  We study single-photon scattering at a giant $\Lambda$-type atom, where both
atomic transitions are coupled with the modes of a single waveguide at two
separated points. The giant-atom structure brings phase-dependent interference
effects to both elastic (frequency-preserving) and inelastic
(frequency-converting) scattering process, and thus leads to a series of
interesting phenomena, such as perfect transmission over the whole frequency
range and total reflection. The condition of the optimal frequency conversion
is also identified and shown to be phase-dependent. We also consider the
combination of the giant-atom interference and the Sagnac quantum interference
by further inserting a Sagnac interferometer at each of the coupling points. It
shows that the two kinds of interference effects are compatible and play their
roles independently, such that efficient frequency conversion with unit
efficiency can be achieved in this case in addition to all the aforementioned
special phenomena.
