Example 1.1. Let’s examine a SS system using binary (1) DS spreading modulation as in (1.22), multiplicative data modulation (1.24), and single- channel phase-synchronous detection (solid line portion of Figure 1.3(a)) over the data symbol duration Tsof Ncchip times, i.e.,TsNcTc. Hence, the pseudorandom quantities which are known to the receiver, but unknown to the jammer, are the DS pulse modulation sequence and the car- rier phase fT. The inner product of two such waveforms with different pseudorandom variables and data modulations is given by
(1.29)
For any particular choice of and fT, and regardless of the val- ues of the data modulation dand d, constants over (0,Ts), the two signals are orthogonal if either T T p/2 or is orthogonal to
. Since the set of real Nc-tuples forms an Nc-dimensional space (and furthermore an orthogonal basis of vectors with 1 entries can be found when Nc is a multiple of 4), and since carrier phase differences of
3c1, pcNc4 3c¿1, pc¿N
c4 3c1, pcNc4
dd¿cos1fTf¿
T2
2 a
Nc
n1
cn c¿
n0Tcƒp1t2ƒ2dt.
dd¿cos1fTf¿T2
2 0Tsc1t2c¿1t2dt 0TsRe5dc1t2ej1vctfT26Re5d¿c¿1t2ej1vctf¿T26dt
c1, c2, pcNc
Energy Gain Calculations for Typical Systemshttp://jntu.blog.com 17
magnitude p/2 cause orthogonality, the jammer is forced to view his wave- form selection problem as being defined for an orthogonal communication system complex with multiplicity factor Kgiven by
(1.30) Example 1.2. Suppose that two independently hopped SS waveforms, c(0)(t) and c(1)(t), of the form (1.20) are employed in a switching modulation scheme as in (1.26) to transmit binary data. The data symbols’ duration Ts spans MDhop times. Frequency-synchronous correlation computations in the receiver are then carried out over individual hop times, and the correlator’s sync clock produces a pulse every Thseconds. The signal parameters, known to the receiver but not to the jammer, are th two pseudorandom hopping
sequences and which represent the two
possible data symbols. Two similar FH waveforms have inner product
(1.31) Orthogonality between two such waveforms is guaranteed, regardless of the values of the phases and data symbols (d,d), provided that
(1.32) where kis any non-zero integer, for all d,din {0, 1}. We assume that two such orthogonal frequency waveforms are used by the communication sys- tem as f0nand f1n, with a different pair for each n, 1 nMD.
If the transmitter and receiver are capable of producing and observing MF distinct orthogonal tones (assume MFis even for convenience), it is clear that during each hop the jammer mmust contemplate combatting a pure SS strat- egy on an orthogonal communication system complex of multiplicity MF/2.
Therefore, during each hop time a single link in the orthogonal system com- plex requires four dedicated orthonormal basis functions (e.g., sines and cosines at two distinct frequencies), and uses 4MDsuch functions over MD hops. By the same reasoning the number of basis functions available to the entire complex is 2MFMD, and hence, the energy gain of (1.18) is given by
(1.33) EG 2MFMD
4MD MF
2 . fd1f¿
d¿1 k Th 1f1, f¿12
sin1f1f¿121cos12p1fd1f¿d¿12Th212 4. 1
4p1fd1f¿
d¿12 3cos1f1f¿12sin12p1fd1f¿d¿12Th2 1
20Thcos12p1fd1f¿d¿12t 1f1f¿122dt 0Thcos12p1fcfd12tf12cos12p1fcf¿d¿12tf¿12dt
f11, f12, p, f1MD f01, f02, p, f0MD
K2Nc2Ts>Tc.
18 A Spread-Spectrum Overview
02_c01 8/16/01 5:59 PM Page 18
http://jntu.blog.com
Example 1.3. One possible hybrid SS communication system employs TH, FH, and DS modulations to produce the wideband waveform
(1.34) in which p(t) is a unit-amplitude rectangular pulse of chip time duration Tc. A total of Ncpulses, modulated by the sequence are concatenated to produce a DS waveform of duration Th, which in turn is frequency-hopped to one of MFfrequencies and time-hopped to one of MTtime intervals within the symbol time Ts. Hence,
(1.35) Modulation by the n-th M-ary data symbol dnis accomplished by switching the DS modulation to the dn-th of Morthogonal vectors
Two such hybrid SS signals have inner product given by
(1.36) Here dzis the Kronecker delta function, which is one if z0 and is zero otherwise. Orthogonality of these waveforms can be achieved, regardless of the data values (d,d) and random phase values , if one or more of the following conditions holds:
1. (1.37)
2. (1.38)
3. (1.39)
The signal variables, known a priorito the receiver, but not the jammer, are the pseudorandom DS chip values {c0md:d1, 2, . . . ,M,m1, 2, . . . ,Nc}, the hop frequency f0, and the time interval index b0. Hence, the receiver must
f0f0¿ k
Tc , for k integer, k0.
b0b¿0. a
Nc
m1
c0mdc¿0md0, for all d, d¿. 1f0, f¿
02 sin1f0f¿021cos12p1f0f¿02Tc212 4a
Nc
m1
c0md0c¿0md¿
0. db0b¿0
# 1
4p1f0f¿
02 3cos1f0f¿02sin12p1f0f¿02Tc2 a
Nc
m¿1
c¿
0m¿d¿0patm¿Tc b¿0
MT Tsb cos12p1fcf0¿2tf¿
02dt
0TSmaNc1
c0md0patmTc b0
MT Tsbcos12p1fcf02tf02
cn1dn, . . ., cnNcdn. TsMTThMTNcTc.
5cnmdn6,
c1d1t221t2 a
n
ej12pfntfn2 a
Nc
m1
cnmdnpatmTc an bn
MTbTsb
Energy Gain Calculations for Typical Systemshttp://jntu.blog.com 19
observe the signal in a 2M-dimensional space whose basis over the interval (0,Ts) consists of Re{c(d)(t) exp(j2pfct)} evaluated for each of the Mvalues of d, with the random hop phase f0set at 0 and at p/2. The jammer, how- ever, must choose his waveform to jam a signal space of dimension 2NcMFMT, whose basis consists of the sines and cosines of Ncorthogonal DSmodula- tions hopped over MF orthogonal tones and MT disjoint time intervals.
Therefore, the nominal energy gain EGof this system is
(1.40) The nominal minimum bandwidth Wssrequired to implement the orthog- onality requirements (1 3) is determined by the minimum hop frequency spacing 1/Tcto be MF/Tc. Using this fact and substituting (1.35) into (1.40) indicates that the energy gain for this hybrid SS system in a closely packed design is TsWss/M.