Finding the impulse response of a system without using Z-Transform

In summary, the impulse response for the given difference equation is a combination of the homogeneous solution and the particular solution, which is affected by the additional terms on the right side. This can be calculated by setting x[n]=δ[n] and observing the system's response, or by using the principle of superposition.
  • #1
jnava
7
0

Homework Statement


Find the impulse response of the following: Assume the system is initially at rest.

y[n] - (1/2) y[n-1] = x[n] + 2 x[n-1] + x[n-2]

The Attempt at a Solution



To find the impulse response y[n]: we know that y[n] = homogenous solution + particular solution

so...

lamda - (1/2) = 0 => lamda= (1/2)

homogenous solution = C1*(1/2)^n

For the particular solution, we assume that a x[n] = delta[n]...am i correct?

so ... y[n] - (1/2) y[n-1] = delta[n] + 2 delta[n-1] + delta[n-2]

so, for n >2, delta[n] = 0

=> y[n] - (1/2) y[n-1] = 0, which means the particular solution is the homogenous solution?

Can someone help me with this?

Thanks
 
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  • #2
jnava said:

Homework Statement


Find the impulse response of the following: Assume the system is initially at rest.

y[n] - (1/2) y[n-1] = x[n] + 2 x[n-1] + x[n-2]



The Attempt at a Solution



To find the impulse response y[n]: we know that y[n] = homogenous solution + particular solution

so...

lamda - (1/2) = 0 => lamda= (1/2)

homogenous solution = C1*(1/2)^n

For the particular solution, we assume that a x[n] = delta[n]...am i correct?

so ... y[n] - (1/2) y[n-1] = delta[n] + 2 delta[n-1] + delta[n-2]
Yes, that's correct.
so, for n >2, delta[n] = 0

=> y[n] - (1/2) y[n-1] = 0, which means the particular solution is the homogenous solution?
No, just because for n>2, the RHS is 0, it doesn't follow the particular solution is the homogeneous solution. If the difference equation were simply

y[n] - (1/2) y[n-1] = x[n]

then the impulse response would be the homogeneous solution. Just try calculating y[0], y[1], y[2], and y[3], and you'll see that's indeed the case. Now, obviously, the two extra terms in the given difference equation will have an effect on how the system responds, so the impulse response for the original equation can't be just the homogeneous solution.
Can someone help me with this?

Thanks
You might try calculating y[0], y[1], y[2], y[3], ... for the original equation when you set x[n]=δ[n] and see how the system responds. It helps not to simplify much so you can see how the delayed terms on the right affect the output.

Alternatively, you can use the principle of superposition to calculate what the output should be in general.
 
  • #3
Thanks for the help...I figured it out...
 

1. What is an impulse response and why is it important in a system?

An impulse response is the output of a system when an impulse input is applied. It is important because it characterizes the behavior and properties of a system, allowing for analysis and prediction of the system's response to different inputs.

2. Can the impulse response of a system be found without using the Z-transform?

Yes, the impulse response can be found using other methods, such as convolution or Fourier analysis. The Z-transform is just one approach that is commonly used.

3. What other methods can be used to find the impulse response of a system?

Other methods include time-domain analysis using convolution or differential equations, frequency-domain analysis using Fourier analysis, and state-space representation using differential equations.

4. How is the impulse response related to the transfer function of a system?

The impulse response and the transfer function are mathematically related, where the transfer function is the Laplace transform of the impulse response. This allows for the use of the transfer function to analyze the system's properties and behavior.

5. Can the impulse response of a system change over time?

Yes, the impulse response can change over time if the system's properties or inputs change. However, for linear time-invariant systems, the impulse response remains constant over time.

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