digital modulation techniques
TRANSCRIPT
DIGITAL MODULATION TECHNIQUES
Digital Modulation FormatsModulation is defined as the process by which some characteristic of a carrier is varied in accordance with a modulating waveWith a sinusoidal carrier, the feature that is used by the modulator to distinguish one signal from another is a step change in amplitude, frequency, or phase of the carrierThe result of this modulation process is
Amplitude-shift keying (ASK)Frequency-shift keying (FSK)Phase-shift keying (PSK)
Digital Modulation Formats
Digital Modulation Formats
The scheme that attains as many of the following design goals as possible
Maximum data rateMinimum probability of symbol errorMinimum transmitted powerMinimum channel bandwidthMaximum resistance to interfering signalsMinimum circuit complexity
Coherent Binary Modulation Techniques
Coherent binary PSKThe pair of signal s1(t) and s2(t), used to represent binary symbols 1 and 0, respectively, are defined
0≤ t ≤ Tb and Eb is the transmitted signal energy per bit
( )tfTE
ts cb
b π2cos2
)(1 =
( ) ( )tfTEtf
TEts c
b
bc
b
b πππ 2cos22cos2)(2 −=+=
Coherent Binary Modulation Techniques
Coherent binary PSKThe basis function
We may expand the transmitted signal s1(t)and s2(t) in terms of Φ1(t)
)2cos(2)(1 tfT
t cb
πφ =
)()( 11 tEts bφ=
)()( 12 tEts bφ−=
bTt ≤≤0
bTt ≤≤0
bTt ≤≤0
Coherent Binary Modulation Techniques
Coherent binary PSKThe coordinates of the message point equal
dtttss bT)()( 10 111 φ∫= bE+=
dtttss bT)()( 10 221 φ∫= bE−=
Coherent Binary Modulation Techniques
Coherent binary PSK
Coherent Binary Modulation Techniques
Coherent binary PSKThe probability of symbol error
⎟⎟⎠
⎞⎜⎜⎝
⎛=
021
NEerfcP b
e
Coherent Binary Modulation Techniques
Coherent binary PSK
Coherent Binary Modulation Techniques
Coherent binary PSK
Coherent Binary Modulation Techniques
Coherent binary FSKIn binary FSK system, symbols 1 and 0 are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount
Where i = 1, 2; symbol 1 is represented by s1(t) and symbol 0 by s2(t)Eb is the transmitted signal energy per bitTransmitted frequency
( )tfTE
ts ib
b
i π2cos0
2)(
⎪⎩
⎪⎨
⎧=
elsewhereTt b≤≤0
b
ci T
inf
+=
Coherent Binary Modulation Techniques
Coherent binary FSKThe most useful form for the set of orthonormal basis functions is
( )tfTt ibi πφ 2cos0
2)(
⎪⎩
⎪⎨⎧
=elsewhere
Tt b≤≤0
Coherent Binary Modulation Techniques
Coherent binary FSKThe coefficient sij for i = 1, 2 and j = 1, 2 is defined
dtttss bT
jiij ∫= 0)()( φ
( ) ( )dttfT
tfTE
ib
i
T
b
bb ππ 2cos22cos2
0∫=
⎪⎩
⎪⎨⎧
=0
bE
jiji
≠=
Coherent Binary Modulation Techniques
Coherent binary FSKThe two message points are defined by the signal vectors
⎥⎦
⎤⎢⎣
⎡=
01bEs
⎥⎦
⎤⎢⎣
⎡=
bE0
2s
Coherent Binary Modulation Techniques
Coherent binary FSKThe observation vector x has two elements, x1 and x2, are defined by, respectively
( ) dtttxx bT)(101 φ∫=
( ) dtttxx bT)(202 φ∫=
Coherent Binary Modulation Techniques
Coherent binary FSK
Coherent Binary Modulation Techniques
Coherent binary FSKDefine a new Gaussian random variable Lwhose sample value l is equal to the difference between x1 and x2
The mean value of L depends on which binary symbol was transmitted
21 xxl −=
Coherent Binary Modulation Techniques
Coherent binary FSKThe conditional mean of the random variable L, given that symbol 1 was transmitted, is
On the other hands,
[ ] [ ] [ ]1|1|1| 21 XEXELE −= bE+=
[ ] [ ] [ ]0|0|0| 21 XEXELE −= bE−=
Coherent Binary Modulation Techniques
Coherent binary FSKThe variance of the random variable L is independent of which symbol was transmittedSince the random variable X1 and X2 are statistical independent, each with a variance equal to N0/2
[ ] [ ] [ ]21 XVarXVarLVar +=
0N=
Coherent Binary Modulation Techniques
Coherent binary FSKSuppose that symbol 0 was transmitted, the corresponding value of the conditional probability density function of random variable L equals
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−=
0
2
0 2exp
210|
NEl
NLf b
L π
Coherent Binary Modulation Techniques
Coherent binary FSKSince the condition x1 > x2, or, equivalently, L > 0, corresponds to the receiver making a decision in favor of symbol 1, the conditional probability of error, given that symbol was transmitted is given by
( )0)0( >= lPPe
( )
( )dl
NEl
N
dllf
b
L
∫
∫∞
∞
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−=
=
00
2
0
0
2exp
21
0|
π
Coherent Binary Modulation Techniques
Coherent binary FSKPut
We may rewrite
zNEl b =
+
02
( )dzzPNEe
b∫∞
−=02
2exp1)0(π
⎟⎟⎠
⎞⎜⎜⎝
⎛=
0221
NE
erfc b
Coherent Binary Modulation Techniques
Coherent binary FSKProbability of symbol error
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=
0221
NbE
erfcPe
Coherent Binary Modulation Techniques
Coherent binary FSK
Coherent Binary Modulation Techniques
Coherent binary FSK
Coherent Binary Modulation Techniques
Coherent quadrature-modulation techniques
The quadrature-carrier multiplexing system produces a modulated wave described as
sI(t) is the in-phase component of the modulated wavesQ(t) is the quadrature component
( ) ( )tftstftsts cQcI ππ 2sin)(2cos)()( −=
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)In QPSK, the phase of the carrier takes on one of four equally space values, such as π/4, 3π/4, 5π/4, 7π/4
i = 1, 2, 3, 4; E is the transmitted signal energy per symbolT is the symbol duration, and the carrier frequency fcequals nc/T for some fixed integer nc
( )⎪⎩
⎪⎨⎧
⎥⎦⎤
⎢⎣⎡ −+=
04
122cos2)(
ππ itfTE
ts ci
elsewhere
Tt ≤≤0
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)We may rewrite
( ) ( )
( ) ( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎦⎤
⎢⎣⎡ −−
⎥⎦⎤
⎢⎣⎡ −
=
0
2sin4
12sin2
2cos4
12cos2
)( tfiTE
tfiTE
tsc
c
i ππ
ππ
elsewhere
Tt ≤≤0
Coherent Binary Modulation Techniques
Quadiphase-shift keying (QPSK)There are only two orhtonormal basis functions, Φ1(t) and Φ2(t), contained in the expansion of si(t)
Tt ≤≤0( )tfT
t cπφ 2cos2)(1 =
( )tfT
t cπφ 2sin2)(2 = Tt ≤≤0
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)There are four message points, and the associated signal vectors are defined by
( )
( ) ⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛ −−
⎟⎠⎞
⎜⎝⎛ −
=
412sin
412cos
π
π
iE
iEis 4,3,2,1=i
Coherent Binary Modulation Techniques
Quadraphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)The received signal, x(t), is defined by
w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density N0/2
)()()( twtstx i +=4,3,2,1
0
=
≤≤
i
Tt
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)The observation vector, x, of a coherent QPSK receiver has two elements, x1 and x2
x1 and x2 are sample values of independent Gaussian random variables with mean values equal to
with common variance equal to N0/2 ]4)12cos[( π−iE
]4)12sin[( π−iE
∫=T
dtttxtx0 11 )()()( φ ( ) 14
12cos wiE +⎥⎦⎤
⎢⎣⎡ −=
π
∫=T
dtttxtx0 22 )()()( φ ( ) 24
12sin wiE +⎥⎦⎤
⎢⎣⎡ −−=
π
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)The probability of correct detection , Pc, equals the conditional probability of joint event x1> 0 and x2> 0, fiven that signal s4(t)was transmitted
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)Since the random variables X1 and X2 are (with sample value x1 and x2, respectively) are independent, Pc also equals the product of the conditional probabilities of the events x1> 0and x2> 0, both given s4(t) was transmitted
( ) ( )2
0
2
2
00
10
2
1
00
2exp12
exp1 dxN
ExN
dxN
ExN
Pc ⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−= ∫∫
∞∞
ππ
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)Let
We may rewrite
zN
ExN
Ex=
−=
−
0
2
0
1 22
( )2
2
2
0
exp1⎟⎠
⎞⎜⎝
⎛ −= ∫∞
−dzzP
NEebπ
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)Since
We have
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−=−∫
∞
−0
2
2
2211exp1
0 NEerfcdzz
NEbπ
2
02211
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ENEerfcPc
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎟⎠
⎞⎜⎜⎝
⎛−=
0
2
0 241
21
NEerfc
NEerfc
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)The average probability of symbol error for coherent QPSK is
ce PP −=1
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
0
2
0 241
2 NEerfc
NEerfc
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)If E/2N0 >> 1, the average probability of symbol error for coherent QPSK as
⎟⎟⎠
⎞⎜⎜⎝
⎛≈
02NEerfcPe
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)In QPSK system, there are two bits per symbol. This mean that the transmitted signal energy per symbol is twice the signal energy per bit, that is
We may expressthe average probability of symbol error in terms of the ratio Eb/N0
bEE 2=
⎟⎟⎠
⎞⎜⎜⎝
⎛≈
0NEerfcP b
e
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Quadriphase-shift keying (QPSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)Consider a continuous-phase frequency-shift keying (CPFSK) signal, which is defined for the interval 0 ≤ t ≤T, as follows
for symbol 1
for symbol 0
Eb is the transmitted sinal energy per bit, and Tb is the bit duration
[ ]
[ ]⎪⎪⎩
⎪⎪⎨
⎧
+
+=
)0(2cos2
)0(2cos2
)(
2
1
θπ
θπ
tfTE
tfTE
ts
b
b
b
b
Coherent Binary Modulation Techniques
Minimum shift Keying (MSK)The phase θ(0), denoting the value of the phase at time t = 0, depends on the past history of the modulation processThe frequencies f1 and f2 are sent in response to binary symbols 1 and 0 appearing at the modulator input, respectively
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)Signal s(t) may be expressed in the conventional form of an angle-modulated wave
where θ(t) is the phase of s(t)when the phase θ(t) is a continuous function of time, the modulated wave s(t) itseft is also continuous at all times, including the inter-bit switching time
[ ])(2cos2)( ttfTEts cb
b θπ +=
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)The nominal carrier frequency fc is chosen as the arithmetic mean of the two frequencies f1and f2
The phase θ(t) of a CPFSK signal increases or decreases linearly with time during each bit period of Tb seconds
( )2121 fff c +=
tThtb
πθθ ±= )0()( bTt ≤≤0
Coherent Binary Modulation Techniques
Minimum Shift KeyingThe phase θ(t) of a CPFSK signal increases or decreases linearly with time during each bit period of Tb seconds
The plus sign corresponds to sending symbol 1, and the minus sign corresponds to sending symbol 0The parameter is referred to as the deviation ratio, measured with respect to the bit rate 1/Tb
tThtb
πθθ ±= )0()(bTt ≤≤0
( )21 ffTh b −=
Coherent Binary Modulation Techniques
Minimum Shift KeyingAt the time t = Tb
for symbol 1for symbol 0⎩
⎨⎧−
=−h
hTb π
πθθ )0()(
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)Using a well-known trigonometic identity, we may express the CPFSK signal s(t) in terms of its in-phase and quadraturecomponents as follows
[ ] ( ) [ ] ( )tftTEtft
TEts c
b
bc
b
b πθπθ 2sin)(sin22cos)(cos2)( −=
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)With deviation ratio h = ½
The plus sign corresponds to symbol 1 and the minus sign corresponds to symbol 0A similar result holds for θ(t) in the interval -Tb≤ t ≤ 0Since the phase θ(0) is 0 or π, depending on the past history of the modulation process, in the interval Tb≤ t ≤+Tb , the polarity of cos[θ(t)] depends only on θ(0), regardless of the sequence of 1s or 0s transmitted before or after t = 0
tT
tb2
)0()( πθθ ±=bTt ≤≤0
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)For this time interval -Tb≤ t ≤ +Tb, the in-phase component, sI(t) consists of a half-cosine pulse defined as follows
the plus sign corresponds to θ(0) = 0, and minus sign corresponds to θ(0) = π
[ ])(cos2)( tTEtsb
b θ=
[ ]
⎟⎟⎠
⎞⎜⎜⎝
⎛±=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
tTT
E
tT
tTE
bb
b
bb
b
2cos2
2cos)(cos2
π
πθ
bb TtT ≤≤−
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)In the interval 0 ≤ t ≤ 2Tb, the quadrature component, sQ(t),consists of a half-sine pulse, whose polarity depends only on θ(Tb)
the plus sign corresponds to θ(Tb) = π/2 and the minus sign corresponds to θ(Tb) = -π/2
[ ])(sin2)( tTEtsb
bQ θ=
[ ]
⎟⎟⎠
⎞⎜⎜⎝
⎛±=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
tTT
E
tT
TTE
bb
b
bb
b
b
2sin2
2sin)(sin2
π
πθ
bTt 20 ≤≤
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)With h = 1/2 , the frequency deviation (i.e., the difference between the two signaling frequencies f1 and f2) equals half of bit rateThis is the minimum frequency spacing that allows the two FSK signals representing symbols 1 and 0, to be cohenrently orthogonal in the sense that they do not interfere with one another n the process of detectionCPFSK signal with a deviation ratio of one-half is referred to as minimum-shift keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)One of four possibilities can arise, as described
The phase θ(0) = 0 and θ(Tb) = π/2, corresponding to the transmission of symbol 1The phase θ(0) = π and θ(Tb) = π/2, corresponding to the transmission of symbol 0The phase θ(0) = π and θ(Tb) = - π/2 (or, equivalently, 3π/2, modulo 2π), corresponding to the transmission of symbol 1The phase θ(0) = 0 and θ(0) = - π/2, corresponding to the transmission of symbol 0
Coherent Binary Modulation Techniques
Minimum Shift KeyingIn MSK signal, the appropriate form for the orthonormal basis functions Φ1(t) and Φ2(t) is as follows
Both Φ1(t) and Φ2(t) are defined for a period equal to twice the bit duration
( )tftTT
t cbb
ππφ 2cos2
cos2)(1 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
bb TtT ≤≤−
( )tftTT
t cbb
ππφ 2sin2
sin2)(2 ⎟⎟⎠
⎞⎜⎜⎝
⎛= bTt 20 ≤≤
Coherent Binary Modulation Techniques
Minimum Shift KeyingCorrespondingly, we may express the MSK signal in the form
The coefficients s1 and s2 are related to the phase states θ(0) and θ(Tb), respectively
)()()( 2211 tststs φφ +=bTt ≤≤0
Coherent Binary Modulation Techniques
Minimum Shift KeyingThe in phase component of s(t)
The quadrature component of s(t)
∫−=T
Tb
dtttss )()( 11 φ
[ ])0(cos θbE= bb TtT ≤≤−
∫=bT
dtttss2
0 22 )()( φ
[ ])(sin bb TE θ−= bTt 20 ≤≤
Coherent Binary Modulation Techniques
Minimum Shift KeyingBoth integrals are evaluated for a time interval equal to twice the bit duration, for which Φ1(t) and Φ2(t) are orthogonalBoth the lower and upper limits of the product integration used to evaluate th coefficient s1 are shifted by Tb seconds with respect to those used to evaluate the coefficient s2The time interval 0 ≤ t ≤ Tb, for which the phase state θ(0) and θ(Tb) are defined is common to both integrals
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift KeyingIn a case of an AWGN channel, the received signal is given by
s(t) is the transmitted MSK signal, and w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density N0/2
)()()( twtstx +=
Coherent Binary Modulation Techniques
Minimum Shift KeyingIn order to decide whether symbol 1 or symbol 0 was transmitted in the interval 0 ≤ t ≤ Tb, we have to establish a procedure for the use of x(t) to detect the phase states θ(0) and θ(Tb)For optimum detection of θ(0), we have to determine the projection of the received signal x(t)onto the reference signal Φ1(t) and Φ2(t)
Coherent Binary Modulation Techniques
Minimum Shift Keying
dtttxx b
b
T
T)()( 11 ∫−= φ
11 ws +=bb TtT ≤≤−
dtttxx bT))(
2
0 22 ∫= φ
22 ws += bTt 20 ≤≤
Coherent Binary Modulation Techniques
Minimum Shift KeyingThe average symbol error for the MSK is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟⎠
⎞⎜⎜⎝
⎛=
0
2
0 41
NEerfc
NEerfcP bb
e
⎟⎟⎠
⎞⎜⎜⎝
⎛≈
0NEerfcP b
e
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Coherent Binary Modulation Techniques
Minimum Shift Keying (MSK)
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationConsider a binary signaling scheme that involves the use of two orthogonal signal s1(t) and s2(t), which have equal energyDuring the interval 0≤ t ≤ T, one of these two signals is sent over an imperfect channel that shifts the carrier phase by an unknown amount
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationLet g1(t) and g2(t) denote the phase-shifted versions of s1(t) and s2(t), respectivelyIt is assumed that g1(t) and g2(t) remain orthogonal and of equal energy, regardless of the unknown carrier phaseWe refer to such a signaling scheme as noncoherent orthogonal modulation
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe channel also introduces an AWGN w(t) of zero mean and power spectral density N0/2We may express the received signal x(t) as
x(t) is used to discriminate between s1(t) and s2(t) , regardless of the carrier phase
⎩⎨⎧
++
=)()()()(
)(2
1
twtgtwtg
tx TtTt
≤≤≤≤
00
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe receiver consists of a pair of filters matched to the basis function Φ1(t) and Φ2(t) that are scaled versions of the transmitted signal s1(t) and s2(t), respectivelyBecause the carrier phase is unknown, the receiver relies on amplitude as the only possible discriminantAccordingly, the matched filter outputs are envelope detected, sampled, and then compared with each other.
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal Modulation
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe quadrature receiver itself has two path
In in-phase path, the receiver signal x(t) is correlated against the basis function Φi(t), representing a scaled version of the transmitted signal s1(t) or s2(t) with zero carrier phase.In the quadrature path, signal x(t) is correlated against another basis function , representing the version of Φi(t) that results from shifting the carrier phase by -900
Naturally, Φi(t) and are orthogonal to each other
)(ˆ tiφ
)(ˆ tiφ
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal Modulation
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe average probability of error for the noncoherentreceiver will be calculated by making use of the equivalence depicted previous pictureSince the carrier phase is unknown, noise at the output of each matched filter has two degrees of freedom, namely, in-phase and quadrature. Accordingly, the noncoherent receiver has a total of four noise parameters that are statistical independent and identically distributed, denoted by xI1, xQ1, xI2, xQ2
The first tow account for degrees of freedom associated with the upper pathThe latter two account for degrees of freedom associated with the lower path
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSince the receiver has a symmetric structure, the probability of choosing s2(t), given that s1(t) was transmitted, is the same as the probability of choosing s1(t), given that s2(t) was transmittedThis means that the average probability of error may be obtained by transmitting s1(t) and calculating the probability of choosing s2(t), or vice versa
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSuppose that signal s1(t) is transmitted for the interval 0≤ t ≤ T, an error occurs if the receiver noise w(t) is such that the output l2 is greater than the output l1
The receiver makes a decision in favor of s2(t) rather than s1(t)
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationTo calculate the probability of error, we must have the probability density function of the random variable L2 (represented by sample value l2).Since the filter in the lower path is matched to s2(t), and s2(t) is orthogonal to the transmitted signal s1(t), it follows that the output of this matched filter is due to noise alone.
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationLet xI2 and xQ2 denote the in-phase and quadrature components of the matched filter output in the lower pathFor i = 2
22
222 QI xxl +=
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe random variables XI2 and XQ2 are both Gaussian distributed with zero mean and variance N0/2
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
22
02 exp1)(
2 Nx
Nxf I
IX I π
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
22
02 exp1)(
2 Nx
Nxf Q
QX Q π
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationRandom variable L2 has the following probability density function
⎪⎩
⎪⎨
⎧⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
exp2)(
0
22
0
2
22 Nl
Nl
lf L
elsewherel 02 ≥
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe conditional probability that l2 > l1, given the sample value l1, is defined
( ) 222112 )(|1
dllflllPl L∫∞
=>
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−=>
0exp|
21
112llllP
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSince the filter in the path is matched to s1(t),and it is assumed that s1(t) is transmitted, it follows that l1 is due to signal plus noiseLet xi1 and xQ1 denote the components at the output of the matched filter (in the upper path) that are in-phase and quadrature to the received signal
21
211 QI xxl +=
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSince is orthogonal to s1(t), it is obvious that xI1 is due to signal plus noise, whareas xQ1 is due to noise alone.
XI1 represented by sample value xI1 is Gaussian distributed with mean and variance N0/2,where E is the signal energy per symbolXQ1 represented by sample xQ1 is Gaussian distributed with zero mean and variance N0/2
)(1̂ ts
E
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe probability density functions of these two independent random variable
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −−=
0
2
1
01 exp1)(
1 NEx
Nxf I
IX I π
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
21
01 exp1)(
1 Nx
Nxf Q
QX Q π
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSince the tow random variable XI1 and XQ1are independent, their joint probability density function is simply the product of the probability density functions of two random variable
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationTo find the average probability of error, we have to average the conditional probability of error over all possible values of l1
Given xI1 and xQ1, an error occurs when the lower path’s output amplitude l2 due to noise alone exceeds l1 due to signal plus noise
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe probability of such an occurrence is
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−=
0
21
21
11 exp),(N
xxxxerrorP QI
QI
( ) )()(, 111111 QQXIIXQI xfxfxxerrorP
[ ]⎭⎬⎫
⎩⎨⎧
+−++−= 2121
21
21
00
)(1exp1QIQI xExxx
NNπ
Noncoherent Binary Modulation Techniques
Non-coherent Orthogonal ModulationSince
22
22)( 2
1
2
12
12
12
121
ExExxExxx QIQIQI ++⎟⎟⎠
⎞⎜⎜⎝
⎛−=+−++
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationThe average probability of error
( ) 11111111 )()(, QIQQXIIXQIe dxdxxfxfxxerrorpP ∫ ∫∞
∞−
∞
∞−=
10
21
1
2
1000
2exp
22exp
2exp1
II dxNx
dxExNN
EN ∫∫
∞
∞−
∞
∞− ⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
π
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationSince
222exp 0
1
2
10
πNdxEx
N II =⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−∫
∞
∞−
22
exp 01
0
21 πN
dxNx
QQ =⎟⎟⎠
⎞⎜⎜⎝
⎛−∫
∞
∞−
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal ModulationAccordingly
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
02exp
21
NEPe
Noncoherent Binary Modulation Techniques
Noncoherent Binary FSKIn the case of binary FSK
( )⎪⎩
⎪⎨
⎧=
0
2cos2
)( tfTE
ts ib
b
iπ
bTt ≤≤0
Noncoherent Binary Modulation Techniques
Noncoherent Orthogonal Modulation
Noncoherent Binary Modulation Techniques
Noncoherent Binary FSKThe noncoherent binary FSK described is a special case of noncoherent orthogonal modulation with
The probability of error
bTT =
bEE =
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
02exp
21
NEP b
e
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)DPSK eliminates the need for a coherent reference signal at the receiver by combining two basic operations at the transmitter
Differential encoding of the input binary wavePhase-shift keying
In effect, to send symbol 0 we phase advance the current signal waveform by 1800, and to send symbol 1 we leave the phase of the current signal waveform unchanged
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)The differential encoding process at the transmitter input starts with an arbitrary first bit, serving as reference, and thereafter the differentially encoded sequence {dk} si generated by
Bk is the input binary digit at time KTb
Dk-1 is the previous value of the differentially encoded bit
kkkkk bdbdd 11 −− +=
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)The receiver is equipped with a storage capability, so that it can measure the relative phase difference between the waveforms received during two successive bit intervalsAssumed that the unknown phase θ contained in the received wave varies slowly (that is, slow enough for it to be considered essentially constant over two bit intervals), the phase difference between waveforms received in two successive bit intervals will be independent of θ
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)Let s1(t) denote the transmitted DPSK signal in the case of symbol 1 at the transmitter input
( )
( )⎪⎪⎩
⎪⎪⎨
⎧
=tf
TE
tfTE
ts
ib
b
ib
b
π
π
2cos22
2cos22
)(1
bb
b
TtTTt2
0≤≤≤≤
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)Let s2(t) denote the transmitted DPSK signal in the case symbol 0 at the transmitter input
( )
( )⎪⎪⎩
⎪⎪⎨
⎧
+=
ππ
π
tfTE
tfTE
ts
ib
b
ib
b
2cos22
2cos22
)(2bb
b
TtTTt2
0≤≤≤≤
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)s1(t) and s2(t) are orthogonal over the two-bit interval 0≤ t ≤ 2Tb
DPSK is a special case of noncoherentorthogonal modulation with
bTT 2=
bEE 2=
Noncoherent Binary Modulation Techniques
Differential Phase-shift Keying (DPSK)The probability of symbol error
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
exp21
NEP b
e
M-Ary Modulation TechniquesIn M-ary signaling scheme, we may send one of M possible signals, s1(t), s2(t), …, sM(t),during each signaling interval of duration T
In almost applications M = 2n
The symbol duration T = nTb
These signal are generated by changing amplitude, phase, or frequency of a carrier in M discrete steps. Thus we have:
M-ary ASKM-ary PSKM-ary FSK
M-Ary Modulation Techniques
M-ary PSKIn M-ary PSK, the phase of the carrier takes one of M possible values, θi = 2iπ/M, where i = 0, 1, …, M-1Accordingly, during each signaling interval of duration T, one of possible signals
E is the signal energy per symbol
Carrier frequency fc = nc/T for some fixed integer nc
⎟⎠⎞
⎜⎝⎛ +=
Mitf
TEts ci
ππ 22cos2)( 1,...1,0 −= Mi
M-Ary Modulation Techniques
M-ary PSKEach si(t) may be expanded in terms of two basis function
)2cos(2)(1 tfT
t cπφ =
)2sin(2)(2 tfT
t cπφ =
Tt ≤≤0
Tt ≤≤0
M-Ary Modulation Techniques
M-ary PSKThe signal constellation of M-ary PSK is two dimensional.The M message points are equally spaced on a circle of radius , and center at the origin
E
M-Ary Modulation Techniques
M-ary PSK
M-Ary Modulation TechniquesM-ary PSK
The optimum receiver for coherent M-ary PSK includes a pair of correlators with reference signals in phase quadratureThe two correlators outputs, denoted as xI and xQ, are fed into a phase discriminator that first computes the phase estimate
The phase discriminator then selects from the set {si(t), i = 0, …, M-1} that particular signal whose phase is closet to the estimate
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
I
Q
xx1tanθ̂
θ̂
M-Ary Modulation Techniques
M-ary PSK
M-Ary Modulation Techniques
M-ary PSKIn the presence of noise, the decision-making process in the phase discriminator is based on the noisy input
Where wI and wQ are samples of two independent Gaussian random variables WI and WQ whose is mean zero and common variance equals N0/2
II wM
iEx +⎟⎠⎞
⎜⎝⎛=π2cos
QQ wM
iEx +⎟⎠⎞
⎜⎝⎛−=π2sin
1,...1,0 −= Mi
1,...1,0 −= Mi
M-Ary Modulation Techniques
M-ary PSKThe message points exhibit circular symmetrBoth random variables WI and WQ have a symmetric probability density functionThe average probability of symbol error Peis independent of the particular signal si(t)is transmitted
M-Ary Modulation Techniques
M-ary PSKWe may simplify the calculation of Pe by setting θi = 0, which corresponds to the message point whose coordinates along the Φ1(t)- and Φ2(t)-axes are and 0, respectivelyThe decision region pertaining to this message point is bounded by the threshold
below the Φ1(t)-axis and the threshold above the Φ1(t)-axis
E
Mπθ −=ˆMπθ +=ˆ
M-Ary Modulation Techniques
M-ary PSKThe probability of correct reception is
is the probability density function of the random variable whose sample value equals the phase discriminator output produced in response to a received signal that consists of the signal s0(t) plus AWGN
∫− Θ=M
Mc dfPπ
πθθ ˆ)ˆ(
)ˆ(θΘf
Θ θ̂
⎟⎟⎠
⎞⎜⎜⎝
⎛
+= −
I
Q
WE
W1tanθ̂
M-Ary Modulation Techniques
M-ary PSKThe probability density function has a known value. Especially, for we may write
)ˆ(θΘfπθπ ≤≤− ˆ
( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=Θ θθθ
ππθ ˆcos
211ˆsinexpˆcosexp
21ˆ
0
2
000 NEerfc
NE
NE
NEf
M-Ary Modulation Techniques
M-ary PSKA decision error is made if the angle falls outside
θ̂MM πθπ +≤≤− ˆ
ce PP −=1
∫− Θ=M
Mdf
π
πθθ ˆ)ˆ(
M-Ary Modulation Techniques
M-ary PSKFor large M and high values of E/N0, we may derive an approximate formula for Pe
For high values of E/N0 and for , we may use the approximation
2ˆ πθ <
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈⎟⎟
⎠
⎞⎜⎜⎝
⎛− θ
θπθ ˆcosexpˆcos
1ˆcos 2
0
0
0 NE
EN
NEerfc
M-Ary Modulation Techniques
M-ary PSKWe get
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈Θ θθ
πθ ˆsinexpˆcos)ˆ( 2
00 NE
NEf
2ˆ πθ <
∫−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−≈
M
Me d
NE
NEP
π
π
θθθπ
ˆˆsinexpˆcos1 2
00
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
MNEerfc πsin
0
M-Ary Modulation Techniques
M-ary PSK
M-Ary Modulation TechniquesM-ary QAM
In an M-ary PSK system, in-phase and quadraturecomponents of the modulated signal are interrelated in such a way that the envelope is constrained to remain constant. This constraint manifests itself in a circular constellation for the message pointsHowever, if this constrained is removed, and the in-phase and quadrature components are thereby permitted to be independent, we get a new modulation scheme called M-ary quadratureamplitude modulation (QAM)
M-Ary Modulation Techniques
M-ary QAMThe signal constellation for M-ary QAM consists of a square lattice of message points.
M-Ary Modulation Techniques
M-ary QAMThe corresponding signal constellations the in-phase and quadaraturecomponents of the amplitude phase modulated wave are shown
M-Ary Modulation Techniques
M-ary QAMThe general form of M-ary QAM is defined by the transmitted signal
E0 is the energy of the signal with the lowest amplitudeai and bi are a pair of independent integers chosen in accordance with the location of the pertinent message point
( ) ( )tfbTE
tfaTE
ts cicii ππ 2(sin2
2(cos2
)( 00 += Tt ≤≤0
M-Ary Modulation Techniques
M-ary QAMThe signal si(t) can be expanded in terms of a pair of basis functions
)2cos(2)(1 tfT
t cπφ =
)2sin(2)(2 tfT
t cπφ =
Tt ≤≤0
Tt ≤≤0
M-Ary Modulation Techniques
M-ary QAMThe coordinates of the ith message point are and , where (ai, bi) is an element of the L-by-L matrix ( )
Eai Ebi
{ }⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−−+−+−+−+−
−−−+−−+−−−−+−−+−
=
)1,1()1,3()1,1(
)3,1()3,3()3,1()1,1()1,3()1,1(
,
LLLLLL
LLLLLLLLLLLL
ba ii
L
MMM
L
L
ML =
M-Ary Modulation Techniques
M-ary QAMFor example, for the 16-QAM whose signal constellation is
{ }
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎥
⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−−−−−−−−
−−−−
=
3,33,13,13,31,31,11,11,3
1,31,11,11,33,33,13,13,3
, ii ba
M-Ary Modulation Techniques
M-ary QAM
M-Ary Modulation Techniques
M-ary QAMSince the in-phase and quadraturecomponents of M-ary QAM are independent, the probability of correct detection for such a scheme may be written as
Where is the probability of symbol error for either component
( )2'1 ec PP −=
'eP
M-Ary Modulation Techniques
M-ary QAMThe signal constellation for the in-phase or quadrature component has a geometry similar to that for discrete pulse-amplitude modulation (PAM) with a corresponding number of amplitude levelsWe may write
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −=
0
0' 11NEerfc
LPe
M-Ary Modulation Techniques
M-ary QAMThe probability of symbol error for M-aryQAM is given
where it is assumed that
ce PP −=1
( )'
2'
2
11
e
e
P
P
≈
−−=
1' <<eP
M-Ary Modulation Techniques
M-ary QAMThe probability of symbol error for M-aryQAM may be written
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −≈
0
0112NEerfc
MPe
M-Ary Modulation Techniques
M-ary QAMThe transmitted energy in M-ary QAM is variable in that its instantaneous value depends on the particular symbol transmittedIt is logical to express Pe in terms of the average value of the transmitted energy rather than E0
Assuming that the L amplitude levels of the in-phase or quadrature component are equally likely, we have
( ) ⎥⎦
⎤⎢⎣
⎡−= ∑
=
2
1
20 122
2L
iav i
LE
E
M-Ary Modulation Techniques
M-ary QAMThe limits of the summation take account of the symmetric nature of the pertinent amplitude levels around zeroWe get
( )3
12 02 EL
Eav−
=
( )3
12 0EM −=
M-Ary Modulation Techniques
M-ary QAMAccordingly, we may rewrite probability of symbol error in terms of Eav
⎟⎟⎠
⎞⎜⎜⎝
⎛
−⎟⎠⎞
⎜⎝⎛ −≈
0)1(23112
NMEerfc
MP av
e
M-Ary Modulation Techniques
M-ary QAM
M-Ary Modulation Techniques
M-ary FSKIn an M-ary FSK scheme, the transmitted signals are defined by
i = 1, 2, …, MThe carrier frequency fc = nc/2T for some fixed integer nc
The transmitted signals are of equal duration T and have equal energy E
( ) ⎥⎦⎤
⎢⎣⎡ += tinTT
Ets ciπcos2)( Tt ≤≤0
M-Ary Modulation Techniques
M-ary FSKSince the individual signal frequencies are separated by 1/2T Hz, the signals are orthogonal
0)()(0
=∫ dttsts j
T
i
M-Ary Modulation Techniques
M-ary FSKFor coherent M-ary FSK, the optimum receiver consists of a bank of M correlatorsor matched filtersAt the sampling time t = KT, the receiver makes decisions based on the largest matched filter output
M-Ary Modulation Techniques
M-ary FSKAn upper bound for the probability of symbol error
For fixed M, this bound becomes increasingly tight as E/N0 is increasedFor M = 2, the bound becomes an equality
⎟⎟⎠
⎞⎜⎜⎝
⎛−≤
02)1(
21
NEerfcMPe
M-Ary Modulation Techniques
M-ary FSKThe probability of symbol error for noncoherent of M-ary FSK is given
The upper bound on the probability of symbol error for noncoherent detection of M-ary FSK
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
−= ∑
−
=
+
0
1
1
1
1exp
11
)1(Nk
kEk
Mk
PM
k
k
e
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−≤
02exp
21
NEMPe
M-Ary Modulation Techniques
Power spectraThe description of a band-pass signal s(t)contains the definitions of ASK, PSK, and FSK signals, depending on the way in which the in-phase component sI(t) and the quadrature component sQ(t) are defined
M-Ary Modulation Techniques
Power SpectraWe may express s(t) in the form
Where Re[.] is real part of the expression contained inside the bracket
)2sin()()2cos()()( tftstftsts cQcI ππ −=
( )[ ]tfjts cπ2exp)(~Re=
M-Ary Modulation Techniques
Power SpectraWe also have
The signal is called the complex envelope of the band-pass signal s(t)The component sI(t) and sQ(t) and thereforeare all low-pass signal
)()()(~ tjststs QI +=
( ) ( )tfjtftfj ccc πππ 2sin2cos)2exp( +=
)(~ ts
)(~ ts
M-Ary Modulation Techniques
Power SpectraLet SB(f) denote the baseband power spectral density of complex envelope We refer to SB(f) as the baseband power spectral densityThe power spectral density, SS(f), of the original band-pass signal s(t) is a frequency-shifted version of SB(f), except for a scaling factor
)(~ ts
[ ])()(41)( cBcBS ffSffSfS ++−=
M-Ary Modulation Techniques
Power Spectra of Binary PSKBaseband power spectral density of binary FSK wave equals
Power spectra of binary PSK
( ) )(sin22
)(sin2)( 2
2
fTcEfT
fTEfS bb
b
bbB ==
ππ
M-Ary Modulation Techniques
Power spectra of binary FSKThe power spectral densities of SB(f) is given
The power spectral density of binary FSK
( )2222
2
14)(cos8
21
21
2)(
−+⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
fTfTE
Tf
Tf
TEfS
b
bb
bbb
bB
ππδδ
M-Ary Modulation Techniques
M-Ary Modulation Techniques
Power Spectra of QPSKThe baseband power spectral density of QPSK signal
The power spectral density of QPSK signal
)2(sin4)(sin2)( 22 fTcETfcEfS bbB ==
M-Ary Modulation Techniques
Power spectra of MSK signalThe baseband power spectral density of MSK signal
The power spectral density of MSK signal
( )( )
2
222 116
2cos322
)(2)(
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−=⎥
⎦
⎤⎢⎣
⎡Ψ=
fT
fTET
ffs
b
bb
b
gB
ππ
M-Ary Modulation Techniques
M-Ary Modulation Techniques
Power spectra of M-ary signalBinary PSK and QPSK are special cases of M-ary PSK signalsThe symbol duration of M-ary PSK is defined by
Where Tb is the bit duration
MTT b 2log=
M-Ary Modulation Techniques
Power spectra of M-ary PSKThe baseband power spectral density of M-ary PSK signal is given by
)(sin2)( 2 TfcEfSB =
)log(sinlog2 22
2 MfTcME bb=
M-Ary Modulation Techniques
M-Ary Modulation Techniques
The spectral analysis of M-ary FSK signals is much more complicated than that of M-aryPSKA case of particular interest occurs when the frequencies assigned to the multilevels make the frequency spacing uniform and the frequency deviation k= 0.5That is, the M signal frequencies are separated by 1/2T, where T is symbol duration
M-Ary Modulation Techniques
Power spectra of M-ary FSKFor k = 0.5, the baseband power spectral density of M-ary FSK signals is defined by
( )⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛++⎟⎟
⎠
⎞⎜⎜⎝
⎛= ∑ ∑∑
= = =
M
i
M
i
M
j j
jji
i
ibB i
iMM
EfS1 1 1
22
2
sinsincos1sin214)(
γγ
γγγγ
γγ
)1(24
4
+−=
⎟⎠⎞
⎜⎝⎛ −=
Mi
bfT
i
ii
α
παγ
Mi ,...,2,1=
M-Ary Modulation Techniques
Power spectra of M-ary FSK
M-Ary Modulation Techniques
Bandwidth efficiencyBandwidth efficiency is defined as the ratio of data rate to channel bandwidth; it is measured in units of bits per second per hertzBandwidth efficiency is also referred to as spectral efficiencyWith the data rate denoted by Rb and the channel bandwidth by B, we may express the bandwidth efficiency, , as
BRb=ρ
ρ
M-Ary Modulation Techniques
Bandwidth efficiency of M-ary PSKThe channel bandwidth required to pass M-ary PSK signals (more precisely, the main spectral lobe of M-ary PSK signals) is given
where T is the symbol durationT
B 2=
M-Ary Modulation Techniques
Bandwidth efficiency of M-ary PSKSince
Channel bandwidth
Channel efficiency of M-ary PSK signals is
MTT b 2log=
MR
B b
2log2
=
2log2 M
BRb ==ρ
M-Ary Modulation Techniques
Bandwidth efficiency of M-ary PSK
M-Ary Modulation Techniques
Bandwidth efficiency of M-ary FSKChannel bandwidth required to transmit M-ary FSK signals as
Channel bandwidth of M-ary FSK signals is
Bandwidth efficiency of M-ary FSK signals is
TMB2
=
MMRB b
2log2=
MM
BRb 2log2
==ρ
M-Ary Modulation Techniques
Bandwidth efficiency of M-ary FSK
M-Ary Modulation Techniques
Bit versus symbol error probabilitiesThus far, the only figure of merit we have used to assess the noise performance of digital modulation schemes has been the average probability of symbol errorWhen the requirement is to transmit binary data, it is often more meaningful to use another figure of merit called the probability of bit error or bit error rate (BER)
M-Ary Modulation TechniquesBit versus symbol error probabilities
Case 1:The mapping from binary to M-ary symbols is performed in
such a way that the two binary M-tuples corresponding to any pair of adjacent symbols in the M-ary modulation scheme differ in only one bit position (Gray code)When the probability of symbol error Pe is acceptably small, we find that the probability of mistaking one symbol for either of the two nearest (in-phase) symbols is much greater than any other kind of symbol errorMoreover, given a symbol error, the most probable number of bit errors is one, subject to the aforementioned mapping constraint
M-Ary Modulation Techniques
Bit versus symbol error probabilitiesCase 1
Since there are log2M bits per symbol, it follows that the bit error rate is related to the probability of symbol error by a formula
Mp
BER e
2log= 2≥M
M-Ary Modulation Techniques
Bit versus symbol error probabilitiesCase 2:
We assume that all symbol errors are equally likely and occur with probability
where Pe is the average probability of symbol errorK = log2M
121 −=
− Kee P
MP
M-Ary Modulation Techniques
Bit versus symbol error probabilitiesCase 2:
There are ways in which k bits out of K may be in errorThe average number of bit errors per K-bit symbol is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛kK
eK
KK
kK
eKk KPPk
122
12)(
1
1 −=
−
−
=∑
M-Ary Modulation Techniques
Bit versus symbol error probabilitiesCase 2
The bit error rate is obtained by dividing the result by K
eK
K
PBER12
12−
=−
ePM
M
BER⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−=
12