  This paper studies the design and optimization of a limited feedback
single-user system with multiple-antenna transmitter and single-antenna
receiver. The design problem is cast in form of the minimizing the average
transmission power at the base station subject to the user's outage probability
constraint. The optimization is over the user's channel quantization codebook
and the transmission power control function at the base station. Our approach
is based on fixing the outage scenarios in advance and transforming the design
problem into a robust system design problem. We start by showing that uniformly
quantizing the channel magnitude in dB scale is asymptotically optimal,
regardless of the magnitude distribution function. We derive the optimal
uniform (in dB) channel magnitude codebook and combine it with a spatially
uniform channel direction codebook to arrive at a product channel quantization
codebook. We then optimize such a product structure in the asymptotic regime of
$B\rightarrow \infty$, where $B$ is the total number of quantization feedback
bits. The paper shows that for channels in the real space, the asymptotically
optimal number of direction quantization bits should be ${(M{-}1)}/{2}$ times
the number of magnitude quantization bits, where $M$ is the number of base
station antennas. We also show that the performance of the designed system
approaches the performance of the perfect channel state information system as
$2^{-\frac{2B}{M+1}}$. For complex channels, the number of magnitude and
direction quantization bits are related by a factor of $(M{-}1)$ and the system
performance scales as $2^{-\frac{B}{M}}$ as $B\rightarrow\infty$.
