What is the difference between energy signal and power signal?

An energy signal has finite total energy (0 < E < ∞) and zero average power. A power signal has finite average power (0 < P < ∞) and infinite energy. To check: compute E = ∫|x(t)|² dt. If finite → energy signal. If infinite, compute P = lim(T→∞) (1/2T)∫|x(t)|² dt. If finite → power signal. Periodic signals are always power signals. Finite-duration bounded signals are always energy signals. A signal cannot be both an energy signal and a power signal simultaneously.

How do you check if a system is linear?

Apply the superposition test: (1) Find y₁ = T{x₁} and y₂ = T{x₂}. (2) Find T{ax₁ + bx₂}. (3) If T{ax₁ + bx₂} = ay₁ + by₂ for all x₁, x₂ and all constants a, b — the system is linear. Common non-linear indicators: squaring (x²), absolute value |x|, multiplication (x₁·x₂), adding a non-zero constant (x+c). Common linear indicators: differentiation, integration, scaling by a constant.

What is the sifting property of the impulse function?

The sifting property states: ∫x(t)δ(t-t₀)dt = x(t₀). The Dirac delta δ(t-t₀) samples (sifts out) the value of x(t) at t = t₀. Key results: ∫δ(t)dt = 1 (unit area), δ(t) = 0 for t ≠ 0, δ(at) = δ(t)/|a|, ∫x(t)δ'(t-t₀)dt = -x'(t₀).

Unit 1 · Signals & Systems

Introduction to Signals
& Systems

Q: How do you check if a system is linear?

Apply the superposition test: (1) Find y₁ = T{x₁} and y₂ = T{x₂}. (2) Find T{ax₁ + bx₂}. (3) If T{ax₁ + bx₂} = ay₁ + by₂ for all x₁, x₂ and all constants a, b — the system is linear. Common non-linear indicators: squaring (x²), absolute value |x|, multiplication (x₁·x₂), adding a non-zero constant (x+c). Common linear indicators: differentiation, integration, scaling by a constant.

Q: What is the sifting property of the impulse function?

The sifting property states: ∫x(t)δ(t-t₀)dt = x(t₀). The Dirac delta δ(t-t₀) samples (sifts out) the value of x(t) at t = t₀. Key results: ∫δ(t)dt = 1 (unit area), δ(t) = 0 for t ≠ 0, δ(at) = δ(t)/|a|, ∫x(t)δ'(t-t₀)dt = -x'(t₀).

B.Tech ECE · Signals & Systems· ~30 min read· 10 GATE Questions· GATE Weightage: 1–2 marks

Contents

1. What is a Signal?
2. Classification of Signals
3. Basic Signal Operations
4. Elementary Signals
5. What is a System?
6. Classification of Systems
7. GATE Questions (10)

1. What is a Signal?

A signal is any physical quantity that is a function of one or more independent variables and carries information. In ECE, signals are typically functions of time.

Examples: voltage across a resistor v(t), audio waveform x(t), ECG trace, digital data stream, image intensity I(x,y).

Key Distinction

Continuous-Time (CT) signal — defined for all values of t ∈ (−∞, ∞). Example: sinusoidal voltage from an oscillator.
Discrete-Time (DT) signal — defined only at integer values of n. Example: digital audio samples x[n] at 44100 Hz sampling rate.

Signal Classification Tree — Overview

Every signal is either CT or DT; independently, either periodic or aperiodic; and separately classified as an energy signal or power signal (but not both — except the trivial x=0 case).

2. Classification of Signals

2.1 Periodic vs Aperiodic

A CT signal x(t) is periodic with fundamental period T₀ if:

Periodicity Condition (CT)

x(t) = x(t + T₀) for all t, where T₀ > 0 is the smallest such value

A DT signal x[n] is periodic with fundamental period N if x[n] = x[n + N] for all integer n.

Critical GATE Fact — DT Periodicity

e^jωn is periodic in DT only if ω/2π is a rational number. In CT, e^jωt is always periodic.
Sum of two periodic CT signals with periods T₁ and T₂ is periodic if T₁/T₂ is rational — period = LCM(T₁,T₂).

2.2 Even and Odd Signals

Any signal can be decomposed into its even and odd parts:

Even / Odd Decomposition

x_e(t) = [x(t) + x(−t)] / 2 ← Even part x_o(t) = [x(t) − x(−t)] / 2 ← Odd part x(t) = x_e(t) + x_o(t)

Even: x(−t) = x(t) | Odd: x(−t) = −x(t). Note: the odd part of any signal is zero at t = 0.

2.3 Energy and Power Signals

Energy and Power — CT Signal

E = ∫₋∞^∞ |x(t)|² dt ← Total energy (Joules) P = lim(T→∞) (1/2T) ∫₋ᵀᵀ |x(t)|² dt ← Average power (Watts)

Energy and Power — DT Signal

E = Σₙ₌₋∞^∞ |x[n]|² ← Total energy P = lim(N→∞) (1/2N+1) Σₙ₌₋ₙᴺ |x[n]|² ← Average power

Signal Type	Energy E	Power P	Example
Energy Signal	0 < E < ∞	P = 0	x(t) = e⁻ᵃᵗu(t), a > 0
Power Signal	E → ∞	0 < P < ∞	x(t) = A·cos(ω₀t)
Neither	E → ∞	P → ∞	x(t) = t (ramp, t > 0)
Note: Periodic signals → always power signals. Finite-duration bounded signals → always energy signals.

Quick Energy Formulas

• x(t) = Ae⁻ᵃᵗu(t): E = A²/2a
• x(t) = A·rect(t/τ): E = A²τ
• x(t) = A·sinc(Wt): E = A²/W (using Parseval's theorem)
• x(t) = A·cos(ω₀t): P = A²/2

2.4 Deterministic vs Random

Deterministic signals are fully described by a mathematical expression — no uncertainty. Random signals are described statistically (mean, variance, power spectral density). In this unit, we deal with deterministic signals only.

3. Basic Signal Operations

3.1 Time Shifting

Time Shifting

y(t) = x(t − t₀) • t₀ > 0 : signal shifted RIGHT (delayed by t₀) • t₀ < 0 : signal shifted LEFT (advanced by |t₀|)

3.2 Time Scaling

Time Scaling

y(t) = x(at) • |a| > 1 : compressed (speeds up) • 0 < |a| < 1 : expanded (slows down) • a < 0 : reversal + scaling

3.3 Time Reversal

Time Reversal

y(t) = x(−t) ← mirror image about t = 0

3.4 Combined Operations — CRITICAL for GATE

For y(t) = x(at + b), the correct procedure to sketch:

Step-by-Step: x(t) → x(at + b)

Method 1: x(t) → x(t + b) [shift] → x(at + b) [scale] Method 2: x(t) → x(at) [scale] → x(a(t + b/a)) [shift by −b/a] Both methods give the same result. Method 1 is usually easier. Example: y(t) = x(2t − 4) Step 1: Shift right by 4: x(t − 4) → intermediate Step 2: Scale by 2: x(2t − 4) ← compress by factor 2

GATE Trap: Scaling of δ(at)

δ(at) = δ(t)/|a| (NOT δ(t))
Example: δ(3t − 6) = δ(3(t − 2)) = (1/3)δ(t − 2)

4. Elementary Signals

4.1 Unit Impulse Function δ(t) — Dirac Delta

Impulse Function — Definition & Properties

δ(t) = 0 for t ≠ 0, ∫₋∞^∞ δ(t) dt = 1 Sifting property: ∫₋∞^∞ x(t) δ(t − t₀) dt = x(t₀) Scaling: δ(at) = δ(t)/|a| Derivative: ∫₋∞^∞ x(t) δ'(t − t₀) dt = −x'(t₀)

Physically: δ(t) is a pulse of infinite height, zero width, and unit area — an idealized impulse.

4.2 Unit Step Function u(t)

Unit Step

u(t) = 1 for t > 0 u(t) = 0 for t < 0 u(0) = 1/2 (conventional) Relation: du(t)/dt = δ(t) and u(t) = ∫₋∞ᵗ δ(τ) dτ DT: u[n] = 1 for n ≥ 0, u[n] = 0 for n < 0

4.3 Unit Ramp r(t)

Unit Ramp

r(t) = t · u(t) = { t for t ≥ 0, 0 for t < 0 } Relation: dr(t)/dt = u(t) and r(t) = ∫₋∞ᵗ u(τ) dτ

4.4 Real and Complex Exponentials

Exponential Signals

Real exponential: x(t) = A·eˢᵗ where s = σ (real) σ > 0 : growing exponential σ < 0 : decaying exponential (energy signal if starts at t=0) σ = 0 : constant (power signal) Complex exponential: x(t) = Ae^(jω₀t) = A[cos(ω₀t) + j·sin(ω₀t)] — always periodic with period T₀ = 2π/ω₀ — |x(t)| = A (constant magnitude) General: x(t) = Ae^(σt)·e^(jω₀t) where s = σ + jω₀

4.5 Sinc Function

Sinc Function

sinc(t) = sin(πt) / (πt), sinc(0) = 1 (by L'Hopital) sinc(t) = 0 at t = ±1, ±2, ±3, ... (integer zeros) FT{sinc(Wt)} = (1/W)·rect(f/W) ← fundamental FT pair

5. What is a System?

A system is any entity that takes an input signal x(t) and produces an output signal y(t) through some transformation T{·}:

y(t) = T{x(t)}

Examples: amplifier, filter, ADC, human ear, digital FIR filter y[n] = Σ h[k]·x[n−k].

6. Classification of Systems

6.1 Linearity

A system is linear if it satisfies superposition (additivity + homogeneity):

Linearity Test

If x₁(t) → y₁(t) and x₂(t) → y₂(t), then: ax₁(t) + bx₂(t) → ay₁(t) + by₂(t) for all a, b Non-linear indicators: x²(t), |x(t)|, x(t)·x(t−1), adding a constant c Linear indicators: d/dt, ∫, multiplication by constant k

6.2 Time-Invariance

A system is time-invariant (TI) if shifting the input shifts the output by the same amount:

Time-Invariance Test

If x(t) → y(t), then x(t − τ) → y(t − τ) for all τ Test procedure: Path 1: apply x(t−τ) as input → get y₁(t) = T{x(t−τ)} Path 2: take output y(t), shift it → y₂(t) = y(t−τ) If y₁(t) = y₂(t) for all τ → Time-Invariant Time-Variant indicators: coefficient depends on t [e.g., t·x(t)], argument scaled [x(2t)], x(−t)

6.3 Causality

Causality

A system is causal if the output at time t depends only on present and past inputs x(τ) for τ ≤ t. CT: impulse response h(t) = 0 for t < 0 DT: impulse response h[n] = 0 for n < 0 Non-causal: y(t) = x(t+1) [needs future], y(t) = x(−t) [for t<0] Causal: y(t) = x(t−1) [only past], y[n] = x[n] + x[n−1]

6.4 BIBO Stability

BIBO Stability

Bounded Input → Bounded Output (BIBO) If |x(t)| ≤ Mₓ < ∞ for all t, then |y(t)| ≤ Mᵧ < ∞ for all t For LTI system — BIBO stable iff: CT: ∫₋∞^∞ |h(t)| dt < ∞ (impulse response absolutely integrable) DT: Σₙ |h[n]| < ∞ (impulse response absolutely summable)

6.5 Memory

Memory vs Memoryless

Memoryless: output depends only on current input Examples: y(t) = x²(t), y[n] = 2x[n] With Memory: output depends on past (or future) inputs Examples: y(t) = x(t−1), y[n] = Σₖ₌₀ⁿ x[k] (accumulator)

System	Linear?	TI?	Causal?	Stable?	Memory?
y(t) = 3x(t)	Yes	Yes	Yes	Yes	No
y(t) = x²(t)	No	Yes	Yes	Cond.	No
y(t) = t·x(t)	Yes	No	Yes	No	No
y(t) = x(t−2)	Yes	Yes	Yes	Yes	Yes
y(t) = x(2t)	Yes	No	No	Yes	No
y(t) = x(−t)	Yes	No	No	Yes	Yes
y[n] = Σx[k] (accum.)	Yes	Yes	Yes	No	Yes

GATE PYQ

GATE Questions — Unit 1

GATE 20211 MarkMCQ

Q1. The signal x(t) = 2e⁻³ᵗ u(t) is:

(A) A power signal with P = 2/3

(B) An energy signal with E = 2/3

(D) Neither an energy signal nor a power signal

Answer: (B)

E = ∫₀^∞ |2e⁻³ᵗ|² dt = ∫₀^∞ 4e⁻⁶ᵗ dt = 4 · [−e⁻⁶ᵗ/6]₀^∞ = 4/6 = 2/3

E = 2/3 < ∞ → Energy signal. Since E is finite, P = 0 (not a power signal).

GATE 20201 MarkMCQ

Q2. The even part of x(t) = e^(−jωt) is:

(A) cos(ωt)

(B) −j·sin(ωt)

(D) sin(ωt)

Answer: (A)

xₑ(t) = [x(t) + x(−t)] / 2 = [e^(−jωt) + e^(jωt)] / 2 = cos(ωt) ← Euler's formula

The odd part: x₀(t) = [e^(−jωt) − e^(jωt)]/2 = −j·sin(ωt)

GATE 20192 MarksMCQ

Q3. The system y(t) = x(t)·cos(ω₀t) is:

(A) Linear and Time-Invariant

(B) Non-linear and Time-Invariant

(D) Non-linear and Time-Variant

Answer: (C)

Linearity: T{ax₁+bx₂} = (ax₁+bx₂)cos(ω₀t) = a·T{x₁} + b·T{x₂} ✓ Linear

Time-Invariance:

Path 1: apply x(t−τ) → y₁(t) = x(t−τ)·cos(ω₀t) Path 2: shift output → y₂(t) = y(t−τ) = x(t−τ)·cos(ω₀(t−τ))

y₁(t) ≠ y₂(t) because cos(ω₀t) ≠ cos(ω₀(t−τ)). → Time-Variant.

GATE 20181 MarkNAT

Q4. δ(3t − 6) = k · δ(t − 2). Find k.

(A) 1/3

(B) 3

(D) 6

Answer: (A) k = 1/3

δ(at − b) = (1/|a|) · δ(t − b/a) δ(3t − 6) = δ(3(t − 2)) = (1/|3|) · δ(t − 2) = (1/3) · δ(t − 2)

So k = 1/3.

GATE 20222 MarksMCQ

Q5. The fundamental period of x[n] = cos(6πn/7) + sin(8πn/7) is:

(A) 7

(B) 14

(D) 4

Answer: (A) 7

For cos(ω₀n): N₁ = 2π·m/ω₀ (smallest integer m giving integer N) ω₀ = 6π/7 → N₁ = 2π·m/(6π/7) = 7m/3 → m=3 → N₁ = 7 For sin(ω₀n): ω₀ = 8π/7 → N₂ = 2π·m/(8π/7) = 7m/4 → m=4 → N₂ = 7 Fundamental period N = LCM(7, 7) = 7

GATE 20172 MarksMCQ

Q6. Given x(t) = t·u(t), y(t) = x(2t − 4). Which of the following is correct?

(A) y(t) = (2t − 4)·u(t)

(B) y(t) = 2(t − 2)·u(t − 2)

(D) y(t) = (t − 2)·u(t − 2)

Answer: (B)

x(t) = t·u(t) y(t) = x(2t − 4) = (2t − 4) · u(2t − 4) = 2(t − 2) · u(2(t − 2)) = 2(t − 2) · u(t − 2) [since u(2(t−2)) = u(t−2) for positive scaling]

This is a ramp starting at t = 2 with slope 2.

GATE 20162 MarksMCQ

Q7. The system y[n] = x[−n] is:

(A) Causal and Time-Invariant

(B) Causal and Time-Variant

(D) Non-causal and Time-Invariant

Answer: (C)

Causality: y[−1] = x[1] (future input). Non-causal.

Time-Invariance:

Path 1: apply x[n−k] → y₁[n] = x[−n−k] Path 2: shift output → y₂[n] = y[n−k] = x[−(n−k)] = x[−n+k]

y₁[n] ≠ y₂[n] (−n−k ≠ −n+k). → Time-Variant.

GATE 20151 MarkMCQ

Q8. For the DT system y[n] = x[n] − x[n−1], the system is:

(A) Linear, Time-Invariant, Causal, and BIBO Stable

(B) Linear, Time-Invariant, Causal, but not BIBO Stable

(D) Non-linear and Causal

Answer: (A)

Impulse response: h[n] = δ[n] − δ[n−1]

Linearity: T{ax₁+bx₂} = ax₁[n]−ax₁[n−1]+bx₂[n]−bx₂[n−1] = aT{x₁}+bT{x₂} ✓ TI: h[n] doesn't depend on n ✓ Causality: h[n]=0 for n<0 ✓ BIBO: Σ|h[n]| = |h[0]|+|h[1]| = 1+1 = 2 < ∞ ✓

GATE 20142 MarksNAT

Q9. The power of the signal x(t) = 3cos(4πt) + 4sin(6πt) is ___ Watts.

(A) 12.5

(B) 25

(D) 7

Answer: (A) 12.5 W

For sinusoidal x(t) = A·cos(ωt), power P = A²/2 P₁ = 3²/2 = 9/2 = 4.5 W P₂ = 4²/2 = 16/2 = 8.0 W Total power = P₁ + P₂ = 4.5 + 8.0 = 12.5 W (Signals at different frequencies → powers add directly)

GATE 20232 MarksMCQ

Q10. The system y(t) = tx(t − 1) is:

(A) Linear and Time-Invariant

(B) Linear and Time-Variant

(D) Non-linear and Time-Variant

Answer: (B)

Linearity: T{ax₁+bx₂} = t(ax₁(t−1)+bx₂(t−1)) = aT{x₁}+bT{x₂} ✓ Linear

Time-Invariance:

Path 1: x(t−τ) → y₁(t) = t·x(t−1−τ) Path 2: y(t−τ) = (t−τ)·x(t−τ−1)

y₁(t) = t·x(t−1−τ) ≠ (t−τ)·x(t−τ−1) = y₂(t). Time-Variant (coefficient t changes with shift).

← Back to

Signals & Systems Hub

Next Unit →

Unit 2: Fourier Series

Introduction to Signals& Systems

1. What is a Signal?

2. Classification of Signals

2.1 Periodic vs Aperiodic

2.2 Even and Odd Signals

2.3 Energy and Power Signals

2.4 Deterministic vs Random

3. Basic Signal Operations

3.1 Time Shifting

3.2 Time Scaling

3.3 Time Reversal

3.4 Combined Operations — CRITICAL for GATE

4. Elementary Signals

4.1 Unit Impulse Function δ(t) — Dirac Delta

4.2 Unit Step Function u(t)

4.3 Unit Ramp r(t)

4.4 Real and Complex Exponentials

4.5 Sinc Function

5. What is a System?

6. Classification of Systems

6.1 Linearity

6.2 Time-Invariance

6.3 Causality

6.4 BIBO Stability

6.5 Memory

GATE Questions — Unit 1

Introduction to Signals
& Systems