1. What is Unit Impulse Response?

The unit impulse response h[n] characterizes a discrete-time system by showing its output when presented with a unit impulse input δ[n]. For geometric series, it represents the cumulative sum of the parameter a raised to increasing powers.

2. How Does the Calculator Work?

The calculator uses the geometric series formula:

Where:

• \( h[n] \) — Unit impulse response at sample n
• \( a \) — System parameter (common ratio)
• \( n \) — Sample number (discrete time index)
• \( k \) — Summation index

Explanation: The response at each sample n is calculated by summing all terms a^k from k=0 to k=n.

3. Importance in Discrete Systems

Details: The unit impulse response completely characterizes linear time-invariant (LTI) systems. It's fundamental for system analysis, convolution operations, and understanding system stability.

4. Using the Calculator

Tips: Enter parameter a (common ratio) and sample number n (must be non-negative integer). The calculator will compute the cumulative sum up to sample n.

5. Frequently Asked Questions (FAQ)

Q1: What does parameter 'a' represent?
A: 'a' is the common ratio in the geometric series, representing how each term relates to the previous one.

Q2: What happens when |a| ≥ 1?
A: The series may diverge (grow without bound) as n increases, indicating an unstable system.

Q3: How is this related to system stability?
A: For stable systems, the impulse response must be absolutely summable (|a| < 1 for geometric series).

Q4: Can this be used for continuous systems?
A: No, this calculator is specifically for discrete-time systems. Continuous systems use integrals.

Q5: What's the closed-form solution?
A: For a ≠ 1, h[n] = (1 - a^(n+1))/(1 - a). For a = 1, h[n] = n + 1.