  Due to the advantage of capacity-achieving, polar codes have been extended to
the block fading channel whereas most constructions involve complex
iterative-calculation. In this paper, we establish a systematic framework to
analyze the error performance of polar codes in the case of block mapping and
random mapping. For both the mappings, by introducing the new concept, named
split polar spectrum, we derive the upper bound on the error probability of
polarized channel which explicitly reveals the relationship between the
diversity order L and the block-wise weight distribution of the codeword. For
the special case L=2 in the block mapping, we design the enumeration algorithm
to calculate the exact split polar spectrum based on the general MacWilliams
identities. For arbitrary diversity order in the random mapping, with the help
of uniform interleaving, we derive the approximate split polar spectrum by
combining the polar spectrum and the probability of fading pattern for a
specific weight. Furthermore, we propose the design criteria to construct polar
codes over the block fading channel. The full diversity criterion is the
primary target so as to achieve the diversity gain and the product distance
criterion requires to maximize the product of the block-wise Hamming distance
whereby obtain the coding gain. Guided by these design criteria, the
construction metric, named polarized diversity weight (PDW) is proposed to
design the polar codes in both mappings. Such a simple metric can construct
polar codes with similar or better performance over those based on traditional
methods in block fading channel.
