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Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.?
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Example: Consider a discrete-time rystem described bythe input-output ...
Understanding Linearity in Systems
To determine if the system described by the relation y[n] = n * x[n] is linear, we need to check two properties: additivity and homogeneity (scaling).
1. Additivity
- Definition: A system is additive if the response to a sum of inputs equals the sum of the responses to each input individually.
- Testing: Let x1[n] and x2[n] be two inputs.
- The output for x1[n]: y1[n] = n * x1[n]
- The output for x2[n]: y2[n] = n * x2[n]
- For the combined input x1[n] + x2[n]:
- y[n] = n * (x1[n] + x2[n]) = n * x1[n] + n * x2[n] = y1[n] + y2[n]
- Conclusion: The system satisfies the additivity property.
2. Homogeneity (Scaling)
- Definition: A system is homogeneous if the response to a scaled input equals the scaled response of the output.
- Testing: Let α be a scaling factor.
- For input α * x[n]:
- y[n] = n * (α * x[n]) = α * (n * x[n]) = α * y[n]
- Conclusion: The system satisfies the homogeneity property.
Final Assessment
Since the system meets both the additivity and homogeneity properties, we conclude:
- The system characterized by y[n] = n * x[n] is linear.
By confirming these properties, we establish that the given discrete-time system adheres to the principles of linearity, allowing for predictable behavior under varying inputs.
To determine if the system described by the relation y[n] = n * x[n] is linear, we need to check two properties: additivity and homogeneity (scaling).
1. Additivity
- Definition: A system is additive if the response to a sum of inputs equals the sum of the responses to each input individually.
- Testing: Let x1[n] and x2[n] be two inputs.
- The output for x1[n]: y1[n] = n * x1[n]
- The output for x2[n]: y2[n] = n * x2[n]
- For the combined input x1[n] + x2[n]:
- y[n] = n * (x1[n] + x2[n]) = n * x1[n] + n * x2[n] = y1[n] + y2[n]
- Conclusion: The system satisfies the additivity property.
2. Homogeneity (Scaling)
- Definition: A system is homogeneous if the response to a scaled input equals the scaled response of the output.
- Testing: Let α be a scaling factor.
- For input α * x[n]:
- y[n] = n * (α * x[n]) = α * (n * x[n]) = α * y[n]
- Conclusion: The system satisfies the homogeneity property.
Final Assessment
Since the system meets both the additivity and homogeneity properties, we conclude:
- The system characterized by y[n] = n * x[n] is linear.
By confirming these properties, we establish that the given discrete-time system adheres to the principles of linearity, allowing for predictable behavior under varying inputs.
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Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.?
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Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.?.
Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Example: Consider a discrete-time rystem described bythe input-output relation. y[n] = nx * [n] show that this system is Linear.?.
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