Signals, Systems & LTI Properties: Types, Graphs & System Analysis

TAKEAWAYS:

• Signals are classified as continuous/discrete-time, deterministic/random, and periodic/aperiodic—each with unique mathematical representations and graphs.
• LTI systems (Linear Time-Invariant) obey superposition and time-shift properties, critical for designing stable communication channels.
• System properties (causality, linearity, stability, memory) determine whether a system can process signals reliably in real-world networks.
• Signal types (step, ramp, impulse, sinusoidal) model real-world phenomena like eSewa transaction spikes (impulse) or Ncell call handoffs (ramp).
• Worked examples tie theory to practice: e.g., calculating the output of an LTI system for a given input signal.
• Exam focus: Graphical illustrations, mathematical definitions, and distinguishing between system properties (e.g., "Is a system causal if its output depends on future inputs?").

1. Signals: Types, Graphs, and Mathematical Relations

Signals carry information in data communication. They are classified based on time domain (continuous/discrete) and nature (deterministic/random).

graph TD;
  A["Continuous-Time Signal"] -->|"e.g., NTC AC Voltage"| B["sinusoidal"]
  A -->|"e.g., eSewa Transaction"| C["impulse"]
  D["Discrete-Time Signal"] -->|"e.g., WhatsApp Voice Samples"| E["step"]
  D -->|"e.g., Ncell Call Traffic"| F["ramp"]

A. Continuous vs. Discrete-Time Signals

Key Signals and Their Graphs:

1. Unit Step Signal ($u(t)$):

   • Definition: $u(t)=\{\begin{array}{ll}1& \text{if }t\ge 0\\ 0& \text{if }t<0\end{array}$
   • Graph: A jump from 0 to 1 at $t=0$.
   • Application: Models sudden changes like eSewa transaction approvals (binary: approved/rejected).

2. Ramp Signal ($r(t)$):

   • Definition: $r(t)=tu(t)$.
   • Graph: Linear increase starting at $t=0$.
   • Application: Represents Ncell call traffic growth over time.

3. Impulse Signal ($\delta (t)$):

   • Definition: $\delta (t)=\{\begin{array}{ll}\infty & \text{if }t=0\\ 0& \text{otherwise}\end{array}$, with ${\int }_{-\infty }^{\infty }\delta (t)dt=1$.
   • Graph: Infinite spike at $t=0$.
   • Application: Models Daraz order spikes at checkout time.

4. Sinusoidal Signal ($\sin (\omega t)$):

   • Definition: $x(t)=A\sin (2\pi ft+\varphi )$.
   • Graph: Oscillates between $-A$ and $A$.
   • Application: Used in NTC power grid signals (AC voltage).

5. Signum Signal ($\text{sgn}(t)$):

   • Definition: $\text{sgn}(t)=\{\begin{array}{ll}1& \text{if }t>0\\ -1& \text{if }t<0\\ 0& \text{if }t=0\end{array}$
   • Graph: Jumps from -1 to 1 at $t=0$.
   • Application: Models bidirectional traffic flow in Kathmandu.

B. Deterministic vs. Random Signals

Worked Example: Model the Khalti transaction delay as a random signal.

• Assumption: Delay $D$ follows a normal distribution $D\sim N(\mu =2,\sigma =0.5)$.
• Why? Real-world delays (e.g., bank processing) are rarely exact.

2. Systems: LTI and Their Properties

A system processes input signals to produce output signals. LTI (Linear Time-Invariant) systems are fundamental in data communication.

sequenceDiagram participant User participant eSewa participant Bank
  User->>eSewa: Impulse Signal (Payment Request)
  eSewa->>Bank: Linear Processing (Verify Balance)
  Bank-->>eSewa: Time-Invariant Response (Approval/Rejection)
  eSewa-->>User: Output Signal (Transaction Status)

A. Definition of LTI System

A system is LTI if it satisfies:

1. Linearity: Superposition and homogeneity hold.
   • If input $x_1(t)\to y_1(t)$ and $x_2(t)\to y_2(t)$, then $a{x}_1(t)+b{x}_2(t)\to a{y}_1(t)+b{y}_2(t)$.
2. Time-Invariance: A time shift in input causes the same shift in output.
   • If $x(t)\to y(t)$, then $x(t-t_0)\to y(t-t_0)$.

Example: An LTI system processes a Pathao ride request signal $x(t)$ (impulse at booking time) to output $y(t)$ (driver assignment delay).

• Linearity: If two requests arrive, the system combines their delays.
• Time-Invariance: A request at 3 PM has the same delay distribution as at 3:01 PM.

B. System Properties

Worked Example: Is the system $y(t)=x^2(t)$ linear?

• Test: Let $x_1(t)=1$, $x_2(t)=2$.
  • $y_1(t)=1$, $y_2(t)=4$.
  • $a{x}_1(t)+b{x}_2(t)=0.5\cdot 1+1\cdot 2=2.5$.
  • $a{y}_1(t)+b{y}_2(t)=0.5\cdot 1+1\cdot 4=4.5$.
  • Since $2.5^2\ne 4.5$, the system is nonlinear.

3. In the Real World

1. eSewa Transaction Processing:

   • Signal: Impulse $\delta (t)$ at payment time.
   • System: LTI system checks balance (linear operation) and updates records (time-invariant).
   • Property Used: Causality (output depends only on past/future inputs, not future states).

2. Ncell Call Handoff:

   • Signal: Ramp $r(t)$ as signal strength decreases during movement.
   • System: LTI filter decides when to handoff to another tower.
   • Property Used: Stability (must handle sudden signal drops without crashing).

3. NTC Power Grid:

   • Signal: Sinusoidal $\sin (2\pi 50t)$ (50 Hz AC).
   • System: LTI transformers and regulators ensure stable voltage.
   • Property Used: Linearity (superposition of loads).

4. Exam Tip

1. Graphs are mandatory: For every signal type (step, ramp, impulse), sketch the graph with axes labeled and mathematical definition.
2. LTI systems: Always verify linearity and time-invariance with examples.
3. System properties:
   • Causality: "Can the output depend on future inputs?" → No.
   • Stability: "If input is bounded, is output bounded?" → Yes for LTI systems.
4. Real-world ties: Relate signals to Nepalese tech (e.g., "eSewa uses impulse signals for transactions").
5. Common pitfalls:
   • Confusing discrete-time $x[n]$ with continuous-time $x(t)$.
   • Forgetting the impulse signal is infinite at $t=0$ but integrates to 1.

sequenceDiagram
    participant User as User (e.g., Khalti App)
    participant System as LTI System (e.g., Bank Server)
    User->>System: Input: Transaction Request (Impulse Signal)
    System-->>User: Output: Approval/Rejection (Step Signal)
    Note over User,System: Linearity: Multiple requests combine.<br/>Time-Invariance: Same delay at any time.

stateDiagram-v2
    [*] --> Stable
    Stable --> Unstable: If input unbounded
    Stable --> Stable: For bounded input (LTI property)
    Unstable --> [*]
    Note over Stable: Bounded Input → Bounded Output