PROBLEMS 177
where the parameterβvaries between 0 and 1. Derive an expression for the
transfer functionHout(f)given by the Fourier transform ofhout(t). Plothout(t)
andHout(f)forβ=0, 0.5, and 1.
4.6 Consider a 0.8-μm receiver with a siliconp–i–nphotodiode. Assume 20 MHz
bandwidth, 65% quantum efficiency, 1 nA dark current, 8 pF junction capaci-
tance, and 3 dB amplifier noise figure. The receiver is illuminated with 5μW
of optical power. Determine the RMS noise currents due to shot noise, thermal
noise, and amplifier noise. Also calculate the SNR.
4.7 The receiver of Problem 4.6 is used in a digital communication system that re-
quires a SNR of at least 20 dB for satisfactory performance. What is the min-
imum received power when the detection is limited by (a) shot noise and (b)
thermal noise? Also calculate the noise-equivalent power in the two cases.
4.8 The excess noise factor of avalanche photodiodes is often approximated byMx
instead of Eq. (4.4.18). Find the range ofMfor which Eq. (4.4.18) can be approx-
imated within 10% byFA(M)=Mxby choosingx= 0 .3 for Si, 0.7 for InGaAs,
and 1.0 for Ge. UsekA= 0 .02 for Si, 0.35 for InGaAs, and 1.0 for Ge.
4.9 Derive Eq. (4.4.22). PlotMoptversuskAby solving the cubic polynomial on a
computer by usingRL=1kΩ,Fn=2,R=1 A/W,Pin= 1 μW, andId=2nA.
Compare the results with the approximate analytic solution given by Eq. (4.4.23)
and comment on its validity.
4.10Derive an expression for the optimum value ofMfor which the SNR becomes
maximum by usingFA(M)=Mxin Eq. (4.4.19).
4.11Prove that the bit-error rate (BER) given by Eq. (4.5.6) is minimum when the
decision threshold is set close to a value given by Eq. (4.5.9).
4.12A 1.3-μm digital receiver is operating at 100 Mb/s and has an effective noise
bandwidth of 60 MHz. Thep–i–nphotodiode has negligible dark current and
90% quantum efficiency. The load resistance is 100Ωand the amplifier noise
figure is 3 dB. Calculate the receiver sensitivity corresponding to a BER of 10−^9.
How much does it change if the receiver is designed to operate reliably up to a
BER of 10−^12?
4.13Calculate the receiver sensitivity (at a BER of 10−^9 ) for the receiver in Problem
4.12 in the shot-noise and thermal-noise limits. How many photons are incident
during bit 1 in the two limits if the optical pulse can be approximated by a square
pulse?
4.14Derive an expression for the optimum gainMoptof an APD receiver that would
maximize the receiver sensitivity by taking the excess-noise factor asMx. Plot
Moptas a function ofxforσT= 0 .2 mA and∆f=1 GHz and estimate its value
for InGaAs APDs (see Problem 4.8).
4.15Derive an expression for the sensitivity of an APD receiver by taking into account
a finite extinction ratio for the general case in which both shot noise and thermal
noise contribute to the receiver sensitivity. You can neglect the dark current.
