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Exercises

CommunicationsTechnologyII

WS2010/2011

Yidong Lang, Henning Schepker

NW1, Room N2350, Tel.: 0421/218-62393

E-mail: lang / [email protected]

Universitat Bremen, FB1

Institut fur Telekommunikation und Hochfrequenztechnik

Arbeitsbereich Nachrichtentechnik

Prof. Dr.-Ing. A. Dekorsy

Postfach 33 04 40

D–28334 Bremen

WWW-Server: http://www.ant.uni-bremen.de

Version December 15, 2010

I WS 2010/2011 “Communications Technology II” – Exercises

Contents

1 Equalization 1

Exercise 1 (eq03): DFE-Equalizer . . . . . . . . . . . . . . . . . . . . . . . 1

Exercise 2 (eq08): Equalizer, T -spaced, T/2-spaced . . . . . . . . . . . . . . 3

Exercise 3 (eq09): Equalizer, T -spaced . . . . . . . . . . . . . . . . . . . . . 4

Exercise 4 (eq11): Non-linear Equalization . . . . . . . . . . . . . . . . . . . 5

Exercise 5 (eq12): Linear Equalizer . . . . . . . . . . . . . . . . . . . . . . . 6

2 Viterbi 7

Exercise 6 (vit01): Viterbi for QPSK . . . . . . . . . . . . . . . . . . . . . . 7

Exercise 7 (vit08): Viterbi-Detection . . . . . . . . . . . . . . . . . . . . . . 8

Exercise 8 (vit10): Viterbi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Exercise 9 (vit14): ISI-Signal Space at QPSK . . . . . . . . . . . . . . . . . 10

Exercise 10 (vit15): Error Event at Viterbi-Detection . . . . . . . . . . . . . 11

3 Mobile Radio Channel 12

Exercise 11 (2007-03-mobrad): Mobile Radio Channel . . . . . . . . . . . . . 12



4 OFDM 15

Exercise 14 (ofdm03): OFDM Error Probability . . . . . . . . . . . . . . . . 15

Exercise 15 (ofdm04): OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Exercise 16 (ofdm05): OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Exercise 17 (2007-10-6): OFDM . . . . . . . . . . . . . . . . . . . . . . . . 18

Exercise 18 (2009-10-ofdm): OFDM . . . . . . . . . . . . . . . . . . . . . . 19

“Communications Technology II” – Exercises WS 2010/2011 II

Conventions and Nomenclature

• All references to passages in the text (chapter- and page numbers) refer to the book:

K.-D. Kammeyer: “Nachrichtenubertragung”, 2.Edition, B. G. Teubner Stuttgart,

1996, ISBN: 3-519-16142-7; References of equations of type (1.1.1) refer to the book,

too, whereas these of type (1) refer to the solutions of the exercises.

• The functions “rect (·)” and “tri (·)” are defined analogous to:

N. Fliege: “Systemtheorie”, 1.Edition, B. G. Teubner Stuttgart, 1991, ISBN: 3-519-

06140-6.

Thus “rect (t/T )” has the temporal expanse T , whereas “tri (t/T )” is not zero for the

length of 2T .

• The letters f and F represent frequencies (in Hertz), ω and Ω angular frequencies (in

rad/s). The following relations are always valid: ω = 2πf resp. Ω = 2πF .

• “δ0(t)” denotes the continuous(!) Dirac-pulse, whereas “δ(i)” represents the time-

discrete impulse sequence.

• So called “ideal” low-, band- and highpassfilter G(jω) have value ’1‘ in the respective

passing range and value ’0‘ in the stop range.

• If a time-discrete data sequence d(i) of rate 1/T stimulates a continues filter with

impulse response g(t), it has to be interpreted as

x(t) =

[

T

∞∑

i=−∞d(i) δ0(t − iT )

]

∗ g(t) = T

∞∑

i=−∞d(i) · g(t− iT ).

Abbreviations

ACF auto-correlation function, sequence ISI inter-symbol interference

BW, BB bandwidth, baseband KKF cross-correlation function, sequence

BP bandpass AF audio frequency

DPCM differential PCM PCM pulse-code modulation

F· Fourier transform PR partial response

H· Hilbert transform S/N = SNR signal-to-noise ratio

HP highpass LP lowpass

Availability on Internet

PDF (or PS) -files of the exercises can be downloaded from:

http://www.ant.uni-bremen.de

1 WS 2010/2011 “Communications Technology II” – Exercises

1 Equalization

Exercise 1 (eq03): DFE-Equalizer

Exam “Communications Technology” (University of Bremen) held on 10/12/00

The following symbol clock model of a transmission system is given:

QPSK

Mod.

n i( )

+y i( )

c i( )d i( )

DFEd i( )

Decod.

QPSK Channel Equalizer

The channel has the symbol clock impulse response c(i) = 1√2[1; 1]. After equalization with

decision feedback (DFE) QPSK decoding is applied (Gray coding). At the receiver input an

Eb/N0 of 10 dB has been measured.

a) Draw the block diagram of the equalizer.

b) Find expressions for the received signal y(i) and the signal yq(i) at the detector input,

considering the source data d(i) and the noise n(i), assuming no wrong decisions have

been made.

c) What is the bit error probability at the output of the system assuming that all previous

decisions were correct?

“Communications Technology II” – Exercises WS 2010/2011 2

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.610

−4

10−3

10−2

10−1

100

x

erfc

(x)


Exercise 2 (eq08): Equalizer, T -spaced, T/2-spaced


The following transmission line is given:

d(i) ∈ +1,−1-

Bitrate 1/Tg(t) - c(t)

LTI-Channel

- h(t) ? -

k T/w

y(k)

The transmit channel has the impulse response c(t) = δ0(t) + δ0(t − T2). The joint impulse

response of transmit filter g(t) and receiver filter h(t) is the following triangular impulse:

-

6g(t) ∗ h(t)

1

T 2T t

@@

@@

@@

(a) Find the total impulse response

f2(k) = g(t) ∗ h(t) ∗ c(t)|t=k T

2

after sampling with double bit rate at the receiver output (w = 2).

(b) The receive signal y(k) is passed through a T/2-spaced equalizer. The impulse response

of this T/2-equalizer is given by

eT/2 = [0.75 − 0.25]T .

Find the total impulse response at the output of the T/2-equalizer.

(c) At the equalizer output sampling with the bit rate is performed. Specify the sampling

phase such that the total impulse response of b) results in a distortionless system (even

or odd k?).

d) As an alternative, a symbol rate equalizer (T -equalizer) is applied. Hence, the receive

filter output is sampled with bit rate 1/T (w = 1). Find the symbol rate impulse

response

f(i) = g(t) ∗ h(t) ∗ c(t)|t=iT .

The coefficients of the T -equalizer are given by

eT = [−0.0008 0.0026 0.6658 − 0.1998]T .

Determine the total impulse response at the output of the T -equalizer.


Exercise 3 (eq09): Equalizer, T -spaced


The symbol rate model of a transmission system is described by the impulse response

f(i) = δ(i) + α · δ(i − 1) ; α∈R , |α| < 1.

The receiver uses a symbol rate equalizer with the impulse response

e(i) = δ(i) − α · δ(i − 1) + α2 · δ(i − 2) − α3 · δ(i − 3).

This design is called “Zero Forcing” solution.

a) Find the total impulse response of the symbol rate model and equalizer.

b) Depict the pole-zero plot of the total system.

c) Determine the output (S/N)ISI of the equalizer, as well as the maximum error, which

is caused by inter-symbol interference with two-level transmission.

d) Increase the order of the equalizer to n by continuing the “Zero Forcing” design.

Determine (S/N)ISI (at the equalizer output) depending on α and the order of the

equalizer n.


Exercise 4 (eq11): Non-linear Equalization


The symbol clock model of a transmission channel is characterized by the impulse response

as

f(i) = δ(i) + 0.5 · δ(i − 2).

The data transmission is bipolar. An equalization by means of quantized feedback is realized

at the receiver.

d(i) x(i)d(i)

y(i)Entzerrer

n(i)

Channel

a) Draw the block diagram of the equalizer.

b) Assume a decision error occurs at i − 2. For this case specify the signal y(i) at the

detector input at the time i.

(Hint: A decision error occurs at i − 2, for d(i − 2) = −d(i − 2))

c) Calculate the probability of a subsequent error, if the additive noise n(i) on the

transmission line is symmetrically distributed and has zero mean. What is the influence

of the power of the noise on this probability?


Exercise 5 (eq12): Linear Equalizer

Exam “Communications Technology” (TUHH, new DPO) held on 10/06/04 (Problem 5)

The figure below illustrates a discrete time model of a digital communication link in the

equivalent baseband domain. QPSK modulated data, given by

d(i)∈1 + j, 1 − j,−1 − j,−1 + j,

is transmitted over a frequency selective channel with the impulse response

h(i) = 1, 0.5 · ejπ/4.

In order to equalize the channel, a linear equalizer e(i) is applied:

e(i)h(i)d(i) x(i) y(i)

w(i)

(a) Which values may the distorted signal x(i) take? Sketch the admissible signal space

points for x(i) in the complex plain.

(b) In order to mitigate the impact of inter-symbol interference (ISI), we apply a linear

filter at the receiver, with the impulse response

e(i) = 1, 0.5 · ej5π/4.

Determine the impulse response of the overall system w(i) = h(i) ∗ e(i).

(c) Which values may the equalized signal y(i) take? Sketch the admissible signal space

points for y(i) in the complex plain.

(d) Determine and sketch the squared magnitude frequency response of the overall system

w(i) = h(i) ∗ e(i) for 0 < Ω < π. What shape would the squared magnitude frequency

response of the overall system have, if the linear filter is an ideal equalizer with respect

to the channel?


2 Viterbi

Exercise 6 (vit01): Viterbi for QPSK


Consider a QPSK transmission at symbol rate T with the symbol alphabet

d(i)∈−1 − j , −1 + j , +1 − j , +1 + j.

Transmit and receive filter are square-root cosine roll-off filter. Both filters fulfill the 1st

Nyquist criterion. The transmission is characterized by a multipath channel with impulse

response

c(t) = δ(t) + δ(t − T ).

The data at the receiver is detected by the Viterbi algorithm.

a) Sketch the Trellis diagram and determine all undistorted signal levels zµν of the state

transistions in a table.

b) Sketch the appropriate path into the Trellis diagram for the symbol sequence given by

d(i) = 1 + j , 1 − j , −1 − j , −1 + j , −1 − j ; i = 1, · · · , 5

and

d(i) = −1 − j ; i ≤ 0 and i > 5.

c) At the receiver we obtain the sampled signal

y(i) = 0.5 + j , 2 + 0.5j , −3j , −2 , −2 + j ; i = 1, · · · , 5

after matched filtering. Calculate the sum metric (euclidian distance, “path cost”) of

the path determined in problem b).


Exercise 7 (vit08): Viterbi-Detection


A data sequence d(i)∈−1, 1 is modulated with a BPSK (symbol length T ) and transmitted

on a multipath channel. The transmission and the reception filter are matched, together they

fulfil the 1st Nyquist criterion. The impulse response of the channel is known as:

c(t) = δ(t) + δ(t − T ) + 0.5 · δ(t − 2 · T ) .

The data is transmitted in blocks, where one block consists of four data bits and two tail bits.

The two tail bits have the value −1. The transmissed data has to be recovered at the receiver

by MLSE.

a) Sketch the appropriate Trellis diagram.

b) After sampling we have the following sequence at the output of the receiver:

s(i) = 1.5;−0.5;−1.5; 2.5;−2.5; 0.5 ; i = 1...6 .

Perform a MLSE using the Trellis diagram from problem a) and mark the approriate

path.

c) Specify the approriate data sequence for i = 1...4.


Exercise 8 (vit10): Viterbi


A signal modulated by a linear modulation scheme is transmitted over a radio link and

distorted by a frequency selective multipath channel. After sampling at symbol rate the

Viterbi algorithm is applied, in order to recover the transmitted symbol sequence at the

receiver. The corresponding Trellis diagram is depicted in the figure below.

S6

S5

S4

S3

S2

S1

S0

S7

(a) Determine the linear modulation scheme according to the depicted Trellis diagram.

How many taps does the channel have? Give the contents of the channel memory for

each state.

(b) Determine the symbol sequence d(i), according to the solid line in the figure.

(c) The dashed line represents an error event. Determine the error vector e and determine

the corresponding S/N loss factor γ2min, if the product of the channel convolution

matrices is given by

FHF =

1 0 −0.7

0 1 0

−0.7 0 1

.


Exercise 9 (vit14): ISI-Signal Space at QPSK


Statistically independent, equally distributed QPSK symbols are transmitted on a channel

with a memory of 1st order. The real-valued symbol clock impulse response of the channel is

given as

f(i) = 0.8 · δ(i) + 0.6 · δ(i − 1) .

Inter-symbol interference (ISI) occurs. The QPSK symbols are taken from the symbol

alphabet

d1 =1 + j√

2, d2 =

−1 + j√2

, d3 =−1 − j√

2, d4 =

1 − j√2

,

which results in the signal space diagram of transmission symbols shown below.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

real →

imag

→

QPSK symbols

a) Sketch the block diagram of the symbol clock model for the transmission channel.

b) Calculate the resulting signal levels w11 and w42 at the output of the channel for the two

combinations of input symbols d(i) = d1 , d(i − 1) = d1 resp. d(i) = d4 , d(i − 1) =

d2.

c) Sketch the signal space diagram at the output of the channel which results from all

possible combinations of input symbols. You can avoid further calculations, if you draw

conclusions on the analogy of problem b).

d) Additional task for the exercise (not in the test):

Calculate the average signal powers at the input and output of the channel.


Exercise 10 (vit15): Error Event at Viterbi-Detection

The Trellis diagram of a 2nd order channel for binary transmission is given, whose data

obviously belong to the alphabet d(i)∈0, 1 .

S0 = 0,0

S1 = 0,1

S2 = 1,0

S3 = 1,1

a) Determine the bit sequence that corresponds to the bold path (true data sequence).

b) At the receiver the data sequence corresponing to the dashed path is detected (in the

last part of the Trellis diagram both paths overlap). Determine the error vector.

c) Calculate the energy ACF of the error vector and then use it to determine the 3 × 3

auto-correlation matrix REee.

d) Calculate the minimum eigenvalue of the matrix. How big is the S/N loss of the Viterbi

detection compared to a transmission on an AWGN channel?


3 MobileRadio Channel

Exercise 11 (2007-03-mobrad): Mobile Radio Channel

Three reflected radio signals are received by a car driving with a velocity of v = 100km/h, as

shown in the figure below. The relative delay of the signals can be neglected at first and the

carrier frequency is f0 = 2 GHz. The reflection coefficients r0, r1, r2 are given in the figure

below.

Hint: Speed of light c0 ≈ 3 · 108m/s

35

v

r0 = 1

r1 = 0.6

r2 = 0.3

a) Calculate the Doppler frequencies fD,ν of the three signals.

b) Sketch the complete spectrum of the received signal in case of a unmodulated signal.

The velocity of the car shall be v = 0km/h now, so that no Doppler effect occurs. The

reflected path components with reflection coefficients r1 and r2 have relative delays τ1, τ2,

with τ2 > τ1, with respect to the direct path (τ0 = 0), with coefficient r0.

c) Sketch die impulse response hK(t) of the multipath channel.

d) Give the expression for the impulse response and calculate the channel transfer function

HK(jω) of the multipath channel.

e) At the receiver the received signal is filtered with an ideal bandpass HBP (jω) with

center frequency f0 and bandwidth B. Calculate the equivalent lowpass description

HTP (jω) of the overall transfer function H(jω) = HK(jω) · HBP (jω).

f) Illustrate the impact of the echos on the absolute transfer function |HTP (jω)|(Short explanation please!).



Consider a BPSK data transmission, with d(i)∈−1, 1, over a flat channel with the time-

variant channel coefficient h(k)

y(k) = h(k) · d(k) + n(k) .

The symbol duration is TBaud = 50 ns, and the signal-to-noise power ratio is Eb

N0

= E|d(k)|2E|n(k)|2 = 7 dB.

The channel h(k) can assume three states, which are characterized by the channel coefficients

h1 = 0, 5 · exp(jπ/4), h2 = 0, 8 · exp(jπ/6), h3 = 0, 1 + j0, 2 .

Furthermore, the states are characterized by an average probability of occurence Pℓ =

Prh(k) = hℓ with P1 + P2 + P3 = 1.

Hint: Assume perfect channel state information and coherent detection at the receiver.

Use the graphic below to solve the following problems.

a) Determine the average bit error probability for uniformly distributed states, P1 = P2 = P3.

b) Determine the average bit error probability for the following probabilities of occurence:

P1 = 0, 6, P2 = 0, 3, P3 = 0, 1 .

c) Asumme perfect channel state transmission at the transmitter. What is the resulting

bit error probability, if the transmitter transmits only during the strongest channel

coefficient?

d) Determie the average bit rate for case c).

0 0.5 1 1.5 2 2.5 3 3.5 410

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

x

erfc

(x)



A mobile radio channel scenario is given in the figure below.

45

ρ1 = 0.5

ρ0 = 1

ℓ1 = 1, 4 km

ℓ0 = 600 mTx Rx

a) Determine the impulse response of the equivalent low-pass channel for a stationary

vehicle. The carrier frequency is f0 = 1 GHz.

Hint: Speed of light c0 = 3 · 108 m/s

b) Determine the channel transfer function of the equivalent low-pass channel. Determine

the minimum and the maximum of its absolute value and the corresponding frequencies.

Sketch the absolute value of the transfer function.

c) Now the vehicle has a speed of v = 150 km/h. Give an expression for the received

equivalent low-pass signal r(t) for an arbitrary transmitted low-pass signal s(t).


4 OFDM

Exercise 14 (ofdm03): OFDM Error Probability


An OFDM system with 2048 active subcarriers is used for wireless transmission. The interval

between two subcarriers is 250 Hz and the guard interval has a length of 2 ms. A BPSK

modulation is used for each subcarrier.

a) Determine the bandwith and transmission rate of the entire system.

b) An IDFT with the length 4096 is used to create the OFDM signal. Determine the

sampling frequency of the output signal and how many samples fall into the guard

interval.

c) The required bandpass energy of an OFDM symbol at the transmitter is EOFDM = 1.4

Ws. White Gaussian noise with the power spectrum N0/2 = 6 · 10−5 Ws is added in

the bandpass. Determine the Eb/N0 ratio (in dB). Specify the bit error rate of the

transmission system. Calculate the average power emitted by the transmitter.

Hint: Take possibly needed values of the erfc-function from the graphic in the textbook.


Exercise 15 (ofdm04): OFDM


An OFDM system is operating within a bandwidth of B = 6 MHz on Nc = 16 subcarriers

with a bandwidth efficiency of u = 0.8. The transmitted data is modulated by a 8-PSK

scheme.

(a) Determine the data rate R of the system.

(b) How large is the maximum delay τmax for the channel? Justify your calculations.

Utilizing the same system parameters, the transmission now uses a data rate of R = 13.5

Mbit/s.

(c) How many of the 16 subcarriers are needed to achieve the given data rate?

A channel estimation yields the impulse response given as:

h(k) = 1 · δ(k) + 0.5 · δ(k − 1)

(d) Which subcarriers do you suggest to be switched off?

Hint: The center frequency of the first subcarrier is located at Ω = 0 .


Exercise 16 (ofdm05): OFDM


For a wireless computer network (WLAN, Wireless Local Area Network) an OFDM system

is used. A transmission rate of 32 Mbit/s shall be achieved. The maximum length of the

channel impulse response is 800 ns.

a) Determine the distance of the subcarriers, if the guard interval is 20% of the overall

symbol duration.

b) Calculate the S/N loss due to the insertion of the guard interval (Violation of the

matched-filter criterion!).

c) The bandwidth of the channel is 20 MHz. Of how many subcarriers does the transmitted

signal consist?

d) From the modulation methods BPSK, QPSK, 8PSK, 16QAM, and 64QAM choose the

one that just reaches the required transmission rate. Give a calculation for justyfication.


Exercise 17 (2007-10-6): OFDM

The upcoming enhancement of the UMTS system is currently specified under the name Long

Term Evolution (LTE), which applies the transmission scheme OFDM. For this technology a

maximum bandwidth of 30.72 MHz and a FFT length of 2048 are specified. The duration of

the OFDM core symbol is specified with 66.67µs, the duration of the cyclic prefix is 16.67µs.

a) State one advantage and one disadvantages of using OFDM as transmission scheme.

b) Determine the subcarrier spacing ∆f and the bandwidth efficiency β. What is the

maximum allowable delay spread of a channel to guarantee that no inter-symbol

interference occurs?

c) How many subcarriers must be switched off or allocated with zeros, if a maximum

bandwidth of 18 MHz must not be exceeded? Determine the maximum data rate that

can be transmitted with this bandwidth, if 64-QAM modulation is applied.

d) To create the OFDM signal at this bandwidth an IFFT of length 2048 is used.

Determine the sampling frequency at the output of the IFFT and how many sampling

points appear in the guard interval respectively.


Exercise 18 (2009-10-ofdm): OFDM

An OFDM system with a minimum bitrate of Rb = 10 Mbit/s is to be realized. Due to the

channel’s frequency selectivity the subcarrier spacing may not exceed ∆f = 10 kHz. An SNR

loss of γ2 = −1 dB due to the cyclic prefix is allowed.

a) Determine the maximal duration of the cyclic prefix.

b) Each subcarrier is modulated using QPSK. The cyclic prefix has the duration calculated

in problem a). Determine the number of subcarriers that are required to achieve the

desired bitrate.

c) Implementation constraints demand an FFT length, which is the number of subcarriers

rounded to the next largest power of two. Which sampling frequency do you need to

provide now?

d) Determine the maximal data rate when the number of subcarriers is identical to the

FFT-length in problem c).

e) The channel transfer function is to be estimated via scattered pilot symbols. Determine

the maximal spacing ∆nPi between subsequent pilot symbols in the frequency direction,

if the maximal echo delay of the channel τmax equals the guard duration.

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