视频讲解：
https://www.bilibili.com/video/BV1Vd4y1e7pj
32
x15
---
160
32
---
480
You’re correct in observing that the column-wise multiplication of two numbers, such as 32 × 15, is a form of convolution. Here’s why:
In the column-wise multiplication of two numbers, we essentially perform a series of shift-and-add operations. For example, when multiplying 32 by 15, we break down the process into:
- Multiply 32 by the least significant digit of 15 (which is 5), and write the result (160) on the first row.
- Multiply 32 by the next digit of 15 (which is 1), and write the result (32) on the second row, shifted one place to the left.
- Add the rows together to get the final result (480).
This process of shifting and adding is analogous to the convolution operation in signal processing and digital filtering.
In convolution, we have two signals or sequences, f(n) and g(n), and the convolution operation is defined as:
(f * g)(n) = Σ f(k) * g(n - k)
Where the sum is taken over all possible values of k.
In the case of multiplying two numbers in column form, we can consider one number as the signal f(n) and the other number as the signal g(n). The process of shifting and adding corresponds to the convolution operation, where each digit of the second number (g(n)) is multiplied by the first number (f(n)), and the results are shifted and added together.
For example, when multiplying 32 by 15:
- f(n) = 32 (the first number)
- g(n) = 15 (the second number)
The convolution operation would be:
(f * g)(n) = 32 * 5 + 32 * 1 * 10^(-1)
= 160 + 32
= 480
This is precisely the result obtained from the column-wise multiplication.
Therefore, the column-wise multiplication of two numbers can be viewed as a specific instance of the convolution operation, where the signals or sequences involved are the digits of the numbers being multiplied.
