a(mpa = sin dt (5.13) where the last identity follows by applying the change of variable + mtnT noting that the expression under the integral will reduce to an odd function of t. Following similar procedures, it can be shown that the real orthogonality (an.k (t), am.1(t)) =0 also holds, when k - l=1 and m-n - is an odd integer, and when k - l -1 and m - n is either an even or an odd integer. Finally, for the cases where |k-11>1, - the real orthog- onality (ankkt),ami(t))R = 0 is trivially confirmed by noting that the underlying basis 5 Filter Bank Multicarrier Modulation functions correspond to filters that have no overlapping bands. The stop-band quality of the frequency response of the prototype filter p(t) determines the accuracy of the equality (ank (t),ami(t))R=0whenk-11> 1. To summarize, the time-frequency phase-space lattice diagram of Figure 5.1 and the subsequent derivations revealed that the set of basis functions an.k (t) that carry the PAM symbols Sk [n] satisfy the real orthogonality condition (5.5) and as a result facilitate extrac- tion of the PAM symbols at the receiver. Furthermore, making use of these results, one arrives at the CMT transmitter and receiver structures that are presented in Figure 5.3. As shown, a synthesis filter bank (SFB) is used to construct the transmit signal. This essentially modulates the basis functions an.k (t) by the PAM symbols sk[n]. At the receiver side, on the other hand, the received signal is passed through an analysis filter bank (AFB) that finds the (real) projections of the received signal over the basis functions an.k(t). Furthermore, it is assumed that there are N subcarrier streams and the data stream of the kth subcarrier at the input to the SFB is represented by the impulse train (5.14) It is also worth noting that, in practice, the analyzed signals at the output of the AFB should be equalized. Here, to keep the presentation simple, we have not included the equalizers. Equalizer details are presented in Section 5.6. 5.3.2 SMT SMT may be thought as an alte