Convolution (convolve) in SciPy - Step-by-Step Execution

Concept Flow - Convolution (convolve)

Start with two arrays

↓

Flip second array

↓

Slide flipped array over first

↓

Multiply overlapping elements

↓

Sum the products

↓

Store sum in output array

↓

Move to next position

Repeat until all positions done↓

End with convolved array

↩Repeat until all positions done

Convolution slides one array over another, multiplies overlapping elements, sums them, and stores the result step-by-step.

Execution Sample

SciPy

import numpy as np
from scipy.signal import convolve

x = np.array([1, 2, 3])
k = np.array([0.5, 1, 0])
result = convolve(x, k, mode='full')

This code convolves two small arrays and stores the full convolution result.

Execution Table

Step	Position	Overlapping Elements (x)	Overlapping Elements (k flipped)	Products	Sum	Output Array So Far
1	0	[1]	[0.5]	[0.5]	0.5	[0.5]
2	1	[1, 2]	[1, 0.5]	[1*1=1, 2*0.5=1]	2	[0.5, 2]
3	2	[1, 2, 3]	[0, 1, 0.5]	[1*0=0, 2*1=2, 3*0.5=1.5]	3.5	[0.5, 2, 3.5]
4	3	[2, 3]	[0, 1]	[2*0=0, 3*1=3]	3	[0.5, 2, 3.5, 3]
5	4	[3]	[0]	[3*0=0]	0	[0.5, 2, 3.5, 3, 0]

💡 All positions covered; convolution complete with output length = len(x)+len(k)-1 = 5

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
Position	N/A	0	1	2	3	4	End
Output Array	[]	[0.5]	[0.5, 2]	[0.5, 2, 3.5]	[0.5, 2, 3.5, 3]	[0.5, 2, 3.5, 3, 0]	[0.5, 2, 3.5, 3, 0]

Key Moments - 3 Insights

Why do we flip the second array before sliding it?

Flipping the second array (kernel) aligns it properly for convolution, as shown in the 'Overlapping Elements (k flipped)' column in the execution_table. Without flipping, the operation would be correlation, not convolution.

Why does the output array length equal len(x) + len(k) - 1?

Each position where the flipped kernel overlaps at least one element of x produces one output value. The execution_table shows 5 steps for arrays of lengths 3 and 3, confirming output length 5.

Why are some overlapping elements zero in the products?

At the edges, the kernel extends beyond the input array, so missing elements are treated as zero. This is why products like 3*0=0 appear in the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at Step 3. What is the sum of products computed?

A 3.5

B 2.5

C 4.0

D 3.0

💡 Hint

Check the 'Sum' column in the execution_table row for Step 3.

At which position does the output array first contain the value 2?

A Position 0

B Position 1

C Position 2

D Position 3

💡 Hint

Look at the 'Output Array So Far' column in the execution_table after Step 2.

If the kernel array k was not flipped, what would happen to the output?

A Output would be the same

B Output length would increase

C Output would represent correlation, not convolution

D Output would be all zeros

💡 Hint

Refer to the key_moments explanation about flipping the kernel.

Concept Snapshot

Convolution slides a flipped kernel over data,
multiplies overlapping elements, sums them,
and stores results in an output array.
Output length = len(data) + len(kernel) - 1.
Edges use zero-padding implicitly.
Use scipy.signal.convolve for easy computation.

Full Transcript

Convolution combines two arrays by flipping the second array, sliding it over the first, multiplying overlapping elements, and summing these products to form a new array. This process repeats for each position where the kernel overlaps the input. The output length is the sum of input lengths minus one. At edges, missing elements are treated as zero. This example uses scipy.signal.convolve to perform convolution on two small arrays, showing each step's overlapping elements, products, sums, and output array growth.