Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 34

≡ √ − with j 1. All x ( n )and y ( k ) can be concatenated in the form of

two N × 1 vectors. Let us also define

e −j 2 N

W N ≡ (2.10)

such that equations (2.8) and (2.9) can be written in the matrix form

y = W − 1 x ,

x = Wy (2.11)

with

⎡
⎣

⎤
⎦

11 1 ··· 1

W N−1
N

W N

1 W N ···

W = (2.12)

. . . . .

W 2(N−1)
N

W (N−1)(N−1)
N

W N−1
N

1 ···

where W is an unitary and symmetric matrix.
Let us choose as an example the case N =2.
Example 2.2:

We then obtain for N =2

W = 11
1 − 1

We see that the columns of W correspond to the basis vectors

w 0 =[1 , 1] T

1] T

w 1 =[1 , −

and, based on them, we can reconstruct the original signal:

1

y ( i ) w i

x =

i=0

Unfortunately, the DFT has the same drawbacks as the continuous-time
Fourier transform when it comes to nonstationary signals: (a) the be-
havior of a signal within a given window is analyzed; (b) accurate repre-
sentation is possible only for signals stationary within a window; and (c)
good time and frequency resolution cannot be achieved simultaneously,
as illustrated by table 2.1.