Introduction To Signals and Systems MCQ Quiz - Objective Question with Answer for Introduction To Signals and Systems - Download Free PDF

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Latest Introduction To Signals and Systems MCQ Objective Questions

Introduction To Signals and Systems Question 1:

Which is the identity for convolution operation?

  1. 1
  2. 0
  3. impulse δ(n)
  4. unit step u(n)

Answer (Detailed Solution Below)

Option 3 : impulse δ(n)

Introduction To Signals and Systems Question 1 Detailed Solution

Explanation:

Identity for Convolution Operation

Definition: The convolution operation is a fundamental mathematical tool widely used in signal processing, systems theory, and various engineering fields. It involves combining two signals to produce a third signal that represents how one signal modifies or overlaps with the other. The identity element for convolution is a special function that, when convolved with any signal, leaves the signal unchanged.

Correct Option: Option 3: Impulse δ(n)

The identity element for the convolution operation is the impulse function, denoted as δ(n). The impulse function has the following properties:

  • Definition: δ(n) is a discrete-time signal defined as:
    • δ(n) = 1 for n = 0
    • δ(n) = 0 for n ≠ 0
  • Convolution Property: When any signal x(n) is convolved with δ(n), the output is the original signal x(n):

    x(n) * δ(n) = x(n)

Explanation: The impulse function serves as the identity for convolution because of its unique characteristics. During convolution, the impulse function effectively "selects" values of the input signal without altering them. This property is crucial in systems analysis and signal processing, where the impulse function is used to analyze and characterize systems.

Importance:

  • The impulse function is used to determine the impulse response of a system, which provides critical insights into the system’s behavior.
  • It is the building block for many mathematical operations in signal processing, such as filtering and system identification.

Correct Option Analysis:

The correct option is:

Option 3: Impulse δ(n)

This option is correct because the impulse function is mathematically proven to be the identity element for convolution. When convolved with any signal, it leaves the signal unchanged, reflecting its role as the identity element.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 1

This option is incorrect. While "1" could be considered an identity element in multiplication or addition in certain mathematical contexts, it is not the identity element for convolution. Convolution is not a simple multiplication operation; it involves summing weighted values of overlapping signals. Therefore, "1" does not satisfy the mathematical requirements to be the identity for convolution.

Option 2: 0

This option is incorrect. The function "0" is the zero element for convolution, meaning that convolving any signal with "0" results in "0." It is not the identity element because it does not leave the original signal unchanged when convolved.

Option 4: Unit Step u(n)

This option is incorrect. The unit step function u(n) is a commonly used signal in signal processing, but it does not act as the identity for convolution. Convolving a signal with u(n) modifies the signal by creating cumulative sums of its values. Thus, it does not satisfy the property of leaving the signal unchanged.

Option 5: (No statement provided)

This option is not applicable due to the absence of a statement or definition. It cannot be considered the identity for convolution operation.

Conclusion:

The impulse function δ(n) is the identity element for the convolution operation. Its unique properties make it a fundamental tool in signal processing and systems theory. By understanding the impulse function and its role in convolution, engineers and scientists can effectively analyze and design systems. Evaluating the incorrect options further reinforces the importance of the impulse function as the identity for convolution.

Introduction To Signals and Systems Question 2:

The equation for voltage waveform v(t) shown below is (where u(t) is unit step input) :

qImage67c001123b6b73a940b30a1e

  1. u(t − 1) + u(t − 2) + u(t − 3) 
  2. u(t − 1) + 2u(t − 2) + 3u(t − 3)
  3. u(t − 1) − u(t − 2) − u(t − 3) + 3u(t − 4) 
  4. u(t − 1) +u(t − 2) +u(t − 3) − 3u(t − 4)

Answer (Detailed Solution Below)

Option 4 : u(t − 1) +u(t − 2) +u(t − 3) − 3u(t − 4)

Introduction To Signals and Systems Question 2 Detailed Solution

Concept:

The voltage waveform can be expressed using unit step functions (u(t)) to represent the changes in voltage at specific time intervals. Each step in the waveform corresponds to the addition or subtraction of a unit step function at the respective time.

Calculation:

Given:

Voltage waveform with the following characteristics:
- Starts at 0V
- Jumps to 1V at t=1
- Jumps to 2V at t=2
- Jumps to 3V at t=3
- Returns to 0V at t=4

Solution:

1. The waveform can be constructed as:

2. This represents:

  • +1V step at t=1
  • +1V step at t=2 (total 2V)
  • +1V step at t=3 (total 3V)
  • -3V step at t=4 (return to 0V)

 

Final Answer:

The correct equation is 4) u(t − 1) +u(t − 2) +u(t − 3) − 3u(t − 4)

Introduction To Signals and Systems Question 3:

A continuous-time signal x(t) is defined as :

x(t)={52t20 otherwise 

The energy of signal x(t) is : 

  1. 0
  2. 50 
  3. 100 
  4. 20

Answer (Detailed Solution Below)

Option 3 : 100 

Introduction To Signals and Systems Question 3 Detailed Solution

Concept:

The energy of a continuous-time signal x(t) is given by: E=|x(t)|2dt

Given:

x(t)={5,2t20,otherwise

Calculation:

Since x(t)=5 from t=2 to t=2, we compute the energy as:

E=22|5|2dt=2225dt=25×(2(2))=25×4=100

 

Introduction To Signals and Systems Question 4:

Which of the following assumptions are required to show consistency, unbiasedness and efficiency of the OLS estimator ?
i. E(ui) = 0
ii. Var(ui) = σ2
iii. Cov(ut, ut-j) = 0 ∀ j and
iv. ut ∩ N(0, σ2)

  1. ii and iv only
  2. i and iii only
  3. i, ii and iii only
  4. i, ii, iii and iv

Answer (Detailed Solution Below)

Option 3 : i, ii and iii only

Introduction To Signals and Systems Question 4 Detailed Solution

The correct answer is: 'Option 3: i, ii, and iii only'.

Key Points

  • i. E(ui) = 0
    • This assumption is correct.
    • The assumption that the expected value of the error term E(ui)=0" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">E(ui)=0 ensures that the OLS estimators are unbiased.
    • If this condition holds, it implies that the error term does not systematically overestimate or underestimate the dependent variable, leading to unbiased parameter estimates.
    • Hence, this assumption is necessary for consistency, unbiasedness, and efficiency.
  • ii. Var(ui) = σ²
    • This assumption is correct.
    • The assumption that the variance of the error term is constant Var(ui)=σ²" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">Var(ui)=σ² is critical for the efficiency of the OLS estimator.
    • This condition, known as homoscedasticity, ensures that the variability of the errors does not change with the level of the independent variable, which is essential for accurate hypothesis testing and confidence intervals.
    • Thus, this assumption is necessary for the OLS estimator to be efficient.
  • iii. Cov(ut, ut-j) = 0 ∀ j
    • This assumption is correct.
    • The assumption that the error terms are uncorrelated over time Cov(ut,utj)=0j" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">Cov(ut,utj)=0j is necessary for ensuring that the OLS estimators are consistent.
    • This condition indicates that there is no autocorrelation in the error terms, which is vital for the reliability of the OLS estimators.
    • Therefore, this assumption is necessary for consistency and efficiency of the estimator.
  • iv. ut ∩ N(0, σ²)
    • This assumption is incorrect.
    • While normality of the error term is a common assumption for inference purposes (such as constructing confidence intervals), it is not strictly required for the consistency, unbiasedness, or efficiency of the OLS estimators.
    • The OLS estimators can still be consistent and efficient even if the error terms are not normally distributed, provided that the other assumptions hold.
    • Hence, this assumption is not necessary.

Additional Information

  • Understanding OLS Estimator Assumptions:
    • Unbiasedness: An estimator is unbiased if its expected value equals the true parameter value. This requires the assumption E(ui)=0" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0">E(ui)=0 .
    • Efficiency: An estimator is efficient if it has the smallest variance among all unbiased estimators. This requires constant variance Var(ui)=σ²" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">Var(ui)=σ² .
    • Consistency: An estimator is consistent if it converges in probability to the true parameter value as the sample size increases, which is supported by the assumption of no autocorrelation in the error terms.

Introduction To Signals and Systems Question 5:

A cellphone tower radiates 1W power while the handset transmitter radiates 0.1 mW power. The correct comparison of the radiation energy received by your head from a tower 100m away (E1) and that from a handset held to your ear (E2) is

  1. E1 >> E2
  2. E2 >> E1
  3. E1 = E2 for communication to be established
  4. insufficient data even for a rough comparison

Answer (Detailed Solution Below)

Option 2 : E2 >> E1

Introduction To Signals and Systems Question 5 Detailed Solution

CONCEPT:

Radiation Energy Comparison

  • The power radiated by a source is distributed over the surface area of a sphere centered at the source. The power per unit area (intensity) decreases with the square of the distance from the source.
  • The formula for intensity (I) at a distance (r) from a source with power (P) is given by:

    I = P / (4πr²)

EXPLANATION:

  • For the cellphone tower:
    • Power, P1 = 1 W
    • Distance from the head, r1 = 100 m
    • Intensity at the head, I1 = P1 / (4πr1²)
    • = 1 W / (4π(100 m)²)
    • = 1 / (4π * 10,000)
    • = 1 / 125,663.7 W/m²
    • = 7.96 x 10⁻⁶ W/m²
  • For the handset:
    • Power, P2 = 0.1 mW = 0.1 x 10⁻³ W
    • Distance from the head, r2 ≈ 0.01 m (assuming the handset is very close to the ear)
    • Intensity at the head, I2 = P2 / (4πr2²)
    • = 0.1 x 10⁻³ W / (4π(0.01 m)²)
    • = 0.1 x 10⁻³ / (4π * 0.0001)
    • = 0.1 x 10⁻³ / 0.001256637
    • = 0.0796 W/m²
  • Comparing the intensities:
    • I2 ≈ 0.0796 W/m²
    • I1 ≈ 7.96 x 10⁻⁶ W/m²
    • I2 >> I1

Therefore, the radiation energy received by your head from the handset (E2) is much greater than that from the tower (E1).

Top Introduction To Signals and Systems MCQ Objective Questions

The value of +etδ(2t2)dt, where δ(t) is the Dirac delta function, is

  1. 12e
  2. 2e
  3. 1e2
  4. 12e2

Answer (Detailed Solution Below)

Option 1 : 12e

Introduction To Signals and Systems Question 6 Detailed Solution

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Concept:

Shifting property of impulse function

x(t)δ(ta)dt=x(a)

Scaling property of impulse function

δ(at)=1|a|)δ(t)

Explanation:

Let:

I=etδ(2t2)dt

I=etδ[2(t1)]dt

Using scaling property of impulse function in the above equation, we'll get:

I=et1|2|δ(t1)dt

Applying Shifting property of impulse function to the above equation, we'll get:

I=12etδ(t1)dt

I=12.et|t=1

12e

 

For a periodic signal v(t) = 30 sin100t + 10 cos300t + 6 sin(500t + π/4), the fundamental frequency in rad/s is _____.

  1. 100
  2. 300
  3. 500
  4. 1500

Answer (Detailed Solution Below)

Option 1 : 100

Introduction To Signals and Systems Question 7 Detailed Solution

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Given, the signal

V (t) = 30 sin 100t + 10 cos 300 t + 6 sin (500t+π/4)

So, we have

ω1 = 100 rads

ω2 = 300 rads

ω3 = 500 rads

∴ The respective time periods are

T1=2πω1=2π100secT1=2πω2=2π300secT3=2π500sec

So, the fundamental time period of the signal is

LCM(T1,T2,T3)=LCM(2π,2π,2π)HCF(100, 300, 500)

as T0=2π100

∴ The fundamental frequency, ω0=2πT0=100 rad/s

Find out even and odd part of signal x(t) = (2 + sin t)2

  1. 4 + sin2t, 4 sin t
  2. 2 + sin2t, 2 sin t
  3. 4, 4 sin t + sin2 t
  4. 2, sin t

Answer (Detailed Solution Below)

Option 1 : 4 + sin2t, 4 sin t

Introduction To Signals and Systems Question 8 Detailed Solution

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Concept:

Any given signal x(t) can be written as the sum of its even part and odd part, i.e.

x(t) = xe(t) + xo(t)

xe(t) = Even part of g(t) calculated as:

xe(t)=x(t)+x(t)2

xo(t) = Odd part of x(t) calculated as:

xo(t)=x(t)x(t)2

Calculation:

Given:

x(t) = (2 + sin t)2

x(t) = 4 + sin2t + 4 sin t

x(-t) = 4 + sin2t - 4 sin t

Even part of g(t) calculated as:

xe(t)=x(t)+x(t)2

=4+sin2t+4sint+4+sin2t4sint2

= 4 + sin2t

Odd part of x(t) calculated as:

xo(t)=x(t)x(t)2

=4+sin2t+4sint4sin2t+4sint2

= 4 sin t

Given f(t) and g(t) as shown below

sigrepor3

EE Signals and Systems mobile Images-Q39.1

g(t) can be expressed as

  1. g(t) = f (2t - 3)
  2. g(t)=f(t23)
  3. g(t)=f(2t32)
  4. g(t)=f(t232)

Answer (Detailed Solution Below)

Option 4 : g(t)=f(t232)

Introduction To Signals and Systems Question 9 Detailed Solution

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Concept:

Time-shifting property: When a signal is shifted in time domain it is said to be delayed or advanced based on whether the signal is shifted to the right or left. 

For example:

F9 Neha B 5-10-2020 Swati D7

Time scaling property: A signal is scaled in the time domain with the scaling factor 'a'. 

If a > 1, then the signal is contracted by a factor of 'a' along the time axis. 

if a < 1, then the signal is expanded by a factor of 'a' along the time axis. 

For example:

F9 Neha B 5-10-2020 Swati D8

Analysis:

We can observe that if we scale f(t) by a factor of ½ and then shift, we will get g(t).

First scale f(t) by a factor of ½ ,

g1(t) = f (t/2)

EE Signals and Systems mobile Images-Q39.2

Shift g1(t) by 3

g(t)=g1(t3)=f(t32)

EE Signals and Systems mobile Images-Q39.3

g(t)=f(t232)

Identify the signal x(2t - 3) given x(t) as 

F9 Neha B 5-10-2020 Swati D2

  1. F9 Neha B 5-10-2020 Swati D1
  2. F9 Neha B 5-10-2020 Swati D3
  3. F9 Neha B 5-10-2020 Swati D4
  4. F9 Neha B 5-10-2020 Swati D5

Answer (Detailed Solution Below)

Option 2 : F9 Neha B 5-10-2020 Swati D3

Introduction To Signals and Systems Question 10 Detailed Solution

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Concept:

Time-shifting property: When a signal is shifted in the time domain it is said to be delayed or advanced based on whether the signal is shifted to the right or left. 

For example:

F9 Neha B 5-10-2020 Swati D7 (1)

Time scaling property: A signal is scaled in the time domain with the scaling factor 'a'. 

If a > 1, then the signal is contracted by a factor of 'a' along the time axis. 

if a < 1, then the signal is expanded by a factor of 'a' along the time axis. 

For example:

F9 Neha B 5-10-2020 Swati D8 (1)

Analysis:

Whenever there is a combination of operations care should be taken while following the sequence of operations.

Correct approach:

x[2(t - 1.5)]

First, time scale the signal by factor 2.

Then, time shift the signal by 1.5 

F9 Neha B 5-10-2020 Swati D13

Incorrect approach:

x[2(t - 1.5)]

First, time shift the signal by 1.5

Then, time scale the signal by 2.

F9 Neha B 5-10-2020 Swati D15

or, 

Correct approach:

x[2t - 3]

First, time shift the signal by 3.

Then, time scale the signal by 2

F9 Neha B 5-10-2020 Swati D13

Incorrect approach:

x[2t - 3]

First, time scale the signal by 2

Then, time shift the signal by 3.

F9 Neha B 5-10-2020 Swati D14 (1)

Consider the system with following input-output relation

y[n]=[1+(1)n]x[n]

where, x[n] is the input and y[n] is the output. The system is

  1. invertible and time invariant
  2. invertible and time varying
  3. non-invertible and time invariant
  4. non-invertible and time varying

Answer (Detailed Solution Below)

Option 4 : non-invertible and time varying

Introduction To Signals and Systems Question 11 Detailed Solution

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Given that, y[n] = (1 + (-1)n) x [n]

y[n - n0] = [1 + (-1)n - n0] x[n - n0]        ----(1)

y[n] = (1 + (-1)n) x[n - n0]       ----(2)

Both the equations 1 and 2 are not equal. Hence y[n] is dependent on time. It is time variant.

For invertible systems, for each unique input x[n], there should be unique output y[n].

If x[n] = δ [n - 1]

y[n] = (1 + (-1)n) δ [n - 1]

y[1] = 0

If x[n] = k δ [n - 1]

y [n] = [1 + (-1)n] k δ [n - 1]

y[1] = 0

Hence for the two different inputs, system producing same output. Hence system is non invertible.

Evaluate etδ(2t2)dt

  1. 0
  2. 12e
  3. 1
  4. 1e

Answer (Detailed Solution Below)

Option 2 : 12e

Introduction To Signals and Systems Question 12 Detailed Solution

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Unit impulse function:

It is defined as, δ(t)={,t=00,t0

The discrete-time version of the unit impulse is defined by

δ[n]={1,n=00,n0

Properties:

1. δ(t)dt=1

2. δ(at)=1|a|δ(t)

3. x(t) δ(t – t0) = x(t0)

4. x(t)δ(tto)dt=x(t0)

5. f(t)δ(at+b)dt=f(t)1|a|δ(t+ba)dt

6. x(t)δn(tto)dt=dnxdtn|t=t0

Calculation:

etδ(2t2)dt

=12etδ(t1)dt

=12e1=12e

An unit impulse function in continuous form is defined to be

  1. δ(t) = t
  2. δ(t) = 1
  3. δ(t)={,t=00,t0
  4. δ(t)={0,t=01,t0

Answer (Detailed Solution Below)

Option 3 : δ(t)={,t=00,t0

Introduction To Signals and Systems Question 13 Detailed Solution

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Unit impulse function:

It is defined as

Properties:

1. δ(t)dt=1

2. δ(at)=1|a|δ(t)

3. x(t) δ(t – t0) = x(t0)

4. x(t)δ(tto)dt=x(t0)

5. f(t)δ(at+b)dt=f(t)1|a|δ(t+ba)dt

6. x(t)δn(tto)dt=dnxdtn|t=t0

The discrete time version of the unit impulse is defined by

δ[n]={1,n=00,n0

Properties:

1. x[n] δ[n] = x[0] δ[n]

2. n=n=x[n]δ[n]=n=n=x[0]δ[n]=x[0]

3. x[n] δ[n – n0] = x[n0] δ[n – n0]

4. n=n=x[n]δ[nn0]=n=n=x[n0]δ[nn0]=x[n0]

5. δ[an] = δ[n]

Consider a signal x(t) = 2r(t) – 2r(t - 2) – 4u(t - 4), evaluate x(t)δ(t3)

  1. 3
  2. 4
  3. 0
  4. -4

Answer (Detailed Solution Below)

Option 2 : 4

Introduction To Signals and Systems Question 14 Detailed Solution

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Concept:

Sampling property of impulse function is:

t1t2x(t)δ(tt0)=x(t0);t1<t0<t2 

Unit ramp function is defined as:

F1 T.S 31.7.20 Pallavi D 6

r(t)={t;t>00;t<0} 

Calculation:

Given:

x(t) = 2r(t) – 2r (t - 2) – 4u(t - 4)

from sampling property;

+x(t)δ(t3)=x(3) 

So,

x(3) = 2r(3) – 2r(1) – 4u(-1)

x(3) = 2 × 3 – 2 × 1 – 4 × 0

x(3) = 4

Identify the nature of the signal shown in the given image.

F30 Shubham B 27-4-2021 Swati D5

  1. Odd signals
  2. Periodic signals
  3. Even signals
  4. Aperiodic signals

Answer (Detailed Solution Below)

Option 1 : Odd signals

Introduction To Signals and Systems Question 15 Detailed Solution

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Odd Signals:

  • A signal x(t) is said to be an odd signal if it satisfies the condition:

          x(-t) = - x(t) for all t (Continuous Time)

          x(-n) = - x(n) for all n (Discrete Time)

  • An odd signal is anti symmetrical about the vertical axis.

 

Analysis:

F30 Shubham B 27-4-2021 Swati D5

Fro the given signal, we can write:

x[-n] = - x[n]

∴ The given signal x[n] is an odd signal.

 

Examples of odd signals:

F1 S.B Madhu 16.03.20 D6

Even Signals:

  • Any signal x(t) is said to be an even signal if it satisfies the condition:

          x(-t) = x(t) for all t (Continuous Time)

          x(-n) = x(n) for all n (Discrete Time)

  • An even signal is identical to its time-reversed signal, i.e. it is symmetrical around the vertical axis.

 

Examples of even signals:

F1 S.B Madhu 16.03.20 D2

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