Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002496/00001
## Material Information- Title:
- Asymptotically optimum low snr msk receiver derivation
- Creator:
- McMillen, Thomas G
- Place of Publication:
- Denver, Colo.
- Publisher:
- University of Colorado Denver
- Publication Date:
- 2002
- Language:
- English
- Physical Description:
- 200 leaves : ; 28 cm
## Thesis/Dissertation Information- Degree:
- Master's ( Master of Science)
- Degree Grantor:
- University of Colorado Denver
- Degree Divisions:
- Department of Electrical Engineering, CU Denver
- Degree Disciplines:
- Electrical Engineering
- Committee Chair:
- Radenkovic, Miloje
## Subjects- Subjects / Keywords:
- Signal processing -- Digital techniques ( lcsh )
Digital communications ( lcsh ) Digital modulation ( lcsh ) Digital communications ( fast ) Digital modulation ( fast ) Signal processing -- Digital techniques ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 199-200).
- General Note:
- Department of Electrical Engineering
- Statement of Responsibility:
- by Thomas G. McMillen.
## Record Information- Source Institution:
- University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 51820811 ( OCLC )
ocm51820811 - Classification:
- LD1190.E54 2002m .M54 ( lcc )
## Auraria Membership |

Full Text |

ASYMPTOTICALLY OPTIMUM LOW SNR MSK RECEIVER DERIVATION
AND SIMULATION by Thomas G McMillen B.S. AMEN, University of Colorado at Denver, 1995 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2002 --j. t i l .^ McMillen, Thomas G. (M.S., Electrical Engineering) Asymptotically optimum low SNR MSK reciver derivation and simulation. Thesis directed by Professor Miloje Radenkovic ABSTRACT This thesis presents the theoretical background supporting the derivation of the optimal symbol-by-symbol detector for communication systems, with and without having memory, whose channel noise is modeled as additive white gaussian noise. The derivation leads to an expression for the asymptotically optimum at low SNR MSK type receiver, which turns out to be a set of average match filters. Demodulation of a MSK signal using AMF is then discussed and implemented in a Matlab simulation for various ratios of sample rate to baud rate, lengths of observation intervals, and values of SNR. Performance of each AMF demodulator simulation is measured in terms of bit error ratio. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed 111 Miloje Radenkovic The thesis for the Master of Science degree by Thomas G. McMillen has been approved by Jan Bialasiewicz Linda Moore DEDICATION I dedicate this thesis to the woman with whom I choose to share life's adventures: my beautiful wife Renee. ACKNOWLEDGEMENT My thanks to Dr. Radenkovic for his direction and support which enabled me to complete my graduate program so far from where it began. My thanks also to Tom Ladd, president of Bit Systems Inc., whose generosity and vision has made completion of my graduate work possible and meaningful. I also give thanks to my parents for nurturing my desire to achieve. Most importantly I thank my wife for her patience and understanding. CONTENTS FIGURES.............................................................................viii TABLES.................................................................................x 1 Introduction to Minimum Shift Keying (MSK) Signaling.............................1 2 MSK Detection....................................................................9 2.1 Optimum Detection for AWGN Channel System Without Memory........................12 2.2 Coherent MSK Detection..........................................................19 2.3 Non-coherent MSK Detection......................................................26 2.4 Asymptotically Optimum Low-SNR MSK Type Receiver................................30 3 Average Match Filter (AMF) Generation...........................................39 4 AMF Demodulation of MSK.........................................................52 4.1 AMF Correlation.................................................................56 4.2 Bit Synchronization.............................................................65 4.3 Measuring BER...................................................................80 5 AMF Demodulator Performance.....................................................88 6 Conclusion......................................................................94 Acronyms..............................................................................96 A1 Evaluation of Cross Correlation for CPM Signal..................................98 A2 Statistics of Match Filter Outputs in Optimum MSK Receiver....................103 A3 Application of Parsevals Theorem to Match Filter Outputs in Optimum MSK Receiverl 12 A4 Matlab Source for Figure 1.1...................................................118 vi A5 Matlab Source for Figure 1.2...................................................120 A6 Matlab Source for Figures 3-1, 3-2, and 3-3.................................122 A7 Matlab Source for make amfm and Figures 3-4, and 3-5........................133 A8 Matlab Source for Figure 4-2....................................146 A9 Matlab Source for makedata.m....................................150 A10 Matlab Source for modbits.m....................................154 A11 Matlab Source for amfdemod.m and Figures 4-4 through 4-8......................163 A12 Matlab Source for baudsync.m and Figures 4-9 through 4-20...................174 A13 Matlab Source for measjber.m and Figures 4-21 through 4-25..................186 A14 Matlab Source for Figure A2.1.................................................192 A15 Matlab Source For Tables 5-1 Through 5-10.....................................195 References........................................................................199 vii FIGURES Figure 1-1 Polar NRZ Signal................................................................2 Figure 1-2 Comparison of PSD for MSK vs. BPSK and QPSK.....................................7 Figure 3-1 Impulse Response of Filter Matched To 10101..................................41 Figure 3-2 Match Filter for "10001" With Pulse Shape Function.............................45 Figure 3-3 Inphase and Quadrature Components of Complex Match Filter......................49 Figure 3-4 Length 5 Average Match Filter For "0..........................................50 Figure 3-5 Length 5 Average Match Filter For "1"..........................................51 Figure 4-1 Flow Chart of MSK Receiver Simulation..........................................53 Figure 4-2 Simulation Message Bits to be MSK Modulated....................................55 Figure 4-3 Block Diagram of AMF Demodulator...............................................58 Figure 4-4 Soft Decision Output of AMF Correlators Without Noise..........................60 Figure 4-5 Soft Decision Output of AMF Correlators with SNR=10db..........................61 Figure 4-6 Soft Decision Output of AMF Correlators with SNR=5db...........................62 Figure 4-7 Soft Decision Output of AMF Correlators with SNR=0db...........................63 Figure 4-8 Soft Decision Output of AMF Corrleators with SNR=-5db..........................64 Figure 4-9 Frequency Domain Response of Baud Rate Filter.................................66 Figure 4-10 Complex Envelope of Baud Rate Filter Time Domain Impulse Response.............67 Figure 4-11 Phase of Filtered Baud Rate and Scaled AMF Output (No Noise)..................70 Figure 4-12 Phase of Filtered Baud Rate and Scaled AMF Output (No Noise)..................71 Figure 4-13 Phase of Filtered Baud Rate and Scaled AMF Output, SNR=5db....................72 viii Figure 4-14 Phase of Filtered Baud Rate and Scaled AMF Output, SNR=0db.......................73 Figure 4-15 Phase of Filtered Baud Rate and Scaled AMF Output, SNR=-5db......................74 Figure 4-16 Rastered Bits for Fs=25 KHz, R=3125 Hz, AMF Length=5, and SNR=10db........75 Figure 4-17 Rastered Bits for Fs=25 KHz, R=3125 Hz. AMF Length=5, and SNR=5db........76 Figure 4-18 Rastered Bits for Fs=25 KHz, R=3125 Hz, AMF Length=5, and SNR=0db........77 Figure 4-19 Rastered Bits for Fs=25 KHz, R=3125 Hz, AMF Length=5, and SNR=-5db........78 Figure 4-20 Rastered Bits for Fs=25 KHz, R=3125 Hz, AMF Length=5, and SNR=-10db.......79 Figure 4-21 Output of Frame Sync Correlator, SNR=10db, BER=0%...............................83 Figure 4-22 Output of Frame Sync Correlator, SNR=5db, BER=0.62%..............................84 Figure 4-23 Output of Frame Sync Correlator, SNR=0db, BER=7.68%..............................85 Figure 4-24 Output of Frame Sync Correlator, SNR=-5db, BER=20.77%...........................86 Figure 4-25 Output of Frame Sync Correlator, SNR=-10db, BER=42.33%..........................87 Figure A2-1: Autocorrelation of White Noise.................................................106 IX TABLES Table 5-1 BER for 2.8 Samples per Baud............................................89 Table 5-2 BER for 2.9 Samples per Baud............................................89 Table 5-3 BER for 3 Samples per Baud.............................................90 Table 5-4 BER for 3.1 Samples per Baud............................................90 Table 5-5 BER for 3.2 Samples per Baud............................................91 Table 5-6 BER for 3.3 Samples per Baud............................................91 Table 5-7 BER for 3.4 Samples per Baud............................................92 Table 5-8 BER for 3.6 Samples per Baud............................................92 Table 5-9 BER for 4 Samples per Baud.............................................93 Table 5-10 BER for 5 Samples per Baud............................................93 x 1 Introduction to Minimum Shift Keying (MSK) Signaling Increasing demand for communication systems within existing radio frequency bands has created a need for digital modulation schemes that are bandwidth efficient and economical to implement. To this end much attention has been given in technical literature to the topic of generalized minimum shift keying (MSK) signals. MSK is a classification referring to continuous phase frequency shift keying (CPFSK) with an index of modulation equal to one half. Such signals are important to designers of digital communication systems due to their power efficient spectral properties, constant complex envelope, and symbol orthogonality [1] [2]. In general continuous phase modulation (CPM) signals may be expressed using the equation s(t) = Acos[2^r + Information contained in the signal is conveyed by the continuous function that maps message symbols to signal phase. In the case of a binary signal such information would consist of a sequence of digital ones and zeros, [I] = [h, h, .... In-!, IJ, where the individual symbols /* e {"0", "1"} are typically represented by 1 positive and negative voltage levels such as those in the baseband polar NRZ waveform in figure 1-1. Baseband signal of alternating 1 's and 0's t Q) > a> Ui m o > -1 0 T_b 2T_b 3T_b 4T_b 5T_b Tim e Figure 1-1 Polar NRZ Signal Such a sequence has left and right discontinuities during the transitions between high and low voltage levels (digital ones and zeros) necessitating integration over the symbol sequence so that the discontinuities are removed. Thus during a symbol interval in which a "1" is transmitted the slope of the phase is positive because the phase function integrates over a voltage of positive one. Similarly integrating over a voltage of negative one results in the phase slope being negative during intervals in which a "0" is transmitted. The extent to which a CPM signals phase increases (positive slope) or decreases (negative slope) from integrating a voltage level over each symbol interval 7), may be modified by assigning a multiplicative weighting /?*-, also 1 "I 1 1 + 1 0 +1 0 + 1 2 referred to as an index of modulation, to each symbol Ik. Further modifications to a CPM signal's phase function can be made by imposing any pulse shape function q(t) satisfying that dictates the phase response over an integer number, L, of symbol intervals each Tb seconds in duration [1]. When the phase response, q(t), changes from zero to one half during only one symbol interval L=1 and the phase function is categorized as being full response; partial response thereby categorizes <(>(t) when q(t) takes L> 1 symbol intervals to change from zero to one half. For the purpose of this thesis L will be considered to be equal to one thereby limiting the scope of discussion to full response CPM signals. Tying these phase modification concepts together allows expression of the CPM signal's phase $() as a function of the transmitted symbol sequence [I]: n [/D=2;rÂ£ I khkq(t-kTb\nTb where 7* is the symbol transmitted, hk is the modulation index used during transmission of the symbol, qO is the pulse shape, and 7 is the duration in time of one message symbol [1][2], The relationship between the index of modulation, hk, for each of the binary symbols 7* e {"0", "1"} and the orthogonality of the CPM message symbols can be found by evaluating the cross correlation ^ (t) over a single baud interval: (') = Â£** *(;[/, tD* [3][5] where [1,1 ] is the sequence of -l previous symbols followed by the n-th symbol In e {"0", "1"} and ^;[A7I)=^cos 2nfct + 2^7khkq(t-kTb)+27dhnq(t-nTb)+ fl-l I t=0 This cross-correlation is evaluated in appendix A1 and results in n a2 sin(4/o+^D R^]-A 2ng(T,iK +*,] where ho is the modulation index associated with 7=0, hi is the modulation index associated withIn=n\", and g(Tt) is the derivative of q(t) referred to in [2] as the frequency response function. The cross-correlation is minimized when the 4 sum of the modulation indicies for the "0" and "1" symbols are non-zero integers such as when ho = h] = 'A. In fact selection of Vz for the modulation indexes results in a cross correlation coefficient of zero thereby implying that each symbol is orthogonal to the other [3][5]. When the derivative of a CPM signals phase response function, q'(t) equivalently known as the frequency response function, g(t)is constant during the L symbol intervals in which it is not zero, the phase response is categorized as L-REC [1]. Thus a full response CPM signal with such a phase response function would be categorized as 1-REC. Imposing a single modulation index of Vz on all symbols in a 1 -REC CPM signal causes the CPM signal's expression to match that of a FSK signal. j(r,[/D = /i-l A cos 2nfct + 2n ^Ikhkq(t-kTh)+ 2rfnhnq(t -nTb)+ Jt=-on \ ) ( = A cos 2n f,+i Jc n 2 27; ^ n-1 +~n 'Yjh +^o k=-x J f 1 \ A cos 2 nt V fc+ i 4nJ + the frequency of s(t) during the -th message symbol I would be^ + l/(4Tb) for 5 one of the binary symbols and fc l/(4Tb) for the other thereby conforming to the representation of a FSK signal. Signals with continuous phase modulation, such as MSK, have been shown to posses power efficient spectra [1] [2]. A qualitative appreciation for the spectral properties of MSK can be reached by examining the power spectral density (PSD) plot in figure 1-2. For comparison the PSD's of both binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) are shown. For frequencies above that of the bit rate, R, MSK clearly has less power than either BPSK or QPSK making MSK more spectrally efficient. 6 PSD's ofMSK. BPSK. and QPSK Figure 1-2 Comparison of PSD for MSK vs. BPSK and QPSK The combination of CPM with FSK and a modulation index of 0.5 provides MSK with extremely desirable attributes: narrow spectra that reduce the likelihood of cross-channel interference [1] [2], orthogonal symbols that minimize identification errors [2] and constant envelope that minimizes the effects of amplifier nonlinearity [1][3][4]. In order to make use of MSK's beneficial 7 properties in a communications system an appropriate receiver must identified. Receiver design has been discussed extensively in the literature covering optimum and sub-optimum designs for both coherent and non-coherent MSK reception. Optimum receivers presented in [1] are noted as being too complex for analysis of error performance leading to sub-optimal receivers for the asymptotic cases of very high and very low signal to noise ratio (SNR). This thesis focuses on the sub-optimum MSK receiver that is asymptotically optimal for very low SNR: the average match filter (AMF). Derivation of the optimum MSK receiver will be discussed for both the coherent and non-coherent detection cases leading to the simplification made for low SNR reception yielding the asymptotically optimum form of the receiver. Implementation of this receiver in Matlab will then be presented followed by an analysis of several simulation trials illustrating the relationship between baud rate, AMF length, and SNR in terms of bit error ratio (BER). 8 2 MSK Detection Digital receivers can be designed to compare a received signal waveform, r(t), observed over a symbol interval, 7*, to a set of expected symbols in order to detect a transmitted symbol. Such a receiver is referred to as a correlation demodulator since it measures the correlation of the received signal, r(t), with each possibly transmitted symbol, sm(t)=s(t;Im) where Im is one of M different symbols, during each symbol interval: J/(*K.{t)dt, wherem = 1,2,...,A/ [2], Linear filters with impulse response functions f(t) could be used instead of correlators resulting in a receiver that evaluates the convolution " r)dt, wherem = 1,2,...,M [2], This is, of course, equivalent to the correlation operation when each impulse response function is equated to a time reversed and shifted version of a transmittable symbol-fn(t) = sm(T J/(r)/*('"0* = ~t + r)dT = Â£*r{j)sm{r)dr, where/ = Tb andm = 1,2,...,M 9 Thus the filters with impulse response fm(t) = sm(Th-t) are considered "matched filters" to sm(t) and thereby have the property of maximizing output SNR when sm(t) is corrupted by additive white gaussian noise (AWGN) [2]. Section 2.1 elaborates on the derivation of this property and presents the mathematical foundations for expressing an optimum receiver. Throughout this thesis usage of the terms correlator and match filter will be considered equivalent and interchangeable. Basing the detection of a symbol on the observation of a single baud interval assumes that the signal observed in any given interval is independent of the signal in all other intervals. Such a modulation scheme is referred to in literature as a memory-less system [1][2]. When symbols in successive intervals do affect one another, as is in the case of MSK, information about the identity of a symbol in one interval may also be present in another. Such signals are referred to in the literature as having memory [1][2]. Limiting the scope of observation to a single baud interval in detecting a MSK symbol does not make use of information present in adjacent symbol intervals; therefore sections 2.2 and 2.3 expand the expression for an optimum receiver to apply to the case of signals with memory. As noted in [1] this form of receiver is too complex to analyze the 10 performance characteristics of hence section 2.4 discusses the constraint placed on the optimum expression resulting in a sub-optimum form of receiver that is implemented in this thesis. 11 2.1 Optimum Detection for AWGN Channel System Without Memory If the transmission channel noise is modeled as additive white gaussian noise (AWGN) then the received signal, r(t), may be expressed as the sum of the transmitted symbol, s,(t), plus noise, n(t): r{t) = s,(t)+ n{t\ where/e {l,2,...,M}[l][2]. In the literature, especially [1] and [2], optimum detectors for communication systems with AWGN channel models are presented as an array of correlators that measure the correlation of the received signal and each element of a set of orthonormal basis functions that span the space containing the set of transmittable signals. Suppose that there are U such orthonormal basis functions fu(t). Then let the output of the array of correlators during the &-th symbol interval be represented by the observation vector rk. r\ 1 'w' 1 rk = rl - _ru_ Allow representation of the sequence of correlator outputs to be the observation vector r such that r = [ri,r2, ..., rk.i, r*, /*+/, ] Identifying the signal 12 waveform transmitted during the &-th symbol interval, s, may then be expressed mathematically as selecting the symbol /, where i e{l,2, or its corresponding signal waveform s,=Sj(t), that maximizes the probability that such symbol was transmitted during the &-th symbol interval given the sequence of observations r: s(0 = j. 3p(5i(i,|r)>p(jm(i)r)vwe {l,2,1,/ + 1,...,A/}. Evaluation of the probability that symbol s, was transmitted conditioned on the received sequence vector r, referred to in literature as the maximum-a- posteriori probability (MAP) criterion, can be restated using Baye's rule as / \ p(r|.s.)/*(?.) PkM =---1-';0i/[l][2]. pin Thus determining the identity of a transmitted symbol is mathematically equivalent to selecting the symbol st that maximizes the ratio of the product of the conditional probability density function (pdf) p{r\st) and the probability that symbol s, was transmitted, P(s,), to the pdf of the observation vector, p(r), for all symbols sm in the set {sj, S2, , sM}- If all symbols have an equal probability of being transmitted then the term P(Sj) does not change value with choice of Sj. Furthermore the denominator is not conditioned on any specific symbol being 13 transmitted; the pdfp(r) is equal to p{r\sm )P(sm )and is thus independent of the identity of the transmitted symbol [2]. Maximizing the probability that the symbol s, was transmitted given the observation vector r may thus be simplified to maximizing the conditional probability density function This is referred to in literature as the maximum-likelihood (ML) decision criterion [1][2]. As stated in [1] the conditional probability density function p{r(t)\si(tj) expresses the probability that the received signal, r(t), is equal to the sum of the transmitted symbol, s,-(t), plus white gaussian noise, n(t): P(r(t) = s: (t) + n(t)). It then follows for the case when symbols in successive time intervals do not affect one another, referred to as a memory-less system [1], that the conditional pdf p(rk |s() expresses the probability that the elements of the observation vector r* are equal to the correlation of s,(t) plus white gaussian noise n(t) and each of the U orthonormal basis function: f fi" Â£1)rk(0 + (0)/,(0^ \ P(r'l\si)=p r2 = Â£r_[)r(s,(0 + (0)/2(0<* \ Tv. ) 14 The pdf of each orthonormal basis function correlator, or equivalently match filter, output conditioned on the transmitted symbol is Pk K )= l{ru = (5,. (t) + n{tj)fu (it)dtJ . Noting that each output ru is a statistically independent gaussian variable, as demonstrated in appendix A2, allows the conditional probability density function of the observation vector rk to be expressed as p(rk k)== {il)T k (0+"(o)/. which by substitution is equivalent to u pk*k)=rW';k)- u=1 Given that the correlator outputs are gaussian random variables, their pdfs J (ru-EVu\f take the form j=e 2cr where the variance, o', is equal to the power o-Jln N spectral density of the noise, and the mean, E[ru\, was shown in appendix A2 to be the correlation of the transmitted symbol, s,, and the orthonorml basis 15 function,/,^, noted as sIU for convenience. Substituting this equation into the conditional pdf of each match filter output, p(ru |st), results in />(r*k)=fl f ( . Jl _L_ e "o V^oJ As noted in [2] the natural logarithm of the above equation, lnP{rkK)=o)" jrZ~su f ^ A0 u1 is a monotonic equation and thus selecting the s, that maximizes it also maximizes p{rk |j(.). By observation it is apparent that only the quadratic term of the logarithmic equation is effected by the selection of s,\ this term is referred to in literature as the distance metric [2], Carrying out the squaring operation in the distance metric results in ZO'. stu f =X(^2 ~2rusiu + s,2) u u U 2>. -2ZrA +Z v u-\ w=l w=i As demonstrated in appendix A3 Parsevals identity can be used to convert the distance metric summation of correlator and mean correlator output terms into the linear combination of integrals 16 kT kT kT jr2(t)dt-2 jr(f)st(t)dt ^s2(t)dt. (k-\)T (A--1 )T (t-l)T Substituting this back into the natural logarithm of the pdf for the observation vector of correlator outputs during the &-th symbol interval, rk, conditioned on symbol st being transmitted yields ln p(rt k) = -y in(aiVo)_ ~rr J(r 2 (0 2r('K (0 + s2 (o)* ^ -o (k-or = -^ln(^0)--i- ]r2(t)dt + ^~ Tjr(t)s,(t)dt~ ]s2{t)dt ^ o (t-or -0 (A--i)r 'o (jt-i)T Examining the integrals involved above focuses the decision criterion on only the term effected by the choice of s,. The integral ^r2(t)dt evaluates the power of the received waveform over one symbol interval and does not depend on Sj. For a communications system where the transmitted symbols all have equal energy, such as in MSK, the integral s,2 (t)dt is identical for all choices of Si. Therefore the only element of the conditional pdf u -f* (rJ(/)-2r(/)s/(/)+5 .(r))rfr , _ - - . NT Ju-dt-' p(Fk\Si) = r i i T $(ru-s*)2 r i ^ - No _ 1^0 J effected by the choice of s, is the term 17 r 2 kT exp fr(t)s,(t)dt (*-l)r where it is observed that the integral expresses the correlation between the received signal and symbol Sj. Thus identifying the symbol transmitted during the k-th symbol interval by selecting the st that maximizes p(rk ^) is equivalent to selecting the s, that maximizes its correlation with the received signal: ' ~ kT \ max{p(r4 U.)} = max- si si exp JK'MO* (*-1)7- ) kT kT s{k) =si(t)B jr(t)si(t)dt> ^r(t)sm(t)dt,^me{\,---,i-l,i + l,---,M} (*-1)7- (*-1)7- As noted in [2] this is referred to in the literature as correlation metrics. 18 2.2 Coherent MSK Detection The optimum detector presented in section 2.1 is only applicable in memory-less systems where symbols transmitted in one interval do not affect symbols transmitted in subsequent intervals. Such is not the case for minimum shift keying (MSK) as the starting phase for the symbol transmitted in the A>th symbol interval is dependent upon the symbols transmitted previous to it. If no consideration is given to which symbols were transmitted prior to the A-th symbol the correlation metric (t)dt might not be maximized with the correct choice of symbol s, because it would be matched to a signal with starting phase different from what was transmitted. Thus for a symbol-by-symbol MSK detector the maximum-likelihood decision criterion needs to be restated. For binary communications systems with memory, maximizing the probability that the received signal waveform r(t) matches the one that would have been received had the symbol s, been transmitted during the Ar-th symbol interval p(r(f (f)) is equivalent to maximizing the sum of the probability that r(t) matches that which would have been received had s, been transmitted in the A-th and 5/ in the k+ 1st symbol intervals and the probability that r(t) matches 19 that which would have been received had s, been transmitted in the k-th and so in the &+lst symbol intervals: In this manner the conditional pdf to be maximized takes into account the effect that symbol Sj, transmitted in the Â£-th symbol interval, has on the starting phase of the following symbol regardless of which symbol was actually transmitted in the k+\st interval. Implementing this using the correlation metrics reached at the end of section 2.1 results in having to evaluate the relation (t-+i)r \r(t)s(r,[l]=['0'',"V'])dt (t-\)T (*+i)r (k-l)T (*+i)r Jr(/>(f;[/]=[T\TD* (i-i)r (i+i)r + jr(r>(r;[/]=[T,"0"]Vt (*-i)r Similarly the maximum-likelihood criterion for a communications system with memory using M symbols can be expressed as max nns (*)(r))}=m^j;f>(K0| s' Lm=l (*+0 Expanding this to account for the effect symbol s has on the starting phase of the k+2nd symbol results in 20 max M M max-^ Z I m=1 m=1 (*+2) P (A + U which in turn can be implemented for the binary case using correlation metrics by evaluating the relation (k+l)T (r(f)s(f;[/]=["0'\T\T])A (i-i)r (*+i)r + Jr(r>(r;[/]=["0",T,"0"]Vr ,<*=-o' U-i)r > (*+i)r < + Jr(r>(/;[/]=["0","0",T])rfr (*-i)r (k+l)T + jV(r>(r; i1) = ["" -. "0"D<* (i-i)r (*+i)r Jr(rMr;[/]=[T,T,T]Vt (i-i)r (k + i)T + fr(t>(r;[/]-[T,T,"0"]Vt (*-i)r (t+i )t + fr(f>(f;[/]=[T\"0",T])rff (k+l)T + jr(t)s(t; [/] = [T',"0","0"])rfi (*-i)r Expansion accounting for the effect on starting phase of subsequent symbols can continue to the point where all symbol intervals of the message are involved in the maximum-likelihood decision criterion; however, implementation would be extremely unpractical. Limiting the consideration of symbol s/^s effects to just the first D-l symbol intervals following the Â£-th symbol interval results in the decision criterion 21 max< M I M z p(r< m=1 m-1 , (*+Â£>-!) in v (**!) (*+n-i [2], By enumerating all possible messages occupying symbol intervals k+1 through k+D-1 using the variable A, where n ranges from 1 to the correlation metric corresponding to the above decision criterion is w(D-ii(*+Â£)-i)r =i (t-i)r i;U,=0 > < "=i (i-i)r This correlation metric can be implemented by using a detector consisting of a bank of match filters similar to the one described in the previous section. Section 2.1 discussed a detector consisting of a bank of filters whose impulse responses were matched to each of the possibly transmitted symbols. The outputs of each filter sampled at integer multiples of the symbol interval comprised an observation vector whose probability density function was conditioned on the transmitted symbol. The same receiver structure can be used to implement detection of a coherent signal with memory by extending the observation interval from one symbol interval to D+l and by matching each 22 filters impulse response to one of the M possibly transmitted waveforms. Thus the pdf of the observation vector conditioned on a message of D symbols is r p(rk\s{f\li,An^)=P v where the set of waveforms [/(, y4n ]) were shown in appendix A1 to constitute a set of orthonormal basis functions that are, obviously, complete with respect to the set of transmitted signals. Following the mathematics in section 2.1 this conditional pdf may be expressed as 23 which may be refined to a form involving the correlation metric: f A v^oy T ( 2 (t+D-l)T exp (it+D-i)r s-i)i i T7 + 77- V^o (t-i)r (*-i)r where the orthonormality of s(t) was used to replace the ^s2{t)dt term in the exponentiation function with ~yNa. As in section 2.1 it is recognized that the only term of the conditional pdf that is affected by the choice of s(t;[Ij,A J) is ' 2 (*+d-\)t \ exp tt fI7.. \N> (.-V ) Thus choosing the sequence of symbols that maximizes the conditional pdf is equivalent to choosing the sequence that maximizes the above term for all possibly transmitted sequences of length D: max Mrk K*; [A. A ]))} = max *(';[/,.4.1) exp f 2 (*+-r1)r (*-i)r yj Returning to the issue of identifying the first of D symbols observed by the detector, maximizing the pdf of observation vector r* conditioned on transmitted symbol s/k)(t) can be expressed as 24 max{p(r(f]|(r))}= maxj Z p(r* |Jfc t7/ A D) ...Ifl-n Z' 2 (i+D-l)r -max Â£ exP TT =i N VJVo (*-i)r For the case of MSK the above decision criterion can be expressed as the likelihood ratio test (o-D ( ^ (i-+z>-i)r Z exp A 1 = kn i*-V________________________ ( ~ (*+z>-i)r Afto-i) f 2 (t+D-\)T Z exP tt Jr(*M*;["0">4,D* N V (*-1)7" where M equals 2 and the /r-th transmitted symbol is identified as a 1 when l is greater than one and as a 0 otherwise [7]. 25 2.3 Non-coherent MSK Detection Non-coherent detection is the identification of transmitted symbols when the carrier phase is unknown [1][2][7], The decision criteria presented earlier in this thesis implicitly assumed that the carrier phase was known; thus no difference between the initial phase of the received signal, at the start of the interval in which symbol identification was made, and each of the match filter impulse responses had to be modeled. If the carrier phase is not known then some account of it must be introduced to represent the fact that the orthonormal basis function expansion of the received signal may not be matched in phase. A solution to the problem of representing the unknown carrier phase is to treat it as a random variable uniformly distributed over the interval [0, 2tt] [1][2][7]. The probability density function of the received signal conditioned on symbol s, being transmitted in the k-th symbol interval must then be averaged over all possible starting phase values: 0 26 Hence selecting the symbol s,- that maximizes the conditional pdf in a manner similar to what was described in the previous section also requires averaging over all possible starting phase values: m/ax| ] X p(r H'; I7.- A D> It should be noted that the observation vector r* covers a range of symbol intervals different from what was discussed in previous sections. In the literature non-coherent detection is presented with the observation interval being odd in length with symbol identification made on the middle baud interval. Thus the observation interval length D consists of D-1 2 symbol intervals preceding the interval in which identification is made and D-1 2 symbol intervals following [1][7]. Justification for this is found in [7] where it is mentioned that the magnitude of complex correlation of two CPFSK waveforms differing in one symbol is minimized when the center baud interval contains the difference. 27 Hence the observation vector rk used to determine the identity of the A-th bit covers symbol intervals A - D-1 through k + D -1 allowing the maximization of the conditional pdf to be expressed as maxi/?lnt|5 ,'*(<))}= max >) 2jt ( O (k+r)T n=l o Z JexP 7T !r(f N \^iV0 (k-T-[)T p{ D-1 was made for ease of notation. Averaging over the random phase Â£ p()d 2 / = I'c n=l N Z'n vvo y m-" 2 A vvo y A/ Z'. where (A+r)r ^ d*-r-l)r y y (A+r)r + J r(/> r;fr.4,lf ^(Ar-r-l)r V 2 - * and 28 ( (A + r)7* y f (k+r)T / \ V -0 n jKrM;tv\4,Jo)* V(*-r-l)7- J /-(/V V.(Ar-l)7- V 2 - dt as noted in [7]. It may be observed that this form of the decision criteria is an envelope detector given that the magnitudes of correlating the received signal with both the inphase, s(t; [/,, An J = f), |