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Given the systems (i) y(n) = n x(n) and (ii) y(n) = ex(n)
- a)(i) and (ii) are non linear
- b)(i) and (ii) are linear
- c)(i) is linear and (ii) is non linear
- d)(i) is non linear and (ii) is linear
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Given the systems (i) y(n) = n x(n) and (ii) y(n) = ex(n)a)(i) and (ii...
Concept: Linearity: Necessary and sufficient condition to prove the linearity of the system is that linear system follows the laws of superposition i.e. the response of the system is the sum of the responses obtained from each input considered separately.
y{ax1[n] + bx2[t]} = a y{x1[n]} + b y{x2[n]}
Conditions to check whether the system is linear or not.
- The output should be zero for zero input
- There should not be any non-linear operator present in the system
Causality: A system is causal, if the output of the system does not depend on future inputs, but only on past input.
Time-Invariance: If the input to a time-invariant system is shifted in time, its output remains the same signal, but is shifted equally in time.
If the output for an input x(t) is y(t), then for a time shift of t0 in the input gives the t0 shift in the output.
x(t) → y(t), then x(t – t0) → y(t – t0)
Application:
(i) y(n) = n x(n)
Here in the system, there is a nonlinear operator and hence it is linear.
(ii) y(n) = ex(n)
Here there is a non-linear operator (exponential) and hence it is nonlinear.
Most Upvoted Answer
Given the systems (i) y(n) = n x(n) and (ii) y(n) = ex(n)a)(i) and (ii...
Linear and Nonlinear Systems:
In the field of electrical engineering, systems can be classified into linear and nonlinear systems based on their mathematical properties. A linear system follows the principles of superposition and homogeneity, while a nonlinear system does not.
Linear System:
A linear system is characterized by the following properties:
1. Superposition: If two inputs, x1(n) and x2(n), produce outputs y1(n) and y2(n) respectively, then for any constants a and b, the input ax1(n) + bx2(n) will produce the output ay1(n) + by2(n).
2. Homogeneity: If an input x(n) produces an output y(n), then scaling the input by a constant a will produce a scaled output, ay(n).
Nonlinear System:
A nonlinear system does not satisfy the properties of linearity mentioned above. This means that the superposition principle and homogeneity do not hold for nonlinear systems.
Analyzing the Given Systems:
Let's analyze the given systems (i) y(n) = n x(n) and (ii) y(n) = ex(n) to determine their linearity.
System (i): y(n) = n x(n)
To check if system (i) is linear, we can apply the properties of linearity and see if they hold.
1. Superposition: Let's assume x1(n) and x2(n) are two inputs that produce outputs y1(n) and y2(n) respectively. According to the system equation, the output for x1(n) is y1(n) = n x1(n) and the output for x2(n) is y2(n) = n x2(n). Now, let's consider the input ax1(n) + bx2(n), where a and b are constants. The output for this input would be y(n) = n(ax1(n) + bx2(n)). Expanding this expression, we get y(n) = a(nx1(n)) + b(nx2(n)). As we can see, the superposition principle holds for this system.
2. Homogeneity: Let's assume x(n) produces an output y(n) = n x(n). Now, if we scale the input x(n) by a constant a, the output for the scaled input would be y(n) = n(ax(n)). As we can see, the homogeneity property also holds for this system.
Based on the analysis, we can conclude that system (i) is linear.
System (ii): y(n) = ex(n)
To check if system (ii) is linear, we can apply the properties of linearity and see if they hold.
1. Superposition: Let's assume x1(n) and x2(n) are two inputs that produce outputs y1(n) and y2(n) respectively. According to the system equation, the output for x1(n) is y1(n) = ex1(n) and the output for x2(n) is y2(n) = ex2(n). Now, let's consider the input ax1(n) + bx2(n), where a and b are constants. The output for this input would be y(n) = e(ax1(n) + bx2(n)). Expanding this expression, we get y(n) = e
In the field of electrical engineering, systems can be classified into linear and nonlinear systems based on their mathematical properties. A linear system follows the principles of superposition and homogeneity, while a nonlinear system does not.
Linear System:
A linear system is characterized by the following properties:
1. Superposition: If two inputs, x1(n) and x2(n), produce outputs y1(n) and y2(n) respectively, then for any constants a and b, the input ax1(n) + bx2(n) will produce the output ay1(n) + by2(n).
2. Homogeneity: If an input x(n) produces an output y(n), then scaling the input by a constant a will produce a scaled output, ay(n).
Nonlinear System:
A nonlinear system does not satisfy the properties of linearity mentioned above. This means that the superposition principle and homogeneity do not hold for nonlinear systems.
Analyzing the Given Systems:
Let's analyze the given systems (i) y(n) = n x(n) and (ii) y(n) = ex(n) to determine their linearity.
System (i): y(n) = n x(n)
To check if system (i) is linear, we can apply the properties of linearity and see if they hold.
1. Superposition: Let's assume x1(n) and x2(n) are two inputs that produce outputs y1(n) and y2(n) respectively. According to the system equation, the output for x1(n) is y1(n) = n x1(n) and the output for x2(n) is y2(n) = n x2(n). Now, let's consider the input ax1(n) + bx2(n), where a and b are constants. The output for this input would be y(n) = n(ax1(n) + bx2(n)). Expanding this expression, we get y(n) = a(nx1(n)) + b(nx2(n)). As we can see, the superposition principle holds for this system.
2. Homogeneity: Let's assume x(n) produces an output y(n) = n x(n). Now, if we scale the input x(n) by a constant a, the output for the scaled input would be y(n) = n(ax(n)). As we can see, the homogeneity property also holds for this system.
Based on the analysis, we can conclude that system (i) is linear.
System (ii): y(n) = ex(n)
To check if system (ii) is linear, we can apply the properties of linearity and see if they hold.
1. Superposition: Let's assume x1(n) and x2(n) are two inputs that produce outputs y1(n) and y2(n) respectively. According to the system equation, the output for x1(n) is y1(n) = ex1(n) and the output for x2(n) is y2(n) = ex2(n). Now, let's consider the input ax1(n) + bx2(n), where a and b are constants. The output for this input would be y(n) = e(ax1(n) + bx2(n)). Expanding this expression, we get y(n) = e
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