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Oct-10 page

Data Transmission Exercises

Prof. Dr. J. Habermann FH Giessen-Friedberg, University of Applied Sciences

Lecture 1: 1. Classify the following signals into the categori es

• energy signals • power signals • neither energy nor power signals.

Calculate, if possible, signal energy or signal pow er

1. )cos()(1 tetx t ⋅= −

2. )sgn()(2 ttx =

3. )2cos()2cos()( 213 tfBtfAtx ππ ⋅+⋅=

2. Determine if the following signals are energy or power signals. Calculate energy or power spectral density and the signal energy or signal power:

1. t

ttctx

)sin()(sin)( ==

2. ∑∞

−∞=

−Λ=n

nttx )2()(

3. White Gaussian noise with zero mean and a power spectral density of N 0/2 is transmitted over a low pass filter with bandwidth B. Calculate the autocorrelation function of the output process Y(t). 4. The output process of a LTI system is stationary if the input process is stationary. Is the opposite also t rue, i.e., can it be assumed that the input process is station ary if the output process is stationary ? 5. Calculate the power spectral density for the fol lowing processes:

1. )2cos()( 0 Θ+⋅= tfAtX π , where A is a constant and θ is a

random variable. θ is uniformly distributed in [0, π/4].

2. X(t) = X + Y, where X and Y are independent . X is uniformly distributed in [-1,1], and Y is uniformly distributed in [0, 1].

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6. What is the equivalent noise bandwidth of a band pass filter with bandwidth W ?

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Lecture 2:

1. Given are the following functions Ψn(t):

1. Show that the functions are orthonormal. 2. Express the signal x(t) as a weighted linear

combination of the above functions, if

≤<−

≤<+

≤≤−

43,1

31,1

10,1

)(

Calculate the weighting coefficients.

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2. Given are the four following functions:

1. Determine the dimensionality of the functions and define a set of basis functions.

2. Give a vector representation of s1, s2, s3, s4. 3. Calculate the minimum distance between any pair

of vectors. 3. Determine a set of orthonormal basis functions f or the following four signals:

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4. The received signal in a binary communication sy stem which applies antipodal signals, is given by

)()()( tntstr += where s(t) is shown in the following figure and n(t ) is AWGN with power spectral density of N0/2 W/Hz.

1. Sketch the impulse response of the filter which

is matched to s(t). 2. Sketch the output signal of the matched filter

(no noise n(t). 3. Calculate the variance of the noise at the

output of the matched filter for t=3.

5. A matched filter is defined by the transfer func tion:

efH

fTj

1)(

2−−=

1. Calculate the impulse response h(t). 2. Calculate the signal to which the filter is mat ched.

6. Sketch the impulse response of the filters which are matched to the following signals. Calculate and ske tch the output signal of the three matched filters.

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7. The three signals m 1(t), m 2(t), and m 3(t) shall be transmitted over an AWGN channel. The power spectra l density of the noise is N 0/2. Die three signals are given by:

≤≤=

otherwise

Tttm

0,1)(1

≤≤−

<≤+

=−=

otherwise

TtT

tmtm

2/,1

2/0,1

)()( 32

1. Determine the dimensionality of the signal space. 2. Find a basis for the signal space (the basis may al so

be found without the Gram-Schmidt-procedure). 3. Sketch the signal space constellation.

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Lecture 3: 1. Given is the signal constellation according to exercise 7 of lecture 2.

1. Calculate and sketch the optimal decision regions Z 1, Z2, and Z 3 .

2. Which of the three signal waveforms is most error-prone, i.e., which of the probabilities P e( z | m i was sent), for i =1,2,3 is largest ?

2. In a binary antipodal signalling system the sig nal waveforms are given by:

≤≤−

<≤

=−=

otherwise

TtTTtA

TtTAt

tsts

2/),/1(2

2/0,/2

)()( 21

The channel is AWGN with S n(f) = N 0/2. The two signals have the a priori probabilities p 1 and p2 = 1- p 1.

1. Determine the structure of the optimal receiver. 2. Calculate the error probability. 3. Optional: Sketch the error probability as a functio n of

p1, for 0 ≤ p1 ≤ 1. 3. Over an AWGN channel with N 0/2 two signal waveforms with equal a priori probability are transmitted:

≤≤=

otherwise

TtTAtts

0,/)(1

≤≤−=

otherwise

TtTtAts

0),/1()(2

1. Determine the optimal receiver structure. 2. Calculate the average error probability.

4. Given is a detector with input signal r = ± A + n where +A and –A occur with equal probability and th e noise is given by a Laplace distribution

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nenp λλ −⋅=2

)(

1. Calculate the average error probability as a functi on

of A and λ. 2. Calculate the SNR which is necessary to obtain a er ror

probability o 10 -5 . Compare with the result of a Gaussian noise distribution.

5. A Manchester coder maps a binary 1 to a 10 and a binary 0 to a 01. The signal waveforms are given in the foll owing figure. Calculate the average error probability if both wav eforms are equally probable.

6. Given is a PAM system with the three signal poi nts -A, 0, +A . Determine

• the input signal of the detector, • the optimal threshold value, which minimises the av erage

error probability, • the error probability.

7. Given is a biorthogonal signal set with M=8 sign al points. All signal points are equally probable. Determine t he union bound of the error probability as a function of E b/N 0. 8. Given is a M-ary DCS, where M = 2 N, and N are the dimensions of the signal space. The signal vectors define the corners of a hypercube, which is centred at the ori gin. Calculate the average probability of a symbol error as a function of E S/N 0, where Es is the symbol energy and N 0/2 is the noise power density of the AWGN channel. All si gnal points are equiprobable. 9. A speech signal is sampled at a rate of 8 kHz. A sample is PCM encoded with 8 bits per sample. The PCM signal is transmitted with a M-ary PAM over an AWGN channel. Determine the bandwidth, if

1. M = 4

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2. M = 8 3. M = 16

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Lecture 4: 1. Binary data at 9600 bit/s are transmitted using 8-ary PAM modulation with a system using a raised cosine roll -off filter characteristic. The system has a frequency response out to 2.4 kHz.

1. What is the symbol rate ? 2. What is the roll-off factor of the filter character istic

? 2. A voice signal in the range 300 to 3300 Hz is sa mpled at 8000 samples/s. We may transmit these samples direc tly as PAM pulses or we may first convert each sample to a PCM and use binary (PCM) waveforms for transmission.

1. What is the minimum system bandwidth required for t he detection of PAM with no ISI and with a filter roll -off characteristic of r = 1 ?

2. Using the same roll-off filter, what is the minimum bandwidth required for the detection of binary (PCM ) PAM waveforms if the samples are quantized to 8 levels ?

3. Repeat part (2) using 128 quantization levels . 3. An analogue signal is PCM formatted and transmit ted using binary waveforms over a channel that is bandlimited to 100 kHz. Assume that 32 quantization levels are used an d that the overall equivalent transfer function is raised cosi ne type with roll-off r = 0.6.

1. Find the maximum bit rate that can be used by this system without introducing ISI.

2. Find the maximum bandwidth of the original analogue signal that can be accommodated with these paramete rs.

3. Repeat parts (1) and (2) for transmission with 8-ar y PAM´waveforms.

4. A transmission channel of length 1000 km is used to transmit data with the aid of binary PAM. Regenera tors are used at distances of 50 km. Each segment of the cha nnel can be assumed to be ideal (no linear distortion) in the f requency range of 0 ≤ f ≤ 1200 Hz. The attenuation is 1 dB/km. The channel is AWGN.

1. What is the maximal (r=0) bit rate, which can be transmitted without ISI ?

2. Calculate the E b/N 0 which is necessary to obtain a bit error probability of 10 -7 within the whole system.

3. Calculate the transmitted power of the regenerative system which is necessary to obtain the E b/N 0 of part (2), when N 0 = 4.1 ⋅10 -21 W/Hz.

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5. A speech telephone channel has a pass band of 3 00 Hz < f < 3000 Hz.

1. Chose a symbol rate as well as a PAM format in ord er to efficiently transmit 9600 bps.

2. Chose the roll-off factor for a raised cosine filte r characteristic. The channel is AWGN.

6. A binary PAM-signal is generated by excitation o f a raised cosine filter with a roll-off factor of r = 0.5. The resulting signal is then multiplied wi th a carrier signal. The data bit rate is 2400 bps.

1. Determine and sketch the spectrum of the modulated binary PAM signal.

2. Sketch a block diagram of the optimal demodulator/detector, if the received signal is com posed of the transmitted signal and AWGN.

7. If the additive noise at the input of the receiv er is coloured, the matched filter to the signal does not maximize SNR any more. In this case a prefilter might be use d, which whitens the coloured noise (prewhitening filter). T he matched filter has then to be adapted to the signal and to the prewhitening process.

1. Calculate the transfer function of the whitening fi lter. 2. Calculate the transfer function of the overall matc hed

filter. 3. Combine these two filters (whitening and overall ma tched

filter) to a generalized matched filter. Calculate the transfer function of this filter.

4. Calculate the SNR at the input of the detector.

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Lecture 5: 1. Binary PAM is used to transmit data over a non-e qualized filter channel. If the symbol a = 1 is transmitted, the noise free output signal of the demodulator is:

−=

otherwise

1,3.0

0,9.0

1,3.0

a. Determine the linear zero-forcing equalizer, such t hat

the output signal of the equalizer is:

±=

1,0

0,1

mqm

b. Determine q m for m = ±2 and ±3 by convolving the impulse

response of the equalizer with the impulse response of the filter characteristics.

2. Transmission of a signal pulse formed by a raise d cosine filter (transmitter and receiver) over a transmissi on channel leads to the following sampled (noise free) signal at the demodulator output:

=−

−=

−=−

otherwise

2,05.0

1,2.0

0,1

1,1.0

2,5.0

a. Determine the coefficients of a zero-forcing equali zer

with three taps. b. Calculate the output signal if a single pulse is

transmitted. Calculate the remaining ISI and its ti me span in symbols.

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3. Given are the two signal constellations of a 4-P SK and a 8-PSK. Determine the radius of the two formats such t hat the minimum distance between two signal points is ident ical. What is the additional power needed for 8-PSK to obtain the same performance as 4-PSK ? 4. The following figure gives two 8-QAM constellati ons.

The minimum distance between two signal points is 2 A. In the left figure the minimum distance to the origin is 2 A; whereas in the right figure there are four points with a di stance of

A7 , two points with A3 , and two points with distance A to the origin. Determine the average transmitted signal power for the two constellations, assuming that the signal points are equally probable. Which constellation is more power efficie nt ? 5. Given is a 16-QAM signal constellation according to the following figure. Determine the decision regions.

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6. Given are the two following signal constellation s.

a. The minimum distance for 8-QAM is A. Calculate the radius a and the radius b.

b. The minimum distance for 8-PSK is also A. Determine the radius r.

c. Calculate the average transmitted power for both constellations. All signal points are equally proba ble. What is the relative power gain of one constellatio n against the other ?

7. Given is the phase coherent demodulator for M-ar y FSK signals:

a. The received signal is given by:

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TttfT

Etr c

S ≤≤Φ+= 0),2cos(2

)( π

Calculate the output signals of the M correlators f or t=T,

if mm Φ≠Φ̂ .

b. Show that the minimal frequency separation is Tf /1=∆ .

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8. Binary on-off keying is defined by the following two signals:

bcb

TttfT

Ets

Ttts

≤≤=

0),2cos(2

)(

0,0)(

The corresponding signals at the output of an AWGN channel are:

bcb

TttntfT

Etr

Tttntr

≤≤++=

≤≤=

0),()2cos(2

)(

0),()(

ϕπ

here n(t) is the additive noise signal and ϕ(t) is the phase of the carrier signal due to the noise.

a. Sketch the block diagram of the optimal non coheren t receiver.

b. Calculate the probability density function of the t wo decision variables at the receiver.

9. Digital information is transmitted over a AWGN c hannel with a bandwidth of 100kHz. Determine the maximum data r ate which can be achieved with:

a. 4-PSK b. 2-FSK (non coherent) c. 4-FSK (non coherent)

10. Derive the carrier phase estimator based on the ML-criterion for binary on-off keying.

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Lecture 6: 1. The lowpass equivalents of three signal sets are given in the following figure.

a. Classify the signals in the sets I to III, i.e., fi nd a category of known modulation formats to which these sets may belong. Determine the signal vectors.

b. Determine the average transmitted signal energy for the three sets.

c. Calculate the average symbol error probability for signal set I with coherent detection.

d. Determine the symbol error probability with the aid of the union bound for signal set II, when coherent detection is applied.

e. Is it possible to use non-coherent detection for si gnal set III (explain) ?

f. Which signal set (or sets) can be used if a relatio n R/W (bitrate to bandwidth) of at least 2 is desired ?

2. A binary PSK-system is used for data transmissio n over an AWGN-channel with noise power density of N 0/2 = 10 -10 W/Hz. The energy of the transmitted signal is E b = A 2T/2, where T is the bit interval and A the signal amplitude. Determine the amplitude which is necessary to obtain a bit error probability of 10 -6 , if the data rate is

a. 10 Kbps b. 100 Kbps

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c. 1 Mbps 3. A digital communication system applies QAM to tr ansmit data over a speech telephone channel with 2400 symbols/s econd. The noise is AWGN.

a. Determine E b/N 0, such that the symbol error rate is 10 -5 at 4800 bps.

b. Repeat a) with 9600 bps. c. Repeat a) with 19200 bps. d. Which conclusions can be drawn from the above resul ts ?

4. Given is the signal constellation from exercise 6/lecture 5.

a. Assign each point of the signal constellation 3 dat a bits, such that neighbouring points only differ in one bit position.

b. Determine the symbol rate, if the desired data rate is 90 Mbps.

c. Compare the SNR of the PSK with the SNR of the QAM if the symbol error rate is identical.

d. Which signal constellation is more resistant agains t phase errors of the carrier signal (explain) ?

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Lectures 7/8: 1. The figure below shows the block diagram of a di rect sequence spread spectrum coder.

Spread spectrum coder HOLD_I: holds a data bit for one period of the spreading sequence. PPGEN1_I: Generator, which generates the spreading sequence. CMPEQ_I: compares both signals. If both signals are

equal, the device outputs a 0, otherwise a 1. The output sequence of the spreader is mapped to a bipolar sequence with: 0 → 1 , 1 → -1. A direct sequence CDMA system is built up of 3 transmitters and the corresponding receivers. Transmitter 1: Data sequence 0 1

Spreading sequence 0 1 1 0 0 1 1 0 Transmitter 2: Data sequence 1 1 0 1

Spreading sequence 1 0 0 1 Transmitter 3: Data sequence 1 0 1 0 Spreading sequence 0 1 0 1 The chip rate of the 3 transmitters is identical an d is 1

Mchip/s. a) Determine the bit rate of each transmitter.

What can be said about the orthogonality of the sig nals ? b) Determine the signals at the output of the 3 transm itters. c) The signal of transmitter 1 will not be delayed in the

channel. The signal of transmitter 2 will be delayed in the channel by 1 chip and the signal of transmitter 3 will be delayed by 2 chips. The signals will be added in the channel; noise has not to be considered.

• Determine the signal at the input of the receiver. • Calculate R1ges[n] at the correlator output of receiver

1 for n = 6 und 8.

• Compare with the nominal values at the decision tim e. • Give an interpretation with respect to the results of

a).

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2. 30 users with the same transmitting power share a communication channel by CDMA. The data rate of eac h user is 10kbps; DSSS is used with 2-PSK. Determine the mini mum chiprate such that a bit error rate of approx. 10 -5 is obtained. The additive Gaussian noise of the receiv er can be neglected. 3. A CDMA system is based on DSSS with a processing gain of 1000 and 2-PSK modulation. Calculate the number of users, if all users transmit the same power and a bit error r ate of approx. 10 -5 should be achieved. Recalculate for a processing gain of 500. 4. A DSSS system is used to resolve the signal com ponents in a two path mobile radio channel. Determine the mini mum chiprate if the two path have a difference in trans mitter to receiver distance of 300 meter. 5. A DSSS system with a 2-PSK modulation uses a pro cessing gain of 500. Calculate the margin against a single tone interference if the desired error rate is 10 -5 .

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Lecture 9: Exercise 1. An OFDM-transmitter is composed of a modulator, an OFDM-coder and a baseband/bandpass conversion.

• The following binary data sequence

11010 00101 01010 01000 01000 01000 of bit rate 1Mbps is transmitted.

• The modulator uses a QAM according to the followin g figure.

• The OFDM-coder uses 3 frequencies with minimal

frequency spacing.

a) Determine OFDM symbol rate and OFDM symbol interval . b) Sketch the spectrum (absolute values) of the first OFDM-

symbol. c) OFDM-coding is performed with the IDFT. Calculate t he

first time sample of the first OFDM-symbol and all time samples of the second OFDM-symbol. Sketch the absolute values of the time samples and mark the boundaries of an OFDM-symbol.

Modulator

OFDM Coder

X I

binary data

X +

cos(ωot)

-sin(ωot)

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Exercise 2. A OFDM receiver according to the following figure is composed of an OFDM-decoder and a demodulator /d etector.

The demodulator/detector uses the following mapping of a 16-QAM signal.

A received OFDM-Symbol is made up of 4 complex values. The first four samples, corrupted by the channel, are:

x(0) = 1 - j x(1) = - 5 - j x(2) = - 9 + j x(4) = - 1 + j * 7

a) The sampling interval between two values x(k) and x (k+1) ist1 µs. Determine the OFDM- symbol rate and the symbol interval. Determine the bit rate.

b) OFDM-Decoding is performed with the DFT. Calculate the complex frequency values and sketch the absolute va lue of the spectrum for the first OFDM-symbol.

Note: For DFT use: ∑∑∑∑−−−−

====

−−−−⋅⋅⋅⋅⋅⋅⋅⋅====1

21 N

N/nkje)k(xN

)n(d π

c) The demodulator/detector uses optimal decision regions

(AWGN/maximum likelihood). - Insert the decision thresholds into the above figur e. - Determine the bit sequence at the output of the

receiver.

complex time samples of received OFDM

binary data detector

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