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What is the impulse response of the system described by the second order difference equation y(n)-3y(n-1)-4y(n-2)=x(n)+2x(n-1)?
- a)[-1/5 (-1)n-6/5 (4)n]u(n)
- b)[1/5 (-1)n – 6/5 (4)n]u(n)
- c)[ 1/5 (-1)n+ 6/5 (4)n]u(n)
- d)[- 1/5 (-1)n+ 6/5 (4)n]u(n)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
What is the impulse response of the system described by the second ord...
Explanation: The homogenous solution of the given equation is yh(n)=C1(-1)n+C2(4)n—-(1)
To find the impulse response, x(n)=δ(n)
now, for n=0 and n=1 we get
y(0)=1 and
y(1)=3+2=5
From equation (1) we get
y(0)=C1+C2 and
y(1)=-C1+4C2
On solving the above two set of equations we get
C1=- 1/5 and C2= 6/5
=>h(n)= [-1/5 (-1)n + 6/5 (4)n]u(n).
To find the impulse response, x(n)=δ(n)
now, for n=0 and n=1 we get
y(0)=1 and
y(1)=3+2=5
From equation (1) we get
y(0)=C1+C2 and
y(1)=-C1+4C2
On solving the above two set of equations we get
C1=- 1/5 and C2= 6/5
=>h(n)= [-1/5 (-1)n + 6/5 (4)n]u(n).
Most Upvoted Answer
What is the impulse response of the system described by the second ord...
To find the impulse response of the system, we need to input an impulse signal as x(n). An impulse signal is a signal where all values are zero, except for one sample which has a value of 1. Therefore, we can set x(n) = δ(n), where δ(n) is the discrete-time unit impulse function.
Substituting x(n) = δ(n) into the given difference equation, we get:
y(n) - 3y(n-1) - 4y(n-2) = δ(n) + 2δ(n-1)
To find the impulse response, we need to solve for y(n) using the initial conditions y(-1) = y(-2) = 0. We can do this using the Z-transform:
Y(z) - 3z^-1Y(z) - 4z^-2Y(z) = 1 + 2z^-1
Y(z)(1 - 3z^-1 - 4z^-2) = 1 + 2z^-1
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
To find the inverse Z-transform and obtain the impulse response, we can use partial fraction decomposition:
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
Y(z) = [(1/5)/(1 + 4z^-1)] - [(6/5)/(1 - z^-1)]
Using the inverse Z-transform tables, we can find that the inverse Z-transform of (1/5)/(1 + 4z^-1) is (-1/5)(-1)^n u(n+1), and the inverse Z-transform of -(6/5)/(1 - z^-1) is (4/5)(-1)^n u(n). Therefore, the impulse response of the system is:
h(n) = (-1/5)(-1)^n u(n+1) + (4/5)(-1)^n u(n)
Simplifying this expression, we get:
h(n) = [(-1/5)(-1)^n - (4/5)(-1)^n] u(n)
h(n) = (1/5)(-1)^n u(n)
Substituting x(n) = δ(n) into the given difference equation, we get:
y(n) - 3y(n-1) - 4y(n-2) = δ(n) + 2δ(n-1)
To find the impulse response, we need to solve for y(n) using the initial conditions y(-1) = y(-2) = 0. We can do this using the Z-transform:
Y(z) - 3z^-1Y(z) - 4z^-2Y(z) = 1 + 2z^-1
Y(z)(1 - 3z^-1 - 4z^-2) = 1 + 2z^-1
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
To find the inverse Z-transform and obtain the impulse response, we can use partial fraction decomposition:
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
Y(z) = [(1/5)/(1 + 4z^-1)] - [(6/5)/(1 - z^-1)]
Using the inverse Z-transform tables, we can find that the inverse Z-transform of (1/5)/(1 + 4z^-1) is (-1/5)(-1)^n u(n+1), and the inverse Z-transform of -(6/5)/(1 - z^-1) is (4/5)(-1)^n u(n). Therefore, the impulse response of the system is:
h(n) = (-1/5)(-1)^n u(n+1) + (4/5)(-1)^n u(n)
Simplifying this expression, we get:
h(n) = [(-1/5)(-1)^n - (4/5)(-1)^n] u(n)
h(n) = (1/5)(-1)^n u(n)
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