GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 1(c) Solution

📖 1. Introduction

When we study systems in subjects like Signals and Systems or Control Systems, it's important to know how a system behaves. We do this by checking certain system properties such as linearity, time-invariance, causality, and dynamicity. These terms might sound difficult at first, but they are just simple ways to understand how a system responds to different inputs.

In this blog post, we’ll look at a very simple system given by this equation:

y(t) = 10x(t) + 5

Even though the equation looks basic, it’s a great example to learn how to check each of these system properties. We’ll explain the meaning of each property, how to test it, and then apply it to this system step-by-step.

If you're a student learning these concepts for the first time, don’t worry—we’ll break it all down in a clear and easy-to-follow way. By the end of this post, you’ll understand what these properties mean and how to analyze any system using them.

❓ 2. Question Statement

In the study of signals and systems, analyzing how a system behaves under different conditions is an essential skill. This includes checking if a system is linear, time-invariant, causal, or dynamic. These properties help us understand whether the system is predictable, stable, suitable for real-time processing, or memory-based.

GTU Sem 4 Signals & Systems Q1(c) Solution Cover Image

Here, we are given a simple system described by the following mathematical equation:

y(t) = 10x(t) + 5

In this equation:

x(t) is the input signal (what we give to the system at time t).
y(t) is the output signal (what we get from the system at time t).

We are asked to carefully analyze this system and determine whether it satisfies the following four properties:

Linearity: Does the system follow the rules of Superposition( additivity and homogeneity )?
Time-Invariance: Does the system behave the same way if the input is delayed?
Causality: Does the system depend only on the current or past input values?
Dynamicity: Does the system have memory or depend on past input values?

Although the system y(t) = 10x(t) + 5 looks very simple, it’s actually a great example for learning how to apply these concepts. By breaking it down and testing each property one by one, we can learn how real-world systems are designed and analyzed.

This type of question is commonly asked in university exams like GTU( Gujarat Technical University ), competitive tests like GATE( Graduate Aptitude Test in Engineering ), and interviews where a basic understanding of systems is expected.

Let’s now move on to understand each of the required terms in detail so we can correctly evaluate the system.

📚 3. Understanding Key Concepts

Before we analyze the system y(t) = 10x(t) + 5, we need to understand four important terms that describe how systems behave. These terms—Linearity, Time-Invariance, Causality, and Dynamicity are used to classify and evaluate systems in the field of signals and systems.

Let’s explore each concept one by one, using simple language and real-life examples to make them easy to understand.

3.1 Linearity

A system is called linear if it obeys the principle of superposition. This principle has two parts:

Additivity: If input A produces output A′, and input B produces output B′, then input A+B should give output A′+B′.
Homogeneity (or Scaling): If we multiply the input by a constant, the output should also get multiplied by the same constant.

If both of these conditions are met, the system is linear. If not, the system is nonlinear.

Real-life example: Imagine you are filling a bucket with water using two taps. If one tap fills the bucket halfway in 1 hour and the other tap does the same, turning on both taps should fill the bucket completely in 1 hour. The total flow adds up—a clear example of a linear system.

3.2 Time-Invariance

A system is time-invariant if its behavior does not change over time. This means that if you delay the input by some time, the output will also be delayed by the same amount, and nothing else will change.

In other words, the system should work the same way no matter when you give it the input.

Real-life example: Think of a coffee vending machine. If you press the button at 10 AM and get a coffee, and press the same button at 5 PM and still get the same coffee, the machine is time-invariant. It doesn’t care about the time it just responds to your input.

3.3 Causality

A system is causal if the output at any time depends only on the current or past values of the input. It should not depend on future inputs.

Causal systems are the ones that make sense in real life because no system can see into the future.

Real-life example: A traffic light turns red only after detecting that the countdown is over. It doesn’t turn red in advance because of something that will happen later. That makes it causal.

Non-causal example (theoretical): Imagine a system that adjusts the heater temperature based on tomorrow’s weather. That would be non-causal—because it relies on a future input.

3.4 Dynamicity

A system is called dynamic if its output depends on past input values or if it keeps memory of what has happened earlier. It usually involves delays, derivatives, or integrals.

If the output depends only on the current input and has no memory, the system is called static.

A real-life example of a dynamic system: A smart thermostat that adjusts your room temperature based on the past few hours of temperature changes is dynamic. It uses memory and past input data.

A real-life example of a static system: A regular electric fan turns on or off depending only on the current switch position. It doesn't remember how long it was running earlier. That makes it static.

🧠 4. Step-by-Step Solution with Explanations

In this section, we’ll check whether the system y(t) = 10x(t) + 5 is Linear, Time-Invariant, Causal, and Dynamic. Don't worry if you're just starting out — we'll break down each concept step by step with easy examples and everyday comparisons.

4.1 Is the System Linear?

To check linearity, we use this formula:

y(t) = F{a₁x₁(t) + a₂x₂(t)} = a₁y₁(t) + a₂y₂(t)

If this holds true for any constants a₁ and a₂, and any inputs x₁(t) and x₂(t), the system is linear.

🧪 Step-by-Step Test

1. Let’s choose:

x₁(t) = 1
x₂(t) = 2
a₁ = 3, a₂ = 4

2. Compute LHS: Plug a₁x₁ + a₂x₂ into the system:

x(t) = a₁x₁ + a₂x₂ = 3(1) + 4(2) = 11
y(t) = F{a₁x₁(t) + a₂x₂(t)} = 10×11 + 5 = 115

3. Compute RHS: Plug x₁ and x₂ into the system first:

y₁(t) = 10×1 + 5 = 15
y₂(t) = 10×2 + 5 = 25
a₁y₁(t) + a₂y₂(t) = 3×15 + 4×25 = 45 + 100 = 145

4. Compare:

LHS = 115
RHS = 145

They are not equal! That means the system does not satisfy linearity.

🎯 Final Answer: Not Linear

💡 Real-Life Analogy: Imagine you're buying train tickets. Each ticket costs ₹10, and there’s always a flat ₹5 booking charge. If you buy tickets separately, you pay the booking fee multiple times. But when you combine them, you only pay once. So, the cost isn’t directly proportional — this breaks the rules of linearity!

4.2 Is the System Time-Invariant?

To check time-invariance, we use this formula:

T{x(t - t₀)} = y(t - t₀)

This means: that if we delay the input by some time (t₀), then the output should also get delayed by t₀. If this holds true for any value of t₀, the system is time-invariant.

🧪 Step-by-Step Test

Original System:
```
y(t) = 10x(t) + 5
```
Delay the Input: Replace x(t) with x(t - t₀):
```
T{x(t - t₀)} = 10x(t - t₀) + 5
```
This is our LHS (Left-Hand Side).
Delay the Output: Delay the whole output by t₀:
```
y(t - t₀) = 10x(t - t₀) + 5
```
This is our RHS (Right-Hand Side).
Compare: Both sides are:
```
10x(t - t₀) + 5
```
LHS = RHS

🎯 Final Answer: The system is Time-Invariant

💡 Real-Life Analogy: Imagine a popcorn machine that always pops kernels in exactly 2 minutes, no matter when you turn it on. Whether you start at 1 PM or 6 PM, the behavior stays the same. This machine’s output is time-independent — that’s what we call a time-invariant system!

4.3 Is the System Causal?

Definition Reminder: A system is causal if it doesn’t “look into the future.” In other words, the output at time t depends only on the values of x at t or earlier — not at any future time.

Let’s examine our system:

y(t) = 10x(t) + 5

It clearly depends only on x(t) — the input at the current time. There’s no x(t+1) or x(t+5) here.

🎯 Final Answer: Causal

Real-Life Example: A room heater turns on based on the current temperature, not the temperature expected in the future. It doesn’t guess. That’s a causal system.

4.4 Is the System Dynamic?

Definition Reminder: A system is dynamic if it uses memory — meaning it depends on past (or sometimes future) values or involves integrals or derivatives. If the output depends only on the current input, it’s called static.

Our system:

y(t) = 10x(t) + 5

No memory. No derivatives. No integration. Just the current value of x(t).

🎯 Final Answer: Not Dynamic (It’s Static)

Real-Life Example: A fan that turns on as soon as you press a switch and turns off when you release it — without remembering anything — is static. No memory involved.

📝 5. Hand-Written Solution

For visual learners, a handwritten version of the solution can often make concepts easier to understand. Below is the scanned image of the full handwritten solution for analyzing the system y(t) = 10x(t) + 5.

This handwritten solution covers:

Linearity check using y{a₁x₁ + a₂x₂} = a₁y(x₁) + a₂y(x₂)
Time-Invariance check using T{x(t - t₀)} = y(t - t₀)
Causality test
Dynamicity test

GTU Sem 4 Signals & Systems Q1(c) Handwritten Solution Page 1

GTU Sem 4 Signals & Systems Q1(c) Handwritten Solution Page 2

GTU Sem 4 Signals & Systems Q1(c) Handwritten Solution Page 3

📌 Note: This is useful for quick revision and exam preparation!

✨ 6. Key Takeaways

Here’s a quick recap of what we learned by analyzing the system y(t) = 10x(t) + 5. These key points will help you remember system properties easily:

Linearity: The system is not linear because it includes a constant (+5) that breaks the superposition rule.
Time-Invariance: The system is time-invariant because shifting the input t₀ results in the same shift in output.
Causality: The system is causal since output at any time depends only on the current or past input, not future values.
Dynamicity: The system is not dynamic (also called static) because it has no memory — it only uses the current input value.

💡 Easy Way to Remember:

If a system adds a constant, it's not linear.
If shifting input = shifting output, it's time-invariant.
If it needs future input, it's not causal.
If it uses memory (past input), it's dynamic.

🎯 Pro Tip: Draw a simple table or flashcards to revise these four properties quickly before exams!

🧾 7. Final Words

Understanding basic system properties like linearity, time-invariance, causality, and dynamicity may seem tricky at first — but with the right approach and examples, they become easy and fun to learn.

The system we studied today — y(t) = 10x(t) + 5 — helped us explore each property in depth, with both logical analysis and real-world comparisons.

Whether you're preparing for your GTU exams or building a strong foundation in signals and systems, keep practicing similar problems and try to visualize them in real life. That's the key to deep learning!

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Enjoyed this explanation? Continue building your understanding with these related GTU PYQ solutions from Signals and Systems:

📘 GTU Sem 4 Signals & Systems Summer 2021 – Question 1(b) Solution

Step-by-step beginner-friendly solution with real-life examples and visuals.

📘 GTU Sem 4 Signals & Systems Summer 2021 – Question 2(a) Solution

Detailed solution to another important S&S PYQ — perfect for revision!

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GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 1(c) Solution