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WIRELESS COMMUNICATION TECHNOLOGIESRutgers University, Department of Electrical and Computer EngineeringCourse No: 16:332:559:01 (Advanced Topics in Communication Engineering)Spring 2001Professor: Narayan Mandayam
Summary by: Andrej Domazetovic
Lectures No: 10 and 11February 21 and 26
CONTINUOUS PHASE MODULATION (CPM)
In order to further improve bandwidth efficiency, some of the mobile radio systems usespecial kind of frequency modulation schemes called continuous phase modulation - CPM.The schemes are particularly attractive because they have constant envelope and excellentspectral characteristics resulting from phase changes in a continuous manner.
The complex envelope of any CPM signal can be represented as:
)()(2
)( tjdnThxkj
AeAetv
t
nfnf
Φ−
=∫∑
= ∞−
ττπ (1)
where,A is the amplitude,kf is the peak frequency deviation,hf(t) is the frequency shaping pulse,{xn} where xn∈{±1, ±3, ..., ±(M-1)} is the source symbol sequence,M is the symbol alphabet size, andT is the symbol duration.
The phase term Φ(t) for the time interval kT ≤ t ≤ (k+1)T can be rewritten as,
∫ ∑ ∫∞−
−
−∞=−+−=Φ
kT K
n
t
kTfkffnf dkThxkdnThxkt
1
)(2)(2)( ττπττπ (2)
where the first term corresponds to accumulated phase and the second term to current phasefrom the beginning of current symbol period up to time t. Sometimes, Φ(t) is called excessphase. The phase is therefore continuous as long as the frequency shaping function hf(t) doesnot contain impulses.
2
The frequency shaping function can be arbitrarily chosen. However, for the purpose of ouranalysis, we will divide it in two possible forms:• full response, when the duration of hf(t) is equal to symbol period T, and• partial response , when the duration of hf(t) is greater than symbol period T, thus is
extended across several symbols.
The equation (2) represents the excess phase for the time interval kT ≤ t ≤ (k+1)T andcombining all such intervals we can write v(t) in the standard form:
∑ −=k
kxkTtbAtv ),()( (3)
where
)(),(
1
))()((tuextb T
txxTj
k
k
nkn∑
=−
−∞=
+ ββ (4)
and
<
≤≤
<
= ∫tTT
Ttdhk
t
tt
ff
,)(
0,)(2
0,0
)(0
β
ττπβ (5)
The first term in the exponent of equation (4) represents the accumulated excess phase,while the second term is the excess phase trajectory for the current source symbol.
Two parameters characterize this type of modulation:• Average frequency deviat ion:
∫=T
ff
f dtthT
kk
0
)( (6)
• Modulation index:
∫ ===T
fff TkdtthT
TkT
h0
2)(1
2)(
πβ
(7)
3
By choosing different shaping pulses hft), modulation indices h and alphabet sizes M we cangenerate an infinite variety of different classes of CPM signals. Here, we will describe fourimportant schemes:• Continuous phase frequency shift keying (CPFSK)• Minimum shift keying (MSK)• Partial response CPM• Gaussian minimum shift keying (GMSK)
Continuous Phase Frequency Shift Key ing (CPFSK)
For CPFSK,
hf(t) = uT(t) – frequency shaping function is rectangular function,_kf = kf – average frequency deviation is equal to peak frequency deviation,
h = 2kfT , and
<=≤≤=
<=
tThTk
TtThttk
t
t
f
f
,2
0,/2
0,0
)(
ππππβ (8)
Since β(t) is continuous function of time, the CPM signals cannot be represented as discretepoints in the signal space diagram. Therefore, the CPM signals are represented by sketchingthe excess phase (9) for all possible symbol sequences {xk}. The plot is called a phase tree .
∑−
−∞=−+=Φ
1
)()()(k
nkn kttxxTt ββ (9)
Note that since the shaping function is rectangular, the phase changes are linear. Forexample, the binary phase tree is shown in Figure 1.
4
t/T64 512 3
+1
-1
π
π
π
π
π
π
0
3h
2h
h
-3h
-2h
-h
Figure 1 : Phase tree of binary CPFSK with an arbitrary modulation index
Minimum Shift Keying (MSK)
MSK is a special case of binary CPFSK, with modulation index h=0.5. For MSK
<=≤≤=
<=
tTTk
TtTttk
t
t
f
f
,2/2
0,2/2
0,0
)(
ππππβ (10)
The carrier phase, Φi(t) = 2πfct + Φ(t), during interval kT ≤ t ≤ (k+1)T is
∑−
−∞=
−++=Φ1
222)(
k
nknci T
kTtxxtft
πππ (11)
Rearranging,
5
∑−
−∞=−++=Φ
1
22)
22()(
k
nkn
kci x
kxt
T
xft
ππππ (12)
Now, during time interval kT ≤ t ≤ (k+1)T, the MSK band-pass signal can be represented as,
=−++= ∑−
−∞=
1
]22
)2
2cos[()(k
nkn
kc x
kxt
T
xfAts
ππππ
∑−
−∞=−++=
1
]22
)4
(2cos[)(k
nkn
kc x
kxt
T
xfAts
πππ (13)
From the above, we can see that in time interval kT ≤ t ≤ (k+1)T MSK signal has one of twopossible frequencies,
Tff cL 4
1−= and T
ff cU 41+=
The difference between these frequencies is T
fff LU 21=−=∆ . This is the minimum
frequency separation required to insure orthogonality with coherent modulation, hence thename minimum shift keying. The power spectral density of MSK signal is depicted in Figure 2.
0 0.5 1 1.5 2 2.5 3 3.5 4-70
-60
-50
-40
-30
-20
-10
0
Normalized frequency fT
Svv
(f) [
dB]
Figure 2: Power density spectra of MSK
6
Partial Response CPM
Partial response CPM signals are a broad class of signals characterized by a frequencyshaping pulse hf(t) of duration greater that symbol period T. If hf(t) has duration KT, then
)()()( tuthth KTff = (14)
where
∑−
=−=
1
0
)()(K
kTKT kTtutu (15)
The advantage of using partial response CPM is an improvement in the spectralcharacteristics of the modulated signal by providing both a narrower main lobe and faster roll-off of side lobes. Combining (14) and (15), the frequency shaping function can be written as:
)()()(1
0
kTtuthth Tf
K
kf −= ∑
−
= (16)
Gaussian Minimum Shift Keying (GMSK)
GMSK is a special case of partial response CPM that uses a low-pass pre-modulation filterwith transfer function:
22ln
2
)(
−
= B
f
efH (17)
where B is the bandwidth of the filter. Since H(f) is the bell-shaped about f=0, the modulationscheme has been named Gaussian MSK. The frequency shaping pulse is usually describedwith BT – normalized filter bandwidth. Note that when BT decreases, the pulse has durationgreater than one symbol period T, thus introducing ISI. For this reason, systems using GMSK(like GSM) must have strong equalizer. The relation between MSK and GMSK is illustratedusing power spectral densities with GMSK BT product equal BT=0.3 sketched in Figure 3.
7
MSK GMSK
0 0.5 1 1.5 2-120
-100
-80
-60
-40
-20
0
Normalized frequency fT
Svv
(f) [d
B]
Figure 3: Comparison between MSK and GMSK PSDs
References:Most of the material presented above is based on information provided during class lectures.Deeper description of some features is taken from[1] ‘Principles of Mobile Communications’, Gordon L. Stuber, Kluwer Academic Publishers,Boston, 1996, and[2] ‘Digital Communications’, John G. Proakis, McGraw-Hill, third edition
DETECTION THEORY
The purpose of this part is to revisit some of the basic concepts of detection of signals innoise, as well as the way to calculate the probability of error for various modulation schemes.
Let us recall the basic principles of any general communication system. The transmittertransmits a sequence of messages from a given alphabet {m1, m2, ... ,mM} of size M. Themessages are represented as signal waveforms s1(t), s2(t),..., sM(t), respectively, for thepurpose of transmission. Each of the waveforms has a duration T called the symbol period.One waveform is transmitted every T seconds. The transmitted waveform depends on therandom sequence that is the actual information encoded for transmission. Recall that if theinput sequence were known in advance, no transmission would be necessary. Therefore,
8
when the input message m equals mi i=0,1,...,M-1, the transmitted signal is si(t). Thus thecorrespondence,
m = mi ⇔ s(t) = si(t) (18)
The a priori probabilities {P(mi)} of the occurrence of each of the input messages specify theinput source. A convenient way to synthesize and represent the transmitting signal waveformsis to use orthonormal signal set defined for the whole transmitting signal set {si(t)}. Then eachand every signal from the given set {si(t)} will be represented as the linear combination oforthonormal waveforms as:
∑=
=N
jjiji tsts
1
)()( φ , i=0,1,...,M-1 (19)
where {φj(t)} are orthonormal, by which is implied:
∫∞
∞−
≠=
=kj
kjdttt jk
kj 0)()(
δφφ for all j and k 0 ≤ j,k ≤ N
where the number N is dimension of the signal space. Note that N ≤ M.
Therefore, we can see that the signal si(t) is completely determined by coefficients sij inequation (19) and therefore can be represented as a vector of these coefficients:
si = [si1, si2, ... ,sim]T, i = 0,1,...,M-1
where
∫∞+
∞−= dtttss kiik )()( φ i = 0,1,...,M-1
Usually, the signals are visualized in N-dimensional space, named signal space, where theset {φi(t)} of orthonormal functions represent the axes. Thus, {sik} are projections of the signalsi(t) onto each of the axes. The axes are, of course, mutually perpendicular.
Recall that using such a representation, the white Gaussian noise has infinite dimensionality.Fortunately, using the theorem of irrelevance, the only noise detectable and relevant for thedetection of signals {si(t)} is the one represented by its projections onto defined orthonormalset, also called basis. Therefore, the relevant additive white Gaussian noise can also berepresented as a vector of same dimension N.
9
Thus, the received signal vector can be represented as
x = si + w
where w is the additive white Gaussian noise (AWGN) vector. It can also be shown that if theGaussian noise process is of zero mean and variance N0/2, also the projections will be a setof statistically independent set of Gaussian random variables with zero mean and samevariance N0/2. This property will significantly reduce the computational effort required for thecommunication system probability of false detection analysis.
Having all this in mind, the design goal is to derive the rule for the detector that by observingthe received vector x would assign the transmitted message to it with minimum probability offalse decision. Ones such a detector is designed, it is called optimum detector. Note that theoptimality criterion is minimization of the probability of false detection.
Stating the problem more formally,
Given x, we form the set of a posterior probabilities defined as P{signal sj was transmitted | x}for all signals from a given alphabet. The decision is then based on selecting the signalcorresponding to maximum a posteriori probability. Such a criterion is called MAP (maximuma posteriori probability).
Then, our estimate m̂=mi would be wrong with probability
Pe (mi , x) = P(mi is not sent | x) (20)=1 – P(mi is sent | x) (21)
Therefore, we have to maximize the probability term in equation (21) or equivalently tominimize the equation (20). Hence, the decision rule is
1,...,1,0,)|()|(ˆ −=≠∀≥= MkkjforsentmPsentmPifmmSet kjj xx
Using Bayes rule, the a posterior probabilities can be expressed as:
)(
)|(][)|(
x
xx
x
x
f
mfmPmP jj
j =
In the above equation, it is evident that the denominator does not depend on transmittedmessage and can therefore be neglected during detection process. A receiver that
determines m̂ by maximizing fx(x|mj) is called ML (maximum likelihood) receiver. Such areceiver is often used when the a priori set of probabilities is not known. Note that when allthe transmitted messages are equally likely, the ML and MAP receivers yield the same result.
10
When the noise is AWGN and the source sequence and noise are statistically independent,we have that:
∑= =
−N
jjx
Nef 1
2
22
1
2/2)2(
1)( σ
πσxn
Also, using some probabilistic identities:
)()|()|( iiii fff mxmmxmx nnx −=−= for all i.
Therefore, the optimum receiver sets m̂=mk whenever
222/][ σiemP i
mx−−
is maximum for i=k. Note that maximizing this expression is equivalent to finding the i thatminimizes
|x – mi|2 - 2σ2 ln P[mi] (22)
This function is visualized geometrically in a signal space. Note that the first term representsthe square of the Euclidean distance in such a space. Whenever the input messages areequally likely, the optimum decision rule is defined as finding the closest vector to thereceived one in a defined signal space.
Using the above reasoning, we can divide the whole signal space into a set of regions Zi
covering the whole space and each corresponding to the message/transmitted signal vectorto which each point of the region is closer than to all other points.
When the input messages are not equally likely, the region boundaries, also called decisionboundaries, are appropriately modified in a way that at the boundary between two of thesignal vectors, the probability of decision mistake is equal.
Before giving the actual structure of optimum receiver, we will first rearrange square term ofthe equation (22):
22
1
22 ||2||)(|| ii
N
jijji mx mmxxmx +•−=−=− ∑
=
Since the |x|2 term is independent of i the decision rule is equivalent to maximizing theexpression:
11
(x.m i) + ci
where
( )22 ||][ln221
ˆ iii smPc −= σ for all i =0,1,...,M-1
and is usually called the bias term. Note that |si|2 is the signal energy.
From the above, there are at least two possible realizations of the optimum receiver:• correlat ion receiver, and• matched filter receiver
The structure of the correlation receiver is depicted in Figure 4, while matched filter receiver isgiven in Figure 5.
Note that the correlation receiver first projects the received signal x(t) onto the orthonormalset, thus forming the received signal vector x, and then performs the ‘dot-product’ operationfor all signal vectors si. Finally, the bias term is added to account for different signal a prioriprobabilities and energies, and the largest result yields the minimum probability of errorestimate of the transmitted message.
Another version of the same receiver type avoids using multiplication. Instead, it makes use ofthe filters matched to the orthonormal functions. By matched, we mean time reversed versionof the desired signal:
φφ1(T-t) is matched to φφ1(t) where T is symbol period
12
X
X
X
integrator
integrator
integrator
Weight ingmatrix
+
+
+
Selectlargest
x1
xN
x2
φ2
φ1
φΝ
c0
cM-1
c1
Figure 4: Correlation receiver structure
Weight ingmatrix
+
+
+
Selectlargest
x1
xN
x2
c0
cM-1
c1
matched to
φΝ
matched to
matched to
φ2
φ1
Figure 5: Matched filter receiver structure
13
By sampling the matched filter output at t=T, we receive the same result as with correlationreceiver. The rest of the detection procedure is the same.
Namely, the filter response would be:
∫∞
∞−+−= ααφα dtTxtu jj )()()(
and sampled at t=T.
∫∞
∞−== jjj xdxTu ααφα )()()(
Another way of showing that these filters yield optimum decision with respect to errorprobability is showing that they also yield the maximum signal to noise ratio (SNR) reception.The formal proof is based on the Schwarz inequality and will not be given here.
Understanding all this, our final goal is to calculate the average probability of error. Recall thatin general case, the error will occur if a symbol mi is transmitted, and the received vector xdoes not lie in a region Zi. Average probability of error is
∑=
=M
iii sentwasmZinlienotdoesPP
1
],[][ xε
Assuming all the messages are equally likely, the above equation can be written as:
∑=
=M
iii sentmZinlienotdoesP
MP
1
]|[1
][ xε
or
∑ ∫=
−=M
i Zii dmf
MP
1
)|(1
1][ xxxε
where the second term corresponds to average probability of a correct decision. In a generalcase, the above integral is very difficult to solve, sometimes computationally intractable. In
14
such cases we can try to estimate the bound of the performance. One of the bound is calledunion bound.
Recall from the probability theory that:
∑==
≤M
kk
M
kk APAP
11
)()(�
with equality when all the events Ak are mutually exclusive. Taking this concept into ourdetection problem, we can denote Aik as event that the observation vector x is closer to skthan to si when mi was sent. Then, the conditional probability of error given that mi was sent is
∑≠=
≠=
≤=M
ikk
ik
M
ikk
ikie APAPmP11
)()()( � for all i=1,2,...,M
where P(Aik) is the probability that the vector x is closer to sk than to si. This will occur whenthe projection of the noise onto a line that joins sk and si is greater than half the distance,assuming that the messages are equally likely. Recall that the projection of the AWGN yieldsthe set of statistically independent random variables with equal mean and variance.Therefore, we are only interested in the noise component that lies on that line. Developingthis, we have
∫∞ −
==2/ 00
)2
(211
0
2
ikd
ikN
u
ik N
derfcdue
NP
π
where dik is the distance between vectors sk and si, and erfc() is a standard complementaryerror function. Using the union bound:
∑≠=
≤M
ikk
ikie N
derfcmP
1 0
)2
(21
)(
Then, using the symmetry and assuming equally likely input messages, the total averageprobability of error can be expressed as:
15
∑ ∑≠=
≠=
=≤M
ikk
M
ikk
ikike N
dQ
N
derfcP
1 1 00
)2
()2
(21
We shall now derive the average probability of error for some generally used modulationschemes.
Coherent Binary Phase Shift Key ing (CBPSK)
With M = 2, the mapping of source symbols mi to the transmitting waveforms si(t), i = 1, 2 isgiven by
tfT
Ets c
b
b π2cos2
)(1 1 =→ for 0 ≤ t ≤ Tb
tfT
Ets c
b
b π2cos2
)(0 2 −=→ for 0 ≤ t ≤ Tb
where fc is the carrier frequency, much greater than 1/Tb, and Eb is energy per transmitted bit.The signal space looks like in Figure 6.
φ1
EbE b-
m2 m1
Z1Z2
Figure 6: The signal space with decision regions for CBPSK
The basis function is: tfT
t cb
πφ 2cos2
)(1 =
The appropriate decision rule for equally likely messages is: Choose m1 when x is greaterthan 0 (in Z1), otherwise choose m2.
16
Then,
∫ ∫ +==b bT T
i dtttwtsdtttxtx0 0
11 )())()(()()()( φφ
The conditional probability of error when m1 is sent is then
)2
()(21
}|0{00
1 N
EQ
N
EerfcsentismPP bb
e ==>= x
Assuming equally likely messages, that is also the total average probability of error forCBPSK system.
Coherent Quadrature Phase Shift Key ing (CQPSK)
Transmitting signal is given by:
( )
=
−+=
otherwise
iitfT
Ets c
i
0
4,3,2,14
122cos2
)(ππ
for 0 ≤ t ≤ T
where fc is the carrier frequency, much greater than 1/T, T is symbol period, and E is energyper transmitted symbol. The signal space looks like in Figure 7.
φ1
m4 (1,1)
Z 4Z 3
Z 2 Z 1
φ2
m3 (0,1)
m2 (0,0) m1 (1,0)
Figure 7: The signal space with decision regions for CQPSK
17
The basis functions are:
tfT
t cπφ 2cos2
)(1 = and tfT
t cπφ 2sin2
)(2 =
The appropriate decision rule for equally likely messages is: Choose mi when x is lies in Zi,where the regions are limited by axes.
Then,
−−
−=
=)
4)12sin((
)4
)12cos((
2
1
π
π
iE
iE
i
ii s
ss for i=1,2,3,4
Then, the received vector x is:
+
+±=
=2
1
2
1
2
2
wE
wE
#x
xx
To calculate the average probability of error, note that QPSK system is equivalent to twocoherent BPSK systems that modulate two orthogonal carriers at the same frequency fc. Notethat E is the symbol energy and that each symbol represents two bits.
Therefore, the average probability of a symbol error for CQPSK system is:
)2
(21
0N
EerfcP s=′
Then, expressing the probability of correct decision as Pc=(1-Pe)2 we can calculate the total
average symbol error probability as:
−=−=
0
2
0 241
)2
(1N
Eerfc
N
EerfcPP ss
cse
18
For large SNR, this can be approximated as:
)2
(0N
EerfcP s
se =
When the Gray encoding is used, as usually is, then one symbol error results most often inonly one bit error. Therefore, the bit error probability for CQPSK system will approximately behalf the symbol error:
Pe ≈ 0.5 Pse
Knowing that there are two bits per symbol and that the Es is symbol energy = 2*Eb, we canexpress total average bit error probability for CQPSK system as:
)2
()(21
00 N
EQ
N
EerfcP bb
e ==
Note that for the same SNR, CBPSK and CQPSK systems have the same probability of error,but with CQPSK we can transmit two times more data than with CBPSK.
M – ary Phase Shift Keying (M -PSK)
Transmitting signal is given by:
)2
)1(2cos(2
)(M
itfT
Ets c
s
si
ππ −+= for 0 ≤ t ≤ Ts
where fc is the carrier frequency, much greater than 1/Ts, Ts is symbol period, and Es isenergy per transmitted symbol.
Note that Es = Eb log2M, Ts=Tblog2M
Using the union bound, the average bit error probability is given as:
19
∑≠=
≤M
ikk
ike N
dQP
1 0
)2
(
Note that as M increases, the distances among signal vectors decrease, thus theperformance deteriorates.
Coherent Binary Frequency Shift Key ing (CBFSK)
For coherent binary frequency shift keying, we have transmitting signals:
tfT
Ets
b
b11 2cos
2)(1 π=→ for 0 ≤ t ≤ Tb
tfT
Ets
b
b22 2cos
2)(0 π=→ for 0 ≤ t ≤ Tb
where the frequencies are chosen as:
b
ci T
inf
+= with i=1,2 and nc being some integer.
The basis functions are:
tfT
tb
12cos2
)(1 πφ = and tfT
tb
22cos2
)(2 πφ =
The transmitted signal vectors can then be represented as:
=
=
b
b
Eand
E 0
021 ss
The signal space representation is shown in Figure 8.
20
φ1
m1
φ2
m2
Z1Z2
Figure 8: Signal space representation for CBFSK
The decision rule is: Choose m1 when x1>x2, otherwise choose m2.
The probability of error is given as:
)()2
(21
00 N
EQ
N
EerfcP bb
e ==
The comparison among several M-PSK modulation schemes with respect to bandwidthefficiency and power efficiency for the required BER of 10-6 is given in the following table.
M 2 4 8 16 32 64ηp = Eb/N0 10.5 10.5 14 18.5 23.4 28.5ηb=Rb/B 0.5 1 1.5 2 2.5 3Table 1: The performance comparison among M-PSK modulation schemes for the required
BER of 10-6
Minimum Shift Keying (MSK)
Recall that the MSK signal can be represented as:
−+
+= ∑
−
−∞=n
k
nn
kc xxt
T
xfAts
2222cos)(
1 ππππ
If we consider interval 0 ≤ t ≤ Tb and denote
b
b
T
EA
2= , then:
21
+
+=
0))0(2cos(2
1))0(2cos(2
)(
2
1
symbolfortfT
E
symbolfortfT
E
ts
b
b
b
b
θπ
θπ
where the phase term depends on the accumulated excess phase. Using phase trellis ande.g. Viterbi decoding scheme, it can be shown that for high SNR, the average bit errorprobability for MSK can be approximated as:
)2
(0N
EQP b
e ≈
References:The material presented above is largely based on information provided during class lectures.Additional information is taken from:[1] ‘Principles of Mobile Communications’, Gordon L. Stuber, Kluwer Academic Publishers,Boston, 1996,[2] ‘Digital Communications’, John G. Proakis, McGraw-Hill, third edition[3] ‘Principles of Communication Engineering’, Wozencraft, Jacobs, Waveland Press, 1990.
