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Digital Communication SystemsLecture 5, Prof. Dr. Habibullah Jamal
Under Graduate, Spring 2008
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Chapter 4: Bandpass Modulation and Demodulation
Bandpass Modulation is the process by which some characteristics of a sinusoidal waveform is varied according to the message signal.
Modulation shifts the spectrum of a baseband signal to some high frequency.
Demodulator/Decoder baseband waveform recovery
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4.1 Why Modulate?
Most channels require that the baseband signal be shifted to a higher frequency
For example in case of a wireless channel antenna size is inversely proportional to the center frequency, this is difficult to realize for baseband signals. For speech signal f = 3 kHz =c/f=(3x108)/(3x103) Antenna size without modulation /4=105 /4 meters = 15 miles - practically
unrealizable Same speech signal if amplitude modulated using fc=900MHz will require
an antenna size of about 8cm. This is evident that efficient antenna of realistic physical size is needed for
radio communication system
Modulation also required if channel has to be shared by several transmitters (Frequency division multiplexing).
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4.2 Digital Bandpass Modulation Techniques
Three ways of representing bandpass signal: (1) Magnitude and Phase (M & P)
Any bandpass signal can be represented as:
A(t) ≥ 0 is real valued signal representing the magnitude Θ(t) is the genarlized angle φ(t) is the phase
The representation is easy to interpret physically, but often is not mathematically convenient
In this form, the modulated signal can represent information through changing three parameters of the signal namely: Amplitude A(t) : as in Amplitude Shift Keying (ASK) Phase φ(t) : as in Phase Shift Keying (PSK) Frequency dΘ(t)/ dt : as in Frequency Shift Keying (FSK)
)](cos[)(cos[)()( 0 tttAttAts
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0( ) ( ) cos( ( )) ( ) cos( )s t A t t A t t
0
)()(
dt
tdti
dtt
i )()(
Consider a signal with constant frequency:
Its instantaneous frequency can be written as:
or
Angle Modulation
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Consider a message signal m(t), we can write the phase modulated signal as
)()( tmKtt pc
)](cos[)( tmKtAts pcPM
)()( tmKdt
dt pci
Phase Shift Keying (PSK) or PM
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In case of Frequency Modulation
0( ) ( )i ft K m t
0( ) [ ( )]t
ft K m t d
0 ( )t
ft K m d
0
0
( ) cos[ ( ) ]
cos[ ( )]
t
FM f
f
s t A t K m d
A t K a t
where:
( ) ( )t
a t m d
Frequency Shift Keying (FSK) or FM
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0 0.05 0.1 0.15-1
-0.5
0
0.5
1
The message signal
0 0.05 0.1 0.15
-1
-0.5
0
0.5
1
Time
The modulated signal
Example
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4.2.1 Phasor Representation of Sinusoid
Consider the trigonometric identity called the Euler’s theorem:
Using this identity we can have the phasor representation of the sinusoids. Figure 4.2 below shows such relation:
00 0cos( ) sin( )j te t j t
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Phasor Representation of Amplitude Modulation
Consider the AM signal in phasor form:
0( ) Re 12 2
m mj t j tj t e e
s t e
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Phasor Representation of FM
Consider the FM signal in phasor form:
0( ) Re 12 2 2
m
m m
j tj t j t j t es t e e e
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Digital Modulation Schemes
Basic Digital Modulation Schemes: Amplitude Shift Keying (ASK) Frequency Shift Keying (FSK) Phase Shift Keying (PSK) Amplitude Phase Keying (APK)
For Binary signals (M = 2), we have Binary Amplitude Shift Keying (BASK) Binary Phase Shift Keying (BPSK) Binary Frequency Shift Keying (BFSK)
For M > 2, many variations of the above techniques exit usually classified as M-ary Modulation/detection
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Figure4.5: digital modulations, (a) PSK (b) FSK (c) ASK (d) ASK/PSK (APK)
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Amplitude Shift Keying
Modulation Process In Amplitude Shift Keying (ASK),
the amplitude of the carrier is switched between two (or more) levels according to the digital data
For BASK (also called ON-OFF Keying (OOK)), one and zero are represented by two amplitude levels A1 and A0
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Analytical Expression:
where Ai = peak amplitude
Hence,
where
00,0
10),cos()(
binaryTt
binaryTttAts ci
)cos(2)cos(2)cos()( 02
00 tAtAtAtsrmsrms
R
VPt
T
EtP
2
00 )cos(2
)cos(2
1,......2,0,00,0
10),cos()(2
)( MibinaryTt
binaryTttT
tEts i
i
i
1,......2,0,)(0
2 MidttsET
i
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Where for binary ASK (also known as ON OFF Keying (OOK))
Mathematical ASK Signal Representation The complex envelope of an ASK signal is:
The magnitude and phase of an ASK signal are:
The in-phase and quadrature components are:
the quadrature component is wasted.
10),cos()()(1 binaryTtttmAts cc 00,0)(0 binaryTtts
)()( tmAtg c
0)(),()( ttmAtA c
)()( tmAtx c
,0)( ty
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• It can be seen that thebandwidth of ASKmodulated is twice thatoccupied by the sourcebaseband stream
Bandwidth of ASK Bandwidth of ASK can be found from its power spectral density The bandwidth of an ASK signal is twice that of the unipolar NRZ
line code used to create it., i.e.,
This is the null-to-null bandwidth of ASK
bb TRB
22
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If raised cosine rolloff pulse shaping is used, then the bandwidth is:
Spectral efficiency of ASK is half that of a baseband unipolar NRZ line code This is because the quadrature component is wasted
95% energy bandwidth
bb RrWRrB )1(2
1)1(
bb
RT
B 33
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Detectors for ASKCoherent Receiver
Coherent detection requires the phase information A coherent detector mixes the incoming signal with a locally generated
carrier reference Multiplying the received signal r(t) by the receiver local oscillator (say
Accos(wct)) yields a signal with a baseband component plus a component at 2fc
Passing this signal through a low pass filter eliminates the high frequency component In practice an integrator is used as the LPF
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The output of the LPF is sampled once per bit period This sample z(T) is applied to a decision rule
z(T) is called the decision statistic Matched filter receiver of OOK signal
A MF pair such as the root raised cosine filter can thus be used to shape the source and received baseband symbols
In fact this is a very common approach in signal detection in most bandpass data modems
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Noncoherent Receiver Does not require a phase reference at the receiver If we do not know the phase and frequency of the carrier, we can
use a noncoherent receiver to recover ASK signal Envelope Detector:
The simplest implementation of an envelope detector comprises a diode rectifier and smoothing filter
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Frequency Shift Keying (FSK) In FSK, the instantaneous carrier frequency is switched between 2 or
more levels according to the baseband digital data data bits select a carrier at one of two frequencies the data is encoded in the frequency
Until recently, FSK has been the most widely used form of digital modulation;Why? Simple both to generate and detect Insensitive to amplitude fluctuations in the channel
FSK conveys the data using distinct carrier frequencies to represent symbol states
An important property of FSK is that the amplitude of the modulated wave is constant
Waveform
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Analytical Expression
General expression is
Where
1,....1,0),cos(2
)( MitT
Ets i
s
si
formAnalog)()(
])([)(
0
0
tmfftdt
df
dmtt
dii
t
di
bsbsi kTTkEEandfiff ,0
1 ii fff
1,....1,0),22cos(2
)( 0 MiftitfT
Ets
s
si
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Binary FSK In BFSK, 2 different frequencies, f1 and f2 = f1 + ∆ f are used to
transmit binary information
Data is encoded in the frequencies That is, m(t) is used to select between 2 frequencies: f1 is the mark frequency, and f2 is the space frequency
bb
s TtfT
Ets 0),(2cos
2)( 110
bb
s TtfT
Ets 0),(2cos
2)( 221
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01)(),cos(
11)(),cos()(
22
11
nc
nc
XortmwhentA
XortmwhentAts
Binary Orthogonal Phase FSK
When w0 an w1 are chosen so that f1(t) and f2(t) are orthogonal, i.e.,
form a set of K = 2 basis orthonormal basis functions
0)()( 21
tt
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General expression is
Where
1,....1,0)],(2cos[2
)( 0 MittfT
Ets i
s
si
Phase Shift Keying (PSK)
1,....1,02
)( MiM
iti
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3. Coherent Detection of Binary FSK
Coherent detection of Binary FSK is similar to that for ASK but in this case there are 2 detectors tuned to the 2 carrier frequencies
Recovery of fc in receiver is made simple if the frequency spacing between symbols is made equal to the symbol rate.
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One of the simplest ways of detecting binary FSK is to pass the signal through 2 BPF tuned to the 2 signaling freqs and detect which has the larger output averaged over a symbol period
Non-coherent Detection
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Phase Shift Keying (PSK)
In PSK, the phase of the carrier signal is switched between 2 (for BPSK) or more (for MPSK) in response to the baseband digital data
With PSK the information is contained in the instantaneous phase of the modulated carrier
Usually this phase is imposed and measured with respect to a fixed carrier of known phase – Coherent PSK
For binary PSK, phase states of 0o and 180o are used
Waveform:
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Analytical expression can be written as
where g(t) is signal pulse shape A = amplitude of the signal ø = carrier phase
The range of the carrier phase can be determined using
For a rectangular pulse, we obtain
( ) ( ) cos[ ( )], 0 , 1,2,....,i c i bs t A g t t t t T i M
bbb
EAassumeandTtT
tg ;0,2
)(
MiM
iti ,....1
)1(2)(
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We can now write the analytical expression as
In PSK the carrier phase changes abruptly at the beginning of each signal interval while the amplitude remains constant
MiandTtM
it
T
Ets bc
b
bi ,....2,1,0,
)1(2cos
2)(
carrier phase changes abruptly at the beginning of each signal interval
Constant envelope
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We can also write a PSK signal as:
Furthermore, s1(t) may be represented as a linear combination of two orthogonal functions ψ1(t) and ψ2(t) as follows
Where
M
it
T
Ets ci
)1(2cos
2)(
tM
it
M
i
T
Ecc
cossin)1(2
sincos)1(2
cos2
)()1(2
sin)()1(2
cos)( 21 tM
iEt
M
iEtsi
]sin[2
)(]cos[2
)( 21 tT
tandtT
t cc
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Using the concept of the orthogonal basis function, we can represent PSK signals as a two dimensional vector
For M-ary phase modulation M = 2k, where k is the number of information bits per transmitted symbol
In an M-ary system, one of M ≥ 2 possible symbols, s1(t), …, sm(t), is transmitted during each Ts-second signaling interval
The mapping or assignment of k information bits into M = 2k possible phases may be performed in many ways, e.g. for M = 4
21
)1(2sin,
)1(2cos)(
M
iE
M
iEts bbi
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A preferred assignment is to use “Gray code” in which adjacent phases differ by only one binary digit such that only a single bit error occurs in a k-bit sequence. Will talk about this in detail in the next few slides.
It is also possible to transmit data encoded as the phase change (phase difference) between consecutive symbols This technique is known as Differential PSK (DPSK)
There is no non-coherent detection equivalent for PSK except for DPSK
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M-ary PSK
In MPSK, the phase of the carrier takes on one of M possible values
Thus, MPSK waveform is expressed as
Each si(t) may be expanded in terms of two basis function Ψ1(t) and Ψ2(t) defined as
MiM
iti ,.....,2,1,
)1(2)(
M
it
T
Etsi
)1(2cos
2)( 0
M
ittgtsi
)1(2cos)()( 0
...........
1616
88
4
2
2
PSK
PSK
QPSK
BPSK
MPSKM k
,cos2
)(1 tT
t cs
,sin2
)(2 tT
t cs
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Quadrature PSK (QPSK)
Two BPSK in phase quadrature QPSK (or 4PSK) is a modulation technique that transmits 2-bit of
information using 4 states of phases For example
General expression:
2-bit Information ø
00 0
01 π/2
10 π
11 3π/2
Each symbol corresponds
to two bits
scs
sQPSK Tti
M
itf
T
Ets
04,3,2,1,)1(2
2cos2
)(
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The signals are:
)cos(2
0 tT
Es c
s
s )sin(2
)2
cos(2
1 tT
Et
T
Es c
s
sc
s
s
)cos(2
)cos(2
2 tT
Et
T
Es c
s
sc
s
s
)sin(2
)2
3cos(
23 t
T
Et
T
Es c
s
sc
s
s
002,0 1800f,cos
2)( andoshiftt
T
Ets c
s
s
003,1 27090f,sin
2)( andoshiftt
T
Ets c
s
s
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scs
sQPSK Tti
M
it
T
Ets
04,3,2,1,
4
)1(2cos
2)(
We can also have:
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One of 4 possible waveforms is transmitted during each signaling interval Ts i.e., 2 bits are transmitted per modulation symbol → Ts=2Tb)
In QPSK, both the in-phase and quadrature components are used The I and Q channels are aligned and phase transition occur once
every Ts = 2Tb seconds with a maximum transition of 180 degrees From
As shown earlier we can use trigonometric identities to show that
M
itf
T
Ets c
s
sQPSK
)1(22cos
2)(
)sin()1(2
sin2
)cos()1(2
cos2
)( tM
i
T
Et
M
i
T
Ets c
s
sc
s
sQPSK
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In terms of basis functions
we can write sQPSK(t) as
With this expression, the constellation diagram can easily be drawn For example:
tfT
tandtfT
t cs
cs
2sin2
)(2cos2
)( 21
)()1(2
sin)()1(2
cos)( 21 tM
iEt
M
iEts ssQPSK
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Coherent Detection1. Coherent Detection of PSK Coherent detection requires the phase information A coherent detector operates by mixing the incoming data signal
with a locally generated carrier reference and selecting the difference component from the mixer output
Multiplying r(t) by the receiver LO (say A cos(ωct)) yields a signal with a baseband component plus a component at 2fc
The LPF eliminates the high frequency component The output of the LPF is sampled once per bit period The sampled value z(T) is applied to a decision rule
z(T) is called the decision statistic
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Matched filter receiver
A MF pair such as the root raised cosine filter can thus be used to shape the source and received baseband symbols
In fact this is a very common approach in signal detection in most bandpass data modems
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2. Coherent Detection of MPSK
QPSK receiver is composed of 2 BPSK receivers one that locks on to the sine carrier and the other that locks onto the cosine carrier
tAt 01 cos)(
tAt 02 sin)(
2
0 0 1 0 0 00 0( ) ( ) ( ) ( cos ) ( cos )
2
s sT TsA T
z t s t t dt A t A t dt L
1 0 2 0 00 0( ) ( ) ( ) ( cos ) ( sin ) 0
s sT Tz t s t t dt A t A t dt
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If
Decision:1. Calculate zi(t) as
2. Find the quadrant of (Z0, Z1)
Output S0(t) S1(t) S2(t) S3(t)
Z0 Lo 0 -Lo 0
Z1 0 -Lo 0 Lo
)45cos()()45cos()( 0201oo tAtandtAt
Output S0(t) S1(t) S2(t) S3(t)
Z0 Lo -Lo -Lo Lo
Z1 Lo Lo -Lo -Lo
dtttrtz i
T
i )()()(0
4cos
2
2
0
sTAL
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A coherent QPSK receiver requires accurate carrier recovery using a 4th power process, to restore the 90o phase states to modulo 2π
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4.3 Detection of Signals in Gaussian Noise
Detection models at baseband and passband are identical Equivalence theorem (for linear systems):
Linear signal processing on passband signal and eventual heterodyning to baseband is equivalent to first heterodyning passband signal to baseband followed by linear signal processing
Where
Heterodyning = Process resulting in spectral shift in signal e.g. mixing
Performance Analysis and description of communication systems is usually done at baseband for simplicity
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4.3.2 Correlation Receiver
T
0
T
0
comparator selectssi(t)
with max zi(t)
Decision Stage
reference signal
......
......
)(1 t
)(tM
dtttrTzT
)()()( 101
dtttrTz M
T
M )()()(0
T
0
T
0
comparator selectssi(t)
with max zi(t)
)(ˆ tsi
Decision Stage)(1 ts
reference signal
)(tsM
......
......
dttstrTzT
)()()( 101
dttstrTz M
T
M )()()(0
)()()( tntstr i
)(ˆ tsi)()()( tntstr i
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4.4 Coherent Detection 4.4.1 Coherent Detection of PSK Consider the following binary PSK example
n(t) = zero-mean Gaussian random process
Where φ : phase term is an arbitrary constant
E: signal energy per symbol
T: Symbol duration Single basis function for this antipodal case:
TttT
Ets 0)cos(
2)( 01
)cos(2
)( 02 tT
Ets )cos(
20 t
T
E
TtfortT
t 0cos2
)( 01
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)()()( 11 ttats ii
Transmitted signals si(t) in terms of ψ1(t) and coefficients ai1(t) are
Assume that s1 was transmitted, then values of product integrators with reference to ψ1 are
)()()()( 11111 tEttats
)()()()( 11212 tEttats
T
dtttntEEszE0 1
211 )()()(|
1
T
dtttntEEszE0 1
212 )()()(|
1
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where E{n(t)}=0
Decision stage determines the the location of the transmitted signal within the signal space
For antipodal case choice of ψ1(t) = √2/T cosw0t normalizes E{zi(T)} to ±√E
Prototype signals si(t) are the same as reference signals ψj(t) except for normalizing scale factor
Decision stage chooses signal with largest value of zi(T)
EdttT
tntET
EszET
0 002
11 cos2
)(cos2
|
EdttT
tntET
EszET
0 002
12 cos2
)(cos2
|
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4.4.2 Sampled Matched Filter
The impulse response h(t) of a filter matched to s(t) is:
Let the received signal r(t) comprise a prototype signal si(t) plus noise n(t)
Bandwidth of the signal is W =1/2T where T is symbol time then Fs= 2W = 1/T
Sample at t =kTs . This allows us to use discrete notation:
Let ci(n) be the coefficients of the MF where n is the time index and N represents the samples per symbol
elsewhere
TttTsth
0
0)()(
,...1,02,1)()()( kiknkskr i
])1[()( nNsnc ii
(eq 4.26)
(eq 4.27)
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Discrete form of convolution integral suggests
Since noise is assumed to have zero mean, so the expected value of a received sample is:
Therefore, if si(t) is transmitted, the expected MF output is:
Combining eq (4.27) and (eq 4.29) to express the correlator outputs at time k = N –1 = 3:
Nmodulo,.....,1,0)()()(1
0
Kncnkrkz i
N
ni
2,1)()( ikskrE i
Nmodulo,.....,1,0)()()(1
0
KncnkskzE i
N
nii
(eq 4.28)
(eq 4.29)
3
1 1 10
( 3) (3 ) ( ) 2n
z k s n c n
3
2 1 20
( 3) (3 ) ( ) 2n
z k s n c n
(eq 4.30a) (eq 4.30b)
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Sampled Matched Filter
Fig 4.10
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4.4.3 Coherent Detection of MPSK The signal space for a multiple phase-shift keying (MPSK) signal set
is illustrated for a four-level (4-ary) PSK or quadriphase shift keying(QPSK)
Fig 4.11
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At the transmitter, binary digits are collected two at a time for each symbol interval
Two sequential digits instruct the modulator as to which of the four waveforms to produce
si(t) can be expressed as:
where:
E: received energy of waveform over each symbol duration T
w0: carrier frequency
Assuming an ortho-normal signal space, the basis functions are:
tT
t 01 cos2
)(
Mi
Tt
M
it
T
Etsi ,...1
0)
2cos(
2)( 0
tT
t 02 sin2
)(
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si(t) can be written in terms of these orthonormal coordinates:
The decision rule for the detector is: Decide that s1(t) was transmitted if received signal vector fall in
region 1 Decide that s2(t) was transmitted if received signal vector fall in
region 2 etc i.e choose ith waveform if zi(T) is the largest of the correlator
outputs The received signal r(t) can be expressed as:
Mi
Tttatats iii ,...1
0)()()( 2211
)(2
sin)(2
cos 21 tM
iEt
M
iE
Mi
Tttntt
T
Etr ii ,...1
0)(sinsincoscos
2)( 00
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The upper corelator computes
The lower corelator computes
dtttrXT
)()( 10
dtttrYT
)()( 20
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The computation of the received phase angle φ can be accomplished by computing the arctan of Y/XWhere:X: is the inphase component of the received signalY: is the quadrature component of the received signal ǿ: is the noisy estimate of the transmitted φi
The demodulator selects the φi
that is closest to the angle ǿ
Or it computes | φi - ǿ | for each φi
prototypes and chooses φi yielding smallest output
Fig 4.13
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4.4.4 Coherent Detection of FSK FSK modulation is characterized by the information in the frequency
of the carrier Typical set of FSK signal waveform:
Where Φ: is an arbitrary constant
E: is the energy content of si(t) over each symbol duration T
(wi+1- wi): is typically assumed to be an integral multiple of λ/T
Assuming the basis functions form an orthonormal set:
Amplitude √2/T normalizes the expected output of the MF
Mi
Ttt
T
Ets ii ,...1
0)cos(
2)(
NjtT
t jj ,....,1cos2
)(
dttT
tT
Ea ji
T
ij )cos(2
)cos(2
0
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Therefore
This implies, the ith prototype signal vector is located on the ith
coordinate axis at a displacement √E from the origin of the symbol space
For general M-ary case and given E, the distance between any two prototype signal vectors si and sj is constant:
otherwise
jiforEaij
0
jiforEssssd jiji 2||||),(
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Signal space partitioning for 3-ary FSK