Digital Signal Processing Reference
In-Depth Information
The very general formula of equation [1.11] fortunately reduces to very simple
elements in most cases.
Here are some of the most widespread examples: orthogonal transformations,
linear time-frequency representations, time scale representations, and quadratic
time-frequency representations.
Orthogonal transformations
This is the special case where m =
n
= 1,
H
[
x
]
= G
[
x
]
= x
and
ψ
(
t
,
u
)
is a set of
orthogonal functions. The undethronable Fourier transform is the most well-known
with
_2
j
π
ft
( )
ψ
tf
,
=
e
.
We may once again recall its definition here:
(
( )
∫
−
j
2
π
ft
()
()
xf
=
FTxt
=
xte
dt
with
f
∈ℜ
[1.12]
ℜ
T
() ( )
and its discrete version for a signal vector
x
=
x
0
…
x N
−
1
of length
N
:
N
−
1
kl
N
−
j
2
π
(
)
∑
[
]
()
()
()
xl
=
DFT xk
=
xke
with
l
∈
0,
N
−
1
[1.13]
k
=
0
If we bear in mind that the inner product appearing in equation [1.12] or [1.13]
is a measure of likelihood between
x
(
t
) and
_2
j
π
ft
e
, we immediately see that the
choice of this function set
( )
ψ
,
tf
is not innocent: we are looking to highlight
complex sine curves in
x
(
t
).
If we are looking to demonstrate the presence of other components in the signal,
we will select another set
( )
t
ψ . The most well-known examples are those where
we look to find components with binary or ternary values. We will then use
families such as Hadamard-Walsh or Haar (see, for example, [AHM 75]).
Linear time-frequency representations
This is the special case where
[] []
(
)
(
τ
−
)
j
2
π
ft
Hx
=
Gx
=
x
and
ψτ
t
, ,
f
=
wt
−
e
leading to an expression of the type:
() ()( )
2
∫
−
j
π
ft
Rt
τ
,
=
xt wt
−
τ
e
t
[1.14]
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