Exercises

5.11 Exercises

5.1. A binary polar signal is linearly generated using $± 1$ symbols and a shaping pulse $p(t)$ of amplitude 1 in the interval $[0, T_{sym}]$ . Neglect ISI and assume AWGN with $N_{0} ∕2 = 1 0^{−3}$ W/Hz at the receiver. A matched filter is used. Find the maximum bit rate that can be sent with a probability of bit error $P_{b} < 1 0^{−3}$ .
5.2. A binary PSK system uses coherent demodulation based on correlators and operates at 100 kbps. The noise at the receiver is AWGN with PSD $N_{0} ∕2 = 2.5 × 1 0^{−7}$ W/Hz. The channel attenuates the signal amplitude by 75% (25% of the transmitted amplitude reaches the receiver). The carrier amplitude at the transmitter is 3 volts and its frequency is 800 kHz. a) Determine the required channel bandwidth if the transmitted signal is based on an ideal Nyquist pulse (“zero roll-off”). Also informe the maximum inferior and minimum superior channel cuttof frequencies. b) Determine the BER for the received signal. c) Suppose that you want to improve the BER by increasing the transmitted signal power, what is the minimum power for a BER less or equal to $1 0^{−7}$ ?
5.3. A BER of $P_{b} = 1 0^{−3}$ is required for a system with a rate of 100 kbps operating on an AWGN channel using M-QAM with coherent detection. The system bandwidth is 50 kHz. Assume that the used shaping pulse is a unity-energy raised cosine with roll-off $r = 1$ and a Gray code is used to map symbols into bits. a) What is the minimum value of $E_{b} ∕ N_{0}$ to achieve the specified BER? b) What is the value of $E_{s} ∕ N_{0}$ under this condition ( $E_{s}$ is the symbol energy)?
5.4. Compare the implementation in Listing 4.2 with Listing 4.3. Check whether or not they lead to the same SER for a significant range of SNR, and estimate the number of arithmetic operations that are saved when using the latter.
5.5. Consider a digital communication with $M = 4$ possible symbols: $m_{0} = −5, m_{1} = 0, m_{2} = 2, m_{3} = 5$ . Assume the vector channel AWGN model, where the output is $y = m + n$ , with $n$ being the noise with zero mean and variance $σ^{2}$ . The threshold for the decision regions used by the receiver are $− 2.5,1$ and 3.5, respectively. For example, the receiver chooses $m_{1}$ in case $y ∈ [−2.5,1[$ . Find the error probability $P_{e}$ in both cases: a) equiprobable symbols, b) symbols with a priori probability $0.3,0.2,0.4$ and 0.1, respectively. c) Calculate $P_{e}$ for case a) assuming $σ = 2$ and the approximation to the Q function given in Eq. (??).
5.6. Assume a generic (not AWGN) vector channel that sends symbols $m ∈ M = {−5,5}$ . The conditional probability of the output vector $r$ given the input $m = −5$ , $p_{r|m=−5}$ is a uniform distribution $U(−12,2)$ with support from $− 12$ to 2, while $p_{r|m=5} = N(0,8)$ is a zero mean Gaussian with $σ = 8$ . Calculate the probability of symbol error assuming the decision threshold adopted by the receiver is $t = 0$ (the decision is +5 if $r > t$ and $− 5$ otherwise) and the a priori probabilities are $p(m = −5) = 0.9$ and $p(m = 5) = 0.1$ .
5.7. A 4-ary digital communication system uses matched filtering and the signals in Figure 5.21, which are equiprobable. a) Calculate the energy and power of each signal. b) Show whether $z_{2} (t)$ and $z_{3} (t)$ are orthogonal, orthonormal or neither. c) Find the union bound on the error probability $P_{e}$ if $E_{b} ∕ N_{0} = 9.6$ dB, where $E_{b}$ is the energy per bit.

Figure 5.21: Signals of a 4-ary digital communication system. The symbol period is $T_{s} = 2$ s and $A = 10$ V. ISI can be neglected and the noise at the receiver is AWGN with PSD level $N_{0} ∕2$ .

5.8. Still considering the digital communication in Figure 5.21, draw the block diagram of an optimal receiver based on matched filtering and explain each block. Specify: a) how is the filtering done (how many filters and the impulse response of each filter including amplitude and duration), b) the synchronization and sampling (what is the rate and the sampling instants), c) the decision system (show the decision regions, etc.)
5.9. Vector channel. A binary PSK system uses coherent demodulation and is contaminated by AWGN noise. The system is modeled as a vector channel with the matched filter output being

z = x + n

, where

n

is distributed according to a zero-mean Gaussian with variance 2. The bits 0 and 1 are represented by

x = −5

and

x = 5

, respectively. The a priori symbol probabilities are 0.2 and 0.8, respectively. a) Assuming the transmitted bit is 0, what is the probability of

z > 1

? b) What is the decision region threshold for ML detection? c) What is the threshold for MAP detection? d) Find

P_{e}

assuming the ML and MAP criteria. e) Even before calculating, which criterion would you expect to lead to the smallest

P_{e}

?
5.10. Consider a PAM constellation

x_{0} = −3, x_{1} = −1, x_{2} = 1, x_{3} = 3

that is used in a AWGN vector channel with variance

σ^{2}

. Find a) the symbol error probability

P_{e}

when using a maximum likelihood (ML) detector and the decisions thresholds; b) using the ML decisions thresholds, calculate

P_{e}

given by the union bound; and c) compare the results in a) and b).