Popular Algorithms for Fast convolution are

• Fast Fourier Transform (FFT) algorithm
• Karatsuba algorithm

Fast Fourier Transform (FFT) algorithm:

• This algorithm uses the properties of complex numbers and trigonometric functions to convert the convolution operation into a point-wise multiplication operation in the frequency domain. 
• This greatly reduces the computational complexity of the operation and makes it possible to perform convolutions of large sequences in a relatively short amount of time. 
• However, the FFT algorithm can be difficult to implement and may not be suitable for all types of problems.

Karatsuba algorithm:

• This algorithm is based on a divide-and-conquer approach and is often used to perform the multiplication of large integers. 
• The Karatsuba algorithm can also be used for convolution by treating the input sequences as two large integers and then applying the multiplication algorithm to the sequences. 
• The Karatsuba algorithm is relatively simple to implement and is often used as a fallback option when other algorithms are not suitable.

Karatsuba vs FFT algorithm:

• The Karatsuba algorithm is an efficient algorithm for multiplying large integers. It reduces the number of multiplications required by breaking the integers into smaller chunks and using a recursive approach.
• FFT (Fast Fourier Transform) is an efficient algorithm for calculating the discrete Fourier transform of a signal. It is widely used in signal processing and other fields to analyze signals and images.
• In terms of performance, FFT is generally considered to be faster than Karatsuba for large inputs. FFT algorithms take advantage of the symmetry and periodicity of the input signal to reduce the number of calculations required. However, the Karatsuba algorithm is more efficient for small inputs.

Convolution is a mathematical operation used in signal processing, image processing, and other fields to combine two functions in order to produce a third function. It is defined as the integral of the product of two functions, one of which is flipped and shifted over time. It is often represented using the symbol “*” and is useful for filtering, smoothing, and other operations on signals and images.