In this article, we will discuss the Numpy convolve function in Python. The convolution operator is a mathematical operator primarily used in signal processing. The convolution of two signals is defined as the integral of the first signal(reversed) sweeping over (“convolved onto”) the second signal. And multiplied (with the scalar product) at each position of overlapping vectors. An array in numpy is a signal. Thus the numpy convolve function performs convolutions over single dimensional arrays.
Contents
Syntax of Numpy convolve
numpy.convolve
(a, v, mode='full')
Parameters:
- a – First one-dimensional input array(N).
- v – Second one-dimensional input array(M).
- mode – {‘full,’ ‘valid,’ ‘same’} (Optional parameter) ‘full’: Mode is ‘full’ by default. This returns the convolution at each point of overlap, with an output shape (N+M-1). At the end-points of the convolution, the signals do not overlap completely, and boundary effects may be seen. ‘same’: returns the output of length—max (M, N). Boundary effects are still visible. ‘valid’: returns the output of length.
max(M, N) - min(M, N) + 1
. The convolution product is only given for points where the signals overlap completely.
Return value of numpy convolve
Returns the discrete, linear convolution of two one-dimensional arrays i.e, of ‘a’ and ‘v’.
Numpy convolve in Python when mode is ‘full’
import numpy as np
a = np.array([3, 7])
print("First vector sequence is: ", a)
v = np.array([1, 2, 5, 7])
print("Second vector sequence is: ", v)
print("\nprinting linear convolution result between v1 and v2 using default 'full' mode:")
print(np.convolve(a, v))
Output:
First vector sequence is: [3 7]
Second vector sequence is: [1 2 5 7]
printing linear convolution result between v1 and v2 using default 'full' mode:
[ 3 13 29 56 49]
Explanation:
In the above example, we see the most basic use of Numpy convolve() method. Firstly, we define two single dimensional arrays as ‘a’ and ‘v’ using the numpy.array() function.
Then, we pass ‘a’ and ‘v’ as parameters to the convolve function. Since the mode is not mentioned, it takes the default value i.e., ‘full’. And calculates the convolution for the default value of mode as a= [3 7] and v= [1 2 5 7] and operation is performed in full mode. Hence, the shape of the output array is: length (M+N-1) here M=2 and N = 4. Therefore the shape of the resultant vector will be 2 + 4 – 1 = 5. ‘a’ is reversed from [3 7] to [7 3], and then we perform the multiplication operation as:
First element: 7 * undefined (extrapolated as 0) + 3 * 1 = 3
Second element: 7*1 + 3*2 = 13
Third element: 7*2 + 3*5 = 29
Fourth element: 7*5 + 3*7 = 56
Fifth element is: 7*7 + 3 * undefined (extrapolated as 0) = 49
Hence, the result is: [3 13 29 56 49]
Numpy convolve when mode is ‘same’
import numpy as np
a = np.array([3, 7])
print("First vector sequence is: ", a)
v = np.array([1, 2, 5, 7])
print("Second vector sequence is: ", v)
print("\nprinting linear convolution result between a and v using 'same' mode:")
print(np.convolve(a, v, mode='same'))
Output:
First vector sequence is: [3 7]
Second vector sequence is: [1 2 5 7]
printing linear convolution result between a and v using 'same' mode:
[ 3 13 29 56]
Explanation:
Firstly, we define two single-dimensional arrays as ‘a’ and ‘v’ using the numpy.array() function. Then, we pass ‘a’ and ‘v’ as parameters to the convolve function. The optional parameter mode(which by default is ‘full’) is set to ‘same’. Thus the Numpy convolve function with mode = ‘same’ computes the convolution as a= [3 7] and v= [1 2 5 7] and operation is performed in the same mode.
The shape of the output array is max(M, N) here M=2 and N = 4. Therefore the shape of the resultant vector will be 4. ‘a’ is reversed from [3 7] to [7 3], and then we perform the multiplication operation as: (does not calculate convolution for the non overlapping points at the end of the array, hence the the fifth element is not calculated)
- First element: 7 * undefined (extrapolated as 0) + 3 * 1 = 3
- Second element: 7*1 + 3*2 = 13
- Third element: 7*2 + 3*5 = 29
- Fourth element: 7*5 + 3*7 = 56
- Hence, the result is: [3 13 29 56].
Convolution when mode is ‘valid’
import numpy as np
a = np.array([3, 7])
print("First vector sequence is: ", a)
v = np.array([1, 2, 5, 7])
print("Second vector sequence is: ", v)
print("\nprinting linear convolution result between a and v using 'valid' mode:")
print(np.convolve(a, v, mode='valid'))
Output:
First vector sequence is: [3 7]
Second vector sequence is: [1 2 5 7]
printing linear convolution result between a and v using 'valid' mode:
[ 3 13 29 56 ]
Explanation:
Just like the above two cases, first, we defined the arrays. numpy.convolve() function uses these two arrays as arguments with mode set to ‘valid’. This computes the convolution as : a= [3 7] and v= [1 2 5 7] and operation valid mode.
The shape of the output array will be given by the formula length max(M, N) – min(M, N) + 1, here M=2 and N = 4, therefore the shape of the resultant vector will be: 4 – 2 + 1 = 3. ‘a’ is reversed from [3 7] to [7 3], and then we perform the multiplication operation as ( convolution product is given only for overlapping values. Since they do not overlap while calculating the first and fifth element, hence they are not given as the output)
Second element: 7*1 + 3*2 = 13
Third element: 7*2 + 3*5 = 29
Fourth element: 7*5 + 3*7 = 56
Hence, the result is: [3 13 29 ]
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Conclusion
In this article we have explicitly discussed about the Numpy convolve function in Python. We have also provided examples with detailed explanations for different modes while computing convolutions of one dimensional arrays. Refer to this article to have a clear knowledge of how the Numpy Convolve works.
However, if you have any doubts or questions do let me know in the comment section below. I will try to help you as soon as possible.
Happy Pythoning!