E303: Communication Systems

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1 E303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London An Overview of Fundamentals: PN-codes/signals & Spread Spectrum (Part B) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 1 / 56

2 Table of Contents 1 Introduction Modelling of PN-signals in SSS Equivalent Energy Unitisation Effi ciency (EUE) Comments 2 Classification of Jammers in SSS 3 Direct Sequence SSS Introductory Concepts and Mathematical Modelling PSD(f ) of a Random Pulse Signal PSD(f ) of a PN-Signal b(t) in DS-SSSs PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) 4 DS-BPSK Spread Spectrum Output SNIR Bit Error Probability with Jamming 5 DS-SSS on the (p e, EUE, BUE)-parameter plane 6 Anti-Jam Margin 7 Frequency Hopping SSSs 8 Appendices Appendix A. Block Diagram of a Typical SSS Appendix B. BPSK/DS/SS Transmitter and Receiver Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 2 / 56

3 Introduction Introduction (a) SSS (jammers): (b) CDMA (K users): Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 3 / 56

4 Introduction Modelling of PN-signals in SSS Modelling of PN-signals in SSS Consider { {α[n]}=a sequence of ± 1 s { } c(t) = an energy signal of duration T c, e.g. c(t) = rect t T c We have seen that the PN signal b(t) can be modelled as follows: DS-SSS (Examples: DS-BPSK, DS-QPSK): b(t) = n α[n].c(t nt c ) (1) FH-SSS (Examples: FH-FSK) b(t) = n exp {j(2πk[n]f 1 t + φ[n])}.c(t nt c ) (2) where {k[n]} is a sequence of integers such that {α[n]} {k[n]} (3) { } and with φ[n] = random: pdf φ[n] = 1 2π rect φ 2π Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 4 / 56

5 Introduction Equivalent Energy Unitisation Effi ciency (EUE) Equivalent Energy Utilisation Effi ciency (EUE) Remember: Jamming source, or, simply Jammer = intentional interference Interfering source = unintentional interference With area-b = area-a we can find N j P j = 2 areaa = 2 areab = N j B j N j = P j B j Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 5 / 56

6 Introduction Equivalent Energy Unitisation Effi ciency (EUE) if then B J = qb ss ; 0 < q 1 (4) EUE J = E b EUE equ = N J E b N 0 + N J = P s.b J P J.r b = PG SJR in q }{{} EUE j = P s.q.b ss P J.B = PG SJR in q (5) (6) ( ) 1 N0 + 1 (7) N j where SJR in P s P J (8) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 6 / 56

7 Introduction Comments Comments EUE equ (or EUE J ): very important since bit error probabilities are defined as function of EUE equ (or of EUE J ) For a specified { performance the smaller the SIRin the better for the signal the larger the SIR in the better for the jammer Jammer limits the performance of the communication system i.e. effects of channel noise can be ignored i.e. Jammer Power P n EUE equ = E b N 0 +N J E b N J = EUE J N.B. exception to this: i) non-uniform fading channels ii) multiple access channels number of possible jamming waveforms There is no single jamming waveform that is worst for all SSSs There is no single SSS that is best against all jamming waveforms. Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 7 / 56

8 Classification of Jammers in SSS Classification of Jammers in SSS Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 8 / 56

9 Direct Sequence SSS Introductory Concepts and Mathematical Modelling Direct Sequence SSS Introductory Concepts & Mathematical Modelling If BPSK digital modulator is employed then the BPSK-signal can be modelled as: BPSK s(t) = A c sin(2πf c t + m(t) π 2 ) (9) where the data waveform can be modelled as follows: m(t) a[n] c 1 (t n T cs ); nt cs t < (n + 1) T cs (10) n }{{} rect{ t n Tcs Tcs } with {a[n]} = sequ. of independent data (message) bits (±1 s) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v.17 9 / 56

10 Direct Sequence SSS Introductory Concepts and Mathematical Modelling Equation (9) can be rewritten as follows: BPSK s(t) = A c m(t) cos(2πf c t) BPSK can be considered as { PM AM (11) remember: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

11 Direct Sequence SSS Introductory Concepts and Mathematical Modelling The PSD(f ) s of m(t) and s(t) are shown below Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

12 Direct Sequence SSS Introductory Concepts and Mathematical Modelling If a DS/BPSK modulator is employed, then the SS-transmitted signal is ±1 {}}{ DS/BPSK : s(t) = A c sin 2πF c t + m(t)b(t) π (12) 2 = A c m(t)b(t) cos(2πf c t) (13) where m(t) a[n] c 1 (t nt cs ); nt cs t < (n + 1)T cs n }{{} rect{ t ntcs Tcs } b(t) = k chip-duration α[k] c 2 (t kt cs ); kt c t < (k + 1) T c }{{} rect{ t ktc Tc } (14) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

13 Direct Sequence SSS Introductory Concepts and Mathematical Modelling BPSK DS/CDMA Transmitter & Receiver TX (see also Appendix-B): Rx (see also Appendix-B): Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

14 Direct Sequence SSS Introductory Concepts and Mathematical Modelling The PN-sequence {α[l]} (whose elements have values ±1) is M times faster than the data sequence {a[n]}. i.e. i.e. T cs = M T c (15) PN-signal Bandwidth data-bandwidth (16) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

15 Direct Sequence SSS Introductory Concepts and Mathematical Modelling Systems which have coincident data and SS code clocks are often said to have a data privacy feature such systems are easy to build and can be combined in single units T cs = 5 MT c Note: chip = T c = smallest time increment Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

16 Direct Sequence SSS Introductory Concepts and Mathematical Modelling If the above data privacy feature is taken into account then m(t) n a[n].c 1 (t n T cs ) nt cs < t < (n + 1) T cs k with T cs = MT c ; = n; b(t) = α[k].c 2 (t kt c ) kt c < t < (k + 1) T c (17) k M where n T cs + k T c t < n T cs + (k + 1) T c k = 0, 1,..., M 1 = k mod M Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

17 Direct Sequence SSS Introductory Concepts and Mathematical Modelling Conclusion DS/BPSK similar to BPSK except that the apparent data rate is M times faster signal spectrum is M times wider Therefore Note: PG = B ss B = M (18) 1 message cannot be recovered without knowledge of PN-sequence i.e. PRIVACY 2 typical: PN-chip-rate several M bits/sec data rate few bits/sec Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

18 Direct Sequence SSS Introductory Concepts and Mathematical Modelling PSD of DS/BPSK/SS Transmitted Signal Tx signal s(t) = m(t) b(t) A c cos(2πf c t) PSD s (f ) = PSD m (f ) PSD b (f ) PSD Ac cos(2πf c t)(f ) PSD s (f ) = PSD m (f ) PSD b (f ) PSD Ac cos(2πf c t)(f ) remember PSD Ac cos(2πf c t)(f ) = A2 c 4 (δ (f F c ) + δ (f + F c )) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

19 Direct Sequence SSS PSD(f ) of a Random Pulse Signal m(t) n a[n].c(t n T cs ) (see "line-codes") PSD m (f ) = FT(c(t)) 2 T cs R aa [0] + + k= k =0 R aa [k] exp( j2πfkt cs ) (19) Note that{ if statistical { } indep. we have: E a 2 R aa [k] = n for k = 0 E {a n } E {a n+k } for k = 0 { µ 2 i.e. R aa [k] = a + σ 2 a for k = 0 where µ a = mean and σ a = std for k = 0 then µ 2 a PSD m (f ) = σ 2 FT(c(t)) 2 a T }{{ cs } Continuous Spectrum + µ2 a Tcs 2 comb 1 ( FT(c(t)) 2 ) Tcs } {{ } Discrete Spectrum Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

20 Direct Sequence SSS PSD(f ) of a Random Pulse Signal PSD(f) of a Random Pulse Signal For a random pulse signal m(t) n a[n].c(t n T cs ) (i.e. a sequence of pulses where there is an invariant average time of separation T cs between pulses) with all pulses of the same form but with random amplitudes a[n] with mean=µ a = E{a[n]} and std=σ 2 a = E{(a[n] µ a ) 2 } statistical independent random time of occurrence, Then: PSD m (f ) = σ 2 FT(c(t)) 2 a T }{{ cs } Continuous Spectrum + µ2 a Tcs 2 comb 1 ( FT(c(t)) 2 ) Tcs } {{ } Discrete Spectrum Note that if µ a = 0 (this is a very common case ) then PSD m (f )=E{a[n] 2 FT(single pulse) 2 } T cs Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

21 Direct Sequence SSS PSD(f ) of a Random Pulse Signal Example: PSD of a random BINARY signal m(t) Consider a random binary sequence of 0 s and 1 s. This binary sequence is transmitted as random signal m(t) with 1 s and 0 s being sent using the pulses shown below. For instance a random binary sequence/waveform could be If 1 s and 0 s are statistically independent with Pr(0) = Pr(1) = 0.5, the PSD of the transmitted signal can be estimated as follows: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

22 Direct Sequence SSS PSD(f ) of a Random Pulse Signal Solution: 2 FT { PSD m (f ) = E a[n] 2}. T cs { = E a[n] 2}. T cs 2 sinc 2 (ft cs ) T cs = E {a[n] 2} T cs sinc 2 {ft cs } }{{} =( A 2 ) 1 2 +(+A2 ) 1 2 = A 2 T cs sinc 2 {ft cs } R mm (τ) = FT 1 {PSD m (f )} =FT 1 { A 2 T cs sinc 2 {ft cs } } { } = A 2 Λ τ T cs Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

23 Direct Sequence SSS PSD(f ) of a PN-Signal b(t) in DS-SSSs PSD(f) of a PN-Signal b(t) in DS-SSSs Autocorrelation function: R bb (τ) R bb (τ) = ( )} Nc +1 N c rep Nc T c {Λ τ T c 1 N c (20) Using the FT tables the PSD b (f )= FT{R bb (τ)} of the signal b(t) is: PSD b (f ) = { Nc +1 comb Nc 2 1 sinc 2 {f T c } } 1 N Nc Tc c δ(f ) (21) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

24 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) PSD(f) of DS/BPSK Spread Spectrum Tx Signal s(t) s(t) = m(t) b(t) A c cos(2πf c t) PSD s (f ) = PSD m (f ) PSD b (f ) A2 c 4 (δ(f F c ) + δ(f + F c )) }{{} term 1 Ignore (for the time being) the effects of m(t) PSD term1 (f ) = A2 c 4 PSD b(f ) (δ(f F c ) + δ(f + F c )) { = A2 c 4 { N c +1 comb Nc 2 1 sinc 2 (f T c ) } 1 } δ(f ) Nc Tc N c (δ(f F c ) + δ(f + F c )) ( = A2 c 4 { Nc +1 comb Nc 2 1 sinc 2 {(f F c )T c } } Nc Tc A2 c 4 + comb 1 Nc Tc 1 N c (δ(f F c ) + δ(f + F c )) { sinc 2 {(f + F c )T c } }) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

25 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) i.e. PSD term1 (f ) = A2 c 4 Nc + 1 Nc 2 A2 c 4 ( comb 1 Nc Tc + comb 1 Nc Tc { sinc 2 {(f F c )T c } } { sinc 2 {(f + F c )T c } }) 1 N c (δ(f F c ) + δ(f + F c )) (22) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

26 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) Above the effects of m(t) have not been taken into account. If m(t) is used, then each discrete frequency in Equation (22) becomes a sinc 2 function. There are two cases: CASE-1: 1 T cs < 1 2N c T c CASE-2: 1 T cs 1 N c T c the peaks will merge into a continuous smooth spectrum Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

27 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) CASE-1: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

28 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) CASE-2: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

29 Direct Sequence SSS PSD(f ) of DS/BPSK Spread Spectrum Tx Signal s(t) N.B.: If then b(t) = random s(t) = A c m(t)b(t) cos(2πf c t) If the effects of m(t) are ignored then PSD s (f ) = A 2 c T 2 c sinc {ft c } 1 4 (δ(f F c ) + δ(f + F c )) PSD s (f ) = T c 4 A2 c i.e. PSD is similar to CASE-2 above ( ) sinc 2 {(f F c )T c } + sinc 2 {(f + F c )T c } Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

30 DS-BPSK Spread Spectrum Output SNIR DS-BPSK Spread Spectrum: Output SNIR Consider the block diagram of a SS-Communication System which employs a BPSK digital modulator: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

31 DS-BPSK Spread Spectrum Output SNIR N.B.: B ss = 1 T c ; B = 1 T cs ; PG = B ss B = T cs T c Then, at point T1, we have s(t) = A c m(t) b(t) cos(2πf c t) (23) and at point T1 : s(t) + n(t) + j(t) (for k = 1) (24) At the input of the receiver the Signal-to-Noise-plus-Interference Ratio (SNIR in ) is: SNIR in = EUE PG (1 + JNR in ) = EUE equ PG (25) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

32 DS-BPSK Spread Spectrum Output SNIR at point T : at point T0 : (s(t) + n(t) + j(t)) b(t τ) 2 cos(2πf c t + θ) (26) P unwanted = P nout + P code-noise + P jout (27) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

33 DS-BPSK Spread Spectrum Output SNIR if τ = 0 and θ = 0 (i.e. the system is synchronized) then: P code-noise = 0 and SNIR out-max = 2EUE equ (28) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

34 DS-BPSK Spread Spectrum Output SNIR However, Therefore, SNIR in = EUE PG (1 + JNR in ) = EUE equ PG (29) SNIR out-max = 2EUE equ = 2.PG.SNIR in (30) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

35 DS-BPSK Spread Spectrum Bit Error Probability with Jamming Bit Error Probability with Jamming A. CONSTANT POWER BROADBAND JAMMER: From the Detection Theory topic we know that the bit-error-probability p e for a Binary Phase-Shift Key (BPSK) communication system is given by: p e = T 2 EUE }{{} SNR out, matched filter where EUE = E b N 0 (31) Consider a DS/BPSK SSS which operates in the presence of a constant amplitude broadband jammer with double sided power spectral density PSD j (f ) = N j 2 (32) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

36 DS-BPSK Spread Spectrum Bit Error Probability with Jamming Then, { 2 } p e = T EUEequ where EUE equ = E b N 0 +N j with N j = P J B ss If we make the assumption that N j N 0 then p e = T { 2 EUE J } (33) where EUE J = E b N j This is known as the BASELINE PERFORMANCE of a DS/BPSK SSS Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

37 B. PULSE JAMMER: DS-BPSK Spread Spectrum Bit Error Probability with Jamming Consider a DS/BPSK SSS which operates in the presence of a jammer which transmits broadband noise with large power but only a fraction of the time. The double-sided power spectral density of the jammer is given by: PSD j (f ) = N j 2ρ (34) where ρ the fraction of time the jammer is on. P j = average jamming power P j ρ = actual power during a jamming pulse duration Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

38 DS-BPSK Spread Spectrum Bit Error Probability with Jamming Let the jammer pulse duration be greater than T cs (data bit time). { Pr(jammer = on ) = ρ Then Pr(jammer = off ) = 1 ρ and the bit-error-probability is given by: { } p e = (1 ρ)t 2 E b E + ρt 2 b N 0 N }{{} 0 + N (35) j ρ 0 (very small) which can be simplified to p e = ρ T { 2ρ EUE J } where EUE J = E b N j (36) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

39 DS-BPSK Spread Spectrum Bit Error Probability with Jamming By plotting the above equation for different values of ρ we get: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

40 DS-BPSK Spread Spectrum Bit Error Probability with Jamming Note that the value of which maximizes p e decreases with increasing values of EUE j there is a value of ρ which maximizes the probability of error p e. This value can be found by differentiating Equation-36 with respect to ρ. That is { dp e dρ = 0 ρ = EUE if EUE j j > if EUE j (37) Therefore { }} p emax = max ρ T { 2ρEUE ρ j } p emax = ρ T { 2ρ EUE j (38) = p emax = { EUE j if EUE j > T { 2EUE j } if EUEj (39) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

41 DS-BPSK Spread Spectrum Bit Error Probability with Jamming N.B.: when jammer pulse length is shorter than a data bit time T cs then the above expression is not valid. However, Equation-39 represents an UPPER BOUND on the bit-error-probability p e. The next graph illustrates the bit-error-prob. plotted against the EUE j for a baseline jammer (i.e. ρ = 1) and the worst case jammer (i.e. ρ = ρ ) for a DS/BPSK SSS. Note the huge difference between the two curves. Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

42 DS-BPSK Spread Spectrum Bit Error Probability with Jamming Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

43 DS-SSS on the (p e, EUE, BUE)-parameter plane DS-SSS on the ( p e, EUE, BUE)-parameter plane Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

44 Anti-Jam Margin Anti-Jam Margin An important parameter of SSS is the ANTIJAM MARGIN which is defined as follows: db(ajm) db(eue equ ) db(eue which corresponds to the p e,pr ) db(ajm) 10 log(eue equ ) 10 log(eue pe,pr ) AJM represents a safety margin against jammer (or against jammer plus noise). Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

45 Frequency Hopping SSSs Frequency Hopping SSSs Consider that the following part of the spectrum has been allocated to a FH/SSS: Let us partition the above spectrum onto L different frequency slots of bandwidth F 1 (or of bandwidth qf 1 where q is a constant). Then B ss = q F 1 L L = 2 m Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

46 Frequency Hopping SSSs Define the following symbols: T c = hop duration (i.e. hop rate r hop = 1 T c ) T cs = message bit duration (i.e. bit rate r b = 1 T cs ) M = number of hops per message bit (i.e. T cs = M T c ) FH/SSS: Fast hop slow hop balance hop r hop > r b r hop < r b r hop = r b The frequency slot is constant in each time chip T c, BUT changes from chip-to-chip. This can be represented by the following diagram: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

47 Frequency Hopping SSSs In general, the No. of different frequency slots L over which the signal may hop, is a power of 2. Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

48 Frequency Hopping SSSs F 1 is, in general, equal to 1 T c, i.e. F 1 = 1 T c (but this is not a necessary requirement). several FH signals occupy a common RF channel FH model of b(t) - (complex representation): { t ntc b(t) = exp {j(2π.k[n].f 1 t + φ n )} rect n T c } where k[n] =f{pn-sequ {α[n]}} k[n] is an integer that is formed by a codeword which is formed by one or more m-sequences Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

49 Frequency Hopping SSSs Transmitter-Receiver path: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

50 Frequency Hopping SSSs A Different Implementation: Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

51 Frequency Hopping SSSs L frequencies are produced by the digital Frequ. Synthesizer, separated by F 1 B s q F 1 L (40) Note: if reception= coherent : more diffi cult to achieve places constraints on the transmitted signal and transmitted medium if reception= non-coherent : PN-gen. can run at a considerably slower rate in this type of system than in a DS system Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

52 Frequency Hopping SSSs FH: non-coherent poorer performance against thermal noise. Performance: Coherent FSK (CFSK) { } p e,cfsk = T EUE Non-Coherent FSK (NFSK) p e,nfsk = 1 ( 2 exp EUE ) 2 Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

53 Frequency Hopping SSSs A serious problem is the DS: severe problem FH: much more susceptible very strong signals at receiver swapping out the effects of weaker signal {}}{ near-far problem : acquisition: much faster in FH than in DS PG = B ss B = it is not very good criterion for FH Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

54 Appendices Appendix A. Block Diagram of a Typical SSS Appendices Appendix A. Block Diagram of a Typical SSS (terrestrial & satellite comm. systems) Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

55 Appendices Appendix B. BPSK/DS/SS Transmitter and Receiver Appendix B. BPSK/DS/SS Transmitter and Receiver BPSK/DS/SS Transmitter Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

56 Appendices Appendix B. BPSK/DS/SS Transmitter and Receiver BPSK/DS/SS Receiver Prof. A. Manikas (Imperial College) E303: PN-signals and SSS (Part B) v / 56

EE401: Advanced Communication Theory

EE401: Advanced Communication Theory Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE.401: Introductory