Matrix Equation for the DTFS
2.5
2002/07/31
2003/07/24 11:15:06.858 GMT-5
Roy
Ha
rha@rice.edu
Prashant
Singh
prash@ece.rice.edu
Mariyah
Poonawala
mariyah@rice.edu
matrix equation
dtfs
fourier
fourier series
discrete-time
discrete time fourier series
standard basis
basis
This module looks at writing our DTFS equations in matrix form to make calculations and displaying the basis easier.
The DTFS
is just a change of basis in
N
.
To start, we have
f
n
in terms of the standard basis.
f
n
f
0
e
0
f
1
e
1
…
f
N
1
e
N
-
1
k
0
n
1
f
k
δ
k
n
f
0
f
1
f
2
⋮
f
N
1
f
0
0
0
⋮
0
0
f
1
0
⋮
0
0
0
f
2
⋮
0
…
0
0
0
⋮
f
N
1
Taking the DTFS, we can write
f
n
in terms of the sinusoidal Fourier basis
f
n
k
0
N
1
c
k
2
N
k
n
f
0
f
1
f
2
⋮
f
N
1
c
0
1
1
1
⋮
1
c
1
1
2
N
4
N
⋮
2
N
N
1
c
2
1
4
N
8
N
⋮
4
N
N
1
…
We can form the basis matrix (we'll call it
W here instead of
B) by stacking the basis vectors
in as columns
W
b
0
n
b
1
n
…
b
N
-
1
n
1
1
1
…
1
1
2
N
4
N
…
2
N
N
1
1
4
N
8
N
…
2
N
2
N
1
⋮
⋮
⋮
⋮
⋮
1
2
N
N
1
2
N
2
N
1
…
2
N
N
1
N
1
with
b
k
n
2
N
k
n
the entry in the k-th row and n-th column is
W
j
,
k
2
N
k
n
W
n
,
k
So, here we have an additional symmetry
W
W
W
W
1
N
W
(since
b
k
n
are orthogonal)
We can now rewrite the DTFS equations in matrix form where we
have:
-
f
= signal (vector in
N
)
-
c
= DTFS coeffs. (vector in
N
)
"synthesis"
f
W
c
f
n
c
b
n
"analysis"
c
W
f
W
f
c
k
f
b
k
Finding (and inverting) the DTFS is just matrix
multiplication.
Everything in
N
is clean: no limits, no convergence questions, just
good ole matrix arithmetic.