2.4 Modulated Signals 2The CSO is similar to the CTB, and can be used to measure the linearity of a broadband system. Again, if we have N signals all at the same power level, we now consider the second-order distortion products of each pair of signals that falls at frequencies w1 ± w2. In this case, the signals fall at frequencies either above or below the carriers rather than right on top of them unlike the triple beat terms, provided that the carriers are not some even multiple of the channel spacing. For example, in Figure 2.11, 49.25 + 55.25 = 104.5 MHz. This is 1.25 MHz above the closest car-rier at 103.25 MHz. All the sum terms will fall 1.25 MHz above the closest carrier, while the difference terms such as 763.25 – 841.25 = 78 will fall 1.25 MHz below the closest carrier at 79.25 MHz. Thus, the second-order and third-order terms can be measured separately. The number of terms that fall next to any given carrier will vary. Some of the w1 + w2 terms will fall out of band and the maximum number in band will fall next to the highest frequency carrier. The number of second-order beats above any given carrier is given by 2(1)2()LBHLffdNNff-+=-- (2.82)where N is the number of carriers, f is the frequency of the measurement channel, fL is the frequency of the lowest channel, fH is the frequency of the highest channel, and d is the frequency offset from a multiple of the channel spacing (1.25 MHz in Figure 2.11).For the case of the difference frequency second-order beats, there are more of these at lower frequencies, and the maximum number will be next to the lowest frequency carrier. In this case, the number of second-order products next to any carrier can be approximated by: (1) 1BHLf dNNffæö-=--ç÷-èø (2.83)Each of the second-order beats is an IP2 tone. Therefore, if each fundamental tone is at a power level of Ps, then the power of the second-order beat (SO) tones will be: IP2IP2SO(dBm)2()s= PPP-- (2.84)Thus, the composite second-order beat product will be given by: IP2IP2CSO(dB)[2() 10log()]ssBPPPPN=---+ (2.85)2.4 Modulated SignalsRadio frequency transceivers are required because it is not feasible to build an an-tenna that will transmit signals at frequencies close to dc. So far in this chapter only sinusoidal tones have been discussed, but in order for these tones to convey any useful information we must change one or more of their properties to convey informa-