Window Functions with a Quasi-Rectangular Spectrum
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RY AND METHODS OF SIGNAL PROCESSING
Window Functions with a Quasi-Rectangular Spectrum Z. D. Lerner* Journal of Communications Technology and Electronics, Moscow, 125009 Russia *e-mail: [email protected] Received August 19, 2019; revised February 19, 2020; accepted March 23, 2020
Abstract—The discrete Fourier transform (DFT) is carried out on a bounded time interval, which is equivalent to multiplying a signal under analysis by a rectangular window whose spectrum has the largest (among the other windows) side lobes. As a result, the effect called spectral leakage occurs. Multiplying the signal by the smoothing window reduces the splatter of its spectral components. There are several dozen variants of window functions as the DFT is used in a variety of problems. Of particular interest is the unique flat-top window, which is employed in broadband applications. In this study, we propose a new adjustable window with a quasi-rectangular spectrum and a method to reduce the level of the spectrum’s side lobes. DOI: 10.1134/S1064226920080070
INTRODUCTION To synthesize the window, we consider, as is customary [1], real-valued even functions g (t ) . Their spectra G ( ω) are also real-valued even frequency functions. According to GOST 24375-80, rectangularity coefficient K r = ( Δf )1 ( Δf )2 of frequency response is a ratio between the filter’s bandwidth with respect to level a1 = 1 2 and its bandwidth with respect to level a2 = 0.1. For a physically impossible ideal filter with rectangular frequency response, K r = 1. Let us extend the concept of rectangularity coefficient to the spectra of window functions. As an example, we consider Gaussian pulse
gG (t ) = 1 exp(−t 2 ) , − ∞ < t < ∞, π whose spectrum is
=
(
)
gG (t )exp ( − j ωt ) dt = exp −ω2 4 .
−∞
F ( f ) = sinc(πf ) 4
d
+ 0.5
k =1
[sinc (π ( f + k )) + sinc (π ( f − k ))] ,
(3)
where
f = ω ; d1 = 1.932; d2 = 1.29; 2π d3 = 0.388; d4 = 0.03.
(4)
(1)
(2)
1. PREREQUISITES FOR THE SYNTHESIS OF A GAUSSIAN–TAYLOR WINDOW The new window, let us call it an Nth order Gaussian–Taylor (GT) window, is given by spectrum k
ω2 N 4 exp −ω2 4 G ( ω) = k! k =0 2 2 = taylor exp ω 4 , 2N + 1 exp −ω 4 .
By solving algebraic equations
GG (ω1) = a1 = 1 2 and GG (ω2 ) = a2 = 0.1, we obtain ω1 = 0.3880. ω2 It can be seen that the rectangularity coefficient is far from one (just as the shape of the pulse’s spectrum is far from rectangular). Kr =
k
The shape of the window’s spectrum is closer to rectangular; however, its rectangularity coefficient is merely K r = 0.56. Let us synthesize an adjustable window with higher K r . All computations in this study were carried out using Matlab 6.5 and Wolfram Alpha.
GG ( ω) = GG ( j ω) ∞
The spectrum of the flat top window ([1], 2.4.3) is
(
)
(5)
( ) ( ) taylor exp ( ω 4 ) , 2N + 1 is a 2N th degree
2 Here, polynomial obtained by truncating the Taylor (Maclaurin) series for exponent exp ω2 4 . This means that taylor (…, 2N + 1) is an operator that is
1010
(
)
1.0 0.9 0.8 0.7 0.6 0.5 0.4
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