89
Wavelet Transform
where Ψ(t) is the continuous mother wavelet, 0 ≤ j ≤ N − 1, and 0 ≤ k ≤ M − 1. Then,
the coefficients of the DWT are calculated as
+∞
W jk =
∫
x t
( )Ψ * jk ( )
t dt
(5.7)
−∞
The aforementioned analysis equation calculates a finite set of discrete coefficients
directly from a continuous signal. This makes the DWT somewhat different from the
DFT that accepts only discrete signals as its input. The beauty of the DWT becomes
clearer from the synthesis equation in the following:
N −1 M −1
x t
( ) = c
W jk Ψ jk ( )
t
(5.8)
∑∑
j =0 k =0
In this equation, c is a constant that depends on the exact choice of the mother wavelet.
The interesting thing about this equation is the fact that we can reconstruct the continuous
signal directly from a set of discrete coefficients. This capability makes the DWT and the
IDWT particularly interesting and useful for the applications where a continuous signal
must be decomposed to and reconstructed from a finite set of discrete values. A closer
look identifies the DWT as the equivalent of the Fourier series as opposed to the DFT.
A relevant question at this point is how to choose the number of basis functions
for a given signal. Specifically, how many shifted and scaled versions of the mother
wavelet are needed to decompose a signal. We start this discussion by describing the
difference between a frame and a basis. A frame is a set of basis functions that can be
used to decompose a signal. This set can be minimal or nonminimal, i.e., if the number
of basis functions in the frame is minimal and any other frame would need the same
number or more basis functions, the frame is called a basis. From the definitions given
earlier, in order to minimize the number of basis functions and therefore the required
computations in calculating the DWT and the IDWT, one would like to use a basis for
the operation as opposed to a nonminimal frame. To see the differences between a
frame and a basis more clearly, consider the energy of a signal x(t):
+∞
E x = x t
( )
2 dt
(5.9)
∫
−∞
Now consider a frame formed based on the functions Ψ jk (t) defined earlier. For
such a frame, it can be proved that there exist some bounded positive values A and
B such that
N −1 M −1
A E x
. ≤
B E x
(5.10)
2 ≤ .
W j k
∑∑
j =0 k =0
Wavelet Transform
where Ψ(t) is the continuous mother wavelet, 0 ≤ j ≤ N − 1, and 0 ≤ k ≤ M − 1. Then,
the coefficients of the DWT are calculated as
+∞
W jk =
∫
x t
( )Ψ * jk ( )
t dt
(5.7)
−∞
The aforementioned analysis equation calculates a finite set of discrete coefficients
directly from a continuous signal. This makes the DWT somewhat different from the
DFT that accepts only discrete signals as its input. The beauty of the DWT becomes
clearer from the synthesis equation in the following:
N −1 M −1
x t
( ) = c
W jk Ψ jk ( )
t
(5.8)
∑∑
j =0 k =0
In this equation, c is a constant that depends on the exact choice of the mother wavelet.
The interesting thing about this equation is the fact that we can reconstruct the continuous
signal directly from a set of discrete coefficients. This capability makes the DWT and the
IDWT particularly interesting and useful for the applications where a continuous signal
must be decomposed to and reconstructed from a finite set of discrete values. A closer
look identifies the DWT as the equivalent of the Fourier series as opposed to the DFT.
A relevant question at this point is how to choose the number of basis functions
for a given signal. Specifically, how many shifted and scaled versions of the mother
wavelet are needed to decompose a signal. We start this discussion by describing the
difference between a frame and a basis. A frame is a set of basis functions that can be
used to decompose a signal. This set can be minimal or nonminimal, i.e., if the number
of basis functions in the frame is minimal and any other frame would need the same
number or more basis functions, the frame is called a basis. From the definitions given
earlier, in order to minimize the number of basis functions and therefore the required
computations in calculating the DWT and the IDWT, one would like to use a basis for
the operation as opposed to a nonminimal frame. To see the differences between a
frame and a basis more clearly, consider the energy of a signal x(t):
+∞
E x = x t
( )
2 dt
(5.9)
∫
−∞
Now consider a frame formed based on the functions Ψ jk (t) defined earlier. For
such a frame, it can be proved that there exist some bounded positive values A and
B such that
N −1 M −1
A E x
. ≤
B E x
(5.10)
2 ≤ .
W j k
∑∑
j =0 k =0