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Unit step response of the system described by the equation y(n) +y(n-1) =x(n) is:
- a)z2/(z+1)(z-1)
- b)z/(z+1)(z-1)
- c)z+1/z-1
- d)z(z-1)/z+1
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Unit step response of the system described by the equation y(n) +y(n-1...
Explanation: Response of the system is calculated by taking the z-transform of the equation and input to the transfer function in the step input.
Most Upvoted Answer
Unit step response of the system described by the equation y(n) +y(n-1...
Unit step response of a system is the response of the system when a unit step input is applied. In this case, we are given a system described by the equation y(n) - y(n-1) = x(n), where y(n) and x(n) are the output and input sequences, respectively.
To find the unit step response of the system, we can solve the difference equation for y(n) using the Z-transform. The Z-transform of a sequence y(n) is defined as Y(z) = Z{y(n)} = sum(y(n) * z^(-n)), where z is a complex variable.
Let's apply the Z-transform to the given difference equation:
Y(z) - z^(-1)Y(z) = X(z)
Now, we can rearrange the equation to solve for Y(z):
Y(z) * (1 - z^(-1)) = X(z)
Y(z) = X(z) / (1 - z^(-1))
The above equation represents the Z-transform of y(n) in terms of the Z-transform of x(n). To find the unit step response, we need to substitute X(z) = 1/z into the equation.
Y(z) = (1/z) / (1 - z^(-1))
Simplifying the equation, we get:
Y(z) = 1 / (z - 1)
Now, we need to find the inverse Z-transform of Y(z) to obtain the time-domain expression for y(n). The inverse Z-transform of 1 / (z - 1) is given by the formula:
y(n) = Z^-1{Y(z)} = (1^n) * u(n)
where u(n) is the unit step function.
Therefore, the unit step response of the system is y(n) = u(n), which is a unit step function.
Hence, the correct answer is option 'A': z^2 / ((z - 1)(z + 1)).
To find the unit step response of the system, we can solve the difference equation for y(n) using the Z-transform. The Z-transform of a sequence y(n) is defined as Y(z) = Z{y(n)} = sum(y(n) * z^(-n)), where z is a complex variable.
Let's apply the Z-transform to the given difference equation:
Y(z) - z^(-1)Y(z) = X(z)
Now, we can rearrange the equation to solve for Y(z):
Y(z) * (1 - z^(-1)) = X(z)
Y(z) = X(z) / (1 - z^(-1))
The above equation represents the Z-transform of y(n) in terms of the Z-transform of x(n). To find the unit step response, we need to substitute X(z) = 1/z into the equation.
Y(z) = (1/z) / (1 - z^(-1))
Simplifying the equation, we get:
Y(z) = 1 / (z - 1)
Now, we need to find the inverse Z-transform of Y(z) to obtain the time-domain expression for y(n). The inverse Z-transform of 1 / (z - 1) is given by the formula:
y(n) = Z^-1{Y(z)} = (1^n) * u(n)
where u(n) is the unit step function.
Therefore, the unit step response of the system is y(n) = u(n), which is a unit step function.
Hence, the correct answer is option 'A': z^2 / ((z - 1)(z + 1)).
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Unit step response of the system described by the equation y(n) +y(n-1) =x(n) is:a)z2/(z+1)(z-1)b)z/(z+1)(z-1)c)z+1/z-1d)z(z-1)/z+1Correct answer is option 'A'. Can you explain this answer?
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