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

Concept: Fourier Transform Symmetry Property

If a time-domain function $f(t)$ is:

• Real: It has no imaginary part

• Even: f(t) = f(-t)" id="MathJax-Element-103-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t) = f(-t)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0"> f(t) = f(-t) $f(t) = f(-t)$ $f(t) = f(-t)$ f(t)=f(−t)

Then it's  Fourier Transform $F(\omega )$ will be:

• Purely real

• Even: F(ω)=F(−ω)" id="MathJax-Element-105-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">

🔍 Example: Cosine Wave

Let’s take: f(t) = cos(ω0​t)

• It is real.

• It is even, because " id="MathJax-Element-107-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">  cos(−ω_0​t)=cos(ω_0​t)

Fourier Transform of " id="MathJax-Element-108-Frame" role="presentation" style="position: relative;" tabindex="0">" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0"> cos(ω_0​t) is: F(ω)=π [δ(ω−ω_0​)+δ(ω+ω_0​)]

This result is:

• Purely real (involves delta functions, no imaginary component)

• Even since it's symmetric around $\omega =0$

Concept:

Sinc pulse shaping is commonly used in digital communication systems to achieve ideal Nyquist pulse shaping, minimizing intersymbol interference (ISI).

The sinc function is defined as:

Explanation:

The sinc function arises as the inverse Fourier Transform of a rectangular function in the frequency domain.

In other words, if the frequency spectrum is a perfect rectangular shape (ideal low-pass filter), its time-domain representation is a sinc function.

Conclusion:

Correct Answer: Option 4) rectangular

Explanation:

Understanding Aliasing in Signal Processing

Definition: Aliasing is a phenomenon that occurs when a signal is sampled at a rate that is insufficient to capture the changes in the signal accurately. Specifically, aliasing happens when the sampling rate is below the Nyquist rate, which is twice the maximum frequency present in the signal. When aliasing occurs, different signals become indistinguishable from each other after sampling, leading to distortion.

Detailed Explanation: In the context of the given problem, a signal contains frequency components up to 15 kHz, and it is sampled using a 20 kHz sampling rate. According to the Nyquist theorem, the sampling rate must be at least twice the maximum frequency present in the signal to avoid aliasing. Therefore, the Nyquist rate for a signal with a maximum frequency of 15 kHz is 30 kHz. Since the given sampling rate of 20 kHz is below this Nyquist rate, aliasing will occur.

When aliasing occurs, the higher frequency components of the signal are "folded" back into the lower frequencies, causing distortion that makes the original signal indistinguishable from its aliased counterpart. This effect results in a loss of information and can significantly degrade the quality of the reconstructed signal.

Illustrative Example: Consider a simple sine wave with a frequency of 15 kHz. If we sample this wave at 20 kHz, the sampling points will not capture the wave's true form accurately. Instead, the samples will represent a different, lower frequency signal, leading to a distorted representation. This misrepresentation can be visualized by plotting the sampled points and observing that the reconstructed signal does not match the original 15 kHz sine wave.

Importance of Proper Sampling: To avoid aliasing, it is crucial to sample signals at a rate that is at least twice the maximum frequency present in the signal. This requirement ensures that the original signal can be accurately reconstructed from the sampled data. In practical applications, engineers often use anti-aliasing filters to remove high-frequency components before sampling, ensuring that the sampled signal adheres to the Nyquist criterion.

Conclusion: In the given problem, the correct answer is option 2 (Aliasing) because the signal with frequency components up to 15 kHz is sampled at a rate of 20 kHz, which is below the Nyquist rate of 30 kHz, leading to aliasing and resulting in distortion.

Important Information:

Let's analyze the other options to understand why they are incorrect:

Quantization error occurs during the process of converting a continuous signal into a discrete digital signal by approximating the signal's amplitude to the nearest value within a finite set of levels. This type of error is related to the precision of the digital representation and is not specifically related to the sampling rate. In the given problem, the primary issue is the sampling rate being too low, leading to aliasing rather than quantization error.

Slope overload is a phenomenon that occurs in delta modulation when the rate of change of the input signal exceeds the ability of the modulator to track it. This results in distortion due to the inability to follow steep slopes in the signal. Slope overload is not related to the sampling rate but to the modulation technique and its parameters. Therefore, it is not applicable to the given problem.

Option 4 suggests that no distortion occurs, which is incorrect. Given that the sampling rate of 20 kHz is below the Nyquist rate for a signal with frequency components up to 15 kHz, aliasing will occur, causing distortion. Thus, this option does not apply to the scenario described in the problem.

• Option 1: Quantization Error
• Option 3: Slope Overload
• Option 4: No Distortion

Explanation:

Condition for Sampling Theorem

Definition: The sampling theorem, also known as the Nyquist-Shannon sampling theorem, is a fundamental principle in signal processing. It states that a continuous-time signal can be completely represented and reconstructed from its samples if the sampling frequency satisfies a specific condition in relation to the highest frequency component of the signal.

Statement: To accurately recover the original signal from its sampled version, the sampling frequency (ω_s) must be at least twice the highest frequency component (ω_m) of the signal. Mathematically, this condition is expressed as:

ω_s ≥ 2ω_m

This condition ensures that no aliasing occurs during the sampling process, allowing the original signal to be reconstructed without distortion.

Correct Option Analysis:

The correct option is:

Option 2: ω_s ≥ 2ω_m

This option correctly represents the Nyquist criterion. According to the theorem, the sampling frequency must be at least twice the highest frequency component of the signal (ω_m) to prevent aliasing and ensure complete reconstruction of the original signal. If this condition is satisfied, the sampled signal contains all the necessary information to recover the original continuous-time signal.

Additional Information

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

Option 1: ω_m s ≤ 2ω_m

This option is incorrect because it suggests that the sampling frequency can be less than twice the highest frequency component (ω_m) and still allow accurate signal reconstruction. However, if ω_s is less than 2ω_m, aliasing occurs, making it impossible to recover the original signal without distortion. The condition ω_s ≤ 2ω_m does not guarantee the Nyquist criterion is met.

Option 3: ω_s m

This option is incorrect because it violates the Nyquist criterion. If the sampling frequency is less than the highest frequency component of the signal, severe aliasing occurs, and the original signal cannot be reconstructed. This condition leads to significant distortion in the sampled signal.

Option 4: ω_s = ω_m

This option is also incorrect. If the sampling frequency is equal to the highest frequency component of the signal, the Nyquist criterion is not satisfied. In such a case, aliasing still occurs, and the original signal cannot be accurately reconstructed. The sampling frequency must be at least twice the highest frequency component to meet the Nyquist criterion.

Conclusion:

The correct condition for accurately recovering the original signal from its sampled version is ω_s ≥ 2ω_m. This ensures that the sampling theorem is satisfied, preventing aliasing and enabling complete reconstruction of the original signal. Other options either violate this condition or fail to guarantee the accurate recovery of the signal, leading to distortion or loss of information.

Z Transform:

The z transform for a discrete signal x[n] is given by:
           X[z] = 

The set of all values of z where X(z) converges to a finite value is called as Radius of Convergence (ROC).
The ROC does not contain any poles.
If x[n] is a finite duration causal sequence or right-sided sequence, then the ROC is entire z-plane except at z = 0.
If x[n] is a finite duration anti-causal sequence or left-sided sequence, then the ROC is the entire z-plane except at z =∞ .

Definition:

Z transform is defined as

Properties:

Differentiation in z domain:

If X(z) is a z transform of x(n), then the z transform of n x(n) is,

Time-shifting:

If X(z) is a z transform of x(n), then the z transform of x(n – n₀) is,

Application:

Let x(n) = 1

Now, by applying the property of differentiation in the z domain,

Now, by applying the property of differentiation in the z domain,

Now, by applying the property of time-shifting,

Concept:

Shifting property of impulse function

Scaling property of impulse function

Explanation:

Let:

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

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

Concept:

Minimum sampling rate to avoid aliasing:

f_s = 2f_m = (Nyquist rate)

Calculation:

Given that, ω_m = 400 π

f_m = 200 Hz = maximum frequency of signal

Sampling frequency f_s = 2 × 200 = 400 Hz

Concept:

Bilateral Laplace transform:

Unilateral Laplace transform:

Some important Laplace transforms:

Calculation:

By applying frequency differentiation property,

Modulation:

• The process in which the characteristics of carrier signal is varied in accordance with baseband message signal to make bandpass signal.

Demultiplexing :

• The process or technique of transmitting multiple analog or digital input signals or data streams over a single channel is called multiplexing.
• The reverse of the multiplexing process. Demultiplexing is a process reconverting a signal containing multiple analog or digital signal streams back into the original signals.

Sampling :

• The process of conversion of continuous-time signals into discrete time signals.

Quantization :

• The process of mapping of continuous-time signals into discrete time signal.
• Hence correct option is "3"

Concept:

Final value theorem:

A final value theorem allows the time domain behavior to be directly calculated by taking a limit of a frequency domain expression

Final value theorem states that the final value of a system can be calculated by

 Where X(s) is the Laplace transform of the function.

For the final value theorem to be applicable system should be stable in steady-state and for that real part of poles should lie in the left side of s plane.

Initial value theorem:

It is applicable only when the number of poles of X(s) is more than the number of zeros of X(s).

Calculation:

Given that, 

Initial value,

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

So, the fundamental time period of the signal is

as 

∴ The fundamental frequency, 

Concept:

Final value theorem:

It states that:

Conditions:

1. It is valid only for causal systems. 

2. Pole of (1 – z-1) X(z) must lie inside the unit circle.

Calculation:

The final value theorem for z-transform is:

= 1/4 × 1 × 2 × 2 = 1

Concept:

The Fourier Transform of a continuous-time signal x(t) is given as:

Analysis:

Given:

x(t) = e^j10t  defined from t = -1 to 1. 

Given:

f_m = 5 kHz, 

ω_c = 2πfc = 2000π 

f_c = 1 kHz

Maximum frequency in x(t) is given as:

f_c + f_m = 6 kHz