  In this paper, we study hierarchical codebook design for channel estimation
in millimeter-wave (mmWave) communications with a hybrid precoding structure.
Due to the limited saturation power of mmWave power amplifier (PA), we take the
per-antenna power constraint (PAPC) into consideration. We first propose a
metric, i.e., generalized detection probability (GDP), to evaluate the quality
of \emph{an arbitrary codeword}. This metric not only enables an optimization
approach for mmWave codebook design, but also can be used to compare the
performance of two different codewords/codebooks. To the best of our knowledge,
GDP is the first metric particularly for mmWave codebook design for channel
estimation. We then propose an approach to design a hierarchical codebook
exploiting BeaM Widening with Multi-RF-chain Sub-array technique (BMW-MS). To
obtain crucial parameters of BMW-MS, we provide two solutions, namely a
low-complexity search (LCS) solution to optimize the GDP metric and a
closed-form (CF) solution to pursue a flat beam pattern. Performance
comparisons show that BMW-MS/LCS and BMW-MS/CF achieve very close performances,
and they outperform the existing alternatives under the PAPC.
