Matrix Review
Recall:
-
Vectors in
ℝN
N
:
∀xi,xi∈ℝ:x=x0x1…x
N
-
1
xi
xi
x
x0
x1
…
x
N
-
1
-
Vectors in
ℂN
N
:
∀xi,xi∈ℂ:x=x0x1…x
N
-
1
xi
xi
x
x0
x1
…
x
N
-
1
-
Transposition:
-
transpose:
xT=x0x1…x
N
-
1
x
x0
x1
…
x
N
-
1
-
conjugate:
xH=x0¯x1¯…x
N
-
1
¯
x
x0
x1
…
x
N
-
1
-
Inner product:
-
real:
xTy=∑i=0N-1xiyi
x
y
i
0
N
1
xi
yi
-
complex:
xHy=∑i=0N-1xn¯yn
x
y
i
0
N
1
xn
yn
-
Matrix Multiplication:
Ax=
a
0
0
a
0
1
…a
0
,
N
-
1
a
1
0
a
1
1
…a
1
,
N
-
1
⋮⋮…⋮a
N
-
1
,
0
a
N
-
1
,
1
…a
N
-
1
,
N
-
1
x0x1…x
N
-
1
=y0y1…y
N
-
1
A
x
a
0
0
a
0
1
…
a
0
,
N
-
1
a
1
0
a
1
1
…
a
1
,
N
-
1
⋮
⋮
…
⋮
a
N
-
1
,
0
a
N
-
1
,
1
…
a
N
-
1
,
N
-
1
x0
x1
…
x
N
-
1
y0
y1
…
y
N
-
1
yk=∑n=0N-1a
k
n
xn
yk
n
0
N
1
a
k
n
xn
-
Matrix Transposition:
AT=
a
0
0
a
1
0
…a
N
-
1
,
0
a
0
1
a
1
1
…a
N
-
1
,
1
⋮⋮…⋮a
0
,
N
-
1
a
1
,
N
-
1
…a
N
-
1
,
N
-
1
A
a
0
0
a
1
0
…
a
N
-
1
,
0
a
0
1
a
1
1
…
a
N
-
1
,
1
⋮
⋮
…
⋮
a
0
,
N
-
1
a
1
,
N
-
1
…
a
N
-
1
,
N
-
1
Matrix transposition involved simply swapping the rows
with columns.
AH=AT¯
A
A
The above equation is Hermitian transpose.
ATkn=Ank
A
k
n
A
n
k
AHkn=A¯nk
A
k
n
A
n
k
Representing DFT as Matrix Operation
Now let's represent the
DFT in vector-matrix notation.
x=x0x1…xN-1
x
x
0
x
1
…
x
N
1
X=X0X1…XN-1∈ℂN
X
X
0
X
1
…
X
N
1
N
Here
xx is the
vector of time samples and
XX is the vector of DFT
coefficients. How are
xx and
XX related:
Xk=∑n=0N-1xnⅇ-ⅈ2πNkn
X
k
n
0
N
1
x
n
2
N
k
n
where
a
k
n
=ⅇ-ⅈ2πNkn=WNkn
a
k
n
2
N
k
n
WN
k
n
so
X=Wx
X
W
x
where
XX is the DFT
vector,
WW is the
matrix and
xx the
time domain vector.
Wkn=ⅇ-ⅈ2πNkn
W
k
n
2
N
k
n
X=Wx0x1…xN-1
X
W
x
0
x
1
…
x
N
1
IDFT:
xn=1N∑k=0N-1Xkⅇⅈ2πNnk
x
n
1
N
k
0
N
1
X
k
2
N
n
k
where
ⅇⅈ2πNnk=WNnk¯
2
N
n
k
WN
n
k
WNnk¯
WN
n
k
is the matrix Hermitian transpose. So,
x=1NWHX
x
1
N
W
X
where
xx is the time
vector,
1NWH
1
N
W
is the inverse DFT matrix, and
XX is the DFT vector.
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