Question

Find the first 10 samples (manual \& code) of the impulse response of the causal system \( y[n]=x[n]+0.5 y[n-1] \) What type of system is it?

Answer 1

In this case, the system equation includes a feedback term
`(0.5y[n-1])`, indicating that it is a recursive or an IIR (Infinite Impulse Response) system.

Step-by-step explanation:

To find the first 10 samples of the impulse response of the causal system
`y[n] = x[n] + 0.5y[n-1]`, we can start by considering the definition of the impulse response.

The impulse response of a system is the output of the system when an impulse input is applied, where an impulse is a signal with a value of 1 at time n = 0 and 0 elsewhere.

To find the first 10 samples manually, we can use the recursive nature of the system equation.

1. For n = 0:

-
`y[0] = x[0] + 0.5y[-1]`

- Since y[-1] is not defined, we assume it to be 0 (since the system is causal and there is no previous output).

- Therefore,
`y[0] = x[0]`

2. For n = 1:

-
`y[1] = x[1] + 0.5y[0]`

- We already know y[0] from the previous step, so we can substitute it.

- Therefore,
`y[1] = x[1] + 0.5x[0]`

3. For n = 2:

-
`y[2] = x[2] + 0.5y[1]`

- We already know y[1] from the previous step, so we can substitute it.

- Therefore,
`y[2] = x[2] + 0.5(x[1] + 0.5x[0])`

We continue this process until n = 9 to find the first 10 samples of the impulse response.

Alternatively, we can use code to compute the impulse response samples.

Here is an example using MATLAB:

```matlab

% Define the system equation

b = [1];

a = [1, -0.5];

% Compute the impulse response

impulse_response = impz(b, a, 10);

% Display the first 10 samples

disp("First 10 samples of the impulse response:");

disp(impulse_response);

```

In this code, we define the system equation using the numerator `b` and denominator `a` coefficients. We then use the `impz` function in MATLAB to compute the impulse response of the system.

Finally, we display the first 10 samples of the impulse response using `disp()`.

Regarding the type of system, we can determine it based on the presence of the feedback term. In this case, the system equation includes a feedback term
`(0.5y[n-1])`, indicating that it is a recursive or an IIR (Infinite Impulse Response) system.