Generating wavelet using amplitude spectra
The complex result of a DFT of a signal codifies amplitude and phase information. When you take a magnitude of the resulting complex vector, you automatically throw away any phase information.

In so many words, the DFT result is informing you of what amplitudes and phases of all its frequency components are required to reconstruct your signal. If you strip away phase information you are lining up all your complex sinusoids and then adding them up coherently. If you lose phase how can you reconstruct your signal? 

That being said, what you can get in this case is the autocorrelation function of $b_1(t)$, by taking the IDFT of $|B_1(f)|^2$, but you need phase information to completely reconstruct the original signal.