The implementation of wireless power transfer in wireless communication systems opens up a new research area, known as wirelessly powered communications (WPC). In next-generation heterogeneous networks where ultra-dense small-cell base stations are deployed, simultaneous-wireless-information-and-power-transfer (SWIPT) is feasible over short ranges. One challenge for designing a WPC system is the severe near-far problem where a user attempts to decode an information-transfer (IT) signal in the presence of extremely strong SWIPT signals. Jointly quantizing the mixed signals causes the IT signal to be completely corrupted by quantization noise and thus the SWIPT signals have to be suppressed in the analog domain. This motivates the design of a framework in this paper for analog spatial cancellation in a multi-antenna WPC system. In the framework, an analog circuit consisting of simple phase shifters and adders, is adapted to cancel the SWIPT signals by multiplying it with a cancellation matrix having unit-modulus elements and full rank, where the full rank retains the spatial-multiplexing gain of the IT channel. The unit-modulus constraints render the conventional zero-forcing method unsuitable. Therefore, the paper presents a novel systematic approach for constructing cancellation matrices. For the single-SWIPT-interferer case, the matrices are obtained as truncated Fourier/Hadamard matrices after compensating for propagation phase shifts over the SWIPT channel. For the more challenging multiple-SWIPT-interferer case, it is proposed that each row of the cancellation matrix is constructed as a Kronecker-product of component vectors, with each component vectors designed to null the signal from a corresponding SWIPT interferer similarly as in the preceding case.