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This paper studies the allocation of feedback resources in the downlink 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 MaxWeight in general require the complete channel-state information (CSI) ( NL link states) for scheduling. Acquiring the complete CSI, however, is a prohibitive overhead in multichannel networks when the number of users is large. In this paper, we consider the scenario where the base station allocates only a limited amount of uplink resources for acquiring channel-state information. We first show that to support a (1-ε) fraction of the full throughput region (the throughput region with the complete CSI), the base station needs to acquire at least Θ((1-ε)L) link states at each time-slot. We then propose a Weight-Based Feedback allocation, named WBF, and show that WBF together with MaxWeight scheduling achieves a (1-ε) fraction of the full throughput region by acquiring Θ(L log[1/(ε)]) link states per time-slot. For i.i.d. on-off channels, we further prove that Θ(Llog[1/(ε)]) link states per time-slot is necessary for achieving a (1-ε) fraction of the full throughput region.