from scipy.special.basic import sinc, j0
from numpy.core import arange, pi, sum, sqrt
import pylab

a = -50.0
b = +50.0
w = 1.0/pi
dt = 0.1;
N = 10;

t = arange(a, b, dt)
t.shape = (-1,1)
y = j0(t)

s = arange(-N,N+1)/w
s.shape = (1,-1)
x = j0(s)
z = sum(x*sinc(w*(t-s)), 1)

golden_mean = (sqrt(5)-1.0)/2.0
pylab.rcParams.update( \
  {'backend': 'ps',
   'ps.usedistiller': 'xpdf',
   'axes.labelsize': 8,
   'text.fontsize': 8,
   'xtick.labelsize': 8,
   'ytick.labelsize': 8,
   'figure.figsize': [ 6.0, golden_mean*6.0 ], })

pylab.figure()
pylab.plot(t, y, color='blue', label='Original function')
pylab.plot(t, z, color='green', label='Approximate reconstruction')
pylab.plot(s, x, 'g.', label='_nolegend_')
pylab.legend()
pylab.savefig('bessel.eps')

t.shape = (-1,1)
s_under = s*2.0
s_over = s/2.0
x_under = j0(s_under)
x_over = j0(s_over)
z_under = sum(x_under*sinc((w/2.0)*(t-s_under)), 1)
z_over = sum(x_over*sinc((w*2.0)*(t-s_over)), 1)
pylab.figure()
pylab.plot(t, y, color='blue', label='Original function')
pylab.plot(t, z_under, color='red', label='Under-sampled')
pylab.plot(t, z_over, color='green', label='Over-sampled')
pylab.plot(s_under, x_under, 'rx', label='_nolegend_')
pylab.plot(s_over, x_over, 'g.', label='_nolegend_')
pylab.legend()
pylab.savefig('bessel-extra.eps')

t = arange(-5, 5, dt)
pylab.figure()
pylab.plot(
  t, sinc(w*t), \
  t, 0.5*sinc(t - 1),
  t, 0.5*sinc(t + 1),
  t, 0.25*sinc(t - 2),
  t, 0.25*sinc(t + 2))
pylab.savefig('basis.eps')

pylab.show()