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250 Fiber Optic Communications

˜
˜
∣ϕ n ( f )∣ ∣n( f )∣

f f
0
f o
B o B o
(a) (b)
Figure 6.2 Fourier transform of (a) the complex noise field, (b) its envelope.

Fig. 6.3 shows the single-sided PSD of the band-pass signal (t).
n
Next let us consider the equivalent low-pass representation. Since n(t) is the low-pass signal, its PSD, by
definition, is
2
< |ñ(f)| >
LP
= lim , (6.14)
ASE,sp T→∞ T
where T is a long time interval. Fig. 6.4 shows the low-pass representation of the ASE PSD. LP can be
ASE,sp
determined by the condition that the noise power in the low-pass representation should be the same as that in
the band-pass representation, as given by Eq. (6.13), i.e.,
N = LP B = ASE,sp o (6.15)
B ,
ASE,sp o
LP
= = n hf (G − 1). (6.16)
ASE,sp ASE,sp sp 0
In other words, the double-sided PSD of the low-pass signal n(t) is the same as the single-sided PSD,
ASE,sp
of (t). From now on, we omit the subscripts and superscripts and denote the PSD of the n(t) as ,
n ASE
LP
≡ = = n hf (G − 1). (6.17)
ASE ASE,sp ASE,sp sp 0

PSD

ρ ASE, sp

f
f o
B o

Figure 6.3 Band-pass representation of ASE PSD.