import numpy as np
from numpy.fft import fft, ifft, ifftshift
import matplotlib.pyplot as plt

def fft_conv_and_center(x0, x1):
    return ifftshift(ifft(fft(x0) * fft(x1))).real

def formula(t, A, B):
    s = lambda x: np.sign(x)
    p = A + B
    m = A - B

    K = (1/4)*s(A*B)
    o0 = (p - 2*t)*s(p - 2*t)  - (m + 2*t)*s(m/2 + t)
    o1 = (m - 2*t)*s(-m/2 + t) + (p + 2*t)*s(p/2 + t)
    return K*(o0 + o1)

def boxcar(A, t):
    A = abs(A)
    return ((t >= -A/2) * (t <= A/2)).astype(int)

N = 4096
A, B = 10, 4
t = np.linspace(-A*B, A*B, N, 0)

x0 = boxcar(A, t)
x1 = boxcar(B, t)

o_fftconv = fft_conv_and_center(x0, x1) * 2 * np.abs(A * B) / N
o_formula = formula(t, A, B)

plot(o_fftconv, w=.7, h=.9)
plot(o_formula, title="FFTconv(x0, x1), formula(x0, x1) | N=%s" % N,
     show=1)
plot(x0, w=.7, h=.9)
plot(x1, title="x0, x1 | A=%s, B=%s" % (A, B))