Exercises Communications technology II WS 2006

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1 Exercises Communications technology II WS 2006 Mark Petermann, Peter Klenner NW1, Room N1350, Tel.: 0421/ , Universität Bremen, FB1 Institut für Telekommunikation und Hochfrequenztechnik Arbeitsbereich Nachrichtentechnik Prof. Dr.-Ing. K. D. Kammeyer Postfach D Bremen WWW-Server: Version November 2, 2007

3 I WS 2006 Communications technology II Exercises Contents 1 Entzerrung / Equalization 1 Exercise 1 (eq04): Linear Equalizer Exercise 2 (eq05): Linear Equalizer Exercise 3 (eq12): linear equalizer Exercise 4 (eq08): Equalizer, T-spaced, T/2-spaced Exercise 5 (eq01): Decision Feedback Equalizer Exercise 6 (eq02): Decision Feedback Equalizer Viterbi 7 Exercise 7 (vit05): Trellis-Diagram for Channel of 3. Order Exercise 8 (vit01): Viterbi for QPSK Exercise 9 (vit04): Viterbi Exercise 10 (vit10): Viterbi Exercise 11 ( ): Viterbi Misc. 12 Exercise 12 ( ber): Bit Error Rate Exercise 13 ( ): Bit Error Rate Exercise 14 ( mobrad): Mobile Radio Channel Exercise 15 ( mobrad): Mobile Radio Channel Exercise 16 (misc03): MAP OFDM 18 Exercise 17 (ofdm03): OFDM Error Probability Exercise 18 (ofdm04): OFDM Exercise 19 (ofdm06): OFDM Exercise 20 (ofdm05): OFDM Exercise 21 ( ): OFDM

4 Communications technology II Exercises WS 2006 II Conventions and Nomenclature All references to passages in the text (chapter- and page numbers) refer to the book: K.- D. Kammeyer: Nachrichten ubertragung, 2.Edition, B. G. Teubner Stuttgart, 1996, ISBN: ; 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: 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. δ 0 (t) denotes the continuous(!) Dirac-pulse, whereas δ(i) represents the timediscrete 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 d(i)δ 0 (t it) g(t) = T d(i) g(t it). Abbreviations i= i= ACF auto-correlation function, -sequence ISI intersymbol-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:

5 1 WS 2006 Communications technology II Exercises 1 Entzerrung / Equalization Exercise 1 (eq04): Linear Equalizer Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 4) The discrete-time model of a digital communication link consisting of transmit filter, receive filter and physical channel is characterized by the impulse response. f(i) = {1, 0.5, 0.5 } (a) Determine the S/I-ratio (S: signal power; I: mean intersymbol interference power) at the receiver in the noiseless case. (b) A linear equalizer with impulse response e(i) = {0.6, 0.2, 0.1} is applied for linear equalization. Determine the S/I-ratio at the equalizer s output in the noiseless case. (c) Suggest a linear equalizer, by which the intersymbol interference can be completely cancelled (no calculations, only keyword).

6 Communications technology II Exercises WS Exercise 2 (eq05): Linear Equalizer Klausur Nachrichtentechnik (Universität Bremen) vom Given is the shown model of a data transmission system. d(i) Σδ 0 (t-it) R Kanal C it d ~ (i) Entzerrer e(i) d^(i) The data is bipolar, i.e. d(i) {1, 1}. The channel is respresented by a RC-module, the time constant determining the module s discharge-curve respectively its pulse response is τ = R C = T, thus identical with the symbol rate. a) Determine the impulse response h(i) of the channel s discrete symbol-clock-model (without equalizer). Due to simplicity no special amplitude-scaling has to be applied; it may be chosen as to provide the (dimensionless) value 1 at t = 0. b) Specify the pole-zero-diagram of the symbol-clock-model. Hint: a n z z a for z > a. c) Calculate the relative vertical eye-aperture at the equalizer s input (hint: geometric series). d) Determine the coefficients of a linear equalizer that is able to completely suppress the intersymbol-interference. (Hint: Consider, that the z-transformed under item b) describes a recursive system.)

7 3 WS 2006 Communications technology II Exercises Exercise 3 (eq12): linear equalizer Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 5) The figure below illustrates a discrete time model of a digital communication link in the equivalent baseband domain. QPSK modulated data 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 e jπ/4 }. In order to equalize the channel, a linear equalizer e(i) is applied. d(i) x(i) y(i) h(i) e(i) { w(i) (a) Which values may the distorted signal x(i) take? Sketch the admissible signal space points in the complex plain. (b) In order to mitigate the impact of intersymbol interference (ISI), at the receiver a linear filter with the impulse response e(i) = {1, 0.5 e j5π/4 } is applied. 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 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?

8 Communications technology II Exercises WS Exercise 4 (eq08): Equalizer, T-spaced, T/2-spaced Klausur Nachrichtentechnik (Universität Bremen) vom The below transmission line is given: LTI-Kanal d(i) {+1, 1} g(t) c(t) Bitrate 1/T k T/w h(t) y(k) The transmit channel has the impulse response c(t) = δ 0 (t) + δ 0 (t T ). Together impulse 2 response of transmit filter g(t) and receiver filter h(t) are resulting in the following triangular impulse. (a) Find the total impulse response 1 g(t) h(t) T 2T f 2 (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-Entzerrers is given by e T/2 = [ ] T. Find the total impulse response at the output of the T/2-equalizer. (c) At the equalizer output a sampling with bit rate is performed. Specify the sampling phase such that the total impulse response of b) reslts in a distortionlesssystem (even or odd k?). d) As an alternative a symbol rate equalizer (T-equalizer) is applied. Hence, the receive filter ouput is samplied at 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 t e T = [ ] T. Determine the total impulse response at the output of the T-equalizer.

9 5 WS 2006 Communications technology II Exercises Exercise 5 (eq01): Decision Feedback Equalizer Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 5) A BPSK modulated data sequence is to be transmitted over a multipath channel. The channel impulse response in base band representation at symbol rate is given by h(i) = {1 1.4}. a) Design a decision feedback equalizer (DFE) of first order. Draw a block diagram of the equalizer. What is the feedback coefficient s value? b) The input of the equalizer which is disturbed by additive white gaussian noise is given by y(i) = { 0.2, 0.5, 0.3, 2.6, 0, 2} ; i = 1,, 6. At time index i = 1 the correct decisions of symbol d(0) = 1 is in the memory of the DFE. Execute a data detection on the basis of the equalizer designed in a). What is the mean power of error before the quantiziser, assuming the decisions are correct? c) Now, assume that at time i = 1 the wrong decision of symbol d(0) = 1 is in the memory of the DFE. Execute a new data detection. How many false decisions are made and what is the power of the mean error before the quantiziser, assuming the decisions of b) are correct.

10 Communications technology II Exercises WS Exercise 6 (eq02): Decision Feedback Equalizer Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 5) BPSK modulated symbols d(i) { 1, +1} are transmitted over a radio link. After sampling at symbol rate, the receiver obtained the discrete time signal in the equivalent baseband x(i) = {0.8, 0.2, 0.6, 1.6, 3.2} for i = 0,, 4. According to the figure below, the transmission system can be characterized by a channel with the impulse response and additive white noise n(i). f(i) = 2δ(i) + 1.1δ(i 1) + 1.1δ(i 2) n( i) d( i) x( i) f( i) a) Design and sketch a decision feedback equalizer (DFE) for the system given above. b) Detect the data using the DFE designed in part a). Assume that all memory elements of the DFE contain 1 at time i = 0. c) How large is the mean noise power σ 2 n assuming that the detected data (part b)) represent the transmitted signal?

11 7 WS 2006 Communications technology II Exercises 2 Viterbi Exercise 7 (vit05): Trellis-Diagram for Channel of 3. Order Klausur Nachrichtentechnik (Universität Bremen) vom A binary transmission takes places on a channel with memory. The pulse response of the channel s symbol clock model consists of four sampled values: f = [f(0),f(1),f(2),f(3)]. Draw the trellis diagram. Apply the nomenclature used in the lectures to describe the different states.

12 Communications technology II Exercises WS Exercise 8 (vit01): Viterbi for QPSK Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 7) 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 1. Nyquist condition. 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 z µν of the state transistions in a table. b) Sketch the appropriate path into the trellis diagram for the symbol sequence d(i) = {1 + j, 1 j, 1 j, 1 + j, 1 j} ; i = 1,, 5 and d(i) = 1 j ; i 0 und i > 5. c) At the receiver we obtain the sampled signal y(i) = {1.5 + j, j, 3j, 2, 2 + j} ; i = 1,, 5 after matched filtering. Calculate the sum metric (euclidian distance, path cost ) of the path obtained in problem b).

13 9 WS 2006 Communications technology II Exercises Exercise 9 (vit04): Viterbi Klausur Nachrichtentechnik (Universität Bremen) vom At an antipodal transmission ( d(i) = 1) with the symbol length T = 15, 625 µs, linear distortions occur because of multi-path spreading. At symbol clock, the channel s pulse response is determined as c(i) = [+1; 2; +1] A Viterbi-receiver shall be used for detection. The data is arranged in bursts with every burst having the following form: d(i) = [d(0);d(1);d(2); 1; 1] The last two symbols are tailbits. There s no break between two neighboring bursts. At the Viterbi-equalizer s input the following received sequence can be found: ŝ(i) = [0; 2; 2; 2; 2] That sequence is assumed to be a complete burst. a) How long is a burst (in s) and what usable data rate is possible? b) Draw the Trellis-diagram which is necessary to equalize a burst. c) Apply a MLSE using the Trellis-diagram found in b). Calculate all necassary path costs and draw the resulting path into the Trellis-diagram. d) What s the according data sequence ˆd(i) for i = 0...2?

14 Communications technology II Exercises WS Exercise 10 (vit10): Viterbi Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 6) 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. S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 (a) Determine the linear modulation scheme according to the depicted Trellis diagram. How many taps has the channel? 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 γ min 2, if the product of the channel convolution matrices is given by F H F =

15 11 WS 2006 Communications technology II Exercises Exercise 11 ( ): Viterbi A data transmission system contains root raised cosine filter at sender and receiver;; the rolloff factor is r = 1. The frequency selective channel in the equivalent baseband has the impulse response h(t) = δ 0 (t) δ 0 (t T/2), T: symbol duration. a) Determine the equivalent impulse response h(i) at symbol clock. b) The transmitter carries out a partial response codiering with coefficients α 0 = 1, α 1 = 1 (without precoding). Determine the impulse response of the total system. c) Decoding shall be done by the Viterbi algorithm. Sketch the corresponding trellis diagram und determine the signal levels for undistorted signals; assume antipodal signaling d(i) { 1, 1}. d) Enter the path for the transmit sequence d(i) = {1, 1, 1, 1, 1, 1, 1} (begin with state 0 ).

16 Communications technology II Exercises WS Misc. Exercise 12 ( ber): Bit Error Rate For a digital communication link a data rate of 50 Mbit/s should be attained. The transmission is corrupted by white gaussian noise with spectral power density N 0 /2 = 0.06 T b. The data is mapped onto the signal space by a linear modulation scheme with mean power d 2 = 1 and subsequently filtered by the rectangular impulse shaping filter 1/T s for 0 < t < T s g (t) = 0 else a) What symbol duration T s in seconds would be necessary, in order to obtain the intended data rate, if the modulation schemes BPSK, QPSK, 8-PSK or 16-QAM were used? b) Which of the modulation forms in a) can be used, if a bit error rate of must not be exceeded? Therefore, calculate the corresponding bit error rates. The values of the erfc-function can be taken from the figure below. Hint: Keep in mind that the following holds E s = d 2 T 2 s g 2 (t) dt

17 13 WS 2006 Communications technology II Exercises erfc(x) x

18 Communications technology II Exercises WS Exercise 13 ( ): Bit Error Rate A bit error probability of P b = 10 3 is measured at the receiver after ideal coherent demodulation of a QPSK signal and transmission over an AWGN channel. a) From this determine the existent E b /N 0 ratio in db if gray coding is applied. Hint: Pick necessary values of the complementary error function erfc ( ) from the figure below erfc(x) x b) How large is the probability of double bit errors using gray-coded QPSK at this E b /N 0 ratio? c) What is the factor the data rate can be increased by using 16-QAM instead of QPSK? Determine the necessary E b /N 0 ratio in db if the same bit error probability is desired.

19 15 WS 2006 Communications technology II Exercises Exercise 14 ( 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 delays of the signals can be neglected at first and the carrier frequency is f 0 = 2 GHz. The reflection coefficients r 0,r 1,r 2 are to be taken from the figure below. Hint: Speed of light c m/s r 1 = 0.6 r 0 = 1 r 2 = v a) Calculate the Doppler frequencies f D,ν of the three signals. b) Sketch the complete spectrum of the received signal in case of a non-modulated signal. The velocity of the car shall be v = 0km/h now, so that no Doppler influence is apparent. The reflected path components with reflection coefficients r 1 and r 2 have relative delays τ 1,τ 2 with τ 2 > τ 1 with respect to the direct path (τ 0 = 0) with coefficient r 0. c) Sketch die impulse response h K (t) of the multipath channel. d) Give the expression for the impulse response and calculate the channel transfer function H K (jω) of the multipath channel. e) Before further processing at the receiver the received signal is filtered with an ideal bandpass H BP (jω) with center frequency f 0 and bandwidth B. Calculate the equivalent lowpass description H TP (jω) of the overall transfer function H(jω) = H K (jω) H BP (jω). f) Illustrate the impact of the echos on the absolute transfer function H TP (jω) (Short explanation please!).

20 Communications technology II Exercises WS Exercise 15 ( mobrad): Mobile Radio Channel Consider a BPSK data transmission, 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 T Baud = 50 ns, and the signal-to-noise power ratio E b N 0 = E{ d(k) 2 } E{ n(k) 2 } = 7 db. The channel h(k) can assume three states, which are characterized by the channel coefficients h 1 = 0, 5 exp(jπ/4), h 2 = 0, 8 exp(jπ/6), h 3 = 0, 1 + j0, 2. Furthermore, the states are characterized by an average probability of occurence P l = Pr{h(k) = h l } with P 1 + P 2 + P 3 = 1. Hint: Assume perfect channel state information at the receiver. Receiver side detection is coherent. Use the graphic below to solve the following problems. a) Determine the average bit error probability for uniformly distributed states, P 1 = P 2 = P 3. b) Determine the average bit error probability for the following probabilities of occurence: P 1 = 0, 6, P 2 = 0, 3, P 3 = 0, 1. c) Asumme perfect channel state transmission at the transmitter. What bit error probability results, if the sender transmits only during the strongest channel coefficient? d) Determie the average bit rate for case c) erfc(x) x

21 17 WS 2006 Communications technology II Exercises Exercise 16 (misc03): MAP An ASK signal written as s m (t) = i d m (i)g T (t it) with d 0 (i) = 0; d 1 (i) = 1 and g T (t) = { 1/T 0 t T 0 else is transmitted over an AWGN channel. The channel amplifies the amplitude by a constant factor a > 0 and causes a phase rotation by ψ 0. The power of the superimposed gaussian noise is given by σ 2 N. (a) Sketch the structure of the MAP-correlation receiver for the case of identical a-priori probabilities P(m = 0) = P(m = 1) = 1/2. (b) Modify the structure of item (a) with respect to different a-priori probabilities P(m = 0) = 0.2 and P(m = 1) = 0.8. (c) In which case is knowledge of the noise power necessary?

22 Communications technology II Exercises WS OFDM Exercise 17 (ofdm03): OFDM Error Probability Klausur Nachrichtentechnik (Universität Bremen) vom 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 takes place on 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. What s the sampling frequency of it s output signal and how many samples fall into the guard interval? c) The bandpass-energy of an OFDM-symbopl needed at the transmitter is E OFDM = 1.4 Ws. White Gaussian noise with the power spectrum N 0 /2 = Ws is added in the bandpass. First, determine the E b /N 0 -ratio (in db). Specify the transmission system s bit error rate. What s the average power that is emitted from the transmitter? Hint: Take possibly needed values of the erfc-function from the graphic in the textbook.

23 19 WS 2006 Communications technology II Exercises Exercise 18 (ofdm04): OFDM Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 4) An OFDM system is operating within a bandwidth of B = 6 MHz on N c = 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 τ max for the channel? Justify your calculations. Utilizing the same system parameters the transmission shall now use a data rate of R = 13.5 Mbit/s. (c) How many of the 16 subcarriers are needed to achieve the above data rate? A channel estimation yields the impulse response given below: h(k) = 1 δ(k) 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.

24 Communications technology II Exercises WS Exercise 19 (ofdm06): OFDM An OFDM-System with bandwidth B = 10 MHz and N subcarriers is considered. (a) Assume N = 12 subcarriers (i) Specify the subcarrier spacing in Hz. (ii) What is the length of the core symbol in seconds? The subcarriers are now divided into groups of four subsequent subcarriers. To transmit data only one subcarrier of each group is allocated with +1 according to the coding pattern in Fig. 1. (The remaining subcarriers remain unallocated carrying 0.) subcarrier index N-4 N-3N-2 N-1 coding st group 2nd group last group Figure 1: OFDM-System An example is given in Fig. 2. The bit sequence {0, 1} shall be transmitted. Hence, the second subcarrier of the respective group of subcarriers is allocated by a coding st group Figure 2: Example (b) The bit sequence {0, 1, 1, 1, 1, 0, 0, 0, 0, 1} shall be transmitted within one OFDMsymbol. (i) How many subcarriers need to be provided? (ii) Which subcarriers are allocated (sketch or subcarrier indices)?

25 21 WS 2006 Communications technology II Exercises Exercise 20 (ofdm05): OFDM Klausur Nachrichtentechnik (TUHH, neue DPO) vom (Aufgabe 8) 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, 64QAM choose the one that just reaches the required transmission rate. Give a calculation for justyfication.

26 Communications technology II Exercises WS Exercise 21 ( ): 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 technique a maximum bandwidth of MHz and a FFT length of 2048 is provided. 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 intersymbol interference appears? 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 still used. Determine the sampling frequency at the output of the IFFT and how many sampling points appear in the guard interval respectively?

Outline Chapter 4: Orthogonal Frequency Division Multiplexing

Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel ISI Single Carrier Equalization Orthogonal Frequency Division Multiplexing Principle