Smoothing Filter: Implement a 1D Moving-Average Smoothing Filter on a Signal

Question

## Problem

Implement a causal moving-average filter that smooths a noisy 1D signal. Given an input array and a window size W, replace each sample with the mean of the W most recent samples (including the current one). Handle the boundary at the start of the signal.

```python
def moving_average(signal: list[float], window: int) -> list[float]:
    # Causal: output[i] = mean(signal[max(0,i-window+1) : i+1])
    pass

def moving_average_fast(signal: list[float], window: int) -> list[float]:
    # O(n) using a running sum; no recomputation per window
    pass
```

**Example:**
```
moving_average([1,3,5,7,9], window=3)
-> [1.0, 2.0, 3.0, 5.0, 7.0]
# Index 0: mean([1])=1, Index 1: mean([1,3])=2, Index 2: mean([1,3,5])=3 ...
```

## Follow-ups

1. What is the time complexity of the naive vs. running-sum approach?
2. How does a Gaussian filter differ from a box (moving-average) filter in the frequency domain?
3. How would you implement an exponential moving average (EMA) for real-time streaming data?
4. How would you extend this to a 2D image smoothing filter?

Smoothing Filter: Implement a 1D Moving-Average Smoothing Filter on a Signal

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