# Difference Between Convolution and Multiplication [duplicate]

I read that multiplication is convolution in frequency domain. I also understand that convolution is just polynomial multiplication. Can somebody explain what are the advantages of doing convolution in frequency domain? Also, i would like to know how to multiply two signals in time domain and what is the significance of it.

## 3 Answers

Multiplication in your first sentence is term-by-term multiplication: $z[n] = x[n]y[n]$ for all $n$. 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 $N^2$ multiplications for sequences of length $N$ instead of the $N$ multiplications of the term-by-term multiplication.

The key point of Fourier analysis is that term-by-term multiplication in one domain is the same as convolution in the other domain. So, in order to calculate the results of a convolution, you can either do it directly, using $N^2$ multiplications, or transform to the other domain, do a term-by-term multiplication, and transform back. This requires two transformations to go from one domain to the other, $N$ multiplications in the other domain, and one inverse transformation to come back to the domain where the convolution result is needed. This more complicated process can, in fact, require less computation because the transformations can be done very efficiently via the Fast Fourier Transform (FFT) algorithm which requires about $N \log N$ multiplications. So, compare the computational effort in calculating the three transforms and doing the term-by-term multiplication to $N^2$ to see if the FFT gives you a more efficient method of computing a convolution.

It is significant that you can do time domain convolution via frequency domain multiplication. This can be done in less time due to existence of highly optimized Fast Fourier Transformation algorithms.

Time domain convolution has great significance in DSP at least because this way we can apply a FIR-filter to a signal.

The convolution operation can be imagined as swiping a credit card ,each instance of both the signals are multiplied with each other. In which case a bar code scanner is more like multiplication ,only two instances of each signals are multiplied .