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

Introduction

Signals are crucial in Communication Systems, Control Systems, and Signal Processing. One fundamental property of any signal is its periodicity—whether it repeats itself after a fixed duration. The minimum duration after which the signal repeats itself is called the fundamental period.

In this post, we will analyze GTU Sem 4 Signals & Systems Summer 2021 PYQ Question 1(a) and determine

1. Whether the given signals are periodic or non-periodic
2. If periodic, we will see how to find their fundamental period.

📚 Table of Contents

• Introduction
• Question Statement
• Understanding Periodicity
  • 1. Continuous-Time Signals
  • 2. Discrete-Time Signals
• Sub-Question 1: Analyzing the Periodicity of x(t)
• Sub-Question 2: Analyzing the Periodicity of x[n]
• Handwritten Solution
• Conclusion
• Related Topics

Question Statement

Find whether the given signals are periodic or not. If yes, give their fundamental period.

1. x(t) = 4 cos(3πt + π/2) + 2 cos(8πt + π/2)
2. x[n] = 10 sin(20n)

Understanding Periodicity

A signal is periodic if it repeats itself at regular intervals. The smallest time interval after which the signal repeats is called the fundamental period T for continuous signals and N for discrete signals.

1. Continuous-Time Signals

A continuous-time signal x(t) is periodic if there exists a fundamental period T such that:

x(t+T)=x(t)    for all t

How can we find if a Continuous Signal is Periodic or not?

Many signals are made up of cosine or sine waves with different angular frequencies ω

x(t)=A1cos⁡(ω1t+ϕ1)+A2cos⁡(ω2t+ϕ2)+…

Each term has an angular frequency:

ωi=2πfi​

And fundamental period:

Ti=2π / ωi​

For the signal to be periodic, the Least Common Multiple (LCM) of all Ti​ values must exist. If it does, the signal is periodic, and the LCM is the fundamental period T. If no LCM exists, the signal is not periodic.

2. Discrete-Time Signals

A discrete-time signal x[n] is periodic if there exists a fundamental period N such that:

x[n+N]=x[n]    for all n

How can we Check if a Discrete-time signal is Periodic or not?

A discrete-time signal is often given in the form:

x[n]=Acos⁡(ωn+ϕ)

Here, ω is the angular frequency in discrete time. Unlike continuous signals, discrete signals are periodic only if:

ω/2π is a rational number p/q​

Where p and q are integers.

The fundamental period N is given as:

N = 2π / GCD( ω, 2π )​

If ω/2π is not a rational number, then the discrete-time signal is not periodic.

Sub-Question 1: Analyzing the Periodicity of x(t)

Step 1: Identify Angular Frequencies

For the given function x(t):First term: ω₁ = 3π, T₁ = 2π/ω₁

T₁ = 2π/3π

T₁ = 2/3

Second term: ω₂ = 8π, T₂ = 2π/ω₂

T₂ = 2π/8π

T₂ = 1/4

 

Step 2: Find the Fundamental Period

Using the LCM formula for fractions:

 LCM(T1​,T2​)=LCM(Numerators) / GCD(Denominators)

Here,

T1=2/3,    T2=1/4

Applying the formula:

T    = LCM( 2/3, 1/4 )    =LCM( 2, 1 ) / GCD( 3, 4 )

LCM(2,1)=2 (Least common multiple of the numerators 2 and 1)
GCD(3,4)=1 (Greatest common divisor of the denominators 3 and 4)


Thus,

T    = 2/1    = 2 seconds

Thus, the fundamental period is 2 seconds, making the signal periodic.

Sub-Question 2: Analyzing the Periodicity of x[n]

Step 1: Identify the Frequency

For x[n] = 10 sin(20n), ω = 20.

ω/2π     = 20/2π     = 10/π

A discrete-time signal is periodic only if ω/2π is a rational number.
Since 10π​ is an irrational number, no integer N can satisfy the periodicity condition x[n+N]=x[n].
Therefore, x[n] is not periodic.

📷 Handwritten Solution

Here is the scanned handwritten solution for your reference:

✍️Sub-Question 1:-

✍️Sub-Question 2:-

Conclusion

• x(t) is periodic with a fundamental period of 2 seconds.
• x[n] is not periodic since its frequency is not a rational multiple of 2π.
• Understanding periodicity is essential for signal processing, communication, and control systems.

🔗 Related Topics

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