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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 TechniquesThree 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)()(

dttdti

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

)()( tmKdtdt 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 es t e

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Phasor Representation of FM

Consider the FM signal in phasor form:

0( ) Re 12 2 2

mm 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

00,010),cos(

)(binaryTtbinaryTttA

ts ci

)cos(2)cos(2)cos()( 02

00 tAtAtAtsrmsrms

RVPt

TEtP

2

00 )cos(2)cos(2

1,......2,0,00,0

10),cos()(2)( Mi

binaryTt

binaryTttTtE

ts ii

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 ASKb

b 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(21)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)( MitTEts is

si

formAnalog)()(

])([)(

0

0

tmfftdtdf

dmtt

dii

t

di

bsbsi kTTkEEandfiff ,0

1 ii fff

1,....1,0),22cos(2)( 0 MiftitfTEtss

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 TtfTEts 0),(2cos2)( 110

bb

s TtfTEts 0),(2cos2)( 221

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01)(),cos(11)(),cos(

)(22

11

nc

nc

XortmwhentAXortmwhentA

ts

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 MittfTEts is

si

Phase Shift Keying (PSK)

1,....1,02)( MiMiti

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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)(

MiMiti ,....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

MiandTtMit

TEts bcb

bi ,....2,1,0,)1(2cos2)(

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

Mit

TEts ci

)1(2cos2)(

tMit

Mi

TE

cc cossin)1(2sincos)1(2cos2

)()1(2sin)()1(2cos)( 21 tMiEt

MiEtsi

]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)(

MiE

MiEts 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

MiMiti ,.....,2,1,)1(2)(

Mit

TEtsi

)1(2cos2)( 0

Mittgtsi

)1(2cos)()( 0

...........161688

422

PSKPSK

QPSKBPSKMPSKM 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 ø00 001 π/2

10 π11 3π/2

Each symbol corresponds

to two bits

scs

sQPSK Tti

Mitf

TEts

04,3,2,1,)1(22cos2)(

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The signals are:

)cos(20 t

TEs cs

s )sin(2)2

cos(21 t

TEt

TEs c

s

sc

s

s

)cos(2)cos(22 t

TEt

TEs c

s

sc

s

s

)sin(2)23cos(2

3 tTEt

TEs c

s

sc

s

s

002,0 1800f,cos2)( andoshiftt

TEts cs

s

003,1 27090f,sin2)( andoshiftt

TEts cs

s

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scs

sQPSK Tti

Mit

TEts

04,3,2,1,

4)1(2cos2)(

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

Mitf

TEts cs

sQPSK

)1(22cos2)(

)sin()1(2sin2)cos()1(2cos2)( tMi

TEt

Mi

TEts 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(2sin)()1(2cos)( 21 tMiEt

MiEts 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 )

2s sT T

sA Tz t s t t dt A t A t dt L

1 0 2 0 00 0( ) ( ) ( ) ( cos ) ( sin ) 0s sT T

z 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()( 0201

oo 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

0sTAL

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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 constantE: signal energy per symbolT: Symbol duration

Single basis function for this antipodal case:

TttTEts 0)cos(2)( 01

)cos(2)( 02 tTEts )cos(2

0 tTE

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

00)(

)(

,...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)(

MiTt

Mit

TEtsi ,...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:

MiTt

tatats iii ,...10

)()()( 2211

)(2sin)(2cos 21 tMiEt

MiE

MiTt

tnttTEtr ii ,...1

0)(sinsincoscos2)( 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 constantE: 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

MiTt

tTEts ii ,...1

0)cos(2)(

NjtT

t jj ,....,1cos2)(

dttT

tTEa ji

T

ij )cos(2)cos(20

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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||||),(

68

Signal space partitioning for 3-ary FSK