903, and I(00) = 0.774. It can be seen that, when a > 1, the com- mon phase error decreases as the subcarrier spacing decreases. To mathematically model the effect of the phase noise, let's consider the noisy output of an oscillator which contains phase noise 4(t) as follows v(t) = Let's further assume that the stochastic variation of the phase can be modeled as the output of a system with a step function impulse response and input perturbation n(t) as follows: Based on the above assumption, the single-sided power spectral density (PSD) of the phase can be written as Sq(f) = Sn(f)/(2r)2 in which Sn(f) denotes the noise PSD function. As an example, if Sn(f) is modeled as white noise, then Sq(f) 2 f-2 and if Sn(f) is modeled as flicker noise, then Sy(f)~f-3. Considering that the PSD of the phase is difficult to observe, one may alternatively look at the PSD of the oscillator's noisy output v(t). It can be shown that the PSD of v(t) can be calculated as follows [30]: where {ak} and {bk} denote the Fourier series coefficients of v(t) and B is a constant. Given that we are only interested in evaluating Sv(f) at fc, the above equation can be simplified as follows: The above function is a Lorentz distribution. We now define function If(f) as the ratio of noise power in 1 Hz bandwidth at offset f from center frequency to carrier power which is expressed in dBc/Hz. As theoretically expected, having a higher phase noise in the signal 11 The Cauchy distribution is a continuous probability distribution which is also known as Lorentz distribution or Cauchy-Lorentz distribution, or Lorentzian function. It describes the distribution of a random variable that is the ratio of two independent standard normal random variables, with the probability density function f(x; 0,1)= [n(1+x2)] 330 Chapter 3 near does fc not and increase the total power. A signal with higher phase noise will have smaller with lower will have a broader spectrum around the center frequency. Conversely, a power signal Therefore su(f) can be expressed as follows