip h(τ) to get h(-τ)
3. Shift by t: h(t-τ)
4. Multiply x(τ)·h(t-τ)
5. Integrate product for each value of t

---

Unit 4: Fourier Series and Transform

Fourier Series (Periodic Signals)

Any periodic signal can be expressed as sum of sinusoids:

Exponential form: x(t) = Σ cₙe^(jnω₀t) cₙ = (1/T)∫ x(t)e^(-jnω₀t)dt

Trigonometric form: x(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)]

Dirichlet Conditions: Signal must be absolutely integrable, have finite maxima/minima and discontinuities in one period.

Fourier Transform (Aperiodic Signals)

FT Pair: X(jω) = ∫ x(t)e^(-jωt)dt (Analysis) x(t) = (1/2π)∫ X(jω)e^(jωt)dω (Synthesis)

Important FT Pairs:

| Signal x(t) | Fourier Transform X(jω) | |---|---| | δ(t) | 1 | | 1 | 2πδ(ω) | | u(t) | 1/jω + πδ(ω) | | e^(-at)u(t) | 1/(a+jω) | | rect(t/τ) | τ·sinc(ωτ/2) | | e^(-a|t|) | 2a/(a²+ω²) |

FT Properties: Linearity, Time shifting, Frequency shifting, Convolution in time ↔ multiplication in frequency, Multiplication in time ↔ convolution in frequency

---

Unit 5: Laplace Transform

For Continuous-Time Systems:

Bilateral: X(s) = ∫ x(t)e^(-st)dt (s = σ + jω)

Unilateral (one-sided): X(s) = ∫₀^∞ x(t)e^(-st)dt

Important LT Pairs:

| x(t) | X(s) | ROC | |---|---|---| | δ(t) | 1 | All s | | u(t) | 1/s | Re(s) > 0 | | e^(-at)u(t) | 1/(s+a) | Re(s) > -a | | tⁿu(t) | n!/s^(n+1) | Re(s) > 0 | | sin(ωt)u(t) | ω/(s²+ω²) | Re(s) > 0 | | cos(ωt)u(t) | s/(s²+ω²) | Re(s) > 0 |

Transfer Function (System Function): H(s) = Y(s)/X(s)

ROC (Region of Convergence): Region in s-plane where Laplace integral converges.

Partial Fraction Expansion: Used to find inverse Laplace transform by breaking complex fractions into simpler ones.

---

Unit 6: Z-Transform (Discrete-Time)

Z-Transform: X(z) = Σ x[n]z^(-n)

Important Z-Transform Pairs:

| x[n] | X(z) | ROC | |---|---|---| | δ[n] | 1 | All z | | u[n] | z/(z-1) | |z| > 1 | | aⁿu[n] | z/(z-a) | |z| > |a| | | naⁿu[n] | az/(z-a)² | |z| > |a| |

Stability in Z-domain: System stable if all poles inside unit circle (|z|<1)

Frequency response: Evaluate H(z) on unit circle: H(e^(jω))

---

Solved PYQ Questions

Q1 (2023): Find convolution of x[n] = 3 with h[n] = 1 Method: y[n] = Σ x[k]h[n-k] y[0] = 1×1 = 1 y[1] = 1×1 + 2×1 = 3 y[2] = 1×1 + 2×1 + 3×1 = 6 y[3] = 2×1 + 3×1 = 5 y[4] = 3×1 = 3 y[n] = 3

Q2 (2023): Find Laplace Transform of x(t) = e^(-3t)u(t) + e^(-5t)u(t) X(s) = 1/(s+3) + 1/(s+5) = (s+5+s+3)/[(s+3)(s+5)] = (2s+8)/[(s+3)(s+5)] ROC: Re(s) > -3

Q3 (2022): Is y(t) = x(2t) time-invariant? Test: Input x(t-t₀) → Output y₁(t) = x(2t-t₀) System output shifted: y(t-t₀) = x(2(t-t₀)) = x(2t-2t₀) y₁(t) ≠ y(t-t₀) → System is NOT time-invariant

---

Quick Revision Checklist

• Signal types: CT vs DT, energy vs power, periodic vs aperiodic
• System properties: linearity, time-invariance, causality, BIBO stability
• Convolution formula and 5-step graphical method
• Fourier series: for periodic; Fourier Transform: for aperiodic
• Key FT pairs: δ(t)↔1, e^(-at)u(t)↔1/(a+jω)
• Laplace: one-sided; key pairs; ROC concept
• Transfer function H(s) = Y(s)/X(s)
• Z-transform: poles inside unit circle → stable