Question

Consider h [n] as in figure 1 and x[n] = u[n] – u [n-3], determine the output y [n] of the LTI system?

This question was addressed to me during an online interview.

My question is from Convolution : Impulse Response Representation for LTI Systems topic in section Time Domain Representation for LTI Systems of Signals and Systems

Answer 1

To find the output \( y[n] \) of an LTI system with input \( x[n] = u[n] - u[n-3] \) and impulse response \( h[n] \), you would use the **convolution sum** formula:

\[

y[n] = (x * h)[n] = \sum_{k=-\infty}^{\infty} x[k] \cdot h[n-k]

\]

Here’s a step-by-step outline of how to proceed:

### 1. **Understand the Input Signal:**

   The input signal \( x[n] = u[n] - u[n-3] \) represents a finite-duration pulse. It is a discrete signal that equals 1 for \( n = 0, 1, 2 \), and 0 otherwise. Mathematically:

   \[

   x[n] = \begin{cases}

   1 & \text{for } n = 0, 1, 2 \\

   0 & \text{otherwise}

   \end{cases}

   \]

### 2. **Impulse Response Representation:**

   You need to know the specific form of the impulse response \( h[n] \) from the figure provided. The impulse response will determine how the system reacts to each input value. Let’s assume \( h[n] \) is known (you should substitute its values accordingly based on the figure provided).

### 3. **Convolution Calculation:**

   With \( x[n] \) and \( h[n] \) defined, perform the convolution sum:

   \[

   y[n] = \sum_{k=-\infty}^{\infty} x[k] \cdot h[n-k]

   \]

   Since \( x[k] \) is nonzero only for \( k = 0, 1, 2 \), the sum reduces to:

   \[

   y[n] = x[0] \cdot h[n] + x[1] \cdot h[n-1] + x[2] \cdot h[n-2]

   \]

   This simplifies further since \( x[k] = 1 \) for \( k = 0, 1, 2 \), so:

   \[

   y[n] = h[n] + h[n-1] + h[n-2]

   \]

### 4. **Final Output:**

   The final output \( y[n] \) is the sum of the impulse response shifted by 0, 1, and 2 time steps. You can calculate this for each value of \( n \) based on the known form of \( h[n] \).

If you provide the figure for \( h[n] \), I can help with the exact output.