Matrix Formulation of the DFT

The DFT can be formulated as a complex matrix multiply, as we show in this section. (This section can be omitted without affecting what follows.) For basic definitions regarding matrices, see Appendix H.

The DFT consists of inner products of the input signal with sampled complex sinusoidal sections : By collecting the DFT output samples into a column vector, we have or where denotes the DFT matrix , i.e., The notation denotes the Hermitian transpose of the complex matrix (transposition and complex conjugation).

Note that the th column of is the th DFT sinusoid, so that the th row of the DFT matrix is the complex-conjugate of the th DFT sinusoid. Therefore, multiplying the DFT matrix times a signal vector produces a column-vector in which the th element is the inner product of the th DFT sinusoid with , or , as expected.

Computation of the DFT matrix in Matlab is illustrated in §I.4.3.

The inverse DFT matrix is simply . That is, we can perform the inverse DFT operation as (6.2)

Since the forward DFT is , substituting from Eq. (6.2) into the forward DFT leads quickly to the conclusion that (6.3)

This equation succinctly states that the columns of are orthogonal, which, of course, we already knew. I.e., for , and : The normalized DFT matrix is given by and the corresponding normalized inverse DFT matrix is simply , so that Eq. (6.3) becomes This implies that the columns of are orthonormal. Such a complex matrix is said to be unitary.

When a real matrix satisfies , then is said to be orthogonal. Unitary'' is the generalization of orthogonal'' to complex matrices.

Next Section:
DFT Problems
Previous Section:
The Length 2 DFT