Design and Improvement of Flattop Windows with Semi-Infinite
Optimization
To Tran, Ingvar Claesson, Mattias Dahl
Blekinge Institute of Technology
Department of Telecommunications and Signal Processing
Box 520, SE-372 25 Ronneby, Sweden
[email protected] [email protected] [email protected]
ABSTRACT
Digital and analog window optimization problems are often characterized by a few number of variables
with many constraints. In some cases the optimization problem becomes semi-infinite, i.e. a finite number
of variables with an infinite set of constraints. This paper presents a method for flattop window design and
enhancement using the Dual Nested Complex Approximation (DNCA) algorithm. Flattop windows can be
used for accurate amplitude measurements in spectral analysis and can also be used to design FIR filters
with very high stopband attenuation. This paper proposes using the DNCA scheme to solve the optimization
problem, due to its low computational complexity and memory consumption. It can be run on any desktop
computer. The framework of the DNCA scheme is presented together with two examples; one concerning
the design of an enhanced version of the ISO 18431-2 flattop window and the other concerns the design of
a flattop window which is comparable with the commercial P-401.
1. INTRODUCTION
Spectral leakage is a contributing factor to errors in spectral analysis. For this cause, a window is normally applied to reduce the effect of leakage. A window can be seen as a function which balances between
amplitude accuracy and frequency resolution. In cases where the frequency resolution is of importance,
the classical windows described in [1] are normally preferred. For accurate amplitude measurements or
instrument calibration, flattop windows are often used. A flattop window is characterized by high sidelobe
attenuation with a wide and flat mainlobe, thereof the name ”flattop”. Classical windows in turn are characterized by a narrow mainlobe and relatively low sidelobe attenuation compared with the flattop windows.
For many problems of engineering interest, the class of signals being sought are periodic, which leads quite
naturally to the use of periodic windows [1].
Beside spectral analysis, windows are commonly used for FIR filter design through the ”window method”.
In short, the window method truncates an ideal impulse response of a filter with infinite duration by multiplying it with a window function to obtain a realizable, finite and linear phase FIR filter. From filter design,
it is known that there are four types of linear phase FIR filter; symmetric or anti-symmetric filters with even
or odd length [2]. The use of symmetric windows in filter design is therefore a necessity to obtain linear
phase characteristic.
The main difference between windows for spectral analysis and windows for FIR filter design, is that
spectral analysis requires a periodic window while filter design requires a symmetric window. A symmetric
window is as its name indicate, symmetric around its midpoint, while a periodic window have a missing
endpoint. The missing endpoint can be considered to be the beginning of the next period if the window is
periodically extended according to the periodic extension property of the DFT [3].
A common way to construct windows, symmetric or periodic, is by using the summation of shifted
Dirichlet kernels. This construction method is convenient because the window shape in the time and fre-
quency domain are mainly characterized by a few number of window coefficients. Given a set of window
coefficients, one can generate a symmetric or periodic window of any length.
The task of window design, is the task of obtaining a set of window coefficients which fulfills a given
specification. In most window design cases, only the magnitude response is specified. Since no information
about the phase is given, it is convenient to assume that the phase is linear, which naturally leads to the
design of symmetric windows where the phase is given a priori [2]. The design of symmetric windows
will in turn lead to a real valued optimization problem with a finite number of constraints. Optimization of
symmetric windows is characterized by a few number of variables and a huge number of constraints.
In contrast, the task of window enhancement assumes that a set of window coefficients is given, which
is the target to the enhancement. In such cases, the design might be based on a periodic window since the
frequency response of the periodic window can be computed given the window coefficients, which yield
amplitude and phase information. The related optimization problem for periodic window enhancement will
be of semi-infinite nature, i.e. the number of variables is finite, but the number of constraints is infinite, due
to the complex valued constraints.
This paper presents a method which can be used to design new flattop windows or enhance existing
flattop windows. The design is based on the minimax criterion (L∞ -norm), which is related to magnitude
response specification. As mentioned previously, the optimization procedure is either semi-infinite or has a
huge number of constraints, depending on whether the optimization is based on a symmetric window or a
periodic window.
This paper uses the Dual Nested Complex Approximation scheme for both design and enhancement,
since it is very efficient in solving conventional and semi-infinite optimization problems. In Section 4, the
framework of the DNCA scheme is presented, which works for finite and semi-infinite, linear and quadratic
optimization problems.
Two examples are included and discussed in Section 5. The first example considers the design of an
enhanced version of the upcoming international standard flattop window, the ISO 18431-2 flattop window.
The design procedure is in this example based on a periodic window, since the ISO flattop window coefficients are public. The second example considers the design of a flattop window which is comparable with
the P-401 flattop window, based on a symmetric window, since the window coefficients for the Hewlett
Packard flattop window P-401 are commercially protected.
Note that even though the design is based on a symmetric window, the obtained window coefficients
can be used to generate a periodic window and vice versa.
2. PROBLEM FORMULATION
A common way to construct windows is by using the summation of shifted Dirichlet kernels. A discrete
time window which is constructed using the summation of shifted Dirichlet kernels is given by
µ
¶
K−1
X
2πkn
k
w(n) =
(−1) ak cos
, n = 0, . . . , N − 1
(1)
Nx
k=0
where K denotes the number of window coefficients ak , N denotes the length of the window and Nx
determines if the window w(n) is symmetric or periodic. If w(n) is symmetric, then Nx = N − 1 and if
w(n) is periodic, then Nx = N .
The Fourier transform of Eq. (1) is given by
Ã
¶!
µ
N
−1 K−1
X
X
2πkn
k
W (ω) =
e−jωn
(−1) ak cos
N
x
n=0
(2)
k=0
By changing the summation order and rearranging the variables, we obtain
¶
µ
K−1
N
−1
X
X
2πkn −jωn
e
W (ω) =
(−1)k ak
cos
Nx
n=0
k=0
(3)
Eq. (3) can be written as
´
2πkn
1 ³ j 2πkn
e Nx + e−j Nx e−jωn
2
n=0
k=0
¶
µ
¶¶
µ µ
K−1
X
2πk
ak
2πk
(−1)k
+D ω+
=
D ω−
2
Nx
Nx
W (ω) =
K−1
X
(−1)k ak
N
−1
X
(4)
(5)
k=0
where D(ω) is known as the Dirichlet kernel [4] presented as
D(ω) = e−jω(
N −1
2
¡ N¢
) sin ¡ω 2 ¢
sin ω 12
(6)
Equation (5) clearly shows that any window constructed using Eq. (1) has a Fourier Transform which
consist of a summation of shifted Dirichlet kernels. Some of the most well known classic windows which
are constructed using Eq.(1) are for example the Hanning, Hamming and the Blackman window.
2.1. Vector Notation
Introducing vector notation, Eq.(3) can be written as
W (ω) = φT (ω)a
where





φ(ω) = 




PN −1
e³−jωn ´
PN −1
2π1n
e−jωn
n=0 (−1) cos
³ Nx ´
PN −1
2π2n
2
e−jωn
n=0 (−1) cos
Nx
..
. ³
´
PN −1
2π(K−1)n
(K−1)
(−1)
cos
e−jωn
n=0
Nx
n=0
1
(7)













, a=


a0
a1
a2
...






(8)
aK−1
where φ(ω) is a response vector and a the window coefficient vector. For a periodic window, i.e. Nx = N ,
φ(ω) in Eq.(8) becomes a complex valued vector. In contrast, if Nx = N −1, corresponding to a symmetric
window, φ(ω) can be made real valued. This is achieved by using the a priori information of the phase for
a length N symmetric window. The phase ϕ(ω) for a length N symmetric window is given a priori by
µ
¶
N −1
ϕ(ω) = −
ω
(9)
2
By removing the phase from from the response vector, φ(ω) can be made real valued. This is achieved by
multiplying the the response vector with e−jϕ(ω) .
3. THE WINDOW DESIGN SPECIFICATION
Consider the following window design specification
½
min ||φT (ω)a − Wd (ω)||∞ , ω ∈ Ωs
|φT (ω)a − Wd (ω)| ≤ σp (ω) ω ∈ Ωp
(10)
where Wd (ω) is the desired window specification, Ωp and Ωs denotes the passband and the stopband,
respectively and σp denotes the tolerance in the passband. The objective of Eq. (10) is to minimize the
max-norm error in the stopband, subject to the constraints given in the passband. A max-norm minimization
in the stopband corresponds to minimizing the peak sidelobe level. The minimization of the peak sidelobe
contributes to reduce the spectral leakage effect and reduces the amplitude error.
By introducing a new variable δ = max |φT (ω)a − Wd (ω)| = ||φT (ω)a − Wd (ω)||∞ , ω ∈ Ωs , Eq.
ω∈Ωs
(10) can be rewritten as

 min δ
|φT (ω)a − Wd (ω)| ≤ δ
ω ∈ Ωs
 T
|φ (ω)a − Wd (ω)| ≤ σp (ω) ω ∈ Ωp
(11)
Note that the extra independent variable δ is active only in the stopband and the amplitude tolerance σp (ω)
is only active in the passband. The non-linear constraints in Eq. (11) makes the problem difficult to solve
as it stands. Depending on whether the design is based on a symmetric or periodic window, Eq.(11) can
reformulated.
3.1. Periodic window, a semi-infinite formulation
In periodic window design, both φ(ω)a and Wd (ω) are complex valued in Eq.(11). The problem formulation in Eq. (11) corresponds to a non-linear optimization problem, which is difficult to solve as it stands.
According to the real rotation theorem [2], a magnitude inequality in the complex plane can be expressed
in the equivalent form
©
ª
|z| ≤ γ ⇐⇒ < zejθ ≤ γ , ∀θ ∈ [0, 2π]
(12)
where <{·} denotes the real part of {·} and the phase θ belongs to the infinite set Θ = [0, 2π]. By making
use of equation (12), the design problem can be formulated as

 min
©¡δ , subject to
¢ ª
< ©¡φT (ω)a − Wd (ω)¢ ejθ ª ≤ δ
ω ∈ Ωs , ∀θ ∈ Θ
(13)

< φT (ω)a − Wd (ω) ejθ ≤ σp (ω) ω ∈ Ωp , ∀θ ∈ Θ
The linear program in equation (13) is categorized semi-infinite since the number of unknown variables are
finite but the constraint set is infinite due to the continuous phase θ ∈ [0, 2π]. For practical implementation
issues, Ωp and Ωs are assumed to be a finite subsets of Ω = [0, π], i.e. ωi = [ω1 , ω2 , . . . , ωI ], i = 1, . . . , I.
In the discrete frequency domain the design problem can be formulated as

 min
©¡δ , subject to
¢ ª
< ©¡φT (ωi )a − Wd (ωi )¢ ejθ ª ≤ δ
ωi ∈ Ωs , i = {1, . . . , I} , ∀θ ∈ Θ
(14)

< φT (ωi )a − Wd (ωi ) ejθ ≤ σp (ωi ) ωi ∈ Ωp , i = {1, . . . , I}, ∀θ ∈ Θ
Observe that the the approximation problem in equation (14) is with respect to the complex valued formulation due to the continuous phase θ ∈ [0, 2π] and is not affected by the discretization of the frequency
domain.
3.2. Symmetric window, a finite-dimensional formulation
In symmetric window design, both φT (ω)a and Wd (ω) are real valued, after removing the constant phase
ϕ(ω), which is determined by the length of the window. By using the fact that an absolute magnitude
inequality constraint can be written as
½
x≤b
|x| ≤ b ⇔
,
(15)
−x ≤ b
eq. (11) becomes

min δ




 (φT (ω)a − Wd (ω))
−(φT (ω)a − Wd (ω))


 (φT (ω)a − Wd (ω))


−(φT (ω)a − Wd (ω))
≤
≤
≤
≤
δ
δ
σp (ω)
σp (ω)
ω
ω
ω
ω
∈ Ωs
∈ Ωs
∈ Ωp
∈ Ωp
(16)
For practical purpose of the implementation of the algorithm, Ωp and Ωs are assumed to be finite subsets of
the continuous frequency domain Ω = [0, π]. In the discrete frequency domain ωi , where i = 1, . . . , I, the
design problem can be formulated as

min δ




≤
δ
ωi ∈ Ωs
 (φT (ωi )a − Wd (ωi ))
−(φT (ωi )a − Wd (ωi )) ≤
δ
ωi ∈ Ωs
(17)

T

(φ
(ω
)a
−
W
(ω
))
≤
σ
(ω
)
ω
∈
Ω

i
d
i
p
i
i
p


−(φT (ωi )a − Wd (ωi )) ≤ σp (ωi ) ωi ∈ Ωp
Equation (17) can now be formulated as a minimax optimization program in standard form, which can
be solved using conventional available linear programming software, such as simplex and interior-point
methods. However, due to the huge number of constraints, conventional linear programming software
might be both time consuming and memory extensive.
4. DUAL NESTED COMPLEX APPROXIMATION ALGORITHM
The Dual Nested Complex Approximation (DNCA) algorithm has shown to be very efficient in solving
semi-infinite linear and quadratic programs [5, 6], i.e. optimization problem with a finite number of variables (unknowns) and infinite number of constraints. Even though the optimization problem as formulated
in Sec.3.2 is not semi-infinite, the DNCA algorithm is still a competitive choice since it is capable to reduce
the computation time significantly and is less memory consuming compared to conventional optimization
software.
This section presents the general framework of the DNCA scheme, which yields for finite and semiinfinite, linear or quadratic optimization problems.
4.1. The Approximation Problem
A general optimization problem can be formulated as

f (x),
 min
x
(P )
 gα (x) ≤ 0, n
x∈X ⊂R
α ∈ A ⊂ Rk
(18)
where x is an N × 1 variable vector, f (x) a convex continuous function, X a convex restriction set, A
an infinite or large index set as a compact subset of Euclidean k-space, and gα (x) a continuous constraint
function which is convex for any fixed index α.
4.2. The DNCA-LP Optimization Algorithm
The Dual Nested Complex Approximation Linear Programming algorithm to solve Eq. (18) is outlined
below. Let A(k) denote a sequence of subsets of the index set A and initialize the algorithm with the subset
A(0) .
1. Given A(k) ⊂ A, solve the subproblem
(
´
³
min f (x), x ∈ X
(k)
x
P
gαk (x) ≤ 0,
αk ∈ A(k)
(19)
yielding the solution vector xk and the Lagrange multiplier vector λk .
2. Reduce the subset by the inactive constraints
n
o
(k)
AR = A(k) \ αk ∈ A(k) |(λk ) = 0
(20)
where the Lagrange multipliers λk = 0 indicates the inactive constraints in the reference set.
3. Define the entering index α̂k and add it to the subset
α̂k = arg max gα (xk ),
α∈A
[
(k)
(k+1)
A
= AR
{α̂k }
(21)
(22)
and return to Step 1.
A practical stopping criteria is when |gα̂k (xk )| ≤ ε(xk ), where ε(xk ) > 0 is a tolerance parameter which
may depend on the current solution xk . If the tolerance is pre-defined, then ε(xk ) may be substituted with
the scalar ε.
The convergence of the algorithm to the optimal solution is non-trivial to show. A theoretical framework
to prove the global convergence of the DNCA algorithm is outlined in [7].
5. EXAMPLES
In this section, the flexibility of the presented design method using the DNCA algorithm is illustrated by
numerical examples. The first example considers the enhancement of the upcoming international standard
flattop window, the ISO 18431-2 flattop window.The second example considers the design of a flattop
window which is comparable with the P-401 flattop window, which can be found in Hewlett Packard’s
frequency analyzers.
5.1. Enhanced ISO Flattop Window
In year 2000, the International Organization for Standardization (ISO) initiated the work of ISO 18431-2,
Mechanical Vibration and Shock - Signal Processing Part 2 - Rectangular, Hanning, and Flattop Windows
for Fourier Transform Analysis [8]. The goal of the project is to standardize the three most commonly
used window functions for frequency analysis. One of the windows to be standardized is a flattop window,
denoted ISO flattop window, for mechanical vibration and shock analysis. The coefficient set for the ISO
flattop window is very similar to the Brüel and Kjær flattop window in the B & K Analyzers [9, 10]. The
ISO flattop window is given by equation (1), where the coefficients are given by

 

a0
1.0
 a1   1.933 

 

 

aISO = 
(23)
 a2  =  1.286 
 a3   0.388 
a4
0.0322
This example aims to design an Enhanced ISO (E-ISO) flattop window, in terms of higher sidelobe
attenuation without effecting the passband properties. Given the window coefficients, one can generate
wISO (n) and WISO (ω) using Eq.(1) and Eq.(3). The time-domain window is a periodic window, i.e.
N x = N since this window is used for sound and vibration measurements. This implies that the design
procedure should be based on a periodic window.
Given the ISO window coefficient in Eq.(23), the desired window specification for the enhancement is
extracted in the following steps:
1. Insert the ISO window coefficients into Eq.(3), to generate a periodic window, i.e. Nx = N .
2. Select a passband edge as the angular frequency ωp as
Wper (ωp ) = 2 − ||Wper (ω)||∞
ω ∈ [0, π]
(24)
3. Select the stopband edge as the angular frequency ωs on the mainlobe where the amplitude is equal
to the peak sidelobe level σs
4. Choose the tolerances in the passband and stopband as
σp (0) = 0
σp (ω) = ||Wper (ω)||∞ − 1
σs (ω) = σs = ||Wper (ω)||∞
ω ∈]0, ωp ]
ω ∈ Ωs
(25)
5. Finally, the desired window response Wd (ω) is defined as

 1
6
Wd (ω) =
ej

0
Wper (ω)
ω=0
ω ∈]0, ωp ]
ω ∈ Ωs
(26)
Note that Wd (ω) is a complex function, representing a specification on both the amplitude and phase within
the passband.
Inserting the determined parameters into Eq. (14), and using the DNCA algorithm described in Section
4, the following set of enhanced window coefficients can be obtained,

aE−ISO


=


1.00000000000000
1.93293488969227
1.28349769674027
0.38130801681619
0.02929730258511






(27)
Figure 1 shows the frequency response of the original ISO flattop window and the enhanced ISO flattop
window. The E-ISO window is slightly (not observable) flatter in the passband compared to the ISO flattop
window. The peak sidelobe level for the ISO window is approximately −84dB while the E-ISO window
peak sidelobe is −89dB, which implies an improvement of ∼ 5dB, see Fig. 2. A more conventional performance measure of windows is the equivalent noise bandwidth, ENBW [1]. The ENBW for the original ISO
and the E-ISO flattop window is 3.770 and 3.765 respectively.
5.2. Comparable P401 Flattop Window
The HP proprietary P401 flattop window is considered to be an outstanding flattop window, due to its high
sidelobe attenuation. The amplitude error for P-401 is around 0.01dB and its highest sidelobe is located
around −94dB. Since the window coefficients for the P-401 flattop window is unknown, the design must
be based on a symmetric window.
¡
¢
Recall the phase for a symmetric window with length N is given a priori by ϕ(w) = − N 2−1 ω. Since
no phase information for the periodic window is available, the most appropriate choice is to design the
window coefficient set based on a symmetric window since the phase is given a priori.
By using the window design specification:
½
Wd (ω) =
and
½
σp (ω) =
1
0
ω ∈ [0, ωp ]
ω ∈ [ωs , π]
0
100.01/20
ω=0
ω ∈]0, ωp ]
(28)
(29)
0
|W (ω)|
ISO
|W
(ω)|
E−ISO
dB
−50
−100
−150
0
0.1
0.2
0.3
0.4
0.5
ω/π
0.6
0.7
0.8
0.9
Fig. 1. Frequency domain representation of the original ISO and the enhanced ISO flattop window for
length N = 256.
−3
−70
x 10
|WISO(ω)|
(ω)|
|W
4
|WISO(ω)|
(ω)|
|W
−75
E−ISO
E−ISO
−80
3
−85
2
dB
dB
−90
1
−95
−100
0
−105
−1
−110
−2
−115
−3
0
0.5
1
1.5
2
ω/π
2.5
3
−120
3.5
−3
x 10
0.05
ω/π
0.1
0.15
Fig. 2. Frequency domain representation of the ISO window |WISO (ω)| and the E-ISO window
|WE−ISO (ω)| for length N = 256 in the passband (left) and at the stopband edge (right).
1
0
|W
(ω)|
Comp−P401
dB
−50
−100
−150
0
0.1
0.2
0.3
0.4
0.5
ω/π
0.6
0.7
0.8
0.9
Fig. 3. Frequency domain representation of the Comp-P401 flattop window |WComp−P 401 (ω)| for length
N = 256.
−3
5
−90
x 10
|WComp−P401(ω)|
|WComp−P401(ω)|
−92
4
−94
3
−96
2
−98
dB
dB
1
0
−100
−1
−102
−2
−104
−3
−106
−4
−108
−5
0
0.5
1
1.5
2
ω/π
2.5
3
3.5
4
−3
x 10
−110
0.05
0.1
0.15
ω/π
0.2
0.25
Fig. 4. Frequency domain representation of the Comp-P401 flattop window |WComp−P 401 (ω)| for length
N = 256 in the passband (left) and at the stopband edge (right).
1
and by setting K = 6, ωp = 0.002π, ωs = 0.02π and N = 256 yields the window coefficient set


1.00000000000000
 1.93774046310203 


 1.32530734987255 

aComp−P401 = 
(30)
 0.43206975880342 


 0.04359135851569 
0.00015175580171
The coefficient set is obtained by inserting the above specification into Eq.(17) and using the DNCA algorithm described in Section 4.
Given the coefficient set, the frequency response of a length N = 256 symmetric window can be
obtained by using Eq. (4), with Nx = N − 1. Figure 3 shows the frequency response of a periodic P401 comparable flattop window. According to available figures of merits for the P-401 flattop window, the
Comp-P401 flattop window have lower amplitude error (∼ 0.0023dB) and higher peak sidelobe attenuation.
The peak sidelobe level for the Comp-P401 windows is approximately ∼ −97dB, as seen in figure 4. For
the Comp-P401 flattop window, the ENBW is found to be ∼ 3.85
Note that even though the design is done on a symmetric window, the periodic window possesses almost
the same characteristics, in both passband and stopband. Therefore, it is concluded that given an amplitude
specification of a window, the engineered coefficients can be used to generate both a symmetric or periodic
window with approximately the same frequency characteristics.
6. CONCLUSIONS
This paper have presented a method for flattop window design and enhancement using the DNCA algorithm.
Two examples are included in this paper. The first example shows that the ISO 18431-2 flattop window
coefficient set is not optimal in the sense that the peak sidelobe can be further attenuated with ∼ 5dB,
without affecting the passband properties. The second example shows that the presented method can be
used to design new flattop windows, given an amplitude specification without explicit knowledge of the
phase. In such case, the design must be based on a symmetric window since the phase information is
absent. It can be shown that given a window coefficient set, the symmetric and periodic window possesses
almost the same frequency characteristics.
The choice of the DNCA optimization algorithm is twofold. Depending on whether the design is based
on a periodic or symmetric window results in two different types of optimization problems, semi-infinite
optimization or finite-dimensional optimization with a huge number of constraints. The DNCA algorithm
is capable to solve both problems. The main reason for the choice of the DNCA algorithm is because it
reduces the computational time significantly and is less memory consuming compared to other conventional
available optimization algorithms.
7. REFERENCES
[1] Harris, F.J. (1978), “On the use of windows for harmonic analysis with the discrete Fourier transform”,
Proc. of the IEEE, Vol. 66, pp. 51-83.
[2] Parks, T.W., Burrus, C.S. (1987), Digital Filter Design, Wiley-Interscience, ISBN 0-471-82896-3.
[3] Oppenheim, A.V., Schafer, R.W. and Buck, J.R. (1998), Discrete-time signal processing, second edition, Prentice Hall, ISBN 0-13-754920-2.
[4] Lanczos, C., (1966), Discourse on Fourier Series, Hafner Publishing Co.
[5] Tran, T., Dahl, M., and Claesson, I. (2003), “A semi-infinite quadratic programming algorithm with
applications to channel equalization”, Proc. of the 7th International Symposium on Signal Processing
and its Applications, ISSPA2003, Paris France, July, Vol. 1, pp. 653-656.
[6] Dahl, M., Claesson, I., and Nordebo, S. (2003), “Antenna array design using dual nested complex
approximation”, Proc. of the 7th International Symposium on DSP for Communication Systems,
DSPCS2003, Coolangatta, Australia, December.
[7] Nordebo, S., Dahl, M., and Claesson, I. (2001), “Complex approximation with applications to antenna
array pattern synthesis”, Proc. of Electromagnetic Computations EMB01, Uppsala, Sweden, November.
[8] Draft International Standard (2003), “Mechanical vibration and shock Signal processing Part 2: Time
domain windows for Fourier transform analysis”, International Organization for Standardization
[9] Gade, S., Herlufsen, H. (1987), “Technical Review - Windows to FFT Analysis (Part I)”, Br ü el &
Kjær Technical Review No.3
[10] Gade, S., Herlufsen, H. (1987), “Technical Review - Windows to FFT Analysis (Part II)”, Br ü el &
Kjær Technical Review No.4