Praise for Fundamentals of WiMAX

120 Chapter 4 • Orthogonal Frequency Division Multiplexingy = h x + h x + … + h xy = h x + h x + … + h xy0 0 0 1 L−1v L−v1 0 1 1 0 v L− v+1⋮=h xL−1 0 L−1+ hx + … + hx .1 L−2 v L−v−1(4.10)From inspecting Equation (4.2), we see that this is exactly the value of y 0, y 1,…,y L −1resultingfrom y = x ⊗ h. Thus, by mimicking a circular convolution, a cyclic prefix that is at least aslong as the channel duration allows the channel output y to be decomposed into a simple multiplicationof the channel frequency response H = DFT{ h}and the channel frequency domaininput, X= DFT{ x}.The cyclic prefix, although elegant and simple, is not entirely free. It comes with both abandwidth and power penalty. Since v redundant symbols are sent, the required bandwith forOFDM increases from B to ( L+ v/ L)B. Similarly, an additional v symbol must be countedagainst the transmit-power budget. Hence, the cyclic prefix carries a power penalty of10log dB in addition to the bandwidth penalty. In summary, the use of the cyclic prefixentails data rate and power losses that are10( L+v/ L)bothLRate Loss = Power Loss = .(4.10)L+vThe “wasted” power has increased importance in an interference-limited wireless system, causinginterference to neighboring users. One way to reduce the transmit-power penalty is noted inSidebar 4.1.It can be noted that for L≫v, the inefficiency owing to the cyclic prefix can be made arbitrarilysmall by increasing the number of subcarriers. However, as the later parts of this chapterexplain, numerous other important sacrifices must be made as L grows large. As with most systemdesign problems, desirable properties, such as efficiency, must be traded off against cost andrequired tolerances.h v h v-1 ... h 1 h 0x L-v x L-v+1 ... x L-1 x 0 x 1 x 2 ... x L-1y 0 = h v x L-v +h v-1 x L-v+1 ... +h 1 x L-1 +h 0 x 0Figure 4.5 Circular convolution created by OFDM cyclic prefix