Response to the following problem:
A Hadamard matrix is defined as a matrix whose elements are ±1 and whose row vectors are pairwise orthogonal. In the case when n is a power of 2, an n × n Hadamard matrix is constructed by means of the recursion given by Equation .
1. Let ci denote the ith row of an n × n Hadamard matrix. Show that the waveforms constructed as
n
si(t) = Σ cikP(t - kTc), i= 1,2,...,n
k=1
are orthogonal, where p(t) is an arbitrary pulse confined to the time interval 0 ≤ t ≤ Tc.
2. Show that the matched filters (or cross-correlators) for the n waveforms {si(t)} can be realized by a single filter (or correlator) matched to the pulse p(t) followed by a set of n cross-correlators using the code words {ci}.