Discrete-Time Signals & Systems: Classification, Operations & Properties

1. Discrete-Time Signals: Definition & Classification

A discrete-time signal is a sequence of numbers $x[n]$ defined only at integer values of $n$ (e.g., $n=0,\pm 1,\pm 2,\dots$). Unlike continuous signals, it exists only at sampled points.

1.1 Classification of Signals

Signals are categorized based on their duration, amplitude, and periodicity:

Visual:

classDiagram
    class Signal {
        +isFinite(): bool
        +isBounded(): bool
        +isPeriodic(): bool
    }
    Signal <|-- FiniteSignal : "1"
    Signal <|-- InfiniteSignal : "1"
    Signal <|-- BoundedSignal : "1"
    Signal <|-- UnboundedSignal : "1"
    Signal <|-- PeriodicSignal : "1"
    Signal <|-- AperiodicSignal : "1"

1.2 Common Discrete-Time Signals

1. Unit Impulse (Delta) Signal $\delta [n]$

   • Defined as: $$\delta [n]=\{\begin{array}{ll}1& \text{if }n=0,\\ 0& \text{otherwise.}\end{array}$$
   • Plot:

   • Use: Used to test systems (like a hammer test in mechanical systems).

2. Unit Step Signal $u[n]$

   • Defined as: $$u[n]=\{\begin{array}{ll}1& \text{if }n\ge 0,\\ 0& \text{otherwise.}\end{array}$$
   • Plot:

   • Use: Models sudden changes (e.g., turning on a switch in a circuit).

3. Exponential Signal $a^{n}u[n]$

   • Plot (for $a=0.5$):

   • Use: Models decaying systems (e.g., capacitor discharge in digital circuits).

4. Sinusoidal Signal $\cos ({\omega }_{0}n+\varphi )$

   • Plot (for ${\omega }_0=\pi /2$):

   • Use: Represents periodic phenomena (e.g., music notes in digital audio).

2. Signal Operations

Operations manipulate signals to extract features or simplify analysis.

2.1 Time Shifting (Delay/Advance)

• Delay by $k$: $x[n-k]$ shifts signal right by $k$ samples.
• Advance by $k$: $x[n+k]$ shifts signal left by $k$ samples.
• Example: If $x[n]=u[n]-u[n-5]$ (rectangular pulse), then $x[n-2]$ delays it by 2 samples.
• Plot:

2.2 Time Reversal (Folding)

• $x[-n]$ flips the signal about $n=0$.
• Example: If $x[n]=nu[n]$, then $x[-n]=-nu[-n]$.

2.3 Time Scaling (Expansion/Compression)

• Expansion by $L$: $x[n/L]$ (only defined for integer $L$).
• Compression by $L$: $x[Ln]$.
• Example: Compressing $x[n]=\cos (n)$ by 2 gives $x[2n]=\cos (2n)$ (higher frequency).

2.4 Amplitude Scaling & Addition

• Scaling: $ax[n]$ multiplies amplitude by $a$.
• Addition: $x_1[n]+x_2[n]$ adds signals pointwise.

3. Discrete-Time Systems

A discrete-time system processes an input signal $x[n]$ to produce an output $y[n]$. It is defined by an input-output relationship: $$y[n]=T\{x[n]\}$$

3.1 System Properties

Systems are classified based on four key properties:

Visual: System Properties Flowchart

flowchart TD
    A["System"] --> B["Causal?"]
    B -->|"Yes"| C["Linear?"]
    B -->|"No"| D["Non-causal"]
    C --> E["Time-Invariant?"]
    C -->|"No"| F["Nonlinear"]
    E --> G["Stable?"]
    E -->|"No"| H["Time-Variant"]
    G -->|"Yes"| I["LTI System"]
    G -->|"No"| J["Unstable"]

3.2 LTI Systems & Convolution

Linear Time-Invariant (LTI) systems are the most important class. Their output is given by convolution: $$y[n]=x[n]\ast h[n]=\sum _{k=-\infty }^{\infty }x[k]h[n-k]$$

Example: Convolution of $x[n]=u[n]$ and $h[n]=\delta [n]-\delta [n-1]$

1. $h[n]$ is a finite impulse response (FIR) filter (differentiator).
2. Convolution: $$y[n]=\sum _{k=-\infty }^{\infty }u[k](\delta [k]-\delta [k-1])=u[n]-u[n-1]$$
   • This is a rectangular pulse of width 1.

Plot:

4. Real-World Applications in Nepal

4.1 eSewa & Khalti: Digital Payment Systems

• Idea Used: Discrete-time sampling of transaction data.
  • When you pay via eSewa, the app samples your fingerprint/OTP as a discrete signal (sequence of binary values).
  • The system processes this signal through an LTI filter (e.g., checksum validation) to detect fraud.
  • Example: If your OTP is 3-8-2-7, the system treats it as $x[n]=[3,8,2,7]$ and applies a moving average filter to smooth out noise (e.g., typos).

4.2 Pathao: Ride-Hailing & Traffic Routing

• Idea Used: Periodic signals for traffic light synchronization.
  • Pathao’s algorithm uses discrete-time models to predict driver arrival times at traffic lights.
  • Traffic lights follow a periodic signal $x[n]=\sin (2\pi n/T)$, where $T$ is the light cycle (e.g., 60 seconds).
  • The system convolves this with driver arrival patterns to optimize green light duration.

4.3 NTC & Ncell: Voice Signal Processing

• Idea Used: Sampling & LTI filtering in mobile calls.
  • Your voice is sampled at 8 kHz (discrete-time signal) and passed through anti-aliasing filters (LTI systems) to remove high-frequency noise.
  • Example: If your voice signal is $x[n]=\cos (2\pi 1000n/8000)$, the filter $h[n]$ removes frequencies above 3.4 kHz (Nyquist limit).

5. Worked Example: Traffic Light Control System

Problem: Design a discrete-time system to model a traffic light that cycles through Green (1s) → Yellow (0.5s) → Red (2s) and repeats.

Solution:

1. Define the periodic signal: The traffic light can be represented as: $$x[n]=\{\begin{array}{ll}1& \text{for }0\le n<8\text{ (Green)},\\ 0& \text{for }8\le n<12\text{ (Yellow)},\\ -1& \text{for }12\le n<28\text{ (Red)},\\ x[n-28]& \text{otherwise (periodic).}\end{array}$$ (Assuming sampling rate = 8 samples/sec: 1s = 8 samples.)

2. Plot:

3. System Properties:

   • Periodic: $x[n+28]=x[n]$.
   • Bounded: $|x[n]|\le 1$.
   • LTI? No, because the output depends on absolute time (not just input history). However, if we model it as a state machine, we can approximate it with LTI components.

4. LTI Approximation: Use a feedback system with a delay line to create periodicity: $$y[n]=x[n]+\alpha y[n-28]$$ where $\alpha$ is a gain to enforce repetition.

6. Exam Tip

What to Focus On:

1. Signal Classification:

   • Memorize the definitions of finite/infinite, bounded/unbounded, and periodic/aperiodic signals.
   • Common pitfalls: Confusing periodic with constant (e.g., $x[n]=5$ is periodic with any period, but trivial).

2. System Properties:

   • Causality: Check if $y[n]$ depends on future inputs (e.g., $y[n]=x[n+1]$ is non-causal).
   • Linearity: Test with superposition (e.g., if $T\{x_1\}+T\{x_2\}\ne T\{x_1+x_2\}$, it’s nonlinear).
   • Time-Invariance: Shift input and see if output shifts similarly.
   • Stability: For LTI systems, check if the impulse response $h[n]$ is absolutely summable ($\sum |h[n]|<\infty$).

3. Convolution:

   • Graphical method: Flip $h[n]$, shift, and multiply-accumulate.
   • Shortcut: For $h[n]=\delta [n-k]$, $y[n]=x[n-k]$ (pure delay).
   • Common mistake: Forgetting to flip $h[n]$ before shifting.

4. Real-World Links:

   • eSewa/Khalti: Think of sampling (discrete signals) and filtering (LTI systems for fraud detection).
   • Pathao: Periodic signals for traffic light synchronization.
   • NTC/Ncell: Sampling rate and anti-aliasing filters.

How to Score Full Marks:

• Draw plots: For every signal operation (shift, scale, convolution), sketch the input/output.
• State properties clearly: For a system, explicitly check all 4 properties (causality, linearity, time-invariance, stability).
• Use examples: Relate convolution to audio effects (e.g., reverb = convolution with impulse response of a hall).
• Avoid memorization: Derive results from definitions (e.g., prove linearity by showing $T\{ax\}=aT\{x\}$).

Key Formulas to Remember