Implement 2D convolution forward pass

Problem

Implement the forward pass of a 2D convolution (conv2d) from scratch (no deep learning libraries).

You are given:

• Input tensor x with shape (N, C_in, H, W) (NCHW layout)
• Filter weights w with shape (C_out, C_in, K_h, K_w)
• Optional bias b with shape (C_out,) (may be None )
• Integer stride s_h, s_w
• Integer padding p_h, p_w (zero-padding applied to height/width)

Compute the output tensor y with shape (N, C_out, H_out, W_out) where:

$H_{out} = \left\lfloor \frac{H + 2p_h - K_h}{s_h} \right\rfloor + 1, \quad W_{out} = \left\lfloor \frac{W + 2p_w - K_w}{s_w} \right\rfloor + 1$

For each output element:

$y[n, c_{out}, i, j] = \sum_{c_{in}=0}^{C_{in}-1} \sum_{u=0}^{K_h-1} \sum_{v=0}^{K_w-1} \; x_{pad}[n, c_{in}, i\cdot s_h + u, j\cdot s_w + v] \cdot w[c_{out}, c_{in}, u, v] + b[c_{out}]$

where x_pad is x padded with zeros by (p_h, p_w).

Requirements

• Return y as a dense numeric array/tensor.
• Do not use existing convolution operators.
• Handle edge cases such as b is None , non-square kernels, and different strides/padding.

Constraints (typical for an interview unit test)

• Sizes are small enough that an $O(N\cdot C_{out}\cdot C_{in}\cdot H_{out}\cdot W_{out}\cdot K_h\cdot K_w)$ implementation passes.
• Inputs are floating point numbers.

Example (shape check)

If x is (1, 3, 32, 32), w is (8, 3, 3, 3), stride (1,1), padding (1,1), then output shape is (1, 8, 32, 32).