ENEE 420 Due 10/01/2002 Homework 04

1. (Fitz 2.5) Certain digital communication schemes use redundancy in the form of an error

control code to improve the reliability of communication. The compact disc recording media

is one such example. Assume a code can correct 2 or fewer errors in a block of N coded bits.

If each bit detection is independent with a probability of error P(E) = 10ô€€€3

(a) Plot the probability of correct block detection as a function of N.

(b) How small must N be so that the probability that the block is detected incorrectly is

less than 10ô€€€6?

2. (P&S 3.5) An information source produces 0 and 1 with probabilities 0.3 and 0.7, respectively.

The output of the source is transmitted via a channel that has a probability of error (turning

a 1 into a 0 or a 0 into a 1) equal to 0.2.

(a) What is the probability that a 1 is observed at the output?

(b) If the output of the channel is observed to be a 1, what is the probability that a 1 was

the output of the source?

3. (Fitz 2.11) In communications, the phase shift induced by propogation between transmitter

and receiver, p, is often modeled as a random variable. A common model is to have

fp() =

( 1

2

ô€€€

0 elsewhere

This commonly referred to as a uniformly distributed random variable.

(a) Find the CDF, Fp().

(b) Find the mean and variance of this random phase shift.

(c) It turns out that a communication system will work reasonably well if the coherent phase

reference at the receiver, ^p, is within 30o of the true value of p. If you implement a

receiver with ^p = 0, what is the probability that the communication system will work?

(d) Assuming you can physically move your system and change the propogation delay to

obtain an independent phase shift, what is the probability that your system will work at

least one out of two times?

(e) How many independent locations would you have to try to ensure a 90% chance of getting

your system to work?

4. (Fitz 8.8) Real valued additive white Gaussian noise, W(t), with a two sided spectral density

N0=2 = 0:1 is input to a linear time-invariant lter with a transfer function of

H(f) =

(

2 jfj W

0 elsewere

The output is denoted N(t).

(a) What is the correlation function, RW( ), that is used to model W(t).

1

(b) What is E fN(t)g?

(c) Calculate the output power spectral density, SN(f).

(d) Select W to give EfN2(t)g = 10.

(e) Give the PDF of one sample, i.e., fN(t1) (n1) when E fN2(t)g = 10.

(f) Give an expression for Pr fN (t1) > 3g when E fN2(t)g = 10.

Additional Problems. Do Not Turn In!!

5. (Fitz 2.7) In a particular magnetic disk drive, bits are written and read individually. The

probability of a bit error is PB(E) = 10ô€€€8. Bit errors are independent from bit to bit. In a

computer application, bits are grouped into 16 bit words. What is the probability that an

application word will be in error (at least one of the 16 bits is in error) when the bits are read

from this magnetic disk drive?

6. (Fitz 2.19) This problem gives a nice insight into the idea of a correlation coecient and

shows how to generate correlated random variables in a simulation. X and W are two zero

mean independent random variables where EfX2g = 1 and E fW2g = 2W. A third random

variable is dened as Y = X +W where is a deterministic constant such that ô€€€1 1.

(a) Prove EfXWg = 0. In other words, prove that independence implies uncorrelatedness.

(b) Choose 2W

such that 2Y

= 1.

(c) Find XY when 2W

is chosen as in part (b).

7. (Similar to P&S 3.44) The random process Z(t) is dened as

Z(t) = X

p

2 cos(2fct) ô€€€ Y

p

2 sin(2fct)

where X and Y are zero-mean independent Gaussian random variables with variances 2X

and

2Y

, respectively, and fc is a known constant.

(a) Find mZ(t).

(b) Find RZ( ) = RZ(t; t ô€€€ ). Is Z(t) wide sense stationary?

(c) Find RZ( ) = RZ(t; t ô€€€ ) when 2X

= 2Y

. Is Z(t) wide sense stationary?

(d) Find the power of Z(t) from part (c).

8. A random process is dened as Y (t) = X(t)+X(tô€€€T), where X(t) is a wide-sense stationary

random process with autocorrelation function RX( ) and power spectral density SX(f).

(a) Find RY ( ) in terms of RX( ).

(b) Find SY (f) in terms of SX(f).

(c) Find and plot SY (f) for T = 0:5 and

RX( ) =

(

5(1 ô€€€ j j) j j 1

0 j j > 1

2

1. (Fitz 2.5) Certain digital communication schemes use redundancy in the form of an error

control code to improve the reliability of communication. The compact disc recording media

is one such example. Assume a code can correct 2 or fewer errors in a block of N coded bits.

If each bit detection is independent with a probability of error P(E) = 10ô€€€3

(a) Plot the probability of correct block detection as a function of N.

(b) How small must N be so that the probability that the block is detected incorrectly is

less than 10ô€€€6?

2. (P&S 3.5) An information source produces 0 and 1 with probabilities 0.3 and 0.7, respectively.

The output of the source is transmitted via a channel that has a probability of error (turning

a 1 into a 0 or a 0 into a 1) equal to 0.2.

(a) What is the probability that a 1 is observed at the output?

(b) If the output of the channel is observed to be a 1, what is the probability that a 1 was

the output of the source?

3. (Fitz 2.11) In communications, the phase shift induced by propogation between transmitter

and receiver, p, is often modeled as a random variable. A common model is to have

fp() =

( 1

2

ô€€€

0 elsewhere

This commonly referred to as a uniformly distributed random variable.

(a) Find the CDF, Fp().

(b) Find the mean and variance of this random phase shift.

(c) It turns out that a communication system will work reasonably well if the coherent phase

reference at the receiver, ^p, is within 30o of the true value of p. If you implement a

receiver with ^p = 0, what is the probability that the communication system will work?

(d) Assuming you can physically move your system and change the propogation delay to

obtain an independent phase shift, what is the probability that your system will work at

least one out of two times?

(e) How many independent locations would you have to try to ensure a 90% chance of getting

your system to work?

4. (Fitz 8.8) Real valued additive white Gaussian noise, W(t), with a two sided spectral density

N0=2 = 0:1 is input to a linear time-invariant lter with a transfer function of

H(f) =

(

2 jfj W

0 elsewere

The output is denoted N(t).

(a) What is the correlation function, RW( ), that is used to model W(t).

1

(b) What is E fN(t)g?

(c) Calculate the output power spectral density, SN(f).

(d) Select W to give EfN2(t)g = 10.

(e) Give the PDF of one sample, i.e., fN(t1) (n1) when E fN2(t)g = 10.

(f) Give an expression for Pr fN (t1) > 3g when E fN2(t)g = 10.

Additional Problems. Do Not Turn In!!

5. (Fitz 2.7) In a particular magnetic disk drive, bits are written and read individually. The

probability of a bit error is PB(E) = 10ô€€€8. Bit errors are independent from bit to bit. In a

computer application, bits are grouped into 16 bit words. What is the probability that an

application word will be in error (at least one of the 16 bits is in error) when the bits are read

from this magnetic disk drive?

6. (Fitz 2.19) This problem gives a nice insight into the idea of a correlation coecient and

shows how to generate correlated random variables in a simulation. X and W are two zero

mean independent random variables where EfX2g = 1 and E fW2g = 2W. A third random

variable is dened as Y = X +W where is a deterministic constant such that ô€€€1 1.

(a) Prove EfXWg = 0. In other words, prove that independence implies uncorrelatedness.

(b) Choose 2W

such that 2Y

= 1.

(c) Find XY when 2W

is chosen as in part (b).

7. (Similar to P&S 3.44) The random process Z(t) is dened as

Z(t) = X

p

2 cos(2fct) ô€€€ Y

p

2 sin(2fct)

where X and Y are zero-mean independent Gaussian random variables with variances 2X

and

2Y

, respectively, and fc is a known constant.

(a) Find mZ(t).

(b) Find RZ( ) = RZ(t; t ô€€€ ). Is Z(t) wide sense stationary?

(c) Find RZ( ) = RZ(t; t ô€€€ ) when 2X

= 2Y

. Is Z(t) wide sense stationary?

(d) Find the power of Z(t) from part (c).

8. A random process is dened as Y (t) = X(t)+X(tô€€€T), where X(t) is a wide-sense stationary

random process with autocorrelation function RX( ) and power spectral density SX(f).

(a) Find RY ( ) in terms of RX( ).

(b) Find SY (f) in terms of SX(f).

(c) Find and plot SY (f) for T = 0:5 and

RX( ) =

(

5(1 ô€€€ j j) j j 1

0 j j > 1

2

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