CONTINUOUS AND DISCONTINUOUS LINEAR APPROXIMATION …
Transcript of CONTINUOUS AND DISCONTINUOUS LINEAR APPROXIMATION …
Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
_________________________________________________________________________________________________________________ Article history: received on May 20, 2011; accepted on June 2, 2011; available online on Sep. 26, 2011.
METROLOGY AND MEASUREMENT SYSTEMS
Index 330930, ISSN 0860-8229 www.metrology.pg.gda.pl
CONTINUOUS AND DISCONTINUOUS LINEAR APPROXIMATION OF THE WINDOW SPECTRUM BY LEAST SQUARES METHOD Józef Borkowski Wroclaw University of Technology, Chair of Electronic and Photonic Metrology, ul. B. Prusa 53/55, 50-317 Wroclaw, Poland (�[email protected])
Abstract
This paper presents the general solution of the least-squares approximation of the frequency characteristic of the data window by linear functions combined with zero padding technique. The approximation characteristic can be discontinuous or continuous, what depends on the value of one approximation parameter. The approximation solution has an analytical form and therefore the results have universal character. The paper presents derived formulas, analysis of approximation accuracy, the exemplary characteristics and conclusions, which confirm high accuracy of the approximation. The presented solution is applicable to estimating methods, like the LIDFT method, visualizations, etc.
Keywords: data window, spectrum approximation, interpolated DFT, LIDFT, spectrum estimation, zero padding.
© 2011 Polish Academy of Sciences. All rights reserved
1. Introduction
The least-squares approximation of the data window frequency characteristics with
appropriate linear functions is used in the linear interpolation of the discrete Fourier transform (LIDFT) [1-3] and the zero padding can also be used with this approximation to increase approximation accuracy [4]. Linear approximation of the spectrum allows for linearizing relationships to determine component frequencies and this feature of the LIDFT method differs it from other methods of signal estimation [5-13].
The piece-wise linear DFT approximation is used in [14] but in the simple form, not by least squares method. In the methods described in [1-4] the approximation function is discontinuous, which can be inadvisable in some applications.
The paper removes discontinuity from the approximation function and is organized as follows. Symbols and basic equations are defined in Section 2. In section 3 the approximation from [4], with zero padding technique, is presented in more general form than in previous papers [1-4]. Afterwards the alternative approximation is analysed in Section 4. Section 5 describes how the formulas for continuous approximation function are obtained and Section 6 presents the exemplary characteristics for a triangle data window.
2. The data window frequency characteristic
The data window frequency characteristic W(λ) is given by equation:
∑−+
=
π−=1
/j20
0
)(Nn
nn
NnnewW λλ , (1)
where: λ = fNT is the normalized frequency (in [bins]), f is the frequency in [Hz], T is the
J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
sampling period, N is the length of data window, which is defined by values wn and n0 denotes the first value of the sample index. Most often n0 = 0, what gives the discrete-time Fourier transform (DtFT) of the data window or n0 = −N/2 what gives shifted DtFT of the data window. In this paper the shifted DtFT is used (n0 = −N/2) because for a symmetric data window, the W(λ) has only real values. The non-shifted standard DtFT is obtained from W(λ) for n0 = −N/2 (and vice-versa) by simple formula:
∑∑−
=
π−π−
−=
π− ==1
0
/j212/
2/
/j2)(N
n
Nnn
jN
Nn
Nnn eweewW λλλλ . (2)
The zero padding technique involves computing (2) for the M-element series {w−N /2, …, wN /2−1, 0, …, 0}NR instead of the series {w−N /2, …, wN /2−1}N. Such a “R-times” zero padding technique (where M = NR) takes the following form from (2):
∑−
−=
π−=12/
2/
/j2)/(N
Nn
MnnewRW λλ . (3)
The values W(λ /R) for integer values of λ are easy to obtain by using the FFT algorithm and (2)-(3):
NRnkRkj
N
Nn
Mnkjn weew }{FFT/
12/
2/
/2 π−
−=
π− =∑ , (4)
where: {wn}NR is the zero padded data window:
}}0,...,0{,}{{}{ )1( −= RNNnNRn ww (5)
and M = NR is the natural power of 2 for radix-2 FFT algorithm. The window spectrum W(λ) defined by (2) is approximated in the LIDFT method by
function )(ˆ λW and in the least squares method the quality of this approximation is given by the mean square approximation error Q:
∫−
−=2/
2/
2|)(ˆ)(|N
N
dWWQ λλλ . (6)
This error has a different form in Sections 3-5, what gives different formulas for the parameters of the function )(ˆ λW .
3. The approximation by linear functions used in the LIDFT method
For data window and the zero padding technique, the LIDFT method assumes an approximation of the frequency characteristic W(λ) from (2) using linear functions, as shown in Fig. 1a [4]. The linear functions kkk baW += λλ)(ˆ approximate the window characteristic W(λ) for λ ∈ [k/R, (k+1)/R]. The error Q from (6) is defined as:
∑ ∫−
−=
+
−=12/
2/
/)1(
/
2)(ˆ)(
NR
NRk
Rk
Rk
k dWWQ λλλ (7)
and is described in the matrix form (on the base of (A7) of Appendix A) by:
AczAzcAccwwAzzzcAzcww HHHTHHT )()( −−+⋅=−−−+⋅= NNQ , (8) where: T
12/2/ ],....,[ −−= NN www , (9)
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Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
⋅⋅=
I0
0IA
3R , (10)
TTT ][ ccc ′′′= , (11)
T12/2/ ],...,[ −− ′′=′ MM ccc ,
++=′ kkk bR
ka
Rc
2/11 , (12)
T12/2/ ],...,[ −− ′′′′=′′ MM ccc , 26R
ac k
k =′′ , (13)
TTT ][ zzz ′′′= , (14)
T12/2/ ],...,[ −− ′′=′ MM zzz , ∫
+
=′Rk
Rk
k dWz/)1(
/
)( λλ , (15)
T12/2/ ],...,[ −− ′′′′=′′ MM zzz , ∫
+
+−=′′Rk
Rk
k dWR
kRz
/)1(
/
)(2/12 λλλ . (16)
The coefficients kz′ , kz′′ from (15)-(16) are given, by use of (2), as:
∑ ∫−
−=
+π−=′
12/
2/
/)1(
/
/2N
Nn
Rk
Rk
Nnjnk dewz λλ , ∑ ∫
−
−=
+π−
+−=′′12/
2/
/)1(
/
/22/12N
Nn
Rk
Rk
Nnjnk de
R
kwRz λλ λ . (17)
The integrals in (17) are given by:
Mnkjjxn
Rk
Rk
Nnj eeR
xde n /2
/)1(
/
/2 sinc π−−+
π− =∫ λλ , (18)
MnkjjxnRk
Rk
Nnj eeR
xjde
R
kn /2
2
/)1(
/
/2
2sinc'2/1 π−−
+π− ⋅=
+−∫ λλ λ , (19)
where:
M
nxn
π= , NRM = , 12/,...,2/ −−= NNn , (20)
n
nn x
xx
sinsinc = , n
nnn
n
nn x
xxx
dx
xdx
/)(sincos)sinc(sinc' −== . (21)
Taking into account (17)-(21), the matrices (15)-(16) are given by:
wWz ′⋅=′ −RR 1 , wWz ′′⋅=′′ −
RR 1 , (22) where: NM
MnkjR e ×
π−= ][ /2W , 12/,...,2/ −−= MMk , 12/,...,2/ −−= NNn , (23)
T12/2/ ],...,[ −− ′′=′ NN www , njx
nnn exww −=′ )(sinc , (24)
T12/2 ][ −− ′′′′=′′ NN/ w,...,ww , njx
nnn exjww −=′′ )sinc'( . (25) Let us notice that: IWW ⋅= MRR
H . (26)
From (24) the matrix z (14) is given by:
′′′
⋅
=
w
w
W0
0Wz
R
R
R
1 . (27)
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J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
The task of determining the coefficients ak, bk of the approximating linear functions that minimize Q defined by (8) can be replaced by an equivalent task of determining the coefficients kc′ , kc ′′ from (12)-(13). They are obtained from:
0zcAc
=−= )(2d
dQ , (28)
which gives: zc = (29)
and then Q has minimum value determined on the basis of (8) and (29):
Azzww HTmin −⋅= NQ . (30)
From (10), (24)-(27) the expression zHAz from (30) is given by:
∑−
−=
+=′′′′+′′⋅=12/
2/
222HHH ])sinc'(3)sinc[(])(3)[(N
Nnnnn xxwNN wwwwAzz (31)
and then (30) has the following form:
∑−
−=
=12/
2/
2min
N
NnnnwcNQ , 22 )sinc'(3)sinc(1 nnn xxc −−= . (32)
Let us define the vectors d′ and d ′′ and coefficients kd′ , kd ′′ , based on the values of
approximating function )(ˆ λkW for λ = k/R, (k+1/2)/R, (k+1)/R (two extreme values and the median value of the range [k/R, (k+1)/R]) as follows:
T12/2/ ],...,[ −− ′′=′ MM ddd , kkkk b
R
ka
R
kWd ++=
+=′ 2/12/1ˆ , (33)
T12/2/ ],...,[ −− ′′′′=′′ MM ddd ,
R
a
R
kW
R
kWd k
kkk =
−
+=′′ ˆ1ˆ . (34)
Using (11)−(13) they can be described as:
cI0
0Iddd ⋅
⋅=′′′=
6][ TTT R (35)
and taking into account (27), (29) one has:
′′⋅′
⋅
=′′′=
w
w
W
0Wddd
60][ TTT
R
R (36)
or in non-matrix form by:
∑−
−=
−−=++=
+=′12/
2/
/π2sinc2/12/1ˆN
Nn
Mnkjjxnnkkkk eexwb
R
ka
R
kWd n , (37)
∑−
−=
−−==
−
+=′′12/
2/
/π2sinc'6ˆ1ˆN
Nn
Mnkjjxnn
kkkk eexwj
R
a
R
kW
R
kWd n . (38)
The coefficients kd′ , kd ′′ can be calculated by FFT from (4). The equations (36)-(38) allow the calculation of the parameters kd′ , kd ′′ of approximating
linear functions, and hence their coefficients ak, bk can be calculated. The graphical interpretation of kd′ , kd ′′ is shown in Fig. 1b.
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Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
R
5.1
R
5.0
R
5.0-
R
5.1-
0.4
0.5
0.6
0.7
0.8
0.9
1.0
[bins]l
R
1-
R
1
R
2-
R
20
00 ba +l
11 ba +l
11 -- +l ba
22 -- +l ba
)(ˆ),( ll WW)(lW
2=R
úû
ùêë
é +Îl+l=l
R
k
R
kbaW kkk
1,for)(ˆ
´W(0)a)
R
5.2
R
5.1
R
5.0
kkkk bR
ka
R
kWd +
+=÷
ø
öçè
æ +=¢
2/12/1ˆ
R
a
R
kW
R
kWd k
kkk =÷ø
öçè
æ-÷
ø
öçè
æ +=¢¢ ˆ1ˆ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0d ¢
1d ¢
2d ¢
1d ¢¢
0d ¢¢
R
1
R
20
[bins]l
)(ˆ
)(
l
l
W
W
2=R´W(0)b)
R
5.1
R
5.0
R
5.0-
R
5.1-
0.4
0.5
0.6
0.7
0.8
0.9
1.0
[bins]l
R
1-
R
1
R
2-
R
20
00 ba +l
11 ba +l11 -- +l ba
22 -- +l ba 22 ba +l
)(ˆ),( ll WW
)(lW
2=R
úû
ùêë
é +-Îl+l=l
R
k
R
kbaW kkk
2/1,
2/1for)(ˆ
´W(0)c)
R
5.2
R
5.1
R
5.0
R
a
R
kW
R
kWd k
kkk =÷ø
öçè
æ --÷
ø
öçè
æ +=¢¢
2/1ˆ2/1ˆ
kkkk bR
ka
R
kWd +=÷
ø
öçè
æ=¢ ˆ
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0d ¢
1d ¢
2d ¢2d ¢¢
1d ¢¢
00 =¢¢d
R
1
R
20
[bins]l
)(ˆ
)(
l
l
W
W
2=R´W(0)d)
Fig. 1. Approximation of data window spectrum by linear functions: a,b) in Section 3; c,d) in Section 4
4. Alternative approximation by linear functions
The approximation alternative to the presented in Section 3 is defined by kkk bλa(λW +=)ˆ for λ ∈ [(k−0.5)/R, (k+0.5)/R] (Fig. 1c). The mean square error Q of such approximation is different from (7), i.e. it has a different range of integration:
∑ ∫−
−=
+
−
−=12/
2/
/)2/1(
/)2/1(
2)(ˆ)(
NR
NRk
Rk
Rk
k dWWQ λλλ (39)
and, on the basis of (A10) of Appendix A, is equal (8) where matrices w , A , c , c ′′ , z are defined by (9)-(11), (13)-(14), and c′ , z′ , z ′′ are defined by:
T12/2/ ],...,[ −− ′′=′ MM ccc ,
+=′ kkk bR
ka
Rc
1 , (40)
T12/2/ ],...,[ −− ′′=′ MM zzz , ∫
+
−
=′Rk
Rk
k dWz/)2/1(
/)2/1(
)( λλ , (41)
T12/2/ ],...,[ −− ′′′′=′′ MM zzz , ∫
+
−
−=′′Rk
Rk
k dWR
kRz
/)2/1(
/)2/1(
)(2 λλλ . (42)
After taking into account (2), the coefficients kz′ , kz′′ from (41)-(42) can be written as:
∑ ∫−
−=
+
−
π−=′12/
2/
/)2/1(
/)2/1(
/2N
Nn
Rk
Rk
Nnjnk dewz λλ , ∑ ∫
−
−=
+
−
π−
−=′′12/
2/
/)2/1(
/)2/1(
/22N
Nn
Rk
Rk
Nnjnk de
R
kwRz λλ λ . (43)
The integrals in both of these dependencies are given by:
Mnkjn
Rk
Rk
Nnj eR
xde /2
/)2/1(
/)2/1(
/2 sinc π−+
−
π− ⋅=∫ λλ , MnkjnRk
Rk
Nnj eR
xjde
R
k /22
/)2/1(
/)2/1(
/2
2sinc' π−
+
−
π− ⋅=
−∫ λλ λ , (44)
where: xn, sinc xn, sinc′xn are defined by (20)-(21). The matrices (41)-(42) can be written as in (22), i.e. wWz ′⋅=′ −
RR 1 , wWz ′′⋅=′′ −RR 1 ,
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J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
where RW is defined by (23), and vectors w′ , w ′′ are defined as:
T12/2/ ],...,[ −− ′′=′ NN www , )(sinc nnn xww =′ , (45)
T12/2 ][ −− ′′′′=′′ NN/ w,...,ww , )sinc'( nnn xjww =′′ . (46)
The task of determining the coefficients ka , kb ( kc′ , kc ′′ ) of the approximating linear functions that minimize Q defined by (39) is obtained by using equations (28)-(29), and similar transformations as in (30)-(31) give the same result for the mean square error of alternative approximation as the (32).
By defining the vectors d′ and d ′′ with coefficients kd′ , kd ′′ , based on the values of
approximating function )(ˆ λkW for λ = (k−1/2)/R, k/R, (k+1/2)/R (two extreme values and the median value of the range [(k−1/2)/R, (k+1/2)/R]) as follows:
T12/2/ ],...,[ −− ′′=′ MM ddd , kkkk b
R
ka
R
kWd +=
=′ ˆ , (47)
T12/2/ ],...,[ −− ′′′′=′′ MM ddd ,
R
a
R
kW
R
kWd k
kkk =
−−
+=′′ 2/1ˆ2/1ˆ , (48)
it is possible, after taking into account (11), (13), (40), (45)-(46), to express them in the matrix form (35)-(36) or in the non-matrix form by:
∑−
−=
π−=+=
=′12/
2/
/2sincˆN
Nn
Mnkjnnkkkk exwb
R
ka
R
kWd , (49)
∑−
−=
π−==
−−
+=′′12/
2/
/2sinc'62/1ˆ2/1ˆN
Nn
Mnkjnn
kkkk exwj
R
a
R
kW
R
kWd . (50)
The coefficients kd′ , kd ′′ can be calculated by FFT from (4). The equations (49)-(50) allow the calculation of the parameters kd′ , kd ′′ of approximating
linear functions, and hence their coefficients ak, bk can be calculated. The graphical interpretation of kd′ , kd ′′ is shown in Fig. 1d.
Both approximations from the Sections 3 and 4 shown in Figure 1 are discontinuous, which may be disadvantageous in some applications. Their modification presented in the next Section, minimizes this discontinuity. 5. Reduction of approximation discontinuity
There is a natural question whether it is possible to eliminate or at least reduce the approximation discontinuity at points of transition to the next line fitting (Fig. 1). To this aim, it is possible to minimize the error (7) taking into account, with the weight µ , an additional component which is the mean square error of discontinuity:
∑−
−=+
+−
++=12/
2/
2
111ˆ1ˆ
NR
NRkkk R
kW
R
kWQQ µ (51)
or in the same way for the error (39):
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Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
∑−
−=+
+−
++=12/
2/
2
112/1ˆ2/1ˆ
NR
NRkkk R
kW
R
kWQQ µ . (52)
In both cases (based on equation (A12) of Appendix A), the 1Q is given by:
AcEIEIEIEIAc ][][ TH1 +−+−+= µQQ , (53)
where: Q , w , A are defined by (8)-(10) and c , z are defined by (11)-(16) for the approximation from Section 3 and by (11), (13)-(14), (40)-(42) for the approximation from Section 4 and, additionally, the matrix E is a circulant matrix defined as:
=
000011000001000010
OOOO
O
O
L
E . (54)
Minimization of error (53) is achieved for the condition:
0AcEIEIEIEIAzcAc
=+−+−+−= ][][2)(2 T1 µd
dQ (55)
and hence: zAcEIEIEIEIc =+−+−+ ][][ Tµ , (56)
zBAc =µ , (57) where: ][][ T11 EIEIEIEIAB +−+−+= −−µ . (58)
Calculation of the matrix 1−A is simple (since A is a diagonal matrix) and in calculating the matrix 1−B can be used specific properties of the matrix B , including the fact that the matrix E appearing in (58) is a cyclic permutation matrix (a kind of circulant matrix), for which the decomposition can be performed to form a diagonal matrix with the use of Fourier matrix W [15]. The solution of equation (57) is (according to equation (A16) of Appendix A) the vector c defined as:
AzW0
0W
ΛΛΛ
ΛΛΛ
W0
0Wzc
⋅
−⋅
−= H
H
2BBA
BA2A4
j
j
M
µ (59)
or by:
zW0
0WA
ΛΛΛ
ΛΛΛI
W0
0Wc
⋅
−−⋅
= H
H
2BBA
BA2A41
j
j
Mµ , (60)
where: ][ /2 Mnkje π−=W is a MM × matrix ( 12/,...,2/, −−= MMkn ), and matrices ][ AA nΛ=Λ , ][ BB nΛ=Λ are M -element diagonal matrices with elements defined by:
2/1
22A sin
31cos121)(sin
−
++⋅=Λ nnnn xxRx µ , M
nxn
π= , (61)
2/1
22B sin
31cos121)(cos
−
++⋅=Λ nnnn xxRx µ , M
nxn
π= , (62)
with the range 12/,...,2/ −−= MMn .
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J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
Using (27) in (60) and the fact that [ ]TH 0I0WW NNR M ×⋅= we have:
′′′
⋅
′−′′⋅−′′⋅′−
⋅
=
w
w
ΛIΛΛ
ΛΛΛI
W
Wc 2
BBA
BA2
A
)(12412)(4
001
RRj
RjR
R R
R
µµµµ
, (63)
where: ][ AA nΛ=′Λ , ][ BB nΛ=′Λ are N -element diagonal matrices with elements defined by (61)-(62) for the range 12/,...,2/ −−= NNn , and vectors w′ , w ′′ are defined by (24)-(25) for the approximation from Section 3 and by (45)-(46) for the approximation from Section 4. Equation (63) can be transformed, using (61)-(62) and dependences for w′ , w ′′ , to the form:
wΛ
Λ
W
Wc ⋅
⋅⋅
=
β
α
)6/1(001
R
R
R, wΛWc αRR
1=′ , wΛWc βRR61=′′ , (64)
where: w is the vector of the data window, diagonal matrices αΛ , βΛ are defined for the
approximation from Section 3 as ][ njxne
−= ααΛ , ][ njxnej −= ββΛ and for the approximation
from Section 4 as ][ nαα =Λ , ][ nj ββ =Λ with the coefficients nα , nβ defined by:
)sin(cos121sinccos121)sinc()( 2
312
nn
nnnn xxR
xxRx
+++=
µµµα ,
)sin(cos121sincsin4'sinc6)( 2
312
2
nn
nnnn xxR
xxRx
++−=
µµµβ . (65)
Taking into account (64) and definitions αΛ , βΛ , (35) has the form:
wΛ
Λ
W
Wd ⋅
⋅
=
β
α
R
R
00
, wΛWd αR=′ , wΛWd βR=′′ (66)
or in the non-matrix form, from (33)-(34) for the approximation from Section 3:
∑−
−=
π−−=++=
+=′12/
2/
/2)(2/12/1ˆN
Nn
Mnkjjxnnkkkk eewb
R
ka
R
kWd nµα , (67)
∑−
−=
π−−==
−
+=′′12/
2/
/2)(ˆ1ˆN
Nn
Mnkjjxnn
kkkk eewj
R
a
R
kW
R
kWd nµβ (68)
and similarly, from (47)-(48) for the approximation from Section 4:
∑−
−=
π−=+=
=′12/
2/
/2)(ˆN
Nn
Mnkjnnkkkk ewb
R
ka
R
kWd µα , (69)
∑−
−=
π−==
−−
+=′′12/
2/
/2)(2/1ˆ2/1ˆN
Nn
Mnkjnn
kkkk ewj
R
a
R
kW
R
kWd µβ . (70)
For 0=µ the Eqs. (67)-(68) are identical with (37)-(38), and (69)-(70) are identical with (49)-(50).
Let us notice that for 0=µ (64) is the same as (27), i.e. )0( == µcz . From this fact and from (64) the expression )()( H zcAzc −− is determined. Using additional relations (30)-(31) the total mean square error Q of approximation defined by (7)-(8) or (39) is obtained:
∑−
−=
⋅=12/
2/
2)(),(N
Nnnn wcNRQ µµ , (71)
where:
386
Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
2222 )]0()([121)]0()([)0(
121)0(1),( nnnnnnn Rc βµβαµαβαµ −+−+−−= . (72)
For 0=µ the Eqs (71)-(72) are the same as (32). The Maclaurin series of the cn(µ) for µ = 0 and µ → ∞ are described by:
)(3274425
1614175
215754
45)0( 12
10864
nnnnn
n xoxxxx
c +−+−==µ , (73)
)(1871116
202514
1894
45)( 12
10864
nnnnn
n xoxxxx
c ++++=∞→µ . (74)
These series and (20) show that for high values of R, the Q(µ → ∞) → Q(µ = 0), what is also confirmed by Fig. 2 and the figures presented in Section 6.
0.0
0.1
0.2
0.3
0.4
0.5
0 2/N2/N-n
)(4 m× ncR
0=m ¥®m
1=R
1=R
2=R
4=R8³R2³R
106 2,...,2=N
Fig. 2. Values of R4cn(µ) versus n as obtained from (72)
6. Exemplary approximations for triangle window
The conclusions obtained at the end of Section 5 and in Fig. 2 are confirmed for the case of
calculation of (71) for the triangle window – the mean square error Q(µ, R) is proportional to R−4: when R is doubled, the value cn(µ) to decrease 16-fold (Fig. 2 and (71)-(72)), which results in the 16-fold reduction of the mean square error Q(µ, R), as is shown in Fig. 3a,b, especially for big R. The value of the Q(µ, R) depends also on the value of parameter µ – for small R this dependence is significant, but the higher the value R, the dependence on µ is lesser (Fig. 2, 3a,b). The influence of the value µ is clearly shown in Fig. 3c in circle zooms – for µ = 0 the discontinuity of the approximation is the largest at analyzed point λ = 0.75 bins (although the mean square error Q has a minimum, as shown in Fig. 3a,b), for µ = 0.1 this discontinuity is more then twice smaller (but the Q is increased), and for µ → ∞ approximation is continuous (but the error Q is significantly larger for small R, as shown in Fig. 3b). The cases for the triangle window, µ = 0 and R = 2 are also shown in Fig. 1.
For the data window, the most important feature is its frequency characteristics |W(λ)|. This characteristic is completed by a graph of the frequency characteristic of the approximation error |)(ˆ)(| λλ WW − (Fig. 4). For 2≥R and triangle window, the envelope of this error is about (2.4R2)-times smaller than the envelope of the triangle window leakage (Fig. 4). The approximation error |)(ˆ)(| λλ WW − is discontinuous for µ = 0 and continuous for µ → ∞. The differences in Fig. 4 are the most visible for small value of R, i.e. R = 2.
387
J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
1 2 3 4 5 6 7 8 9 101.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0),(
2
4
RQN
Rm
R
0=m
1.0=m
3.0=m
1=m
¥®m
106 2,...,2=Nx 10
-3
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0=m
0=m
¥®m
)(lW
)(lW
)(lW
)(ˆ lW)(ˆ lW
)(ˆ lW
)(ˆ
)(
l
l
W
W
´W(0)
10 10 10 10 10 10 10 10 10-4 -3 -2 -1 0 1 2 3 4
),(2
4
RQN
Rm
2=R
3=R
4=R
8=R
128=Rm
106 2,...,2=N5.1=R
b) c)
[bins]l
2=R
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
x 10-3
a)
Fig. 3. The quality of approximation (example for a triangle window): a,b) dependence of the mean square error
Q from µ and R in a graph R4N−2Q versus µ and R, c) different degree of the approximation discontinuity at a point λ = 0.75 bins magnified in three circle zooms for µ = 0, 0.1, ∞
0=m
]bins[l
|)(| lW
|)(ˆ)(| ll WW -2=Rfor
16=R
|)(ˆ)(| ll WW -for
0 1 2 3 4 5 6 7-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
[dB])4.2(log20 210 R×
|)(| lW |)(ˆ)(| ll WW -,]dB[
64=N
1.0=m
]bins[l
|)(| lW
|)(ˆ)(| ll WW -2=Rfor
16=R
|)(ˆ)(| ll WW -for
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
0 1 2 3 4 5 6 7
|)(| lW |)(ˆ)(| ll WW -,]dB[
64=N
¥®m
]bins[l
|)(| lW
|)(ˆ)(| ll WW -2=Rfor
16=R
|)(ˆ)(| ll WW -for
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
0 1 2 3 4 5 6 7
64=N|)(| lW |)(ˆ)(| ll WW -,]dB[
Fig. 4. Graph of |)(| λW and |)(ˆ)(| λλ WW − for a triangle window and µ = 0, 0.1, ∞
7. Conclusions
The main result of the paper are the Eqs. (65)-(70) for the parameters kd′ , kd ′′ of the
approximating linear functions (and hence their coefficients ka , kb ) for the given value of the parameter µ , which defines the degree of the approximation discontinuity. The special cases are obtained for 0=µ (largest discontinuity, smallest mean square error of approximation) and ∞→µ (continuity approximation, largest mean square error). The values of coefficients of approximating linear functions are easy to calculate by FFT algorithm with the use of zero padding technique. The obtained formulas are valid for both kinds of approximation: when approximating linear functions are defined for the range ]/)1(,/[ RkRk + bins (Fig. 1a,b), and for the range ]/)2/1(,/)2/1[( RkRk +− bins (Fig. 1c,d). Results obtained in the paper are valid for any data window, not only symmetric like the triangle window used as an example, because all formulas are obtained without any assumption of this type. The formulas for parameters of approximating functions are given for the shifted DtFT, but it is possible to use them also for the non-shifted DtFT, by simple multiplication by the function )/exp( Rkjπ− , as follows from (2)-(4). Appendix A. Selected mathematical equations
R
dRk
Rk
1/)1(
/
=∫+
λ , 2
/)1(
/
2/1R
kd
Rk
Rk
+=∫+
λλ , 33
2/)1(
/
2
121)2/1(RR
kd
Rk
Rk
++=∫+
λλ , (A1)
R
dRk
Rk
1/)2/1(
/)2/1(
=∫+
−
λ , 2
/)2/1(
/)2/1( R
kdRk
Rk
=∫+
−
λλ , 33
2/)2/1(
/)2/1(
2
121RR
kdRk
Rk
+=∫+
−
λλ , (A2)
)]()([)]()([)])([(|)(|
])(][)([)(ˆ)(2
2
λλλλλλλλλ
λλλλλλ∗∗∗∗∗∗
∗∗∗
+−+−+++=
−−−−=−
WbWaWbWababaW
baWbaWWW
kkkkkkkk
kkkkk . (A3)
Based on (2), (9) there is:
388
Metrol. Meas. Syst., Vol. XVIII (2011), No. 3, pp. 379-390.
wwT12/
2/
22/
2/
2|)(| ⋅== ∑∫−
−=−
NwNdWN
Nnn
N
N
λλ . (A4)
Based on (A1), (10)-(13) there is:
AccH12/
2/
/)1(
/
))(( =++∑ ∫−
−=
+∗∗
M
Mk
Rk
Rk
kkkk dbaba λλλ . (A5)
Based on (10)-(16) there is:
AzcH12/
2/
/)1(
/
)()( =+∑ ∫−
−=
+∗
M
Mk
Rk
Rk
kk dWba λλλ . (A6)
Based on (A3)-(A6) there is:
AczAzcAccww HHHT12/
2/
/)1(
/
2)(ˆ)( −−+⋅=−∑ ∫
−
−=
+
NdWWM
Mk
Rk
Rk
k λλλ . (A7)
Based on (A2), (10)-(11), (13), (40) there is:
AccH12/
2/
/)2/1(
/)2/1(
))(( =++∑ ∫−
−=
+
−
∗∗M
Mk
Rk
Rk
kkkk dbaba λλλ . (A8)
Based on (10)-(11), (13)-(14), (40)-(42) there is:
AzcH12/
2/
/)2/1(
/)2/1(
)()( =+∑ ∫−
−=
+
−
∗M
Mk
Rk
Rk
kk dWba λλλ . (A9)
Based on (A3)-(A4), (A8)-(A9) there is:
AczAzcAccww HHHT12/
2/
/)2/1(
/)2/1(
2)(ˆ)( −−+⋅=−∑ ∫
−
−=
+
−
NdWWM
Mk
Rk
Rk
k λλλ . (A10)
For any T12/2/ ],...,[ −−= MM xxx , T
12/2/ ],...,[ −−= MM yyy meeting the condition kMk xx =+ , the matrix E from (54) and identity matrix I , there is:
ExyH12/
2/1 =∑
−
−=
∗+
M
Mkkk yx , yEx TH
12/
2/1 =∑
−
−=
∗+
M
Mkkk yx . (A11)
For any T12/2/ ],...,[ −−= MM xxx , T
12/2/ ],...,[ −−= MM yyy meeting the conditions kMk xx =+ ,
kMk yy =+ , TTT ][ yxz = , the matrix E from (54) and identity matrix I , based on (A11), there is:
zEIEIEIEIz ][][)()( TH12/
2/
211 +−+−=+−−∑
−
−=++
M
Mkkkkk yyxx . (A12)
For diagonal matrices iΛ ( 3,2,1=i ), orthogonal matrix W meeting the condition IWWWW ⋅== MHH there is for the following block matrices:
IWΛΛΛWΛWΛΛΛWΛ
WΛΛΛWΛWΛΛΛWΛ
WWΛWWΛ
WWΛWWΛ 2H12
3211H12
3213
H123213
H123212
H2
H3
H3
H1
)()()()(
M=
+++−+
⋅
− −−
−−
. (A13)
For the matrix ][ /2 Mnkje π−=W ( 12/,...,2/, −−= MMkn ), matrix E from (54) and diagonal matrix ][ /2 Mnj
e e π−=Λ there is:
HWWΛE eM =⋅ , HT WWΛE ∗=⋅ eM , ][ /2 Mnje e π−=Λ . (A14)
389
J. Borkowski: LINEAR APPROXIMATION OF THE DATA WINDOW SPECTRUM BY LEAST SQUARES METHOD
For the matrices: E from (A14), A from (10) and identity matrix I there is:
+++−−−+−+
=
+−+−+=
∗−∗
∗∗−
−−
H1H
HH1
T11
)]())3(2[()()()]())(2[(1
][][
WΛΛIWWΛΛW
WΛΛWWΛΛIW
EIEIEIEIAB
eeee
eeee
R
R
M µµ
µ. (A15)
For )())(2( 11
∗− +−+= eeR ΛΛIΛ µ , )())3(2( 12
∗− +++= eeR ΛΛIΛ µ , ∗−= ee ΛΛΛ3 in (A15) there is from (A13):
AW0
0W
ΛΛΛ
ΛΛΛ
W0
0WAA
W0
0WA
ΛΛΛ
ΛΛΛAI
W0
0WB
−
−=
⋅
−−⋅
=−
H
H
2BBA
BA2A
2
H
H
2BBA
BA2A1
4
4
j
j
M
j
j
M
µµ
µµ
, (A16)
where: AΛ , BΛ are defined by (61)-(62). References [1] Borkowski, J. (2000). LIDFT – The DFT linear interpolation method, IEEE Trans. Instrum. Meas., 49,
741-745.
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[12] Chen, K.F., Jiang, J.T., Crowsen, S. (2009). Against the long-range spectral leakage of the cosine window family, Computer Physics Communications, 180, 904-911.
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