(contd. )

In the Fourier integrals of (1) and (2), the coordinate positions in the signal and frequency domains can now be expressed by the respective mapping functions such that,

	(3.2.6)

and

	(3.2.7)

The signal amplitudes shall not be changed by the coordinate mapping i.e.

	(3.2.8)

The spectral values must also uniquely be mapped, but may be subject to a constant scaling of amplitudes,

	(3.2.9)

Then, comparing eqn(3.2.4) ,(3.2.5) to (3.2.9) we get.

and as

we have,

For the 2D case imposes four conditions on the basis vectors:

;

It is evident that each basis vectors in spatial domain has its orthogonal counterpart in frequency domain s.t. for ;.

For ; the inner product of vectors is unity. We say that the basis systems described by and are bi-orthogonal is the dual matrix of .

(continued in the next slide)