Difference Between Convolution and Multiplication [duplicate]

1. Is multiplication the same as convolution?
2. Why convolution is used instead of multiplication?
3. Is multiplication of two sequences is same as convolution of two sequences?
4. What is the relationship between a convolution and a matrix multiplication?

Is multiplication the same as convolution?

Algebraically, convolution is the same operation as multiplying polynomials whose coefficients are the elements of u and v . w ( k ) = ∑ j u ( j ) v ( k − j + 1 ) .

Why convolution is used instead of multiplication?

Convolution, for discrete-time sequences, is equivalent to polynomial multiplication which is not the same as the term-by-term multiplication. Convolution also requires a lot more calculation: typically N2 multiplications for sequences of length N instead of the N multiplications of the term-by-term multiplication.

Is multiplication of two sequences is same as convolution of two sequences?

This property states that multiplication of two DFTs is equivalent to circular convolution of their sequences in time domain. This means multiplication of two sequences in time domain results in circular convolution of their DFTs in frequency domain.

What is the relationship between a convolution and a matrix multiplication?

convolutions can be mapped as matrix multiplication operations by flattening and rearranging the weights and input features. As illustrated in Figure 2, 64 × 3 kernels with a size of 3 × 3 are mapped to a rearranged matrix with dimensions of 64 × (3 × 3 × 3).