def fft_for_MajorA_signature(a_arr, MajorA_2d_sorted_arr, num_of_majorA ):
binary_flag = 1
MajorA_flag_arr = seperate_a_arr_in_MajorA_vs_MinorA(a_arr, MajorA_2d_sorted_arr, num_of_majorA, binary_flag)
MajorA_flag_fft_arr = np.fft.fft(MajorA_flag_arr)
target_num = 0 # not in use
plt.figure()
plot_one_list(MajorA_flag_arr, target_num, a_arr, 'b+')
plt.title('MajorA_flag_arr')
plt.figure()
x_arr = range(0, len(MajorA_flag_fft_arr))
plot_one_list(MajorA_flag_fft_arr.real, target_num, x_arr, 'b+')
plt.title('MajorA_flag_fft_arr')
return MajorA_flag_fft_arr, MajorA_flag_arr
F1aN_aN_t2, a_arr2 = sum_log_epa(N, M, arc2)
tiny_adjustment = 0.000000001
print 'prepare_MajorA_2d_sorted_arr_for_plotting'
arc_separate_pts_x_list, arc_signature_list, num_of_majorA = prepare_MajorA_2d_sorted_arr_for_plotting(MajorA_2d_sorted_arr, tiny_adjustment)
# if we plot arc_signature_list directly, often it is too small, hard to see it
# We adjust it by multiply N/10.
# arc_signature_list_adjust_by_N = (N/10)*np.array(arc_signature_list)
arc_signature_list_adjust_by_N = 1*np.array(arc_signature_list)
v_bound_arr = vino_bound_arr(a_arr, N)
# since the difference for t1 and t2 is small, we use data for t1, which is F1aN_aN_t1, in the following plotting.
F1aN_aN_in_MajorA, F1aN_aN_in_MinorA = seperate_data_in_MajorA_vs_MinorA(F1aN_aN_t1, a_arr, MajorA_2d_sorted_arr)
a_flag_arr = seperate_a_arr_in_MajorA_vs_MinorA(a_arr, MajorA_2d_sorted_arr, num_of_majorA)
print "======= before plotting ========= "
plot_F1aN_for_diff_two_t(F1aN_aN_t1, F1aN_aN_t2, N, M, a_arr, a_arr2, arc_signature_list_adjust_by_N, arc_separate_pts_x_list )
plot_F1aN_MajorA_MinorA(N, M, a_arr, arc_signature_list_adjust_by_N, arc_separate_pts_x_list, F1aN_aN_in_MajorA, F1aN_aN_in_MinorA, F_N_a_q_x_list, a_q_x_list, F_N_a_q_list, a_q_list, v_bound_arr)
plot_arc_signature_list(N, M, a_arr, arc_separate_pts_x_list, arc_signature_list, a_flag_arr)
plot_phases(F1aN_aN_t1, F1aN_aN_t2, N, M, a_arr, a_arr2, arc_signature_list_adjust_by_N, arc_separate_pts_x_list, F1aN_aN_in_MajorA, F1aN_aN_in_MinorA, F_N_a_q_x_list, a_q_x_list)
plot_fft(F1aN_aN_t1, F1aN_aN_in_MajorA, F1aN_aN_in_MinorA, N)
verify_fft_conv_relation(F1aN_aN_t1, F1aN_aN_in_MajorA, F1aN_aN_in_MinorA, N, M, a_arr, MajorA_2d_sorted_arr, num_of_majorA)
print "All Done !"
