Window Functions with a Quasi-Rectangular Spectrum

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.

Appendices

Hereinafter, the listing format of the programs is preserved.

APPENDIX A

Hereinafter, the listing format of the programs is preserved.

The Roots of Generalized Laguerre Polynomial $L_{N}^{{\left( {\frac{1}{2}} \right)}}\left( {{{t}^{2}}} \right)$

laguerreL(3,0.5,t^2)/laguerreL(3,0.5,0) = –0.0761905t^6 + 0.8 t^4 – 2. t^2 + 1

${{r}_{{1,2}}}$ = ±0.816288, ${{r}_{{3,4}}}$ = ±1.67355, ${{r}_{{5,6}}}$ = ±2.65196

laguerreL(4,0.5,t^2)/laguerreL(4,0.5,0) = 1/945(16. t^8 – 288. t^6 + 1512. t^4 – 2520.      t^2 + 945)

${{r}_{{1,2}}}$ = ±0.723551, ${{r}_{{3,4}}}$ = ±1.46855, ${{r}_{{5,6}}} =$ ±2.26658, ${{r}_{{7,8}}}$ = ±3.19099

laguerreL(5,0.5,t^2)/laguerreL(5,0.5,0) = –0.0030784 t^10 + 0.0846561 t^8 – 0.761905     t^6 + 2.66667 t^4 – 3.33333 t^2 + 1

${{r}_{{1,2}}}$ = ±0.65681,    ${{r}_{{3,4}}}$ = ±1.32656,    ${{r}_{{5,6}}}$ = ±2.02595,    ${{r}_{{7,8}}}$ = ±2.78329,    ${{r}_{{9,10}}}$ = ±3.66847

laguerreL(6,0.5,t^2)/laguerreL(6,0.5,0)= 0.0004736 t^12 – 0.0184704 t^10 + 0.253968      t^8 – 1.52381 t^6 + 4. t^4 – 4. t^2 + 1

${{r}_{{1,2}}}$ = ±0.605764,    ${{r}_{{3,4}}}$ = ±1.22006,    ${{r}_{{5,6}}}$ = ±1.85311,    ${{r}_{{7,8}}}$ = ±2.51974,     ${{r}_{{9,10}}}$ = ±3.24661, ${{r}_{{11,12}}}$ = ±4.10134

laguerreL(7,0.5,t^2)/laguerreL(7,0.5,0) = –0.0000631467 t^14 + 0.0033152 t^12 –      0.0646465 t^10 + 0.592593 t^8 – 2.66667 t^6 + 5.6 t^4 – 4.66667 t^2 + 1

${{r}_{{1,2}}}$ = ±0.56507, ${{r}_{{3,4}}}$ = ±1.13612, $~{{r}_{{5,6}}}$ = ±1.71999, ${{r}_{{7,8}}}$ = ±2.32573,     ${{r}_{{9,10}}}$ = ±2.96717, ${{r}_{{11,12}}}$ = ±3.66995, ${{r}_{{13,14}}}$ = ±4.49999

laguerreL(8,0.5,t^2)/laguerreL(8,0.5,0) = 1/135 (0.00100292 t^16 – 0.0681985 t^14 +    1.79021 t^12 – 23.2727 t^10 + 160. t^8 – 576. t^6 + 1008. t^4 – 720. t^2 + 135)

${{r}_{{1,2}}}$ = ±0.531633, ${{r}_{{3,4}}}$ = ±1.06765, $~{{r}_{{5,6}}}$ = ±1.61292, ${{r}_{{7,8}}}$ = ±2.1735,     ${{r}_{{9,10}}}$ = ±2.75776, ${{r}_{{11,12}}}$ = ±3.37893, ${{r}_{{13,14}}}$ = ±4.06195, ${{r}_{{15,16}}}$ = ±4.87135

laguerreL(9,0.5,t^2)/laguerreL(9,0.5,0) = –7.82003 × 10^–7 t^18 + 0.0000668612 t^16 –      0.00227328 t^14 + 0.0397824 t^12 – 0.387879 t^10 + 2.13333 t^8 – 6.4 t^6 + 9.6 t^4 –      6. t^2 + 1

${{r}_{{1,2}}}$ = ±0.50352, ${{r}_{{3,4}}}$ = ±1.01037, ${{r}_{{5,6}}}$ = ±1.52417, ${{r}_{{7,8}}}$ = ±2.04923,    ${{r}_{{9,10}}}$ = ±2.59113, ${{r}_{{11,12}}}$ = ±3.15785, ${{r}_{{13,14}}}$ = ±3.76219, ${{r}_{{15,16}}}$ = ±4.42853,    ${{r}_{{17,18}}}$ = ±5.22027

laguerreL(10,0.5,t^2)/laguerreL(10,0.5,0) = (0.000258061 t^20 – 0.0270964 t^18 +    1.15837 t^16 – 26.2564 t^14 + 344.615 t^12 – 2688. t^10 + 12320. t^8 – 31680. t^6 +    41580. t^4 – 23100. t^2 + 3465)/3465

${{r}_{{1,2}}}$ = ±0.479451, ${{r}_{{3,4}}}$ = ±0.9615, $~{{r}_{{5,6}}}$ = ±1.44893, ${{r}_{{7,8}}}$ = ±1.94496,     ${{r}_{{9,10}}}$ = ±2.45355, ${{r}_{{11,12}}}$ = ±2.97999, ${{r}_{{13,14}}}$ = ±3.53197, ${{r}_{{15,16}}}$ = ±4.122,    ${{r}_{{17,18}}}$ = ±4.77399, ${{r}_{{19,20}}}$ = ± 5.55035

laguerreL(11,0.5,t^2)/laguerreL(11,0.5,0) = –6.47621 × 10^–9 t^22 + 8.19241 × 10^–7      t^20 – 0.0000430102 t^18 + 0.00122579 t^16 – 0.0208384 t^14 + 0.218803 t^12 –    1.42222 t^10 + 5.5873 t^8 – 12.5714 t^6 + 14.6667 t^4 – 7.33333 t^2 + 1

${{r}_{{1,2}}}$ = ±0.458538, ${{r}_{{3,4}}}$ = ±0.919151, ${{r}_{{5,6}}}$ = ±1.38404, ${{r}_{{7,8}}}$ = ±1.85568,    ${{r}_{{9,10}}}$ = ±2.33702, ${{r}_{{11,12}}}$ = ±2.8318, ${{r}_{{13,14}}}$ = ±3.34513, ${{r}_{{15,16}}}$ = ±3.88447,    ${{r}_{{17,18}}}$ = ±4.46209, ${{r}_{{19,20}}}$ = ±5.10153, ${{r}_{{21,22}}}$ = ±5.86431

laguerreL(12,0.5,t^2)/laguerreL(12,0.5,0) = (2.12161 × 10^ – 6 t^24 – 0.000318241 t^22    + 0.0201288 t^20 – 0.704506 t^18 + 15.0588 t^16 – 204.8 t^14 + 1792. t^12 – 9984.     t^10 + 34320. t^8 – 68640. t^6 + 72072. t^4 – 32760. t^2 + 4095)/4095

${{r}_{{1,2}}}$ = ±0.440147, ${{r}_{{3,4}}}$ = ±0.881983, ${{r}_{{5,6}}}$ = ±1.32728, ${{r}_{{7,8}}}$ = ±1.778,    ${{r}_{{9,10}}}$ = ±2.23642, ${{r}_{{11,12}}}$ = ±2.70532, ${{r}_{{13,14}}}$ = ±3.18829, ${{r}_{{15,16}}}$ = ±3.69028,

${{r}_{{17,18}}}$ = ±4.21861, ${{r}_{{19,20}}}$ = ±4.78532, ${{r}_{{21,22}}}$ = ±5.41364, ${{r}_{{23,24}}}$ = ±6.16427

APPENDIX B

Programs (Matlab 6.5)

1. Spectra G(w) (see Fig. 1).

syms w G;

figure(1)

hold on

for k = 1:6

G(k,:) = taylor(exp(w.^2/4),12*(k – 1) + 1)*exp(–w.^2/4);

ezplot(G(k,:),0,14)

end

grid on

w = 0:0.01:14;

G = (1 + 1/4*w.^2 + 1/32*w.^4 + 1/384*w.^6).*exp(–w.^2/4);

plot(w,G)

grid on

xlabel('w')

ylabel('G ( w )')

title('W i n d o w s G (N = 0) a n d G T (N = 3, 6, 12, 18, 24, 30')

$2.~~{\text{The computation of}}~~{{w}_{1}}~~{\text{ for }}~~{{a}_{1}} = {1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }}$.

syms w G;

G = taylor(exp(w^2/4),13)*exp(–w^2/4);

solve(G–1/(2^0.5),w)

solve is an operator for solving the algebraic equation (G – ${1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }}$ = 0)

3. The logarithm of the modulus of normalized function g(t) (N = 9)${{\log }_{{10}}}\left| {{{g\left( t \right)} \mathord{\left/ {\vphantom {{g\left( t \right)} {g\left( 0 \right)}}} \right. \kern-0em} {g\left( 0 \right)}}} \right|~$.

Fs =1024;

t = –5.22027:1/Fs: 5.22027;

g = exp(–t.^2).*(–7.82003*10^–7* t.^18 + 0.0000668612* t.^16 – 0.00227328* t.^14 + 0.0397824* t.^12 – 0.387879* t.^10 + 2.13333* t.^8 – 6.4* t.^6 + 9.6* t.^4 – 6* t.^2 + 1);

plot(t,log10(abs(g)));

title('W i n d o w G T (N = 9 ) i n T i m e D o m a i n (L o g S c a l e' ))

xlabel('T i m e ( t )')

ylabel('abs(g (t ))');

4. The spectrum of function g(t) (N = 5) for various interval lengths.

Fs = 1024;

t1 = –1.32656: 1/Fs: 1.32656;

t2 = –2.02595: 1/Fs: 2.02595;

t3 = –2.78329: 1/Fs: 2.78329;

t4 = –3.66847: 1/Fs: 3.66847;

g1 = exp(–t1.^2).*(–0.0030784 *t1.^10 + 0.0846561* t1.^8 – 0.761905* t1.^6 + 2.66667* t1.^4 – 3.33333* t1.^2 + 1);

g2 = exp(–t2.^2).*(–0.0030784 *t2.^10 + 0.0846561* t2.^8 – 0.761905* t2.^6 + 2.66667* t2.^4 – 3.33333* t2.^2 + 1);

g3 = exp(–t3.^2).*(–0.0030784 *t3.^10 + 0.0846561* t3.^8 – 0.761905* t3.^6 + 2.66667* t3.^4 – 3.33333* t3.^2 + 1);

g4 = exp(–t4.^2).*(–0.0030784 *t4.^10 + 0.0846561* t4.^8 – 0.761905* t4.^6 + 2.66667* t4.^4 – 3.33333* t4.^2 + 1);

n = 131072;

F1 = fft(g1,n);

F2 = fft(g2,n);

F3 = fft(g3,n);

F4 = fft(g4,n);

f = Fs*(0:(n/2))/n;

P1 = 20*log10(abs(F1/F1(1)));

P2 = 20*log10(abs(F2/F2(1)));

P3 = 20*log10(abs(F3/F3(1)));

P4 = 20*log10(abs(F4/F4(1)));

plot(f,P1(1:n/2 + 1),f,P2(1:n/2 + 1), f,P3(1:n/2 + 1),f,P4(1:n/2 + 1)),grid

title('W i n d o w s G T (N = 5) i n F r e q u e n c y D o m a i n')

xlabel('F r e q u e n c y ( f )')

ylabel('G (f ), d B');

5. The GT window functions (N = 3, N = 6, and N = 12) (see Fig. 4, curves 1–3) and their respective spectra (see Fig. 5).

Fs = 1024;

x = –0.5:1/Fs:0.5;

u = exp(–(x*5.3039).^2).*(–1696.38* x.^6 + 633.146* x.^4 – 56.2648 *x.^2 + 1);

u1 = exp(–(x*2.51974*2).^2).*(1 – 101.585* x.^2 + 2579.9* x.^4 – 24960* x.^6 + 105649* x.^8 – 195134* x.^10 + 127069* x.^12);

u2 = exp(–(x*2.70532*2).^2).*(2.05295*10^8* x.^24 – 1.0519*10^9 *x.^22 + 2.27267*10^9* x.^20 – 2.71711*10^9 *x.^18 + 1.98388*10^9 *x.^16 – 9.21631*10^8 *x.^14 + 2.75466*10.^8* x.^12 – 5.24248*10^7* x.^10 + 6.15577*10^6 *x.^8 – 420548* x.^6 + 15083.7* x.^4 – 234.2* x.^2 + 1);

n = 131072;

F = fft(u,n);

F1 = fft(u1,n);

F2 = fft(u2,n);

f = Fs*(0:(n/2))/n;

P = 20*log10(6.5*abs(F/Fs));

P1 = 20*log10(9*abs(F1/Fs));

P2 = 20*log10(12*abs(F2/Fs));

plot(f,P(1:n/2+1),f,P1(1:n/2 + 1),f,P2(1:n/2 + 1)), grid

title('Windows GT (N = 3, N = 6, N = 12) in Frequency Domain’)

xlabel('Frequency (y)')

ylabel('F(y),dB')

6. The comparison of the GTM and GT windows (see Fig. 6) and their spectra (see Fig. 7).

Fs = 1024;

x = –0.5:1/Fs:0.5;

v = x.*exp(–1.68294*x.^2–0.6341278*x.^4);

u =exp(–(x*5.3039.*(1 + 1.5*x.^6 + 2*x.^4)).^2).*( –1696.38*v.^6 + 633.146*v.^4 – 56.2648 *v.^2 + 1);

u1 = exp(–(x*5.3039).^2).*(–1696.38* x.^6 + 633.146* x.^4 – 56.2648 *x.^2 + 1);

plot(x,u,x,u1),grid;

title('W i n d o w s G T M (N = 3) a n d G T (N = 3) i n T i m e D o m a i n' )

legend('G T M','G T')

xlabel('T i m e (x)')

ylabel('u (x)');

n = 131072;

F = fft(u,n);

F1 = fft(u1,n);

f = Fs*(0:(n/2))/n;

P = 20*log10(abs(F/F(1)));

P1 = 20*log10(abs(F1/F1(1)));

plot(f,P(1:n/2 + 1),f,P1(1:n/2 + 1)),grid

title('W i n d o w s G T M (N = 3) a n d G T (N = 3) i n F r e q u e n c y D o m a i n' )

legend('G T M','G T')

xlabel('F r e q u e n c y ( y )')

ylabel('F ( y ), d B');

Fs = 1024;

x = –0.5:1/Fs:0.5;

v = x.*exp(–$0.540777$*x.^2–2.918446*x.^4);

u = exp(–(x*5.56658.*(1 + 1.5*x.^6 + 2*x.^4)).^2).*(–87944.8*v.^10 + 78048.7*v.^8 – 22669*v.^6 + 2560.49*v.^4 – 103.289*v.^2 + 1);

u1 = exp(–(x*5.56658).^2).* (–87944.8*x.^10 + 78048.7*x.^8 – 22669*x.^6 + 2560.49*x.^4 – 103.289*x.^2 + 1);

plot(x,u,x,u1),grid ;

title('W i n d o w s G T M (N = 5) a n d G T (N = 5) i n T i m e D o m a i n')

legend('G T M','G T')

xlabel('T i m e ( x )')

ylabel('u ( x )');

n = 131072;

F = fft(u,n);

F1 = fft(u1,n);

f = Fs*(0:(n/2))/n;

P = 20*log10(abs(F/F(1)));

P1 = 20*log10(abs(F1/F1(1)));

plot(f,P(1:n/2 + 1),f,P1(1:n/2 + 1)),grid

title('W i n d o w s G T M (N = 5) a n d G T (N = 5) i n F r e q u e n c y D o m a i n')

legend('G T M','G T')

xlabel('F r e q u e n c y (y)')

ylabel('F ( y ), d B');

Fs = 1024;

x = –0.5:1/Fs:0.5;

v = x.*exp(–0.413772*x.^2–3.172456*x.^4);

u = exp(–(4.6515*x.*(1 – 0.413772*x.^2 + 4*x.^4)).^2).*( –140169* v.^14 + 340116* v.^12 – 306531* v.^10 + 129867*v.^8 – 27010.1* v.^6 + 2621.56 *v.^4 – 100.97* v.^2 + 1);

u1 = exp(–(x*4.6515).^2).*( –140169* x.^14 + 340116* x.^12 – 306531* x.^10 + 129867* x.^8 – 27010.1* x.^6 + 2621.56 *x.^4 – 100.97* x.^2 + 1);

plot(x,u,x,u1),grid ;

title('W i n d o w s G T M (N = 7) a n d G T (N = 7) i n T i m e D o m a i n')

legend('G T M','G T')

xlabel('T i m e (x)')

ylabel('u (x)');

n = 131072;

F = fft(u,n);

F1 = fft(u1,n);

f = Fs*(0:(n/2))/n;

P = 20*log10(abs(F/F(1)));

P1 = 20*log10(abs(F1/F1(1)));

plot(f,P(1:n/2 + 1),f,P1(1:n/2 + 1)),grid

title('W i n d o w s G T M (N = 7) a n d G T (N = 7) i n F r e q u e n c y D o m a i n')

legend('G T M','G T')

xlabel('F r e q u e n c y (y)')

ylabel('F (y), d B');

Fs = 1024;

x = –0.5:1/Fs:0.5;

v = x.*exp(–1.357794*x.^2 – 1.284411*x.^4);

u = exp(–(x*5.4106.*(1 + 1.5*x.^6 + 2*x.^4)).^2).*(2.05259*10^8 *v.^24 – 1.05172*10^9*v.^22 + 2.27233*10^9*v.^20 – 2.71674*10^9*v.^18 + 1.98365*10^9 *v.^16 – 9.21535*10^8*v.^14 + 2.75441*10^8*v.^12 – 5.2421*10^7*v.^10 + 6.15541*10^6 *v.^8 – 420529*v.^6 + 15083.2*v.^4 – 234.197*v.^2 + 1);

u1 = exp(–(x*5.4106).^2).* (2.05259*10^8 *x.^24 – 1.05172*10^9*x.^22 + 2.27233*10^9*x.^20 – 2.71674*10^9*x.^18 + 1.98365*10^9 *x.^16 – 9.21535*10^8* x.^14 + 2.75441*10^8*x.^12 – 5.2421*10^7*x.^10 + 6.15541*10^6 *x.^8 – 420529*x.^6 + 15083.2*x.^4 – 234.197*x.^2 + 1);

plot(x,u,x,u1),grid ;

title('W i n d o w s G T M (N = 12) a n d G T (N = 12) i n T i m e D o m a i n')

legend('G T M','G T')

xlabel('T i m e (x)')

ylabel('u (x)');

n = 131072;

F = fft(u,n);

F1 = fft(u1,n);

f = Fs*(0:(n/2))/n;

P = 20*log10(abs(F/F(1)));

P1 = 20*log10(abs(F1/F1(1)));

plot(f,P(1:n/2 + 1),f,P1(1:n/2+1)),grid

title('W i n d o w s G T M (N = 12) a n d G T (N = 12) i n F r e q u e n c y D o m a i n')

legend('G T M','G T')

xlabel('F r e q u e n c y (y)')

ylabel('F (y), d B');

APPENDIX C

The Results of Computing Certain Integrals to Evaluate the Parameters of Window Functions (Wolfram Alpha)

integral_(–0.5)^0.5 (exp(–(5.3039 x)^2) L_3^0.5((5.3039 x)^2))/(L_3^0.5(0)) dx = 0.152991

integral_(–0.5)^0.5 ((exp(–(5.3039 x)^2) L_3^0.5((5.3039 x)^2))/(L_3^0.5(0)))^2 dx = 0.130915

integral_(–0.5)^0.5 (exp(–(5.3039 x)^2) L_3^0.5((5.3039 x)^2) cos(π x))/(L_3^0.5(0)) dx = 0.152713

integral_(–0.5)^0.5 (exp(–(4.53316 x)^2) L_4^0.5((4.53316 x)^2))/(L_4^0.5(0)) dx = 0.159561

integral_(–0.5)^0.5 ((exp(–(4.53316 x)^2) L_4^0.5((4.53316 x)^2))/(L_4^0.5(0)))^2 dx = 0.138604

integral_(–0.5)^0.5 (exp(–(4.53316 x)^2) L_4^0.5((4.53316 x)^2) cos(π x))/(L_4^0.5(0)) dx = 0.158743

integral_(–0.5)^0.5 (exp(–(5.56658 x)^2) L_5^0.5((5.56658 x)^2))/(L_5^0.5(0)) dx = 0.117523

integral_(–0.5)^0.5 ((exp(–(5.56658 x)^2) L_5^0.5((5.56658 x)^2))/(L_5^0.5(0)))^2 dx = 0.103945

integral_(–0.5)^0.5 (exp(–(5.56658 x)^2) L_5^0.5((5.56658 x)^2) cos(π x))/(L_5^0.5(0)) dx = 0.117639

integral_(–0.5)^0.5 (exp(–(5.0395 x)^2) L_6^0.5((5.0395 x)^2))/(L_6^0.5(0)) dx = 0.119713

integral_(–0.5)^0.5 ((exp(–(5.0395 x)^2) L_6^0.5((5.0395 x)^2))/(L_6^0.5(0)))^2 dx = 0.107035

integral_(–0.5)^0.5 (exp(–(5.0395 x)^2) L_6^0.5((5.0395 x)^2) cos(π x))/(L_6^0.5(0)) dx = 0.119961

integral_(–0.5)^0.5 (exp(–(4.6515 x)^2) L_7^0.5((4.6515 x)^2))/(L_7^0.5(0)) dx = 0.120908

integral_(–0.5)^0.5 ((exp(–(4.6515 x)^2) L_7^0.5((4.6515 x)^2))/(L_7^0.5(0)))^2 dx = 0.109086

integral_(–0.5)^0.5 (exp(–(4.6515 x)^2) L_7^0.5((4.6515 x)^2) cos(π x))/(L_7^0.5(0)) dx = 0.121318

integral_(–0.5)^0.5 (exp(–(4.0985 x)^2) L_9^0.5((4.0985 x)^2))/(L_9^0.5(0)) dx = 0.12202

integral_(–0.5)^0.5 ((exp(–(4.0985 x)^2) L_9^0.5((4.0985 x)^2))/(L_9^0.5(0)))^2 dx = 0.111704

integral_(–0.5)^0.5 (exp(–(4.0985 x)^2) L_9^0.5((4.0985 x)^2) cos(π x))/(L_9^0.5(0)) dx = 0.122793

integral_(–0.5)^0.5 (exp(–(5.4106 x)^2) L_12^0.5((5.4106 x)^2))/(L_12^0.5(0)) dx = 0.0812328

integral_(–0.5)^0.5 ((exp(–(5.4106 x)^2) L_12^0.5((5.4106 x)^2))/(L_12^0.5(0)))^2 dx = 0.0749071

integral_(–0.5)^0.5 (exp(–(5.4106 x)^2) L_12^0.5((5.4106 x)^2) cos(π x))/(L_12^0.5(0)) dx = 0.0813022