1. S-72.227 Digital Communication Systems The Viterbi algorithm, optimum coherent bandpass modulation

2. Timo O. Korhonen, HUT Communication Laboratory Topics today u Revision of convolutional codes state diagrams u Viterbi decoding: phase trellis and surviving path, ending the decoding u Principle of convolutional code error rate bound determination u Bandpass digital transmission –ASK, QAM, PSK, FSK, MSK –waveforms (LP-presentation) and constellation diagrams –modulator blocks –spectral properties, transmission BW –binary and M-ary cases u Optimum coherent detection –Matched filter and correlator principle –Matched filter impulse response

3. Timo O. Korhonen, HUT Communication Laboratory Representing convolutional code compactly: code trellis and state diagram Shift register states 0 0 1 -> 1 1 0 1 1 -> 0 1 1 1 0 -> 0 1 1 0 1 -> 0 0...

4. Timo O. Korhonen, HUT Communication Laboratory Exercise: State diagrams u,,,,

5. Timo O. Korhonen, HUT Communication Laboratory a b c d e f g h a=000 b=001 c=010 d=011 e=100 f=101 g=110 h=111 111 100 011 101 For instance from d to h you go with the input mj=1, thus xj’=1, xj’’=1+d’’=0, and xj’’’=d’+d’’’=1

6. Timo O. Korhonen, HUT Communication Laboratory The Viterbi algorithm u Exhaustive maximum likelihood method must search all paths in phase trellis for 2 k bits for a (n,k,L) code u By Viterbi-algorithm search depth can be decreased to comparing surviving paths where 2 L is the number of nodes and 2 k is the number of branches coming to each node (see the next slide!) u Problem of optimum decoding is to find the minimum distance path from the initial stage back to initial stage (below from S 0 to S 0 ). The minimum distance is the sum of all path metrics that is maximized by the correct path u The Viterbi algorithm gets its efficiency via concentrating into survivor paths of the trellis Channel output sequence at the RX TX Encoder output sequence for the m:th path

7. Timo O. Korhonen, HUT Communication Laboratory The survivor path u Assume for simplicity a convolutional code with k=1, and up to 2 k = 2 branches can enter each stage in trellis diagram u Assume optimal path passes S. Metric comparison is done by adding the metric of S into S1 and S2. At the survivor path the accumulated metric is naturally smaller (otherwise it could not be the optimum path) u For this reason the non-survived path can be discarded -> all path alternatives need not to be considered u Note that in principle whole transmitted sequence must be received before decision. However, in practice storing of states for input length of 5L is quite adequate

8. Timo O. Korhonen, HUT Communication Laboratory Example of using the Viterbi algorithm u Assume received sequence is and the (n,k,L)=(2,1,2) code shown below. Determine the Viterbi decoded output sequence! (Note that for this encoder code rate is 1/2 and memory depth L = 2)

9. Timo O. Korhonen, HUT Communication Laboratory The maximum likelihood path The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming distance to the received sequence is 4 and the respective decoded sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path. (Black circles denote the deleted branches, dashed lines: '1' was applied) 1 1 Smaller accumulated metric selected First depth with two entries to the node After register length L+1=3 branch pattern begins to repeat (Branch Hamming distance in parenthesis)

10. Timo O. Korhonen, HUT Communication Laboratory How to end-up decoding? u In the previous example it was assumed that the register was finally filled with zeros thus finding the minimum distance path u In practice with long code words zeroing requires feeding of long sequence of zeros to the end of the message bits: wastes channel capacity & introduces delay u To avoid this path memory truncation is applied: –Trace all the surviving paths to the depth where they merge –Figure right shows a common point at a memory depth J –J is a random variable whose magnitude shown in the figure (5L) has been experimentally tested for negligible error rate increase –Note that this also introduces the delay of 5L!

11. Timo O. Korhonen, HUT Communication Laboratory Error rate determination of convolutional codes u Error rate depends on –channel SNR –input sequence length, number of errors is scaled to sequence length –code trellis topology u These determine which path in trellis was followed while decoding u Assume all-zero sequence is transmitted and so far no errors have not been occurred. Hence the maximum likelihood path having the minimum distance d free is followed. u Now, all the paths producing errors must have a distance that is larger than the all-zero path distance (d free ), e.g. there exists the bound Number of paths at the Hamming distance d Probability of the d:th path at the Hamming distance d

12. Timo O. Korhonen, HUT Communication Laboratory Selected convolutional code error rates u Probability of the d:th path at the Hamming distance d depends on decoding method. For antipodal (polar) signaling it is bounded by that can be further simplified for low error probability channels by remembering that then the following bound works well: u Here is a table of selected convolutional codes and their associative code gains R C d f /2 (d f = d free ): We return to both convolutional code and block code error rates after discussing bandpass modulation where

13. Timo O. Korhonen, HUT Communication Laboratory Bandpass digital transmission u Carrier wave modulation is required to transmit messages via suitable, usually long distance medium as air, copper or coaxial cable, fiber class or even water u The message reserves a transmission band around the allocated carrier that depends on message bandwidth or amount of information u Discuss –modulated carrier spectral properties –amplitude, frequency and phase shift keying –binary and M-ary signaling –coherent and noncoherent detection u Compare various methods with respect of their –spectral efficiency –error rate performance in AWGN channel –hardware complexity

14. Timo O. Korhonen, HUT Communication Laboratory CW binary waveforms

15. Timo O. Korhonen, HUT Communication Laboratory Spectral analysis of CW signals u Apply the quadrature-carrier (complex envelope) form that separates the slow and fast varying parts of the carrier: u The spectra can be decomposed by using modulation theorem to the following four components:

16. Timo O. Korhonen, HUT Communication Laboratory M-ary signal equivalent low-pass spectrum: general expression u The respective equivalent lowpass spectra is u M-ary (M-level) baseband signal with the rate r=1/D is represented by u For which the spectra is can be shown to be where relate to inter-symbol correlation properties for the transmitted symbols a k by u For rectangular NRZ-pulses with Fourier transform yields the PSD: Pulse PSD

17. Timo O. Korhonen, HUT Communication Laboratory M-ary amplitude shift keying (ASK) u Take the I-component to be an unipolar NRZ signal, hence u For this signal the mean and variance are Spectral efficiency: Note the carrier component that does not convey information hence Transmission BW: Spectral width inversely proportional to the number of bits For high spectral efficiency strive to get a rapid decay

18. Timo O. Korhonen, HUT Communication Laboratory Binary Quadrature Amplitude Modulation (QAM) u Note that the orthogonal branch rates are half of the data rate Hence for u Therefore QAM is twice as spectral efficient as ASK Also, impulse that wastes power is missing

19. Timo O. Korhonen, HUT Communication Laboratory Binary phase reversal keying (PRK) u For two phases PRK is called as binary phase shift keying (BPSK) u Modulated carrier can be expressed by u This is in quadrature carrier form u The phases are u Note that phase shift keying has always constant envelope, still for N=1, M=4, phase constellation of PRK and QAM are similar u PRK has however better overall error rate performance due to missing carrier component

20. Timo O. Korhonen, HUT Communication Laboratory PRK constellations u Below PRK with M=4 and M=8 and QAM constellations QAM constellation N=1, no constant envelope

21. Timo O. Korhonen, HUT Communication Laboratory Example u Draw the signal constellation and spectrum for a 2-PSK signal with Note the unnecessary DC-component I Q

22. Timo O. Korhonen, HUT Communication Laboratory Determining variance in Maple ®

23. Timo O. Korhonen, HUT Communication Laboratory Frequency Shift Keying (FSK) u Two frequency modulation methods can be used: u M-ary FSK signal is defined by u Adjacent frequencies are space by 1/T s =2f d u Phase continuity can be obtained by selecting generator frequencies as multiples of data rate r=1/D: Discrete generator M-ary FSKContinuous phase FSK

24. Timo O. Korhonen, HUT Communication Laboratory Example of continuous discrete generator M-ary FSK signals

25. Timo O. Korhonen, HUT Communication Laboratory Binary FSK (Sunde’s FSK) u For Sunde’s FSK select u Assume rectangular data-pulses: u The lowpass i and q components are obtained from the general FSK expression (constant envelope!): constant rate: produces at two sided spectra impulses at

26. Timo O. Korhonen, HUT Communication Laboratory Sunde’s FSK PSD u PSD was defined by where now For high spectral efficiency strive to get a rapid decay

27. Timo O. Korhonen, HUT Communication Laboratory Continuous phase FSK u The baseband waveform is defined by and the modulated carrier is u Substituting x(t) into the integral yields then by using piecewise integration u Thus the CPFSK can be expressed as Why does this term enables continuous phase?

28. Timo O. Korhonen, HUT Communication Laboratory Minimum-shift keying (MSK) u Analyze CFSK by MSK that is its frequently used form. Now and its PSD can be shown to be u Note that continuous carrier phase can be illustrated as a phase trellis:

29. Timo O. Korhonen, HUT Communication Laboratory Coherent binary systems: Error rate analysis u Coherent systems utilize carrier phase information to recover data, thus t optimum error rate can be obtained t carrier reconstruction required at the receiver t carrier reconstruction must be precise u Non-coherent systems decode data without carrier phase reference, thus t error rate is deteriorated t detection easier when carrier phase recovery related circuits omitted u A good compromise of the coherent and non-coherent techniques are the differentially coherent systems u Concentrate first on AWGN system only u Focus on OOK, FSK, PSK u Band-limited channels are considered later. Then techniques are introduced to alleviate or remove produced Inter Symbolic Interference (ISI) u Important special case are fading channels that are characterized by statistical multipath propagation

30. Timo O. Korhonen, HUT Communication Laboratory Optimum binary detection u Any bandpass signal can be presented by u This can be expressed by using different waveforms for ‘0’ and ‘1’ bits: u Received waveforms, that indicate the transmitted bits, are recovered coherently by using matched filtering or correlation receiver:

31. Timo O. Korhonen, HUT Communication Laboratory Bases of optimum detection u Received signal consist of bandpass filtered signal and noise, that is then sampled at the time instants t k : u Assuming that the BPF has the impulse response h(t), the signal at the sampling instant is then expressed by u How the bandpass filter impulse response should be selected to maximize received SNR at the time instant of sampling? Let’s first have a look on optimum binary error rate: Note how this expression shows the MF and correlator reception!

32. Timo O. Korhonen, HUT Communication Laboratory Optimum binary error rate u Assuming ‘0’ and ‘1’ reception is equally likely, error happens when the H 0 signal hits the dashed region (or H 1 its left-hand side). The decision threshold is at Therefore for equally likely ‘0’ or/and ‘1’ the error rate is For optimum performance we wish to maximize the SNR

33. Timo O. Korhonen, HUT Communication Laboratory Impulse response of matched filtering u The signal part of the SNR expression is the difference signal after the bandpass filter (z 1 and z 0 are convoyed by s 1 and s 0 respectively): u The noise component of the SNR after the bandpass filter is u And the SNR after the matched filter is:

34. Timo O. Korhonen, HUT Communication Laboratory Using Schwarz’s inequality for optimum filtering u Schwarz’s inequality: u Therefore SNR is maximized at the time instant of sampling by using

35. Timo O. Korhonen, HUT Communication Laboratory Matched filtering and correlator reception u Note that both circuits fulfil the expression Correlator Matched filter