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UDC 621.372 Articles
doi:10.31799/1684-8853-2021-1-55-65
Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples
O. V. Ponomarevaa, Dr. Sc., Tech., Professor, orcid.org/0000-0002-7311-3108, ponva@mail.ru
A. V. Ponomareva, PhD, Econ., Associate Professor, orcid.org/0000-0002-3746-9289
aKalashnikov Izhevsk State Technical University, 7, Studencheskaya St., 426069, Izhevsk, Russian Federation
Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.
Keywords — discrete Fourier transform, two-dimensional signal, Fourier processing, effects of discrete Fourier transform, basis, variable parameter.
For citation: Ponomareva O. V., Ponomarev A. V. Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2021, no. 1, pp. 55-65. doi: 10.31799/1684-8853-2021-1-55-65
Introduction
Fourier-processing of finite discrete two-dimensional (FDTD) signals (including images) in informational control (IC) systems is the most important method for studying processes and analyzing information [1-8]. The theoretical basis of Fourier-processing of FDTD signals is two-dimensional direct and inverse discrete Fourier transforms (2D DFT, 2D IDFT) [9-15] which can be represented in form of:
— algebraic form 2D DFT
SN1,N2 (k1, k2 ) =
1 N1 -1N2-1
=nN-: I>("I- »2wN:1 ■ <r.
N1N2 n1=o n2=0
where Sn1 ,n2 (k, ^ ) are coefficients (bins) 2D
DFT; k1 = 0, (N1 -1), k2 = 0, (N2 -1) are spatial frequencies; x(nv n2) is 2D signal; »1 = 0, N1 -1,
»2 = 0, N2 -1; wN» = exp[- jf k» j; WN» = = eXP (- f k2»2
— algebraic form 2D IDFT
n1 -1 n2-1
xin-1, m) = l l sn1,n2(ki, h)w-nkni •wnk2n2. k1=0 k2=0
The practice of using DFT and 2D DFT, on the one hand, confirmed their efficiency, on the other hand, revealed a number of effects: aliasing effect, scalloping effect, picket fence effect, negatively affecting on the results of analysis and information processing [16-21].
To eliminate these negative effects of DFT and 2D DFT, the zero-padding operation (ZPO) the FDTD signal has found wide application. ZPO can significantly reduce the impact of negative effects on the results of Fourier-processing [22, 23]. However, effective use of ZPO requires solving the problem of Fourier-processing of FDTD of this kind of signals. The essence of the problem lies in the fact that in Fourier-processing of signals subjected to ZPO, on the one hand, it is necessary to provide the corresponding IC systems with a significant amount of additional memory, on the other hand, the IC systems must perform unproductive computations with zero samples, which significantly increases time of Fourier-processing. The paper pro-
poses and investigates new discrete Fourier transforms, which allow efficient and effective analysis and processing of two-dimensional signals padded with zero samples.
The role of the zero-padding operation of FDTD signals in two-dimensional Fourier-processing
The systems analysis of Fourier-processing theory of FDTD signals made it possible to formulate its axiomatic basic provisions:
— determination of FDTD signals on a finite two-dimensional reference plane, which is interpreted as a two-dimensional fundamental period
SA
N1 xN2
(2D period). 2D period is set by horizontal
and vertical periods;
— determination of the shift of a two-dimensional discrete signal in the form of a cyclic shift, carried out by cyclic permutation of its samples on the final reference area SANi xn2;
— definition of a complete two-dimensional basis system
defn ,n2 (ki, щ, k2, п2 ) = w^1
W,
k2n2
n2
where ni = 0,Ni -1; n2 = 0,N2 -1; ki = 0,(Ni -1);
k2 = 0,(N2 -1).
As a result of the discreteness and periodicity of 2D signals in the spatial domain, the periodicity and discreteness of 2D Fourier spectra in the spatial-frequency domain, the mathematical operations of convolution and correlation are cyclical. However, the analysis, design, and modeling of iso-planatic systems requires the results of linear operations with 2D signals.
The method, which allows obtaining the results of linear operations using cyclic operations, consists in expanding the reference regions with zero samplex of the convoluted signals by applying ZPO to them.
If the reference area SAV xy of signal x(nx, n2) and the reference area SAqi xq2 of signal y(n1, n2) are specified, then the size of the reference area, padded with zeros to obtain linear convolution
hlinear(nV n2^ should be
SA(Vl +Qi )X(V2+Q2)'
where n1 = 0, (V1 + Q1 -1); n2 = 0, (V2 + Q2 -1).
And the size of the reference area for obtaining linear correlation CL(n1, n2) should be
SA
2vi x2V2 ,
where ni = 0, (2Vi -1); n2 = 0, (2V2 -1).
Therefore, the algorithm for obtaining 2D linear convolution based on 2D cyclic convolution consists of the following operations:
1. Pad 2D signals x(n1, n2) and y(n1, n2) with Q1, Q2 and F1, V2 zero samples respectively, which sets new 2D signals x0(n1, n2), y0(n1, n2) with horizontal N2 and vertical N1 periods according to the ratios
N1 > (V1 + Q1 -1); N2 > (V2 + Q2 -1).
2. Perform 2D DFT of 2D signals x0(n1, n2) and yo(np n2):
F
x0 (n1, n2 )-* X0,N ,N2 (k1, k2 );
F
V0 (n1, n2 )-> Yo,N1 ,N2 (k1, k2 ),
F
where-> is the 2D DFT execution symbol.
3. Perform 2D IDFT product
X0, N1,N2 (k1, k2 ) •Y0,N1,N2 (k1, k2 ).
The algorithm for obtaining a linear 2D correlation function based on a cyclic 2D correlation function is easy to obtain from the previous algorithm. Fig. 1, a and b illustrates the differences between cyclic CC(n1, n2) and linear CL(n1, n2) correlation functions of a finite unit 2D signal.
According to the two-dimensional version of the Wiener — Khinchin theorem, Fourier transform of the linear 2D correlation function allows one to obtain the energy spectrum of a 2D signal. There is a so-called direct method for obtaining the energy spectrum of a 2D signal x(n1, n2), bypassing the stage of obtaining the correlation function:
I I2
GN1,N2 ^ k2 ) = N1N2 |SN1,N2 (k1, k2 ) .
A significant drawback of this definition of the energy spectrum of a 2D signal x(n1, n2) is insufficient detailing Gn1,n2 (k1, k2), for example, to fulfill the conditions of Pugachev canonical signal decomposition. The method of increasing the detail GN N2 (k1, k2) is carried out by padding the 2D signal x(n1, n2) with zeros at least twice. Fig. 2, a and b illustrates the detailing of the energy spectrum of a finite single 2D signal.
As noted in the introduction, the effective application of the ZPO requires a solution to the problem of Fourier-processing of FDTD signals padded with zero samples. The essence of the problem lies in the fact that in Fourier-processing of signals subjected to the ZPO, on the one hand, it is necessary to provide the corresponding IC system with a significant additional amount of RAM (storage), on the other hand, IC system must perform a lot of non-productive calculations with zero samples, which significantly increases the time of Fourier-processing.
■ Fig. 1. Cyclic (a) and linear (b) correlation 2D functions of a finite single 2D signal
■ Fig. 2. Energy spectrum of a finite single 2D signal with zero frequency in the center of the spectrum: a — a finite single 2D signal N1 = 8, N2 = 8; b — a finite single 2D signal, padded with zero samples to N1 = 32, N2 = 32
Let us consider a generalization of 2D DFT in the form of a 2D DFT with a variable parameter, which makes it possible to efficiently analyze and process two-dimensional signals subjected to ZPO.
Two-dimensional DFT with variable parameter
Let two 2D signals be given: a signal Xn1 xn2 and a signal On1xn2 with zero samples. To perform the linear transformations considered in the previous section, it is necessary to pad (supplement) the horizontal period of the 2D signal Xn1 xn2 with (r2 - 1) zero matrices On1 xn2 , which leads to a block matrix:
X,
0
[X N^Nz
(Г2 - 1)
O,
O,
J
(1)
Taking into account the separability property of the 2D DFT, Fourier transform of a signal XN1X(N2r2) in matrix form can be represented as
:>h>k2
7(2)
n1x(n 2^2) n1n2 n1xn1
X
F(1)
n1 x(n2r2) (n2^2 )x(n2v2 )
(2)
where
F(2)
F N1 xN1
(N1 -1)
W,
0
0-0
N,
W
1-0
N,
W
0-1
N,
W
1-1
N1
w(n!-1)-0 W(N! -1)-1
N1 N1
(N1 -1)
0-(N!-1)
W
N1
W
1-(Nj -1)
N1
W
n1
•(N1 -1)(Ni -1)
N1
X
N!x(N2r2 )
(Ni - 2) (Ni -1)
0
x(0, 0) x(1, 0)
x((Ni - 2), 0) x((Ni -1), 0)
(N2 -1) x(0, (N2 -1)) x(1, (N2 -1))
N2 0 0
x((N1 - 2), (N2 -1)) 0 x((N1 -1), (N2 -1)) 0
(N2 Г2 - 1) П2 0 0
F«
r(N2r2 )x(N2r2 )
(N2^2 -1) П2
0
W 00
WN2
1 • 0
W
n2
W
N2
W 01
WN2
W
1 • 1
N,
(N2r2 -1) • 0 W(N2r2 -1) • 1
N2
(N2^2 -1) 0-(N2r2-1)
W
N2
W
1-(N2r2-1)
N2
W
(N2r2 -1)(N2r2-1) J
N2
(3)
Let us interrogate the structure of matrix equation (2). It is easy to see that the multiplication of matrices
X
N1x(N2r2 )
and F
<1)
(N2r2 )x(N2r2 )
leads to a rectangular matrix
N1x(N2r2)• A matrix Y^x^) can be interpreted
r(1)
as the product of a matrix Xf1Xf2 by a matrix Ff x(f r )'
where
Y — X F(1)
^x^^) XN1xN2 FN2x(N2r2Г
X
N1xN 2
(N1 - 2) (N1 -1) n1
0
x(0, 0) x(1, 0)
x(0, 1) x(1, 1)
x((N1 - 2), 0) x((N - 2), 1) x((N1 -1), 0) x((N1 -1), 1)
F(1)
FN2 x( N2V2 )
(N2 - 1)
n2
0
W
0- 0
n2
W
0-1
n2
W
1 • 0
N2
W
1 1
N2
W (N2-1)^ 0 W (N2-1)^ 1
W N2 WN2
(N2 -1) x(0, (N2 -1)) x(1, (N2 -1))
x((N1 - 2), (N2 -1)) x((N1 -1), (N2 -1))
(N2^2 -1)
0^ ( N2^2-1)
"2
W
k2
N2
W
1 ( N2^2-1)
N2
W
(N2-1)(N2 Г2 -1) J
n2
(4)
(5)
Comparing matrices FN2r2)x(N2r2) (3) and FNv2x(N2r2) (5)' we may see that the matrix FN2*(N2r2) iS the result
T(1)
7(1)
of truncating N2(r2 - 1) the rows of the matrix F(
.(1)
column numbers by A...:
(N2r2 HN2r2 )
According to [22], "we denote the set of matrix
A : A — {0, 1, 2, ..., (N2^1 -1)}.
n
1
k
2
We apply to the set A the relation of comparability modulo r2.
It is known that the relation of comparability in modulus m is an equivalence relation and has the properties of reflexivity, symmetry and transitivity."
Also we know that "the relation of comparability modulo r2 divides the set A into r2 classes of residues modulo r2:
A0 ={0, r2, ..., (N2-1)r2};
-1) ={2 -1), ..., (N2^2 -1)}; r -1
Ai А^ П Aj = 0; U Ai = A.
1Ф j i=0
(6)
The matrix F'
.(1)
N2x(N2r2)
, applying the partition (6) of the set A into r2 residue classes modulo r2, can be rep-
resented in the form of r2 matrices of size N2 x N2" [22]:
F(1)
fn2xn2,e2
(N2 -1) П2
W,
W,
0
0-(0+62) n2 1-(0+02)
N2
W
0-(1+e2)
W
N2
1-(1+e2)
N2
W
(N-1)(0+e2) W(n2-1)(1+e2)
N,
N,
W
W
(N2 -1)
0-(n 2-1+e2) k2 N2
1'(n2-1+e2)
N2
W
(N2-1)( N 2-1+e2)
N2
(7)
where 82 = 0; 1/r2, ..., (^ - 1)/r2.
Discrete two-dimensional exponential functions of the form
defffP,N1,N2 (k1, "1, k2, п2, e2) = wn\n • wnk+6a" =
I -2я ,
exp I -1-«1П1
1 N1 1 1
exp I - j~N''N- (k2 +e2 )"2
cos| куП1 I- j sin 1k1"1
N1 J {N1
COs| N" (k +e2 )"2 J-/sin ^ NN2 (k> +e2 ) I "2
= COS | k1"1 + (k2 +e2 )^2 ]-/sin I + (k2 +e2 )"2
N1 N2 J ^ N1 N2
(8)
where k = 0, N1 -1; k2 = 0, N2 -1; 0 < 82 < 1, will be called two-dimensional discrete exponential functions with a variable parameter — 2D DEF-VP (Figs. 3-5).
■ Fig. 3. Two-dimensional exponential function with variable parameter at N1 = 32, N2 = 64; k1 = 1, k2 = 1; 82 = 1/2: a — a real part; b — an imaginary part
■ Fig. 4. Two-dimensional exponential function with variable parameter at N1 = 32, N2 = 64; k1 = 1, k2 = 2; 02 = 1/3: a — a real part; b — an imaginary part
■ Fig. 5. Two-dimensional exponential function with variable parameter at N1 = 32, N2 = 64; k1 = 2, k2 = 3; 02 = 1/3: a — a real part; b — an imaginary part
The introduction of discrete exponential functions with a variable parameter makes it possible to generalize the concept of periodicity of the DEF-VP system. Recall that the periodicity of the DEF system in the classical DFT is understood as a periodic continuation of the DEF system outside the interval of N samples. Moreover, the system of discrete basis functions in the classical DFT does not contain discontinuities. In the case of discrete Fourier transform with a variable parameter (DFT-VP) (9), for the DEF-VP system to be inseparable, the periodicity should be understood as parametric periodicity. The parametric periodicity of discrete exponential functions with a variable parameter is understood as their periodic continuation with rotation in complex space by an angle of 2tc6. Note that the introduced concept of parametric periodicity is valid for 1D and 2D real and complex functions.
Consider the main properties of two-dimensional discrete exponential functions of 2D DEF-VP.
Main properties of 2D DEF-VP
Each of the two-dimensional discrete exponential functions with a variable parameter has its own
spatial frequencies k1, k2, which determine its place in a particular basic system. The set of 2D DEF-VP makes its basic system of two-dimensional discrete Fourier transform with a variable parameter (2D DFT-VP) in space iN.
For each value of the parameter 62 we can say that:
1. 2D DEF-VP are complex functions by definition.
2. The basis system 2D DEF-VP is a generalization of the basis system 2D DEF and is equal to it at 92 = 0.
3. 2D DEF-VP are two-dimensional functions of four equivalent variables k1, k2, n1, n2, and one variable parameter 62:
def
HP,N1,N2 (k 1' n1' k2' n2' ) _
- Wk1n1 • W(k2 +02)n2
N
N,
4. 2D DEF-VP are periodic in variables k1 and n1 with a period N1 and a variable with a period N2:
defffp,n1,n2 ((k ± lNi), (ni + qNi), (h ±mN2),
n2, ®2) = defHP,N1,N2 (k 1' n1' k2' n2' ®2), where l, m, q are integers.
5. 2D DEF-VP are parametrically periodic in a variable n2 with a period N2:
def
HP,N1,N2 (k 1, ni, k2, n2 + pN2, 62) = defVP,N1,N2 (k1, n1, k2, ^ 02)'WN2
■q2n2 p
where p is integer.
A parametric shift of a two-dimensional signal Xn1Xn2 two-dimensional cyclic parametric shift of the form of
in the horizontal direction is understood as a
C
H.Sh
0 1
2
(N2 -1)
[XNixN2 ] ' HH.Sh,N2xN2 T 1
[XNixN2 ] ' HH.Sh,N2xN2 T 2
[XNixN2 ] ' HH.Sh,N2xN2
[X nT H(N2-1) [XNixN2 J HH.Sh,N2xN2
where HH sh n2Xn2 is two-dimensional identity matrix, expression H"H sh N2XN2, m = 0, (N2 -1) means raising to the power m of the matrix of two-dimensional parametric shift:
H
H.Sh, N2 xN2,6 2
1 2 0 1 0 0 0 0
0
0 0
(N2 - 2) (N2 -1) exp(-2^62 ) 0 0 n2
(N2 - 2) (N2 -1) n2 0 0 0
6. The basis system 2D DEF-VP in the variables k1, k2 is not multiplicative:
defHP,N1,N2 (k 1, n1, k2, n2, 82 ) • defHP,N1,N2 (k3 , k4, n2, 82) * * defhp,n1,n2 ((k1 + k3 )modn1, n1, (k2 + k4 )modn2, n2, 82 ).
7. The basis system 2D DEF-VP in the variables n1, n2 is multiplicative:
defHP,N1,N2 (k 1, k2, n2, 82) • defH,N1 ,N2 (k1 n3, k2, n4, 82) = = defHP,N1,N2 (k1, (n1 + n3 )modN1, k2, (n2 + n4 )modN2, 82 ).
8. Average value of 2D DEF-IP with spatial frequencies k1*0, k2*0 is equal to zero:
N1 -1 N2 -1
N1 -1
S S defHP,N1,N2 (k1, n1, k, n2, 62 ) = S Wvi
N 2k1n1 xN2
ni=0 n2=0
1 - W
n1
1 - wk1 1 N1
1 - W
n1 =0
k +62 ) N2 N2
n2 -1
S WNi
N2 (k2 +6i )n2
xN2
1 - W,
n2
= 0.
9. The basic system 2D DEF-VP is an orthogonal basis system with respect to variables k1, k2:
1 N:1 N:-1 w (N2k1»1 + N1 ( k2 +62 )»2 )
: : WN1XN2 x
N N2
1 2 n1=0 n2=0 (Njfeg«! +Nj(k4 +92)n2)* = il, if ki = kg, k2 = k4 ;
N1XN2 ={0, if k1 * kg , k2 Ф k4 '
and with respect to variables n1, n2:
-, N1 -1N2-1
1 V V w№vl +N1 (k2 +92)«2) x
N1N2 £0 £0 n1xn2
(N2k1na+N1 (k2+9 2)n4)* {1, if n1 = n3, n2 = n4 n1 xN2 ={0, if n1 * n3, n2 * n4 ,
W
where the symbol * means complex conjugation. 10. 2D DEF-VP can be represented by two unit
vectors, which represent Wj1"1 and Wj2 2^ 2 .
The unit vectors rotate discontinuously (discretely) in perpendicular complex planes. On the interval
N1, the unit vector which displays Wp"1, passes
the angle of h radians, making h revolutions, and on the interval N2, the unit vector, represent-
(k2 +6 2)n2
ing WN
N2
2, passes the angle 2л (k2 +92 ) radi-
ans, making (k2 +62) revolutions. The unit vectors representing the complex conjugate DEF-VP:
W-Vi = W(Ni -ki)ni and
wn1 wn1 and
W-(k2 +92)n2 = w(N2 —(k2 +9 2))n2 wn2 - Wn2
0.5
-0.5
-1 -1
-0.5
■ Fig. 6. Representation of a two-dimensional discrete exponential function with a variable parameter in the spatial domain
and make (N1 - k1) and (N2 - (k2 +62)) revolutions respectively.
Figure 6 illustrates such a 2D DEF-VP representation, where the angles 2%k1/N1 and 2^(k2 + 62) /N2 are marked with the corresponding points.
11. The basis system 2D DEF-VP is complete in
tN
space I2 .
Expansion in basis systems of the form (8) is defined as a 2D DFT-VP. Algebraic form of 2D DFT-VP
SNX ,N2 (k1, k2, 62 ) =
=j^ZNL-1x(n1, "2Wj"1 -Wj2+62)n2, (9)
N1N2 n=0 n2=0
where k,, k2are spatial frequencies, % = 0, (N1 -1),
k2 = 0, (N2 -1); 62 is a parameter, 0 < 62 < 1; x(n,, n2) — two-dimensional signal, »1 = 0, N1 -1,
n2 = 0, N2 -1; SN N2 (k1, k2, 62) are bins of 2D
DFT-VP (two-dimensional vector spatial-frequency spectrum of the signal x(n1, n2) in the basic 2D DEF-VP system).
The algebraic form of direct 2D DFT-VP, taking into account the property of separability of the kernel (core) of 2D DFT-VP, can be represented as
N1 —l
Zi
n1 =0
1
= Z wkini
N1 Z Wni
SN1 ,N2 (k1, k2, 92 ) =
N2 —l
— Z x(nl, n2 WN
N
2 n2=0
(k2 +92 )n2 N,
. (10)
It can be seen that formula (10) makes it possible to step-by-step calculation of the direct 2D DFT-VP by the method of sequential calculation of two DFT-P (parametric DFT). Note that the calculation of the DFT-P can be carried out by methods of parametric fast Fourier transform (FFT-P) [1].
There is an inverse 2D DFT-VP (2D IDFT-VP):
x(ni, n2 ) =
N1 —l N2 —l
= Z Z SNi,N2 (kl, k2, 92 )WNkini • W k1=0 k2=0
k1n1 w"(k2 +92 )n2 N2 '
where n1 = 0, N1 -1; n2 = 0, N2 -1.
Using the separability property of the 2D DFT-VP kernel, we can introduce the matrix form of the direct 2D DFT-VP:
S = A. f(2) A.
sn1xn2,9 2 ni nixni n2
where 0 < 92 < 1;
X F(i)
xn1xn2 fn2 xN2,92
1
0
1
F(2)
F N xNj
W 00
W
10
N,
Xn1 xN2 is given by relation (4);
0 1
F(1)
rN2xN2,e2
W,
0
0^ (0+e2)
N
W
1 • (0+e2)
N
(N2 -1) L wN
n2
W 04
WNl
W
01
(N1 -1) L wnni -1) •0 WN h 1
N1
(N1 -1) • 1
W
1
0^ (1+e2)
N2
W
1 (1+e2)
N2
AN 2-1)(0+e2)
N2
W
(N -1)(1+e2)
N2
(N1 -1)
0 • (N-1)
W
n1
N1
W
1 (N -1)
N1
W(N1 -1)(N-1) J WN
W
(N2 -1)
0 • ( n2-1+e2)
N2
W
1 (n2-1+e2)
N2
W
h
(n2-1)(n2-1+e2)
N2
(11)
We note the difference between matrices (11) and (7), which lies in the nature of the parameter 82 change. The inverse 2D DFT-VP in matrix form is determined by the matrix equation
X = — F(2)* JL re F(1)* i
xn1xn2 " n N1xN1 'n 1 xN2,82 ' fN2xn2,82 J,
where 0 < e2 < 1.
It can be shown that the theorems of linearity, shift, correlation and Parseval's equality are valid for 2D DFT-VP. For 2D DFT-VP, similar to 2D DFT, the concepts of power spectrum PN N2 (%, ¿2, 82) and energy spectrum G^, n2 (¿1, ¿2, 82) can be introduced
Pnn(k1, k2, e2) = |sn1,n2(hi, h, e2) ; Gn1,n2(hi, h, e2) = Pn1,n2h, e2)/Af; af = 1 /(N1 N2).
Let us estimate the efficiency of increasing the detailing of the two-dimensional energy spectrum Gn n2 (k1, k2) using 2D DFT-VP in comparison with the classical 2D DFT.
Evaluation of the efficiency of Fourier-processing of signals padded with zero samples in 2D DFT-VP basis
The increase in the detailing of the two-dimensional energy spectrum Gn1,n2 (¿1, ¿2) by r2 times is carried out by padding the horizontal period of the 2D signal XNi xN2 with (r2 - 1) zero matrices On1 xN2 (1). Padding the horizontal period of a 2D signal Xn1 xn2 with (r2 - 1) zero matrices O^ xn2 makes it possible to obtain a new 2D signal Xn^n^) from a 2D signal X^ xn2 .
Applying the 2D DFT in algebraic form to the 2D signal XN1X( N2r2), we obtain the number of coefficients (bins) of 2D DFT SN1,N2r2 (k1, k2), which is r2 times greater than with 2D DFT of the signal XN1xN2. However, obtaining a r2 times more detailed energy spectrum GN1,N2r2 (¿1, ¿2) by a method based on the separability of the 2D DFT kernel, will require additional (r2 - 1)N1N2 cells for storing zero samples and implementing N1N2r2(N1 + N2r2) additional complex operations.
Obtaining an r2 times detailed energy spectrum GN1,N2r2 (¿1, ¿2) by a method based on the separability property of the 2D DFT-VP kernel does not require additional RAM (storage) for storing zero samples and requires N1N2r2(N1 + N2) complex operations. Thus, the use of 2D DFT-VP instead of the classic 2D DFT allows:
N1 + N2r2
— decrease number of complex operations by y = —1-— times;
N1 + N2
— decrease storage size by r2 times;
— parallelize the process of detailing the two-dimensional energy spectrum Gn n2 (¿1, ¿2), thus reducing the execution time of the 2D DFT by
r2 times.
Conclusions
Discrete Fourier transforms with a variable parameter have been developed. These transforms make it possible to efficiently process two-dimensional signals, the horizontal periods of which are padded with zero samples. The generalization of classical two-dimensional discrete Fourier transform is based on a new method of splitting the rectangular matrix of discrete Fourier transform into square matrices. The splitting of rectangular matrices into square matrices is carried out by using the ordering of the columns of rectangular matrices using the equivalence relation — the relation of comparability in modulus. The properties of the bases of the proposed transformations are investigated. The valid-
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