eir common properties: The modulated bandwidth never exceeds twice the message spectrum. The transmission spectrum is basically the transferred message spectrum. The destination S/N is never better than if the baseband transmission was used (no modulation at all). This means that the noise power added to the transmitted signal on the line is detected in the receiver together with the desired modulating signal and the S/N is not improved in detection. The exponential modulation methods of frequency modulation (FM) and phase modulation (PM) that we will discuss next differ on all of these counts. 4.2.3 In contrast to linear modulation, exponential modulation is a nonlinear process and therefore the modulated spectrum is not related to the message spectrum in a simple fashion. The modulated waveform after exponential modulation can be expressed by the following equation: t)= co+p()] (4.2) where (t) represents the varying phase or the frequency containing the mes- sage, A is the constant amplitude, w = 2rf is the angular frequency of the carrier wave, and Re means that we take the real part of the exponential func- tion in brackets. As we can see, the message is inserted into the angle of the carrier wave or in the exponent of the function describing a cosine wave. This Introduction to Telecommunications Network Engineering is why these modulation methods are called either angle or exponential modulations. In FM the instantaneous frequency of the carrier is varied according to the message and its amplitude is kept constant. Figure 4.6 shows an example in which the frequency of the carrier is increased when the value of the modulating message is increased and vice versa. We can assume that FM has good noise performance, because if the amplitude is distorted we can cut it back to the constant value in the receiver, thus eliminating most of the external disturbances. In the detector of the receiver only the instants when the signal curve crosses zero voltage need to be detected. The distur- bances are highly attenuated 