Question

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

Answer 1

The system described is a first-order IIR filter and using an impulse input, the first 10 samples of the impulse response are manually calculated by iteratively applying the given formula. a python code snippet is also provided to compute the samples programmatically.

Step-by-step explanation:

The equation given, y[n] = x[n] + 0.5y[n−1], describes a discrete-time linear system with feedback. to find the first 10 samples of the impulse response of this system, we need to input an impulse function which is defined as x[n] = δ[n] where δ[n] is 1 at n=0 and 0 otherwise. the resulting impulse response h[n] is the output when x[n] = δ[n].

• At n = 0: h[0] = 1, since y[n] = x[n] when y[n-1] is not defined.
• At n = 1: h[1] = 0 + 0.5 * h[0] = 0.5.
• At n = 2: h[2] = 0 + 0.5 * h[1] = 0.25.
• Continue this process to find h[n] for all n up to 10.

This system is a first-order IIR (Infinite Impulse Response) filter because the output depends on both the input and previous outputs indefinitely.

Manual Calculation:

1. h[0] = 1
2. h[1] = 0.5
3. h[2] = 0.25
4. h[3] = 0.125
5. h[4] = 0.0625
6. h[5] = 0.03125
7. h[6] = 0.015625
8. h[7] = 0.0078125
9. h[8] = 0.00390625
10. h[9] = 0.001953125

Code Implementation:

Here's a simple Python code snippet that can be used to find the first 10 samples of the impulse response:

h = [1] # Initial impulse
for i in range(1, 10):
h.append(0.5 * h[i-1])
print(h)