Multipath channel estimation in single carrier modulation
What's missing to answer this is the spectral occupancy of x, specifically the modulated version that is convolved with the channel h. Since the convolution is a linear operation, and h is a linear channel, we will only be able to determine h for those frequencies that are occupied by x, and with a relative accuracy that will be proportional to the power at each frequency (We will be able to make a better estimate of the frequency response of h for those portions of the spectrum with higher SNR).
If x does not sufficiently occupy all the channels of interest, we will not be able to completely estimate the linear channel.