We now have a way of computing the spectrum for an arbitrary signal: The Discrete Fourier Transform (DFT) computes the spectrum at NN equally spaced frequencies from a length- NN sequence. An issue that never arises in analog "computation," like that performed 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 needed processing. The number of steps, known as the complexity, becomes equivalent to how long the computation takes (how long must we wait for an answer). Complexity is not so much tied to specific computers or programming languages but to how many steps are required on any computer. 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 the amount demanded.

For example, consider the formula for the discrete Fourier transform. For each frequency we chose, we must multiply each signal value by a complex number and add together the results. For a real-valued signal, each real-times-complex multiplication requires two real multiplications, meaning we have 2⁢N 2N multiplications to perform. To add the results together, we must keep the real and imaginary parts separate. Adding NN numbers requires N−1 N1 additions. Consequently, each frequency requires 2⁢N+2⁢(N−1)=4⁢N−2 2N 2 N1 4N 2 basic computational steps. As we have NN frequencies, the total number of computations is N⁢(4⁢N−2) N 4N 2 .

In complexity calculations, we only worry about what happens as the data lengths increase, and take the dominant term—here the 4⁢N2 4 N2 term—as reflecting how much work is involved in making the computation. As multiplicative constants don't matter since we are making a "proportional to" evaluation, we find the DFT is an O⁢N2ON2 computational procedure. This notation is read "order NN-squared". Thus, if we double the length of the data, we would expect that the computation time to approximately quadruple.

Content actions

Add module to:

My Favorites Login Required (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own Login Required (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Footer

This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Elizabeth Gregory on Aug 4, 2004 4:41 pm -0500.