Definition
An N-point DFT is expressed as an N-by-N matrix multiplication as, where is the original input signal, and is the DFT of the signal.
The transformation of size can be defined as, or equivalently:
where is a primitive th root of unity in which . This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum and the sign of the exponent in ω are merely conventions, and differ in some treatments. All of the following discussion applies regardless of the convention, with at most minor adjustments. The only important thing is that the forward and inverse transforms have opposite-sign exponents, and that the product of their normalization factors be 1/N. However, the choice here makes the resulting DFT matrix unitary, which is convenient in many circumstances.
Fast Fourier Transform algorithms utilize the symmetries of the matrix to reduce the time of multiplying a vector by this matrix, from the usual . Similar techniques can be applied for multiplications by matrices such as Hadamard matrix and the Walsh matrix.
Read more about this topic: DFT Matrix
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (17721834)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than thisdevoted and obedient. This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (18201910)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)