Clod-Form Discrete Fractional and Affine Fourier
Transforms
Soo-Chang Pei,Fellow,IEEE,and Jian-Jiun Ding
Abstract—The discrete fractional Fourier transform(DFRFT)
is the generalization of discrete Fourier transform.Many types
of DFRFT have been derived and are uful for signal processing
applications.In this paper,we will introduce a new type of
DFRFT,which are unitary,reversible,and flexible;in addition,
the clod-form analytic expression can be obtained.It works
in performance similar to the continuous fractional Fourier
transform(FRFT)and can be efficiently calculated by FFT.Since
the continuous FRFT can be generalized into the continuous affine
Fourier transform(AFT)(the so-called canonical transform),we
also extend the DFRFT into the discrete affine Fourier transform
(DAFT).We will derive two types of the DFRFT and DAFT.Type
1will be similar to the continuous FRFT and AFT and can be
ud for computing the continuous FRFT and AFT.Type2is the
improved form of type1and can be ud for other applications of
digital signal processing.Meanwhile,many important properties
continuous FRFT and AFT are kept in clod-form DFRFT and
DAFT,and some applications,such as the filter design and pattern
recognition,will also be discusd.The clod-form DFRFT we
introduce will have the lowest complexity among all current
DFRFT's that are still similar to the continuous FRFT.
Index Terms—Affine Fourier transform,discrete affine Fourier
transform,discrete Fourier transform,discrete fractional Fourier
transform,Fourier transform.
I.I NTRODUCTION
T HE continuous fractional Fourier transform(FRFT)[1],
[2],which is the generalization of Fourier transform,is
defined as
is constrained in the range of
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the continuous FRFT.Although,in this ca,the DFRFT
can work very similarly to the continuous ca and has a
fast algorithm,but the transform kernel will not be orthog-
onal and additive.Besides,many constraints,including
the input signal constraint,should be satisfied.
3)Linear combination-type DFRFT.In[6]–[8],and[24],the
discrete fractional Fourier transform is derived by using
the linear combination of identity operation,DFT,time
inver operation,and IDFT.In this ca,the transform
matrix is orthogonal,and the additivity property and the
reversibility property will satisfy for this type of DFRFT.
However,the main problem is that the transform results
will not match to the continuous FRFT.Besides,it will
work very similarly to the original Fourier transform or
the identity operation and lo the important character-
istic of“fractionalization.”
4)Eigenvectors decomposition-type DFRFT.In[9]–[11],
and[16],the authors derive another type of discrete frac-
tional Fourier transform by arching the eigenvectors
and eigenvalues of the DFT matrix and then compute the
fractional power of the DFT matrix.This type of DFRFT
will work very similarly to the continuous FRFT and
will also have the properties of orthogonal,additivity,
and reversibility.In[11],they have further improved this
type of DFRFT by modifying their eigenvectors more
similarly to the continuous Hermite functions,which are
the eigenfunctions of the FRFT.The types of DFRFT’s
lack the fast computation algorithm,and the eigenvectors
cannot be written in a clod form.
5)Group theory-type DFRFT.In[13],the concept of group
theory[15]is ud,and the DFRFT as the multiplication
of DFT and the periodic chirps are derived.The DFRFT
derived will satisfy the rotation property on the Wigner
distribution,and the additivity and reversible property
will also be satisfied.However,this type of DFRFT can
be derived only when the fractional order of the DFRFT
equals some specified angles,and when the number of
points
is a periodic,equally spaced
impul train,and if the number of impuls in a period
is.Besides,the
value of is limited and must be a rational number
(
.
Although many types of the discrete fractional Fourier trans-
form(DFRFT)have been derived recently,no discrete affine
Fourier transform(DAFT)has yet been derived.
In this paper,we will derive a new type of DFRFT,and then
extend it into the discrete affine Fourier transform(DAFT).The
DFRFT and DAFT we derived come from the proper sampling
of the continuous FRFT and AFT.The DFRFT introduced in
[5]is also derived from the sampling of the continuous FRFT.
Here,however,we will sample the continuous FRFT and affine
Fourier transform by some proper intervals,and therefore,the
transform matrix will be orthogonal and reversible.It can be
written in the clod form so that many properties can be de-
rived,and the fast algorithms can be achieved.Our idea comes
from the[12]and[22].In the papers,when we sample the
fractional Fourier transform properly,we will obtain an unitary
transform.We will improve upon the ideas.
In this paper,our focus is on the practical applications.
反比例函数公式Thus,although our DFRFT/DAFT em neat in concepts and
sacrifice the additivity property,they are very suitable for the
practical applications due to the simpler and clod form of
discrete fractional convolution and correlation introduced in
Section II-D and the advantages listed in Section II-E.Our
DFRFT/DAFT will have the lowest complexity among all the
current the DFRFT/DAFT's that still have the similar properties
as the continuous FRFT/AFT.
Due to the orientation of practical usage,we will derive two
types of DFRFT/DAFT.The two types of DFRFT/DAFT are
esntially the same but different in parameterizations.The first
type we derive has the parameters that are more directly linked
to the continuous FRFT/AFT and suits the applications of com-
puting the continuous FRFT/AFT.On the other hand,type2
has the simpler parameters t and allows more elegant expres-
sion for the operator kernels.It is suitable for other applications
of DFRFT/DAFT,such as the filter design,pattern recognition
(described in Section IV),and the u for the pha retrieval
discusd in[12]and[22]can also be improved by the type2
DFRFT/DAFT propod in this paper.
In Section II,we will give the derivation and definitions of
our new types of DFRFT and DAFT.For different applications,
we will u different parameterizations to define2types of
DFRFT/DAFT.In Section III,we will discuss their properties
and their transform results for some special signals.In Section
IV,we discuss their applications.Finally,in Section V,we
make a conclusion.
II.D ERIV ATION OF C LOSED-F ORM D ISCRETE F RACTIONAL
AND A FFINE F OURIER T RANSFORMS
A.The Clod-Form Discrete Fractional Fourier Transform
of Type1
1)The Derivation:To derive the DFRFT,we first sample
the input function of the
FRFT[e(1)]by the interval
(6)
where,and
.Instead,we try to make the DC component in the center.
From(6),we can convert(1)as
The above equation can be written as the form of transformation
matrix
(8)
where
(9)
in order for(8)to be reversible.We will try to make the inver
transform to be the Hermitian(conjugation and transpo)
of
(10)
Then,from(8)and
(9)
(11)
If we want the summation
for
(12)
where.In this ca,(9)
becomes
(13)
and
sgn
sgn
DFRFT of type1:
1)
when is
integer
when is
integer
(18)
Additionally,the constraints
that
and
and
.In fact,in the cas,we can
just u the following definitions:
3)
is just the forward continuous
FRFT with order.In fact,this property will also exist for
海螺做法
the DFRFT defined as(17)–(21).Since,from(11),the inver
of
(23)
that is,the DFRFT of order with the sampling
interval
with the sampling
interval
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As the continuous FRFT,the DFRFT of type1also has the
periodic property,that is
(24)
The DFRFT of type1will have the period of
is real
for the DFRFT with the
parameter and u for the DFRFT with the param-
eter(The sampling interval in both time domains is the same
and fixed).Then,from(16),we find
where
is some integer such that
and do the DFRFT defined as(17)or
(18)with the order
(29)
where
Modification form of the DFRFT of type1when
(30)
where
,where
is the length of the output.Among all types of
DFRFT,the linear combination type DFRFT[6]–[8],[24]will
have the least complexity and only require multi-
plication operations.However,it does not match the continuous
FRFT and lacks many of the characteristics of the continuous
FRFT.For example,it is hard to filter out the chirp noi with
this type of DFRFT.For most of the other types of DFRFT,such
as the improved directly sampling-type DFRFT[5],we need
multiplication operations becau there is one
convolution operation and two chirp multiplication operations
required.The DFRFT we introduce will have the lowest com-
plexity among all types of DFRFT that still work similarly to
the continuous FRFT.
3)Applications for Calculating the Continuous FRFT:We
can u the DFRFT of type1to calculate the continuous FRFT.
When using the DFRFT for this application,we first sample the
input continuous function into a discrete quence,do the for-
ward DFRFT,and get the output of DFRFT as the sampling of
the transform results of continuous FRFT.We note that becau
when we derive the DFRFT of type1,we have normalized the
unitary[from(14)to(15)].Thus,when using the DFRFT of type 1to implement the continuous FRFT,we must consider this nor-malization factor,that is,
if
and
is chon as
打坐的好处
for
(34)
where
when
)will be not required here.
Not all the input functions of the FRFT will be time limited,
as is the above experiment.If the input function has very long
time duration,we will modify the above process a little.We will
cut the input function into veral subctions with short time
duration and sample
them
(35a)
where(35b)
and input them into DFRFT.We will u the shifting property
for the continuous FRFT
[2]
is some integer.
Then,
from and the relation of(19),we e
that
where
sgn
)
from
(39)
as the input of continuous FRFT.The transform result
of
(39a)
We will
choo
,,
and
.We then u the
DFRFT to compute the transform result
of
and.We plot the result in the Fig.3.Then,
we also u(34)to compute the error and obtain
err
err
for
Fig.1.DFRFT for the rectangular function x (n )=5(n=225),i.e.,f (t )=
5(t=4:5).Upper left: =0:05.Upper right: =0:2.Lower left: =0:4,Lower right: = =4
.
Fig.2.DFRFT for the triangular function x (n )=3(n=125),i.e.,f (t )=3(t=2:5)).Upper left: =0:05.Upper right: =0:2.Lower left: =0:4.Lower right: = =4.
When we u the DFRFT to calculate the continuous FRFT,we should consider its precision.There are two main constraints
绞丝旁加乞念什么that must be satisfied for precision.First,the value
of
can be ignored
outside
if the bandwidth
of
is
(the effective
width of the input signal)increas,
then
(the bandwidth
of
is fixed,
then should be incread.
3)
When
increas,
<,
,
and
will be incread.
B.Clod-Form Discrete Affine Fourier Transform of Type 1We can u a similar way to derive the DAFT that is analogous to the continuous ca in (3).Similar to the process to derive the DFRFT,we find that
if
(41)
and
,then the transform
matrix (42)
and
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DAFT of type 1
1)
when
when
.
The reversible property for the DAFT of type 1
is
(45)
where for parame-
ters
,we also find that the
DAFT fails to be defined from (43)and (44)becau the right
side of (41)will be 0.However,
for
,the continuous affine Fourier transform results will
be
(46)
or
is the Fourier transform
of
,as follows.
1)
is an integer.(48)
