Let's break down the problem step by step and compute the impulse response of the system.

Problem:

We are given $x[n] = 2(u[n] - u[n-3])$, where $u[n]$ is the unit step function.

This can be interpreted as a discrete-time signal that has a value of 2 for $0 \leq n < 3$, and 0 elsewhere.

Now, to compute the impulse response $h[n]$, we need to assume that the system is linear and time-invariant (LTI). In this case, the impulse response is the system's output when the input is an impulse $\delta[n]$.

Step 1: Expression of $x[n]$

Rewriting the signal:

[ x[n] = 2(u[n] - u[n-3]) ]

This signal represents a rectangular pulse of length 3, scaled by 2. Explicitly, this can be written as:

\begin{cases} 2, & 0 \leq n \leq 2 \\ 0, & \text{otherwise} \end{cases}$$ ### Step 2: Compute the Z-transform of $$x[n]$$ To compute the impulse response, we usually take the Z-transform of both the input and the system transfer function $$H(z)$$. However, we need to calculate the Z-transform of $$x[n]$$: $$X(z) = 2\left( \sum_{n=0}^{2} z^{-n} \right)$$ Expanding the sum: $$X(z) = 2(1 + z^{-1} + z^{-2})$$ Thus, the Z-transform of $$x[n]$$ is: $$X(z) = 2 \left( \frac{1 - z^{-3}}{1 - z^{-1}} \right)$$ ### Step 3: Impulse Response via Partial Fraction Decomposition To find the impulse response, we also need the transfer function $$H(z)$$ of the system. If the system is defined by a difference equation, we would compute the Z-transform and use partial fraction decomposition. However, in this case, since we do not have a specific system transfer function provided, we cannot apply partial fraction decomposition directly to compute the impulse response. The problem only involves calculating the input signal $$x[n]$$, which has already been expressed. Thus, **partial fraction decomposition is not necessary** in this scenario unless we are explicitly provided with a system's transfer function. --- ### Conclusion: For the given input $$x[n] = 2(u[n] - u[n-3])$$, the impulse response cannot be computed without knowing the system's characteristics (i.e., its transfer function). Partial fraction decomposition would only be relevant if we were working with a specific system function, which isn't provided here. Do you have any further details on the system or the transfer function? If so, we can proceed with further steps like using partial fractions. Would you like to explore more about this topic? --- ### 5 Related Questions: 1. How do you derive the Z-transform of the unit step function $$u[n]$$? 2. What is the significance of the impulse response in a linear time-invariant (LTI) system? 3. How do we determine the transfer function $$H(z)$$ of a system? 4. When is partial fraction decomposition used in signal processing? 5. How does the convolution sum relate to the impulse response of a system? ### Tip: In signal processing, the Z-transform is a powerful tool to analyze discrete-time signals and systems. Always consider it when dealing with convolution or system response analysis.