In this section, we derive pairwise error probability (PEP) and outage probability for the CDD cases.
4.1. Pairwise Error Probability
PEP in this section is the probability of decoding to an error symbol instead of the transmitted symbol . Let the Euclidean distance between these two symbols be The received signal is rewritten in a matrix form as
where , is a transpose operation, is an diagonal matrix with the as its main diagonal. The channel vector represents the effective channel transfer function which is corresponding to symbol intervals. The vector is a noise vector. With maximum likelihood decoder, the decision rule is
where denotes the Frobenius norm.
Follow the same line of derivation in , let us define a correlation matrix of the effective channel, . is a conjugate transpose operation. The PEP of CDD of a UWB-OFDM system is readily obtained as [12 , Theorem 2]
where is eigenvalue of the matrix The integration over the variable comes from using an alternate representation of function .
To find is written as
where . is an phase matrix written as
where is a phase vector, , is a all-zero vector. Therefore, the correlation matrix is found from
where is an identity matrix, is a Kronecker product and
which corresponds to the correlation matrix of the channel transfer function in the single antenna case. Each element can be found from 
where From the PEP, SER can be computed using a well-known union bound, that is, by summing the PEP corresponding to each incorrect symbol.
4.2. Outage Probability
Outage probability for CDD case is defined as the probability that the combined effective SNR falls below a specified threshold The combined SNR in this case is
where is the SNR per transmitted symbol. Therefore, the outage probability can be expressed as
where and is the probability density function of Following the approach in , we have to find the moment-generating function (MGF) and do inverse Laplace transform in order to obtain and then find . Having done so, we may arrive at the MGF and outage probability, respectively, as 
However, for (18) to be valid, all must be distinct. This is a striking difference between CDD and the case in . For CDD case, we have to do it is possible that some eigenvalues will be the same and repeated roots will appear in the denominator of (17). In such cases, partial fraction and higher-order inverse Laplace transform according to the obtained eigenvalues case by case.
For example, in the case of 128-point FFT, two transmit antennas and coding across four subcarriers, that is, , , , , , we will have and The can be readily obtained as
Other cases can be similarly derived with some tedious manipulations.