5.7 Dft: computational complexity

We now have a way of computing the spectrum for an arbitrary signal: The Discrete Fourier Transform (DFT) computes the spectrum at $N$ equally spaced frequencies from a length- $N$ sequence. An issue that never arises in analog "computation," like thatperformed by a circuit, is how much work it takes to perform the signal processing operation such as filtering. In computation,this consideration translates to the number of basic computational steps required to perform the neededprocessing. The number of steps, known as the complexity , becomes equivalent to how long the computation takes (how long must we wait for ananswer). Complexity is not so much tied to specific computers or programming languages but to how many steps are required on anycomputer. Thus, a procedure's stated complexity says that the time taken will be proportional to some function of the amount of data used in the computation and theamount demanded.

For example, consider the formula for the discrete Fourier transform. For each frequency we choose, we must multiply each signalvalue by a complex number and add together the results. For a real-valued signal, each real-times-complex multiplication requirestwo real multiplications, meaning we have $2N$ multiplications to perform. To add the results together, we must keep the real and imaginary partsseparate. Adding $N$ numbers requires $N-1$ additions. Consequently, each frequency requires $2N+2(N-1)=4N-2$ basic computational steps. As we have $N$ frequencies, the total number of computations is $N(4N-2)$ .

In complexity calculations, we only worry about what happens as the data lengths increase, and take the dominant term—here the $4N^2$ term—as reflecting how much work is involved in making the computation. As multiplicative constants don't mattersince we are making a "proportional to" evaluation, we find the DFT is an $O(N^2)$ computational procedure. This notation is read "order $N$ -squared". Thus, if we double the length of the data, we would expect that the computationtime to approximately quadruple.