HOMEWORK 1

Problem 1:

a) What are the energies in electron volts (eV) of light at wavelengths 850, 1310,

1490, and 1550 nm?

b) Consider a 1-ns pulse with a 100-nW amplitude at each of these wavelengths.

How many photons are in such a pulse at each wavelength?

Problem 2:

A WDM optical transmission system is designed so that each channel has a spectral

width of 0.8 nm. How many wavelength channels can be used in the C-band?

Problem 3:

What is the duration of a bit for each of the following three signals which have bit

rates of 64 kb/s, 5 Mb/s, and 10 Gb/s?

Problem 4:

a) Convert the following absolute power levels to dBm values: 1 pW, 1 nW, 1 mW,

10 mW, and 50 mW

b) Convert the following dBm values to power levels in units of mW: -13 dBm, -6

dBm, 6 dBm, and 17 dBm.

Problem 5:

A signal travels from point A to point B:

a) If the signal power is 1 mW at point A and 0.125 mW at point B, what is the

attenuation in dB?

b) What is the signal power at point B if attenuation is 15 dB?

Problem 6:

A 50-km long optical fiber has a total attenuation of 24 dB. If 500 W of optical

power get launched into the fiber, what is the output optical power level in dBm and W?

Problem 7:

10 mW of optical power enters a fiber. Compute (in a table) and plot (using any

programming language, C, Matlab, Scilab, Mathematica or even excel is OK) the output

power (in mW) as a function of fiber loss for losses in the range from 0-50 dB. Use

linear, log scale for the vertical axis.

Problem 8:

a) What are the frequencies of the optical wavelengths 800 nm, 1310 nm and 1550

nm?

b) What is the frequency spacing between optical wavelengths 1510 nm and 1520

nm?

Problem 9:

In a binary communication system (as in the figure), a 0 or 1 is transmitted. Because

of channel noise, a 0 can be received as 1 and vice versa. Let m0 and m1 denote the events

of transmitting 0 and 1, respectively. Let r0 and r1 denote the events of receiving 0 and 1,

respectively. Let P(m0) = 0.5, P(r1|m0) = p = 0.2, and P(r0|m1) = q = 0.1.

(1) Find P(r0) and P(r1)

(2) If a 0 is received, what is the prob. that a 0 was sent?

(3) If a 1 was received, what is the prob. that a 1 was sent?

(4) Calculate the prob. of error Pe

(5) Calculate the prob. that the transmitted signal is correctly read as the receiver

Problem 10:

Digital transmission hierarchy used in the North American telephone network.

As can be seen the Fig., at each multiplexing level some overhead bits are added for

synchronization purposes. What are the overhead for T1 and T3?

Problem 11:

An observation shows that a number of customers visiting a telephone cabin within every

hour is a Poisson random variable with =10. Find the prob. that

a) There are exactly 4 customers visiting that cabin in one hours

b) There are more than 12 customers visiting that cabin in one hours

Problem 12:

X and Y are independent Poisson radom variable (r.v). Prove that Z=X+Y is also a

Poisson r.v. Find lamda_Z.

Problem 13:

Let Y=aX + b, where a, b are constants. Prove that

a) E(Y) = E (aX + b) = aE(X) + b

b) Var (Y) = Var (aX + b) = a2 Var (X)