## 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 :

*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

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

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

*inverse*DFT matrix is simply , so that Eq.(6.3) becomes

*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.

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DFT Problems

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The Length 2 DFT