This paper studies feedback resource allocation in the down-link of a Frequency Division Duplex (FDD) multichannel wireless system. We consider a downlink network with a single base station, L shared channels and N mobile users. Throughput optimal algorithms like theMaxWeight scheduling in general require the complete channel state information (with N × L link states) for scheduling, which could be unaffordably expensive when the number of users is large. In this paper, we consider the scenario where the base station allocates only limited uplink resource for acquiring channel state information. We first show that to support a fraction (1-ε) of the full throughput region (the throughput region with full channel state information), the base station needs to acquire at least Θ(L) link states at each time slot. We then propose a Weight Based Feedback allocation, named WBF, and show that WBF together with the MaxWeight scheduling achieves a fraction (1 - ε) of the full throughput region by acquiring at most Θ ( Llog 1/ε ) link states per time slot.