Nonlinear Distortion Analysis of Directly Modulated ... - 연세대학교

p(t)==×∞∞∑∑⎨Re⎡∫ ⋅⋅⋅∫n=1∏r=1∞n∑∑n=1kkZ⎧⎩krp( fnfr−n+1n!2m ! ⋅⋅⋅m) e( t)−Ki2πftrK! ⎢⎣−∞⎤ ⎛dfr ⎥exp⎜i2π⎦ ⎝∞−∞n∑r=1Hfkrn( f1⎞⎤t ⎟⎥⎠⎦−fk1,..., fn−fkn)2-41where the k under the summation indicates that the sum includes all thedistinct sets {k 1 , …, k n } with k r =-K, …, K and m k =0, …n is the number oftimes each distinct k r occurs in this set, such thatK∑k=−Km = n2-42kBy comparison of eqn. 2-39 and 2-41, it can be concluded that when the inputcurrent consists of a sum of narrow-band signals, the laser generates newnarrow-band components centered at all carrier intermodulation frequencies.P nf (t) represents the photon density waveform centered at frequencyf =n∑f=K∑krr= 1 k = −Kmkfk2-43generated by intermodulation of the input signal components centered at f k1 ,…, f kn which has complex envelope q nf (t)pnfn⎧ ⎛ ⎞⎫( t)= Re⎨qnf( t)exp⎜i2π ∑ fkrt⎟⎬2-44⎩ ⎝ r=1 ⎠⎭given by−n+1n!2qnf( t)=m ! ⋅⋅⋅m!n−K∏r=1ZkrKr∫( f ) e∞−∞⋅⋅⋅i2πftr∫df∞−∞rHn( f1−fk1,..., fn−fkn) ×2-45Hence p nf is the IMP due to the input signals with carriers at frequencies f k1 , …,15