04

Here is Note 4. This is the highest-yield theoretical and mathematical section in Chapter 1. Examiners use these exact derivations and numericals to separate the A+ students from the rest.

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Note 4: Signal Classification Part II (The Heavy Math)

Focus: Mathematical categorization of signals based on their time-domain boundaries, symmetry, and energy/power distribution.

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4.1 Causal, Anti-causal, and Non-causal Signals

This classification is determined entirely by where the signal exists on the time axis ($t=0$ boundary).

• Causal Signal: A signal that only exists in positive time. It starts at or after $t=0$.
  • Mathematical Condition: $x(t)=0$ for $t<0$.
  • Tip: Any signal multiplied by a unit step $u(t)$ is automatically forced to be causal.
• Anti-causal Signal: A signal that only exists in negative time. It ends exactly at or before $t=0$.
  • Mathematical Condition: $x(t)=0$ for $t>0$.
  • Tip: Any signal multiplied by $u(-t)$ is anti-causal.
• Non-causal Signal: A signal that exists on both sides of the $t=0$ boundary (e.g., a standard unshifted sine wave).

[Graph: Three separate axes. 1. A decaying exponential curve strictly on the right side of the y-axis (Causal). 2. A rising curve strictly on the left side of the y-axis (Anti-causal). 3. A bell-curve crossing both the left and right sides of the y-axis (Non-causal)]

🔥 Exact PYQs for Topic 4.1:

• State whether the following signals are causal, anticausal or noncausal. (i) $x(t)=e^{-2t}u(t)$; (ii) $x(t)=tu(t)$. (2025, 2020)
  • Solution: Both signals are multiplied by $u(t)$. By definition, $u(t)=0$ for $t<0$. Therefore, both $x(t)=0$ for $t<0$, making them strictly Causal signals.

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4.2 Even and Odd Signals (Symmetry)

Any arbitrary signal $x(t)$ can be decomposed into an Even component $x_e(t)$ and an Odd component $x_o(t)$.

• Even Signal: Symmetrical across the y-axis. Condition: $x(t)=x(-t)$.
• Odd Signal: Anti-symmetrical across the origin. Condition: $x(t)=-x(-t)$.
• Decomposition Formulas: $x_e(t)=\frac{x(t)+x(-t)}{2}$ $x_o(t)=\frac{x(t)-x(-t)}{2}$ $x(t)=x_e(t)+x_o(t)$

[Graph: An arbitrary pulse x(t). Below it, its even component x_e(t) showing y-axis symmetry, and its odd component x_o(t) showing origin symmetry]

🚨 MUST-MASTER PROOF: The Energy of Even/Odd Components

Question: If $x_e(t)$ and $x_o(t)$ are even and odd parts of $x(t)$, show that ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt$.

Proof:

1. We know that $x(t)=x_e(t)+x_o(t)$.
2. Squaring both sides and integrating over all time: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }[x_e(t)+x_o(t){]}^{2}dt$ ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt+2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt$
3. Analyze the cross-product term: $x_e(t)\cdot x_o(t)$.
   • The product of an Even function and an Odd function is always an Odd function.
   • The definite integral of any Odd function over symmetrical limits ($-\infty$ to $\infty$) is mathematically equal to zero.
4. Therefore, $2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt=0$.
5. Substituting this back into the equation leaves: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt\text{(Proved)}$

🔥 Exact PYQs for Topic 4.2:

• If $x_e(t)$ and $x_o(t)$ are even and odd parts of $x(t)$, show that ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt$. (2024, 2023, 2021)
• Find the odd and even components for each of the following signals: (i) $x(t)=u(t)$; (ii) $x(t)=e^{-\alpha t}u(t)$. (2020)
  • Solution snippet for (i): $u_e(t)=\frac{u(t)+u(-t)}{2}=\frac{1}{2}$ (for $t\ne 0$). $u_o(t)=\frac{u(t)-u(-t)}{2}=\frac{1}{2}\text{sgn}(t)$.
• Find the Even and Odd components of the given signal: $x(t)=\cos (20\pi t-5\pi )$ and also, sketch the signal. (2018, 2015)
• Determine the even and odd components of the signal $x(t)=\sin (\sqrt{2}\pi t)+\sin (15\pi t)$. (2019)

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4.3 Energy and Power Signals

This tests whether a signal’s amplitude eventually dies out (Energy) or continues forever (Power).

1. Energy of a Signal ($E$): Total area under the squared magnitude of the signal.

• Continuous-Time: $E={\int }_{-\infty }^{\infty }|x(t){|}^{2}dt$
• Discrete-Time: $E={\sum }_{n=-\infty }^{\infty }|x(n){|}^2$

2. Average Power of a Signal ($P$): The time-average of the signal’s energy.

• Continuous-Time: $P={\lim }_{T\to \infty }\frac{1}{T}{\int }_{-T/2}^{T/2}|x(t){|}^{2}dt$
• Discrete-Time: $P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=-N}^N|x(n){|}^2$

The Golden Rules:

1. Energy Signal: Has finite energy ($0<E<\infty$) and zero average power ($P=0$). Most transient, finite-duration, or exponentially decaying pulses are energy signals.
2. Power Signal: Has finite average power ($0<P<\infty$) and infinite total energy ($E=\infty$). All periodic signals and infinite-duration steps (like $u(t)$) are power signals.
3. A signal cannot be both.
4. If a signal grows towards infinity (e.g., $e^{at}$, $tu(t)$), both $E$ and $P$ are infinite, so it is neither.

PYQ Master-Class: How to solve the Numerical traps

Trap 1: The Zero-Energy Trick

• Question: Determine whether $x(n)=\{\begin{array}{llc}\sin (\pi n);& n\ge 00,& \text{otherwise}\end{array}$ is energy or power signal and also find its corresponding energy or power. (2025)
• Solution: Because $n$ is an integer (discrete-time), $\sin (\pi n)$ is exactly $0$ for all values of $n$ (i.e., $\sin (0)=0,\sin (\pi )=0,\sin (2\pi )=0$). Therefore, the sequence $x(n)=0$. Energy $E=0$ and Power $P=0$. It is technically neither an energy nor power signal.

Trap 2: Discrete Cosine Power Signal

• Question: Determine whether $x(n)=\{\begin{array}{llc}\cos (\pi n);& n\ge 00,& \text{otherwise}\end{array}$ is energy or power signal… (2023, 2019)
• Solution: For integers $n$, $\cos (\pi n)=(-1{)}^n$.
  • Energy: $E={\sum }_{n=0}^{\infty }|(-1{)}^n{|}^2={\sum }_{n=0}^{\infty }1=\infty$. (Not an energy signal).
  • Power: $P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=0}^N|(-1{)}^n{|}^2={\lim }_{N\to \infty }\frac{N+1}{2N+1}$. Using L’Hopital’s or dividing by $N$, $P=\frac{1}{2}$.
  • Result: It is a Power Signal with $P=0.5$ Watts.

Trap 3: The Continuous-Time Step

• Question: Determine the power and energy of the continuous time signal $x(t)=u(t)$. (2022)
• Solution:
  • $E={\int }_{-\infty }^{\infty }|u(t){|}^{2}dt={\int }_0^{\infty }1dt=\infty$.
  • $P={\lim }_{T\to \infty }\frac{1}{T}{\int }_0^{T/2}1dt={\lim }_{T\to \infty }\frac{T/2}{T}=\frac{1}{2}$.
  • Result: Power signal with $P=0.5$ Watts.

Trap 4: Bounded Discrete Range

• Question: Determine whether $x(n)=\{\begin{array}{llc}\cos (\pi n);& -4\le n\le 40,& \text{otherwise}\end{array}$ is energy or power signal… (2018)
• Solution: The signal has finite duration (only exists from $n=-4$ to $4$). Any bounded, finite-duration signal is strictly an Energy Signal.
  • $E={\sum }_{n=-4}^4|\cos (\pi n){|}^2={\sum }_{n=-4}^{4}1=9$ Joules. ($P=0$).

🔥 Remaining Exact PYQs for Topic 4.3:

• Determine whether $x(n)=\{\begin{array}{l}n;0\le n<510-n;5\le n\le 100;otherwise\end{array}$ is energy or power… (2017) (Hint: Finite duration, so it is an Energy signal. Sum the squares of the values).
• Consider a signal $x(t)=5\cos (\pi t)+\sin (5\pi t)$. Determine if it is an energy signal or a power signal and also find its energy/power. (2016) (Hint: Sum of periodic signals is periodic. Thus, Power signal. Power of a sinusoid $A\cos (\omega t)$ is $A^2/2$. Add the powers of independent frequencies: $5^2/2+1^2/2=13$ W).
• Compute the energy of $x(t)=\sin (\sqrt{2}\pi t)+\sin (15\pi t)$. (2015) (Hint: If a question asks for the energy of an infinite sum of sines, the energy is infinite. It is a power signal).