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Drahtlose Kommunikation
Wintersemester 2016/2017
Prof. Hannes Frey / Dr. Jovan Radak
Assignment 3
voluntary submission until Sunday 2017-01-08
as PDF via mail to [email protected]
Name
Email Address
Drahtlose Kommunikation – WS 2016/17 – Assignment 3
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Exercise 1
a)
Given is a communication channel with maximum channel capacity of 100 Mbps. The
average signal power is 1.024 W and there is an interfering signal with an average power
of 0.5 mW. Calculate the bandwidth according to Shannon. Show your calculations.
b)
To achieve a data rate of 270 Mbit/s using the new WLAN standard IEEE 802.11n, a 40
MHz wide channel and an SNR of 34.5 dB is required 1. Which capacity does this channel
have?
Note: Pay attention to correct usage of linear and logarithmic units.
c)
Assume a communication channel with signal to noise ratio being SNR = -100 dB. Is it still
possible to transmit data over this channel?
Exercise 2
a)
Assume the data rate on a communication channel is doubled:
(1) How does Eb/N0 change?
(2) How does the SNR change?
b)
Assume the number of symbols in the set of symbols is doubled. How does the bitrate
change?
1
Quelle: IEEE Wireless Coexistence Working Group, doc.: IEEE 802.11-06/0338r4
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Exercise 3
a)
Assume you want to send a message D(x). To protect it from bit errors you want to encode
it with CRC by using the polynomial P(x).
D(x) = 1010001101 and P(x) = 110101
Calculate the codeword T(x).
b)
The following codeword T(x) = 10101100111001 has been received by a receiver and was
encoded using the CRC polynomial P(x) = 110101. Check whether T(x) was received
correctly. If yes, determine the message D(x). Otherwise state the rest C(x).
c)
Show how the codeword T(x) (e.g. from Exercise 3a) can be altered such that the error
can not be detected by CRC.
d)
Assume that a given CRC generator polynomial P(x) surely detects up to n bit errors,
where n is an even number. Can n+1 bit errors (hence odd number) also be surely
detected, if:
(1) after applying CRC an additional parity bit “even parity” is appended to the CRCencoded message T(x)?
(2) first a parity bit “even parity” is appended to the message D(x) and then CRC is applied
to the concatenation of message and parity bit?
e)
Someone wants to use CRC with the generator polynomial P(x) = 101010. What can you
say about the distribution of the last bit of the messages T(x) that are generated with this
polynomial?
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Exercise 4
What minimum Hamming distance is needed to be able to:
(1) Detect 10 bit errors?
(2) Correct 10 bit errors?
Show your calculation.
Exercise 5
a)
Given is the following received message T(x)=1110011 (x6+x5+x4+x+1). It was encoded
using the CRC generator polynomial P(x)=1011 (x3+x+1). There is one bit error in the
message. Find and correct this bit error in two steps:
(1) Calculate the syndrome table for the given generator polynomial P(x) and fill out the
table.
(2) Use the table to correct T(x). Show the decoded and corrected message D(x) without
CRC bits.
error pattern
syndrome
0000001
0000010
0000100
0001000
0010000
0100000
1000000
b)
A generator polynomial C(x) has a length of n bits. Thus, each message D(x) is extended
by n-1 redundancy bits during encoding. What is the maximum length of a message D(x),
such that each one bit error in the encoded message T(x) can be corrected using the
syndrome table? Give a short explanation.
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Exercise 6
Given is an (n,k,K) encoder for convolutional codes with k=1, v n1=un un-2 and
vn2= un-1 un-2.
a)
Determine the variables n and K of the encoder according to the above definition. Give a
short explanation.
b)
Draw the register implementation of the given encoder.
c)
Draw the corresponding state diagram for the given encoder.
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Exercise 7
Given is the following (2,1,3) encoder for convolutional codes that is depicted by the state
diagram. It holds that vn1=un un-2 and vn2=un un-1 un-2.
a)
Encode the following data word using the given encoder. Begin with the most
left bit (order: left to right).
→ 1 1 0 1 0 0 0 12
b)
Draw the complete Trellis diagram for this encoder, i.e. at least until the state transitions
begin to repeat.
00
01
10
11
c)
The following code sequence → 11 01 01 10 00 11 2 was encoded using the convolutional
code was received with bit errors. Decode and correct the sequence (order: left to right)
using the Viterbi decoding algorithm together with the above shown state diagram. Draw
the decoding paths into the Trellis diagram and choose the correct path using the metrics.
Finally give the corrected data word.
00
01
10
11
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Exercise 8
Given is the following static arrangement of wireless nodes A – E. The dashed circle
around the nodes shows their transmission range. All nodes transmit on the same
frequency and use CSMA to avoid collisions.
The following table shows various scenarios for data exchange between pairs of nodes.
Indicate in the table what happens, if the data exchange (2) starts while (1) is already in
progress. Chose one of the given answers for each scenario.
Example: In scenario a) node A sends data to node C ((1) A → C). During this
transmission the node E also wants to transmit data to node C ((2) E → C). State whether
this results in a hidden terminal problem, an exposed terminal problem, or whether the
data are sent without collision or if the ongoing transmission is detected and the
transmission from E to C is not started immediately (backoff).
Mark the appropriate case with an (X) in the corresponding cell.
(2) recognizes (1)
Hidden
Exposed
(1) and (2) send
Scenario
and avoids
Terminal Problem Terminal Problem without collision
collision
a)
(1) A → C
(2) E → C
b)
(1) A → C
(2) D → C
c)
(1) B → C
(2) D → C
d)
(1) B → A
(2) D → E
e)
(1) A → C
(2) D → B
f)
(1) A → B
(2) E → D
g)
(1) B → A
(2) C → D
h)
(1) A → B
(2) D → E