Energy vs Power Signals, Z-Transform Properties and Signal Basics

Energy Signals and Power Signals

Q1. Differentiate between energy signals and power signals. Can a signal be both? Find the energy of x(t) = e⁻²ᵗu(t). (5 marks)

Energy of a signal: E = ∫|x(t)|² dt over (−∞, ∞). For a discrete signal, E = Σ|x(n)|² over all n. Average power: P = lim (T→∞) (1/2T) ∫|x(t)|² dt over (−T, T).

Difference:

• Energy signal: 0 < E < ∞ and P = 0. It is finite in duration or decaying. Examples: rectangular pulse, e⁻ᵃᵗu(t) for a > 0.
• Power signal: 0 < P < ∞ and E = ∞. It is periodic or random and lasts forever. Examples: sine wave, constant (DC), unit step u(t).

Properties of the Z-Transform

Q4. Explain with proof any two properties of the z-transform. (5 marks)

Property 1: Linearity. If x₁(n) ↔ X₁(z) and x₂(n) ↔ X₂(z), then a·x₁(n) + b·x₂(n) ↔ a·X₁(z) + b·X₂(z). The ROC contains at least the intersection of the two ROCs.

Proof: Z{a·x₁(n) + b·x₂(n)} = Σ [a·x₁(n) + b·x₂(n)] z⁻ⁿ = a·Σ x₁(n)z⁻ⁿ + b·Σ x₂(n)z⁻ⁿ = a·X₁(z) + b·X₂(z)

Property 2: Time shifting. If x(n) ↔ X(z), then x(n − k) ↔ z⁻ᵏX(z). The ROC stays the same, except that z = 0 or z = ∞ may be added or removed.

Proof:

• Z{x(n − k)} = Σ x(n − k) z⁻ⁿ
• Let m = n − k, so n = m + k: = Σ x(m) z⁻⁽ᵐ⁺ᵏ⁾
• = z⁻ᵏ·Σ x(m)z⁻ᵐ = z⁻ᵏ·X(z)

A delay of k samples multiplies the transform by z⁻ᵏ.

Other properties (name them for short questions):

• Time reversal: x(−n) ↔ X(1/z)
• Scaling in z: aⁿx(n) ↔ X(z/a)
• Convolution: x₁(n) * x₂(n) ↔ X₁(z)·X₂(z)
• Differentiation in z: n·x(n) ↔ −z·dX(z)/dz
• Initial value: x(0) = lim (z→∞) X(z). Final value: x(∞) = lim (z→1) (z − 1)X(z)

Classification of Signals

Q5. Write the major classification of signals. (5 marks)

A signal is a function that carries information, usually varying with time. Signals are classified as follows.

• Continuous-time and discrete-time: defined for every value of time t, or only at integer instants n. Examples: x(t) = sin t; x(n) = {1, 2, 3}.
• Analog and digital: amplitude takes any value in a range, or only a finite set of levels. Examples: speech from a microphone; a binary stream.
• Deterministic and random: fully described by a formula, or unpredictable and described statistically. Examples: x(t) = 5 cos 2πt; thermal noise.
• Periodic and aperiodic: x(t + T) = x(t) for some T > 0, or no such T exists. Examples: cosine; e⁻ᵗu(t).
• Even and odd: even if x(−t) = x(t), odd if x(−t) = −x(t). Examples: cos t is even; sin t is odd.

Basics of Signal Transformation

Define the basics of signal transformation. (3 marks)

Signal transformation means changing a signal by modifying its independent variable (time) or its amplitude, so that a new signal is obtained from the original x(t) or x(n).

• Time shifting: x(t − t₀). Delay: moves the graph right by t₀ if t₀ > 0. Advance: moves it left if t₀ < 0.
• Time reversal (folding): x(−t). Mirror image about the vertical axis.
• Time scaling: x(at). Compression if a > 1, expansion if 0 < a < 1.
• Amplitude scaling: A·x(t). Amplified if A > 1, attenuated if 0 < A < 1, inverted if A < 0.

Fourier Series and Fourier Transform

T2. Differentiate between Fourier series and Fourier transform. (3 marks)

• Applies to: the series is for periodic signals. The transform is for aperiodic signals (also periodic ones through impulses).
• Spectrum: the series gives a discrete spectrum (lines at nω₀). The transform gives a continuous spectrum in ω.
• Formula: series: x(t) = Σ cₙe^(jnω₀t), with cₙ = (1/T)∫x(t)e^(−jnω₀t) dt over one period. Transform: X(ω) = ∫x(t)e^(−jωt) dt over (−∞, ∞).
• Condition: the series needs the Dirichlet conditions over one period. The transform needs x(t) to be absolutely integrable.
• Signal type: the series is for power signals. The transform is for energy signals.
• Output: the series gives a set of coefficients cₙ. The transform gives a function X(ω).

The Fourier transform can be seen as the limit of the Fourier series as the period T tends to infinity, when the lines of the spectrum merge into a continuum.

Signal Representation Techniques

What are the different representation techniques available for signals? (3 marks)

• Functional (mathematical): a formula, for example x(t) = e⁻²ᵗu(t), or x(n) = n for 0 ≤ n ≤ 3 and 0 otherwise.
• Graphical: a plot of amplitude against time. A continuous curve for x(t), and a stem plot for x(n).
• Tabular: a table of n against x(n).
• Sequence: a list of values, for example x(n) = {1, 2, 3, 4}, with an arrow marking the sample at n = 0.
• Using elementary signals: a signal written as a sum of shifted and scaled impulses, x(n) = Σ x(k)δ(n − k), or built from unit steps.
• Transform domain: frequency domain (Fourier), s-domain (Laplace) or z-domain, for example aⁿu(n) ↔ 1/(1 − az⁻¹).

The Laplace Transform

T4. Explain the Laplace transform with a suitable example. (3 marks)

The Laplace transform converts a time-domain signal x(t) into a function of the complex variable s = σ + jω: X(s) = ∫x(t)e⁻ˢᵗ dt (bilateral: −∞ to ∞; unilateral: 0⁻ to ∞). It exists for those values of s for which the integral converges. That set is the region of convergence (ROC).

Example: x(t) = e⁻ᵃᵗu(t)

X(s) = ∫ from 0 to ∞ of e⁻ᵃᵗe⁻ˢᵗ dt = ∫ from 0 to ∞ of e⁻⁽ˢ⁺ᵃ⁾ᵗ dt

X(s) = 1/(s + a), ROC: Re(s) > −a

Other standard pairs: δ(t) ↔ 1, u(t) ↔ 1/s.

Uses: Solves linear differential equations by turning them into algebraic equations.

Describes an LTI system by its transfer function H(s). The poles decide stability (all poles in the left half of the s-plane for a stable causal system).

Short Notes on Key Terms

Write short notes on: (a) unit step function, (b) sequence representation, (c) impulse, (d) linear, (e) causal signals. (5 marks)

(a) Unit step function: u(t) = 1 for t ≥ 0 and 0 for t < 0. In discrete time, u(n) = 1 for n ≥ 0 and 0 for n < 0. It is the running sum of the unit impulse, u(n) = Σ δ(k) for k ≤ n. Multiplying a signal by u(t) switches it on at t = 0, which is how a signal such as e⁻ᵃᵗ is made causal.

(b) Sequence representation: a discrete-time signal is written as an ordered list of values, with an arrow under the sample at n = 0, for example x(n) = {1, 2, 3, 4} with the arrow under 2. It can also be written as a sum of shifted impulses, x(n) = Σ x(k)δ(n − k).

(c) Impulse: δ(t) is zero for t ≠ 0 with unit area. It has the sifting property ∫x(t)δ(t − t₀) dt = x(t₀), and its output through an LTI system is the impulse response h(t).

(d) Linear: a system is linear if it obeys superposition, T{a·x₁ + b·x₂} = a·T{x₁} + b·T{x₂}. Example: y = 2x is linear, y = x² is not.

(e) Causal signals: a signal is causal if x(t) = 0 for t < 0, so it starts at or after t = 0, for example u(t) or e⁻ᵗu(t). A causal system is one whose output depends only on present and past inputs, never on future ones, so h(t) = 0 for t < 0.

Energy Signals, Power Signals and Whether a Signal Can Be Both

What is the difference between energy signals and power signals? Can a signal be both?

Energy signal: the total energy over all time is finite, and its average power is zero. A signal that dies out, or lasts only for a short time, is an energy signal. Energy: E = ∫|x(t)|² dt over (−∞, ∞), and for an energy signal 0 < E < ∞. Examples: a rectangular pulse, e⁻ᵃᵗu(t) with a > 0.

Power signal: the signal goes on forever, so its total energy is infinite. But its average power is finite and non-zero. Average power: P = lim (T→∞) (1/2T) ∫|x(t)|² dt over (−T, T), and for a power signal 0 < P < ∞. Examples: sine wave, constant (DC), unit step u(t), any periodic signal.

Can a signal be both? No. The two classes exclude each other:

• If the energy E is finite, then the average power is P = lim E/2T = 0. A finite number divided by an ever-growing time tends to zero, so the signal cannot be a power signal.
• If the power P is finite and non-zero, the energy keeps adding up over infinite time, so E = ∞ and it cannot be an energy signal.

Some signals are neither: if both E and P are infinite, as with the ramp r(t) = t·u(t) or the growing exponential eᵗ, the signal falls in neither class.

Quick examples: x(t) = e⁻²ᵗu(t) has E = ∫ e⁻⁴ᵗ dt from 0 to ∞ = 1/4 J, which is finite, so it is an energy signal. A sine wave A sin(ωt) has P = A²/2, which is finite and non-zero, so it is a power signal.

Linear Time-Invariant (LTI) Systems and the Superposition Principle

Define linear time-invariant (LTI) systems. Why is the superposition principle important for linearity?

Definition

A system is a rule T that converts an input x(n) into an output y(n) = T{x(n)}. An LTI system is one that is both linear and time-invariant.

Linear: the system satisfies two conditions.

• Additivity: T{x₁(n) + x₂(n)} = T{x₁(n)} + T{x₂(n)}
• Homogeneity: T{a·x(n)} = a·T{x(n)}

Time-invariant: if x(n) gives y(n), then a delayed input x(n − k) gives the delayed output y(n − k). The behaviour of the system does not change with time.

Examples: y(n) = 2x(n) is LTI. y(n) = x²(n) is not linear. y(n) = n·x(n) is not time-invariant.

Superposition principle

Additivity and homogeneity combined: if x₁(n) gives y₁(n) and x₂(n) gives y₂(n), then for any constants a and b, a·x₁(n) + b·x₂(n) gives a·y₁(n) + b·y₂(n).

• It is the test for linearity. A system is linear if and only if it obeys superposition. If it fails for even one pair of inputs, as y = x² does, it is not linear. Check: for x₁ and x₂, the output of the sum is (x₁ + x₂)² = x₁² + 2x₁x₂ + x₂², but the sum of the outputs is x₁² + x₂².
• It lets a complex input be broken into simple parts. The input is split into simple signals such as shifted impulses or complex exponentials. The response to each part is found separately and the responses are added.
• It leads to the convolution sum. Write the input as x(n) = Σ x(k)δ(n − k). By superposition each impulse gives a scaled response, and by time-invariance the response to δ(n − k) is h(n − k). Hence y(n) = Σ x(k)h(n − k) = x(n) * h(n).
• It is the basis of transform analysis. Fourier, Laplace and z-transforms work only because the system is linear. They turn convolution in the time domain into multiplication in the transform domain.
• It gives complete system description. Because of superposition, one function, the impulse response h(n), is enough to predict the output for every possible input.