Graduate Program School of Electrical and Computer Engineering
Chapter 3: Characterization of Communication Signals and Systems
Overview
• Pulse amplitude modulation • Phase modulation • Quadrature amplitude modulation • Multidimensional signals • Biorthogonal signal
Digital Communications – Chapter 3: Communication Signals & Systems 2
Sem. I, 2012/13
Linearly Modulated Digital Signals
• Linear digitally modulated signals are expanded in terms of two orthonormal basis functions of the form
• If the low-frequency representation is desired
• Modulator maps blocks of k=log2M binary digits at a time from the information sequence {ak} and selecting one of M=2k deterministic and finite energy waveform {Sm(t), m=1,2,…..M} for transmission over the channel
Digital Communications – Chapter 3: Communication Signals & Systems 3
Sem. I, 2012/13
tfT
tfandtfT
tf cc ππ 2sin2)(2cos2)( 21 ==
signal modulating therepresents)( and )( where)()()()()()()( 21
tytxtftytftxtSthentjytxS
ll
llmlllm +=+=
Memoryless Modulation – PAM Signals
• Assume that the sequence of binary digits at the input of the modulator occurs at the rate of R bits/s
• In pulse-amplitude modulation (PAM) signals
• 0 ≤ t ≤ T and {Am, 1 ≤ m ≤ M} denotes the M possible amplitudes corresponding to M=2k possible k-bit blocks or symbols
• Am takes discrete values or levels, Am = (2m-1-M)d where 2d is the distance between two adjacent signal amplitudes
• g(t) is a real valued signal pulse whose shape influences the spectrum of the transmitted signal
Digital Communications – Chapter 3: Communication Signals & Systems 4
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[ ] M1,2,...mtπf2cosg(t)Aeg(t)ARe(t)S cmtπf2j
mmc ===
Memoryless Modulation – PAM Signals …
• Symbol rate for PAM signals is R/k, the rate at which changes occur in the amplitude of the carrier • Bit interval Tb = 1/R and Symbol interval T = k/R = kTb
• PAM signals have energies
• Note that these signals are one dimensional (N=1) and hence can be represented by the general form
• Where and
Digital Communications – Chapter 3: Communication Signals & Systems 5
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g(t) pulse theofEnergy 21)(
21)( 2
0
22
0
2
−
=== ∫∫g
gm
T
m
T
mm AdttgAdttS
ε
εε
)()( tfStS m=
2cos)(2 tftgf(t) cg
πε
= MmAS gmm ,.......2,1
2==
ε
Memoryless Modulation – PAM Signals …
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Signal space diagram for digital PAM for M=2, 4, and 8
Memoryless Modulation – PAM Signals …
• The preferred assignment of k information bits to the M=2k
possible signal amplitudes is one in which the adjacent signal amplitudes differ by only one binary digit (Gray Encoding)
• Note the Euclidean distance between any pair of signal points is
• The minimum distance between a pair of adjacent signal point occurs when
Digital Communications – Chapter 3: Communication Signals & Systems 7
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nmε2dAAε21)S(Sd gnmg
2nm
emn −=−=−=
;1=− nm ge dd ε2min =
Memoryless Modulation – PAM Signals …
• The above DSB signal requires a bandwidth BW=2BWLP
• Alternatively, one may use SSB whose representation is
• Where is the Hilbert transform of • The BW of SSB signal is half that of DSB signal
Digital Communications – Chapter 3: Communication Signals & Systems 8
Sem. I, 2012/13
)(tg∧
MmetgjtgAS tfjmm
c ,....2,1};)()(Re{ 2 =
±=
∧π
)(tg
Memoryless Modulation – PAM Signals …
• For transmission over channels that does not require carrier modulation
• This is called baseband signal
Digital Communications – Chapter 3: Communication Signals & Systems 9
Sem. I, 2012/13
MmtgAtS mm ,...,2,1),()( ==
Overview
• Pulse amplitude modulation • Phase modulation • Quadrature amplitude modulation • Multidimensional signals • Biorthogonal signal
Digital Communications – Chapter 3: Communication Signals & Systems 10
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Memoryless Modulation – Phase Modulated Signals
• In digital phase modulation, the M signal waveform are
• Where are the M possible
phases of the carrier T • These signal waveforms have equal energy given by
Digital Communications – Chapter 3: Communication Signals & Systems 11
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tfmM
tgtfmM
tg
TtmM
tftg
MmeetgtS
cc
c
tfjMmj
mc
ππππ
ππ
ππ
2sin)1(2sin)(2cos)1(2cos)(
.0;)1(22cos)(
.,...2,1;)(Re)( 2)1(2
−−−=
≤≤
−+=
=
=
−
g
TT
m dttgdttS εε21)(
21)(
0
2
0
2 === ∫∫
)1(2andpulsesignal)( −=− mM
tg mπθ
Phase Modulated Signals …
• PM signal is also represented as a linear combination of two orthonormal waveforms f1(t) & f2(t) such that
• Alternatively these two dimensional vectors may be expressed as
Digital Communications – Chapter 3: Communication Signals & Systems 12
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[ ]
tftgtf
andtftgtf
SSStfStfStS
cg
cg
mmmmmm
πε
πε
2sin)(2)(
2cos)(2)(Where
);()()(
2
1
21
_
2211
=
=
=+=
[ ]
−−== )1(2sin
2)1(2cos
221
_m
Mm
MSSS gg
mmmπεπε
Phase Modulated Signals …
• The phase of the carrier signal is used for modulation (carrying information)
• Every symbol (k bits) is mapped into a given phase • The total phase is divided equally among all possible
symbols • The signal space is two dimensional with signals having as
coordinates
Digital Communications – Chapter 3: Communication Signals & Systems 13
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[ ]
Mm
mM
mM
SSS ggmmm
..........,.........2,1
)1(2sin2
)1(2cos221
_
=
−−==
πεπε
Phase Modulated Signals …
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Signal space diagrams for PSK signals
Overview
• Pulse amplitude modulation • Phase modulation • Quadrature amplitude modulation • Multidimensional signals • Biorthogonal signal
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Quadrature Amplitude Modulation (QAM)
• Bandwidth efficiency can be obtained by simultaneously impressing two separate k-bit symbols from the information sequence {an} on the amplitude of the two quadrature carriers cos2πfct and sin2πfct such that
• Amc and Ams are the information bearing signal amplitudes of the quadrature carriers and g(t) is the signal pulse
Digital Communications – Chapter 3: Communication Signals & Systems 16
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[ ]tftgAtftgA
MmetgjAAtS
cmscmc
tfjmsmcm
c
ππ
π
2sin)(2cos)(,......,2,1,)()(Re)( 2
−==+=
Quadrature Amplitude Modulation …
• Alternatively, QAM signal waveform is represented by
• Where
• Note: This may be viewed as a combined amplitude and
phase modulation • If we take M1 as the PAM levels M2 as the phases, we can
construct an M=M1M2 combined PAM-PSK signal constellation
Digital Communications – Chapter 3: Communication Signals & Systems 17
Sem. I, 2012/13
)2(cos)()( mcmm tftgVtS θπ +=
=+= −
nc
msmmsmcm A
AAAV 122 tan and θ
Quadrature Amplitude Modulation …
• M1 = 2n and M2 = 2m PAM-PSK signal constellation results in the simultaneous transmission of m+n = log2M1M2 binary digits occurring at the symbol rate of R/(m+n)
• Examples of combined PAM-PSK signal space diagrams
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Quadrature Amplitude Modulation …
• Several Signal Space Diagrams for Rectangular QAM
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Quadrature Amplitude Modulation …
• Like PSK signals, QAM signals may also be represented as a linear combination of two orthonormal signal waveforms f1(t) and f2(t) such that
• And the Euclidean distance between any pair of signal vectors is given by
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[ ] ;22
2sin)(2)(2cos)(2)(Where
);()()(
21
_
21
2211
==
==
+=
gms
gmcmmm
cg
cg
mmm
AASSS
tftgtfandtftgtf
tfStfStS
εε
πε
πε
21
22)( ))()((2
−+−=−= nsmsncmc
gnm
emn AAAASSd
ε
Quadrature Amplitude Modulation …
• Where the signal amplitudes take discrete values {2m-1-M}, m = 1,2,…M and the signal space diagram is rectangular
• Note that the minimum distance between adjacent pair of signal points is
• Which is the same as for PAM
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ge dd ε2)(
min =
Overview
• Pulse amplitude modulation • Phase modulation • Quadrature amplitude modulation • Multidimensional signals • Biorthogonal signal
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Multidimensional Signals
• Consider a set of N-dimensional signal vectors • For any N subdivide a time interval T1 = NT into N
subintervals of length T = T1 /N • In each subinterval we can use PAM to transmit an
element of an N-dimensional signal vector
• For N even, we can use quadrature carriers to transmit two components independently • Thus, N-dimensional signal vectors are transmitted in NT/2 sec.
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Multidimensional Signals …
• Alternatively, a frequency band of width NΔf may be subdivided into N frequency slots each of width Δf
• N-dimensional signal vector can be transmitted by simultaneously modulating amplitudes of N carriers, one in each frequency slot
• Δf must be chosen such that there will not be cross-talk interference among the signals
• Using QAM, N-dimensional signal vectors (for N even) are transmitted in N/2 frequency slots • Thus reducing the channel bandwidth by a factor of two
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Multidimensional Signals …
• In general one may use both time and frequency domains jointly to transmit an N-dimensional signal vector
• The figure below demonstrates this principle • A 24-dimensional signal can be transmitted using QAM • Or 12-dimensional signal can be transmitted using PAM
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Orthogonal Multidimensional Signals
• As a special case of constructing a multidimensional signals, consider M equal-energy orthogonal signals that differ in frequency
• Where the equivalent low-pass signal waveforms are
• This modulation is called frequency-shift keying (FSK)
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TtandMmeT
tS tfmjlm ≤≤== ∆ 0,...2,1;2)( 2πε
[ ])22(cos2
0,...2,1;)(Re)( 2
tfmtfT
TtandMmetStS
c
tfjlmm
c
∆+=
≤≤==
ππε
π
Orthogonal Multidimensional Signals …
• These frequency modulated signals have equal energy and cross-correlation coefficients given by
• Whose real part can be expressed as
• Note that ρr = 0 when Δf =1/2T and m ≠ k • Since |m - k |=1 corresponds to adjacent frequency slots,
Δf =1/2T represents the minimum frequency separation between adjacent signal for orthogonality of the M signals
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fkmTjT
ftkmjkm e
fkmTfkmTdte ∆−∆−
∆−∆−
== ∫ )(
0
)(2
)()(sin
22/2 ππ
ππ
εερ
[ ]Δfk)πT(m2Δfk)πT(m2sin)ρRe(ρ kmr −
−==
SHOW!
Orthogonal Multidimensional Signals …
• Plots of ρr and |ρkm| versus frequency are shown in the figure below
• Also note that |ρkm| = 0 for multiples of 1/T whereas ρr=0 for multiples of 1/2T
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Cross-correlation coefficient as a function of frequency separation
for FSK signals
Orthogonal Multidimensional Signals …
• For the case in which Δf =1/2T, the M FSK signals are equivalent to the N-dimensional vectors
S1 = [√ε 0 0 ……………0 0] S2 = [ 0 √ε 0 .. ………….0 0]
. . .
. . . SN = [ 0 0 0 ……………..√ε]
• Where N = M and the distance between pairs of signals is
• for all m, k which is also the minimum distance
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ε2)( =ekmd
Orthogonal Signals for M = N = 3 and M = N = 2
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Overview
• Pulse amplitude modulation • Phase modulation • Quadrature amplitude modulation • Multidimensional signals • Biorthogonal signal
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Biorthogonal signal
• A set M of biorthogonal signals can be constructed from 1/2M orthogonal signals by augmenting with negatives of the orthogonal signals • This requires N = ½ M dimensions
• The correlation between any pair of signals is either -1 or 0 • The corresponding distances are d = 2√ε or √2ε, the latter
being the minimum distance
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Biorthogonal signal …
• Signal space diagrams for M = 4 and M = 6 biorthogonal signals
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Simplex Signals
• Consider M orthogonal waveforms {Sm(t)} or their vector representation {Sm} whose mean is
• Construct another set of signals by subtracting the mean from each of the M orthogonal signals; i,.e
• The effect of this subtraction is to translate the origin of the m orthogonal signals to the point
• The resulting signal waveform is called simplex signal
Digital Communications – Chapter 3: Communication Signals & Systems 34
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∑=
=M
mmS
MS
1
_ 1
Mm ...,.........2,1; =−=_
m'm SSS
_S
Simplex Signals …
• It can be shown that, simplex signals have the following properties (See text)
• Energy per waveform is
• Cross-correlation between any pair of signals in
• Hence, simplex waveforms are equally correlated & requires less energy (factor of 1-1/M) than orthogonal waveforms
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nmallforMM
Mmn ,
11
/11/1)Re(
−−=
−−
=ρ
−=
MSm
11 ' ε
Simplex Signals …
• Signal space diagrams for M-ary simplex signals
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Signal Waveform from Binary Codes
• Signal space diagrams for signals generated from binary codes
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Signal Waveform from Binary Codes …
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Baseband signals