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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 set x(n) = δ(n), where δ(n) is the discrete-time unit impulse function. Substituting into the difference equation, we get:
y(n) - 3y(n-1) - 4y(n-2) = δ(n) + 2δ(n-1)
We can find the solution to this difference equation using the z-transform. Taking the z-transform of both sides, we get:
Y(z) - 3z^-1 Y(z) - 4z^-2 Y(z) = 1 + 2z^-1
Solving for Y(z), we get:
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
We can use partial fraction decomposition to express this in terms of simpler fractions:
Y(z) = A/(1 + z) + B/(1 + 4z)
where A and B are constants to be determined. Multiplying both sides by the denominator, we get:
(1 - 3z^-1 - 4z^-2)Y(z) = A(1 + 4z) + B(1 + z)
We can solve for A and B by setting z = -1 and z = -1/4, respectively:
A = (1 + 2(-1)^-1)/(1 + 4(-1)) = -1/5
B = (1 + 2(-1/4)^-1)/(1 + 4(-1/4)) = 4/5
Substituting back into the partial fraction decomposition, we get:
Y(z) = -1/5/(1 + z) + 4/5/(1 + 4z)
Taking the inverse z-transform, we get:
y(n) = -1/5 (-1)^n + 4/5 (-4)^n u(n)
where u(n) is the discrete-time unit step function. Therefore, the impulse response of the system is:
h(n) = -1/5 (-1)^n + 4/5 (-4)^n u(n)
Option (a) is the closest answer, but it has a mistake in the second term. It should be 4/5 (4)^n u(n) instead of 4/5 (4)n u(n). Option (b) is not correct.
y(n) - 3y(n-1) - 4y(n-2) = δ(n) + 2δ(n-1)
We can find the solution to this difference equation using the z-transform. Taking the z-transform of both sides, we get:
Y(z) - 3z^-1 Y(z) - 4z^-2 Y(z) = 1 + 2z^-1
Solving for Y(z), we get:
Y(z) = (1 + 2z^-1)/(1 - 3z^-1 - 4z^-2)
We can use partial fraction decomposition to express this in terms of simpler fractions:
Y(z) = A/(1 + z) + B/(1 + 4z)
where A and B are constants to be determined. Multiplying both sides by the denominator, we get:
(1 - 3z^-1 - 4z^-2)Y(z) = A(1 + 4z) + B(1 + z)
We can solve for A and B by setting z = -1 and z = -1/4, respectively:
A = (1 + 2(-1)^-1)/(1 + 4(-1)) = -1/5
B = (1 + 2(-1/4)^-1)/(1 + 4(-1/4)) = 4/5
Substituting back into the partial fraction decomposition, we get:
Y(z) = -1/5/(1 + z) + 4/5/(1 + 4z)
Taking the inverse z-transform, we get:
y(n) = -1/5 (-1)^n + 4/5 (-4)^n u(n)
where u(n) is the discrete-time unit step function. Therefore, the impulse response of the system is:
h(n) = -1/5 (-1)^n + 4/5 (-4)^n u(n)
Option (a) is the closest answer, but it has a mistake in the second term. It should be 4/5 (4)^n u(n) instead of 4/5 (4)n u(n). Option (b) is not correct.
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