Page 125 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

3. Numerical Linear Algebra 113

11.1 Computation of Transformation Matrix
for the Fourier Transformation

Consider that the vector space 1 is spanned by the 4 basis as given below.

1 0 0 0
v1= 0 v2= 1 v3= 0 v4= 0
0 0 1 0
0 0 0 1

Every vector in the space spanned by the basis vector given above is
mapped to the vector in vector space 2 called as Fourier space. Fourier
transformation of the basis vectors v1, v2, v3, v4 are given as

T
T(v1) = [1 1 1 1 ]

T
T(v2) = [1.0000 0 - 1.0000i -1.0000 0 + 1.0000i ]

T
T(v3) = [1 -1 1 -1]

T
T(v4) = [1.0000 0 + 1.0000i -1.0000 0 - 1.0000i ]

The basis vectors of the Fourier space is given as

T
f1 = (1/2)* [1 1 1 1 ]

T
f2 = (1/2) * [1.0000 0 - 1.0000i -1.0000 0 + 1.0000i ]

T
f3 = (1/2) * [1 -1 1 -1]

f4 = (1/2)* [1.0000 0 + 1.0000i -1.0000 0 - 1.0000i ] T

The transformed vector T(v1) is represented as the linear combination of
f1, f2, f3, f4. Note that the basis are orthonormal basis