Hani Mehrpouyan 1, Outline

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1 Hani Mehrpouyan 1, Department of Electrical and Computer Engineering, Lecture 20 (Error Probability) February 20 th, Some of the lectures notes here reproduced are taken from course textbooks: Digital Communications: Fundamentals and Applications B. Sklar. Communication Systems Engineering, J. G. Proakis and M Salehi, and Lecture Notes for Digital Communication, Queen s University, Canada, S. Yousefi. 1 Outline Error Rate for M-PSK Differential Encoding and Differential PSK (DPSK): Error Probability for Coherent PSK Error Probability for DPSK 2

2 Error Rate for M-PSK For binary case (M = 2), we have already obtained both bit and symbol error rates (BPAM): To extend the results for the M-ary case, we will consider two general methods of detection (demodulation) each with own level of complexity and performance: 3 Error Rate for M-PSK Coherent or Phase-Coherent detection: This method of detection requires that the received signal r(t) and the correlating waveforms in the demodulator ({φ n (t)} N n=1) be perfectly synchronized with each other in time/carrier phase. Non-coherent detection: otherwise. In practice, due to propagation delays in the medium as well as non-idealities in the local oscillators in Tx and Rx, the phase of cos(2πf c t) (carrier phase) generated in the Rx might not be locked to that of r(t) (i.e., phase of Tx oscillator). There are a number of techniques to cause the carrier to be phase-locked to the received signal. One remedy is to use a Phase-Locked Loop (PLL). A PLL will lock the phase of the Rx to that of Tx. We do not discuss the PLL techniques/circuits in this course. 4

3 Error Rate for M-PSK Other alternative to the use of PLL circuits: transmit a replica of the carrier signal with the information signal, e.g., in the AM case (we add a strong carrier replica to the DSB-SC signal to construct an AM signal). This carrier component is referred to as the pilot signal. The pilot signal can be filtered from the received signal and be used for coherent demodulation. What is the price to pay here? 5 Differential Encoding and Differential PSK (DPSK): Differential encoding is another method to combat ambiguities in the phase of the received signal. This is a modulation scheme with memory. In PSK: we adopt a phase for a given block of information (bits). In DPSK: we alter the carrier phase with respect to the previous (thus, differential) signaling interval (block of information). To show this very simple concept, let us compare QPSK and DQPSK: 6

4 DPSK In QPSK 7 In DQPSK: DPSK Di-bit 00 from the source shift the phase by 0 o Di-bit 01 from the source shift the phase by 90 o Di-bit 11 from the source shift the phase by 180 o Di-bit 10 from the source shift the phase by 270 o Assume we have the binary sequence to transmit. As the schemes are both quaternary, we need to parse the date into di-bits. 8

5 DPSK The transmitted signal s(t) is obtained by varying the phase of a carrier according to the mappings discussed for QPSK and DQPSK That is, the information is really stored in the phase difference of two successive signaling intervals. 9 DPSK Implication: system will not be sensitive to phase ambiguities as long they are stationary. Consider a phase jitter of θ degrees stationary over a few successive intervals. Then, if the perceived phases are: Through differential detection: which is the correct phase for the 2nd signaling interval. 10

6 Error Probability for Coherent PSK For the coherent case, all the arguments used in the introduction of demodulation and detection are valid: The demodulator is perfectly in synchronization with the received signal and the observation vector r has the Gaussian PDF presented before. The detector will implement an appropriate detection using r (say MED, for equiprobable set in AWGN). 11 Error Probability for Coherent PSK For an equiprobable case, MED involves simple Voronoi regions. Considering the perpendicular bisectors of lines joining the neighboring signal points in an M-PSK signal set, the decision regions turn out to be wedges with tips or apices at the origin. D 1 is the decision region for s 1 (t): a wedge with a 2π/M rad angle. Then, using the geometry of the constellation and Voronoi regions, the error rate will simplify to: 12

7 Error Probability for Coherent PSK Comparison of symbol error rates for M-ary PSK (P M = P s (e)): We will show later that the case of QPSK is an easier one where we can find closed-form expressions for the probability of error. 13 Error Probability for DPSK Using the differential phase modulation and demodulation discussed, we have the following DPSK demodulator: 14

8 Error Probability for DPSK Coherent PSK naturally performs better than the simpler DPSK variant. Comparison of BPSK and DBPSK: we do not discuss the derivations of error probability for the differential PSK here. For the binary case, the error rate is given by: 15 16

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Digital Modulation Schemes

Digital Modulation Schemes 1. In binary data transmission DPSK is preferred to PSK because (a) a coherent carrier is not required to be generated at the receiver (b) for a given energy per bit, the probability