  Electrons, when scattered by static random disorder, form standing waves that
can be imaged using scanning tunneling microscopy. Such interference patterns,
observable by the recently developed technique of Fourier transform scanning
tunneling spectroscopy (FT-STS), are shown to carry unique fingerprints
characteristic of the electronic order present in a material. We exploit this
feature of the FT-STS technique to propose a test for the nature of the
enigmatic pseudogap phase in the high-$T_c$ cuprate superconductors. Through
their sensitivity to the quasiparticle spectra and coherence factors, the
FT-STS patterns in principle carry enough information to unambiguously
determine the nature of the condensate responsible for the pseudogap
phenomenon. We argue that the next generation of FT-STS experiments, currently
underway, should be able to distinguish between the pseudogap dominated by the
remnants of superconducting order from the pseudogap dominated by some
competing order in the particle-hole channel. Using general arguments and
detailed numerical calculations, we point to certain fundamental differences
between the two scenarios and discuss the prospects for future experiments.
