the CR length. For a length K + 1 filter, defined as f = [fo,f1 ..,f]T, occupying a subband of width equivalent to M subcarriers in a channel of N subcarriers, the energy rate consisting in the k middle samples is defined as 50.00 (2.24) Then, the number of overhead can be chosen to ensure that the minimum amount of energy captured by the CR is greater than a predefined value. We define the following proposition: Proposition 2.2 Proposition 2.2: Consider a single-band f-OFDM system depicted in Figure 2.1 with the transmitter filter, the channel, and the receiver filter being defined in (2.4), 2.3 In-Band Interference Analysis and Discussion (2.8), and (2.11), respectively. It is considered as a nearly in-band interference-free system under the condition of perfect synchronization at the receiver if and Ncs Kq, (2.25) where Kp and Kq are selected to satisfy (2.26) with a being a predefined value, e.g. a = 0.99. The received data symbol can then be approxi- mated by (2.19) with trivial interference small enough to be ignored, and the effective channel is nearly diagonal. When the condition in (2.25) is met, it can be seen from the numerical results in Figure 2.5 that the power of effective interference, i.e. the maximum total power of ICI, f-ISI, and b-ISI, reduces to the level very close to -30 dB. Equation (2.25) is named as a nearly in-band interference-free condition of f-OFDM systems. The solutions to the optimization problems in (2.26) can be obtained using a linear search in a sorted list (1, 2, , K). In particular, the linear search sequentially checks each element of the list and evaluates S in (2.24) until it finds the first CP length that satisfies the specified condition in (2.26). The denominator of S is a constant value (only calculated once), and the numerator is an accumulated term. Therefore, the calculation of S at each iteration comprises an addition, a multiplication, and a division, i.e. the computation complexity of each iteration is constant. In the worst cases, it makes K comparisons. Howeve