Unit Pulse Response for a discrete time system

[Schniz2]

Homework Statement

Compute the unit-pulse response h[n] for n= 0,1,2,3 for the following discrete time system:
y[n+2] + 1/2y[n+1] + 1/4y[n] = x[n=1] - x[n]

Homework Equations

I think i am supposed to replace the functions of x with delta functions, which are zero at all except n=0, however i can't work out what to do with the y[2], y[1] functions?... do i just leave them in that form or is there a way to evaluate them? I've been searching my textbook but its so hard to follow what is going on! Its Fundamentals of signals and systems by Kamen and Heck... which is the prescribed text for uni but dos anyone know of one which is written in a more reader-friendly way? :P.

The Attempt at a Solution

See attached image

 

[KataKoniK]

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Hello! You are correct in thinking that you need to replace the x[n] terms with delta functions. This is because the unit-pulse response is the output of the system when the input is a unit impulse, which is represented by a delta function.

For the y[n] terms, you can use the definition of the unit-pulse response: h[n] = y[n] when x[n] is a unit impulse. So for n=0,1,2,3, you can simply replace y[n] with the corresponding values given in the system equation.

For example, when n=0, you have y[2] + 1/2y[1] + 1/4y[0] = delta[n=1] - delta[n]. Since x[n] is a unit impulse at n=0, you can set delta[n] = 0 and solve for y[0]. This gives you the value of y[0] for the unit-pulse response at n=0.

I hope this helps! As for a more reader-friendly textbook, I recommend "Signals and Systems" by Alan Oppenheim and Alan Willsky. It's a great resource for understanding the fundamentals of signals and systems. Good luck!

 

[piercas]

I would approach this problem by first understanding the concept of a unit-pulse response for a discrete time system. The unit-pulse response represents the output of the system when a single unit pulse input is applied at a specific time. This response is important in understanding the behavior and characteristics of the system.

To solve this problem, we can use the given equation and the definition of a unit pulse to compute the unit-pulse response h[n]. The definition of a unit-pulse function is a function that is equal to 1 at n=0 and 0 otherwise. Therefore, we can replace the x[n] term with a unit pulse function.

Next, we can use the properties of a discrete time system to simplify the equation. By shifting the indices, we can rewrite the equation as y[n] + 1/2y[n-1] + 1/4y[n-2] = δ[n-1] - δ[n]. This simplification allows us to solve for h[n] by isolating the y[n] term on one side of the equation.

Using this method, we can compute the unit-pulse response for n=0,1,2,3. For n=0, h[0]=1; for n=1, h[1]=-1/2; for n=2, h[2]=1/4; and for n=3, h[3]=-1/8. These values represent the output of the system at each time step when a unit pulse input is applied at that time.

In terms of evaluating the y[2] and y[1] terms, we can use the same approach of shifting indices and simplifying the equation to isolate these terms. However, since we are only interested in the output of the system at n=0,1,2,3, we can simply substitute the values of h[0], h[1], h[2], and h[3] into the equation instead of solving for y[2] and y[1] separately.

In terms of finding a more reader-friendly textbook, I would recommend searching for other textbooks on signals and systems that may be more suitable for your learning style. It is important to find a resource that is easy to understand and follow, as this subject can be complex and challenging. Additionally, seeking out additional resources such as online tutorials or videos can also be helpful in understanding the material.