1D Convolution

Perform 1D convolution between an input signal and a kernel with zero padding and centered kernel.

Let $r = \frac{K-1}{2}$. Out-of-bounds accesses to $\text{A}$ are treated as zero.

$$\text{C}[i] = \sum_{j=0}^{K-1} \text{A}[\,i + j - r\,] \cdot \text{B}[j]$$

The convolution operation slides the kernel over the input signal, computing the sum of element-wise multiplications at each position. Zero padding is used at the boundaries.

Input:

• Vector $\text{A}$ of size $\text{N}$ (input signal)
• Vector $\text{B}$ of size $\text{K}$ (convolution kernel)

Output:

• Vector $\text{C}$ of size $\text{N}$ (convolved signal)

Notes:

• $\text{K}$ is odd and smaller than $\text{N}$
• Use zero padding at the boundaries where the kernel extends beyond the input signal
• The kernel is centered at each position, with $(K-1)/2$ elements on each side
• Output size is $N$ (same as input) due to padding
• This matches PyTorch torch.nn.functional.conv1d(..., padding=K//2) (cross-correlation, kernel is not flipped)
• This problem is adapted from KernelBench