1-D Convolution

Intuition

Slide a small window (the kernel) along a signal; at each position, take the dot product of the kernel with the part of the signal under it. A kernel [−1, 1] detects increases; [1/3]×3 smooths.

Precise explanation

Deep-learning libraries implement cross-correlation (no kernel flip) and still call it convolution:

y_i = b + \sum_{j=0}^{k-1} x_{is + j}\, w_j,

after zero-padding $p$ values on both ends. Output length (CS231n formula):

n_\text{out} = \left\lfloor \frac{n + 2p - k}{s} \right\rfloor + 1.

NumPy's np.convolve flips the kernel (mathematical convolution); np.correlate(x, w, "valid") matches the DL definition for stride 1, no padding.

Worked example

x = [1, 2, 3, 4, 5], w = [1, 0, −1], stride 1: [1−3, 2−4, 3−5] = [−2, −2, −2]; length (5 − 3)/1 + 1 = 3. With stride 2: positions 0 and 2 → [−2, −2]. With padding 1: x becomes [0,1,2,3,4,5,0] → length 5: [−2, −2, −2, −2, 4].

Common mistakes

Flipping the kernel (using np.convolve).
Off-by-one in the number of windows.
Padding only one side.

Why this matters

Convolutions power audio models and text CNNs, and their 2-D version is the core of computer vision (CV-10).

Library implementation

With stride 1, np.correlate(np.pad(x, p), kernel, mode="valid") computes the same values (NumPy's np.convolve flips the kernel; deep-learning "convolution" does not). torch.nn.functional.conv1d adds batches, channels and strides.

References

CS231n: Convolutional Neural Networks notes — Stanford University CS231n (S015)
Dive into Deep Learning — Aston Zhang, Zachary C. Lipton, Mu Li, Alexander J. Smola (S013)