section 3 1 digital mod tech v7

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Digital modulation techniquesSignal Space Analysis

Receiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound

Passband modulation

Chapter 3: Physical-layer transmission techniques

Section 3.1: Digital modulations

Mobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 1

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Passband modulation

Advantages over analog modulationMain considerations in digital modulation techniquesTypical types of digital modulation techniques

Advantages over analog modulation

The advances over the last several decades in hardware anddigital signal processing have made digital transceivers muchcheaper, faster, and more power-efficient than analogtransceivers.

More importantly, digital modulation offers a number of otheradvantages over analog modulation, including:

higher data rates,powerful error correction techniques,resistance to channel impairments,

more efficient multiple access strategies, andbetter security and privacy.


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Passband modulation


Advantages over analog modulation (cont.)

Digital transmissions consist of transferring information in theform of bits over a communications channel.

The bits are binary digits taking on the values of either 1 or 0.These information bits are derived from the informationsource, which may be a digital source or an analog source thathas been passed through an A/D converter.

Both digital and A/D converted analog sources may becompressed to obtain the information bit sequence.


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Passband modulation


Main considerations in digital modulation techniques

Digital modulation consists of mapping the information bitsinto an analog signal for transmission over the channel.

Detection consists of determining the original bit sequencebased on the signal received over the channel.

The main considerations in choosing a particular digitalmodulation technique are:

high data ratehigh spectral efficiency (minimum bandwidth occupancy)high power efficiency (minimum required transmit power)robustness to channel impairments (minimum probability of biterror)low power/cost implementation


Di i l d l i h i

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Passband modulation


Typical types of digital modulation techniques

Often the previous ones are conflicting requirements, and thechoice of modulation is based on finding the technique thatachieves the best tradeoff between these requirements.

There are two main categories of digital modulation:

amplitude/phase modulationfrequency modulation

Frequency modulation typically has a constant signal envelopeand is generated using nonlinear techniques, this modulation

is also called constant envelope modulation or nonlinearmodulation


Di it l d l ti t h i

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Passband modulation


Typical types of digital modulation techniques (cont.)

Amplitude/phase modulation is also called linear modulation.

Linear modulation generally has better spectral propertiesthan nonlinear modulation, since nonlinear processing leads tospectral broadening.

However, amplitude and phase modulation embeds theinformation bits into the amplitude or phase of thetransmitted signal, which is more susceptible to variationsfrom fading and interference.


Digital modulation techniques

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Passband modulation


Typical types of digital modulation techniques (cont.)

In addition, amplitude and phase modulation techniquestypically require linear amplifiers, which are more expensiveand less power efficient than the nonlinear amplifiers that canbe used with nonlinear modulation.

Thus, the general tradeoff of linear versus nonlinearmodulation is one of better spectral efficiency for the formertechnique and better power efficiency and resistance tochannel impairments for the latter technique.


Digital modulation techniques Rational

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Passband modulation

RationalSignal and system modelGeometric representation of signalsPractical examplesSignal space representation

Rational

Digital modulation encodes a bit stream of finite length intoone of several possible transmitted signals.

Intuitively, the receiver minimizes the probability of detectionerror by decoding the received signal as the signal in the set of possible transmitted signals that is closest to the one received.

Determining the distance between the transmitted andreceived signals requires a metric for the distance betweensignals.



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Passband modulation


Rational (cont.)

By representing signals as projections onto a set of basisfunctions, we obtain a one-to-one correspondence between theset of transmitted signals and their vector representations.

Thus, we can analyze signals in finite-dimensional vectorspace instead of infinite-dimensional function space, usingclassical notions of distance for vector spaces.

In this section we show:

how digitally modulated signals can be represented as vectors

in an appropriately-defined vector space, andhow optimal demodulation methods can be obtained from thisvector space representation.

This general analysis will then be applied to specificmodulation techniques in later sections.



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Passband modulation


Transmitted signal

Transmitter Receiver

+

n(t)

AWGN Channel

s(t)i 1 K

m ={b ,...,b } ^1 K

m ={b ,...,b }^ ^r(t)

Figure 1: Communication system model over AWGN channel (i.e., aspecial case of wireless channel).

Consider a communication system model as shown in theabove figure.

Every seconds, the sytem sends = log2 bits of information through the channel for a data rate of = / bits per second (bps).



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g qSignal Space Analysis


Passband modulation

Signal and system modelGeometric representation of signalsPractical examplesSignal space representation

Transmitted signal (cont.)

There are = 2 possible sequences of bits and each bitsequence of length comprises a message = {1,..., } ∈ , where = {1,..., } is the set of all such messages.

The message has probability of being selected fortransmission, where

=1 = 1.

Suppose that message is to be transmitted over theAWGN channel during the time interval [0, ). Since the

channel is analog, the message must be embedded into ananalog signal for channel transmission.



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g qSignal Space Analysis


Passband modulation


Transmitted signal (cont.)

Therefore, each message ∈ is mapped to a uniqueanalog signal () ∈ = {1(),..., ()} where () isdefined on the time interval [0, ) and has energy

= 0

2 (), = 1,...,. (1)

When messages are sent sequentially, the transmittedsignal becomes a sequence of the corresponding analog signals

as follows() =

( − ). (2)



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gSignal Space Analysis


Passband modulation


Transmitted and received signals

In the aforementioned model, the transmitted signal is sentthrough an AWGN channel where a white Gaussian noiseprocess () of power spectral density /2 is added to formthe received signal

() = () + (). (3)

T0 2T 3T 4T

s (t)1 1 1

2s (t−T)

s (t−2T) s (t−3T)

s(t)

...

m

1

m

1

m

1

m2


Digital modulation techniquesS S

RationalS

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Signal Space AnalysisReceiver Structure and Sufficient Statistics

Error Probability Analysis and the Union BoundPassband modulation


Received signal

Given (), the receiver must determine the best estimate of which () ∈ was transmitted during each transmissioninterval [0, ).

This best estimate of () is mapped to a best estimate of

the message () ∈ and the receiver produces this best

estimate = 1,..., of the transmitted bit sequence.

The goal of the receiver design in estimating the transmittedmessage is to minimize the probability of message error

= =1

( ∕= ∣ sent) ( sent) (4)

over each time interval [0, ).


Digital modulation techniquesSi l S A l i

RationalSi l d d l

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Introduction

By representing the signals {(), = 1,..., } geometrically,one can solve for the optimal receiver design in AWGNchannels based on a minimum distance criterion.

Note that, wireless channels typically have a time-varying

impulse response in addition to AWGN. We will consider theeffect of an arbitrary channel impulse response on digitalmodulation performance in the next sections.

The basic premise behind a geometrical representation of

signals is the notion of a basis set.


Digital modulation techniquesSi l S A l sis

RationalSi l d s st d l

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Basis function representation of signals

Specifically, using a Gram-Schmidt orthogonalizationprocedure, it can be shown that any set of real energysignals = {1(),..., ()} defined on [0, ) can berepresented as a linear combination of

≤ real

orthogonal basis functions {1(),..., ()}.We say that these basis functions span the set .Each signal {() ∈ } can be represented by

() =

=1 , (), 0≤

< , (5)

where

, =

0

() () (6)

is a real coefficient representing the pro jection.Mobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 16


RationalSignal and system model

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Basis function representation of signals (cont.)

These basis functions have the following property 0

() () =

1 = ,

0

∕= .

(7)

The basis set consists of the sine and cosine functions

1() =

2

cos (2 ) (8)

and

2() =

2

sin(2 ) . (9)

where 2 is used to obtain

0 2 () = 1, = 1, 2.




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Basis functions in linear passband modulation techniques

With these basis functions, one only obtain an approximationto (7), since

0

21() =

2

0

0.5 [1 + cos (4 )] = 1+sin (4 )

4 (10)

The numerator in the second term of (10) is bounded by 1,and for ≫ 1 the denominator of this term is very large.As a result, this second term can be neglected.




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Basis functions in linear passband modulation (cont.)

With these basis functions, one can have 0

1()2() =2

0

0.5sin(4 ) =− cos(4 )

4 ≈ 0

(11)where the approximation is taken as an equality as ≫ 1.

With the basis set 1() =

2/ cos (2 ) and2() =

2/ sin (2 ), the basis function representation

(5) corresponds to the complex representation of () in

terms of its in-phase and quadrature components with anextra factor of

2/ as follows

() = ,1

2

cos (2 ) + ,2

2

sin(2 ) . (12)




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Basis functions in linear passband modulation (cont.)

In practice, the basis set may include a baseband pulse-shapingfilter () to improve the spectral characteristics of the transmittedsignal:

() = ,1() cos (2 ) + ,2() sin (2 ) (13)

where the simplest pulse shape that satisfy (7) is the rectangularpulse shape () =

2/ , 0 ≤ < .




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g p yReceiver Structure and Sufficient Statistics


g yGeometric representation of signalsPractical examplesSignal space representation

Definitions used in signal space representation

We denote the coefficients {,} as a vectors = [,1,...,, ] ∈ ℛ which is called the signal

constellation point corresponding to the signal ().

The signal constellation consists of all constellation points

{s1, ..., s }.

Given the basis functions {1(),..., ()} there is aone-to-one correspondence between the transmitted signal() and its constellation point s.

The representation of () in terms of its constellation points ∈ ℛ is called:

its signal space representation andthe vector space containing the constellation is called thesignal space .




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Passband modulation

Geometric representation of signalsPractical examplesSignal space representation

Definitions used in signal space representation (cont.)

A two-dimensional signal space is illustrated in the belowfigure, where we show s ∈ ℛ2 with the th axis of ℛ2

corresponding to the basis function (), = 1, 2.

=4, K=2

0011

01

10

M=8, K=3

000

001

011

110

100

010

110

101

si1

si2

si1

si2




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Passband modulation



With this signal space representation we can analyze theinfinite-dimensional functions () as vectors s infinite-dimensional vector space ℛ2.

This greatly simplifies the analysis of the system performance

as well as the derivation of the optimal receiver design.

Signal space representations for common modulationtechniques like MPSK and MQAM are two-dimensional(corresponding to the in-phase and quadrature basis

functions).In order to analyze signals via a signal space representation,we need to use some definitions for the vector characterizationin the vector space ℛ .



R i S d S ffi i S i i

RationalSignal and system modelG i i f i l

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Passband modulation



In particular, the length of a vector in ℛ is defined as

∥s∥ =

⎷

=12,. (14)

The distance between two signal constellation points s and s

is thus

∥s − s∥ = ⎷ =1

(, − ,)2 = 0

(() − ())2 .

(15)



R i St t d S ffi i t St ti ti

RationalSignal and system modelG t i t ti f i l

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Passband modulation



Finally, the inner product ⟨(), ()⟩ between two realsignals () and () on the interval [0, ) is defined as

⟨(), ()

⟩=

0

()(). (16)

Similarly, the inner product ⟨s, s⟩ between two real vectors is

⟨s, s

⟩= ss =

0

()() =

⟨(), ()

⟩. (17)

It is noted that two signals are orthogonal if their innerproduct is zero.



Receiver Structure and Sufficient StatisticsGeneral resultsProofs of sufficient statistics for optimal detection

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Passband modulation

Proofs of sufficient statistics for optimal detectionDecision regions and criterion

Receiver structure and sufficient statistics

Given the channel output () = () + (), 0 ≤ < , wenow investigate the receiver structure to determine whichconstellation point s or, equivalently, which message , wassent over the time interval [0, ).

A similar procedure is done for each time interval[ , ( + 1) ).

We would like to convert the received signal () over eachtime interval into a vector, since it allows us to work in

finite-dimensional vector space to estimate the transmittedsignal.

However, this conversion should not compromise theestimation accuracy. For this conversion, consider the receiverstructure shown in the next figure.




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Passband modulation


Receiver structure and sufficient statistics (cont.)

)()()( t nt st r i

³T

dt 0()

³

T

dt 0 ()

111, r n si

)(1 t I

)(t N

I

N N N ir n s ,

Find ii

mm ˆ

As shown in the above figure, the components of signal andnoise vectors are determined by

, = 0

() (), (18)

and

=

0

() (). (19)




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Passband modulation


Receiver structure and sufficient statistics (cont.)

We can rewrite () as

() =

=1(, + ) () + () =

=1 () + (),

(20)where = , + and () = () −

=1 ()denotes the remainder noise.




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Passband modulation


Proofs of sufficient statistics for optimal detection

If we can show that the optimal detection of the transmittedsignal constellation point s given received signal () does notmake use of the remainder noise (), then the receiver canmake its estimate of the transmitted message as afunction of r = (1,..., ) alone.

In other words, r = (1,..., ) is a sufficient statistic for ()in the optimal detection of the transmitted messages.

Let exam the distribution of r. Since () is a Gaussian

random process, if we condition on the transmitted signal() then the channel output () = () + () is also aGaussian random process and r = [1,..., ] is a Gaussianrandom vector.




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pDecision regions and criterion

Proofs of sufficient statistics for optimal detection (cont.)

Recall that = , + . Thus, conditioned on thetransmitted constellation s, we have

∣s = [ ∣s] = [, + ∣,] = , (21)

since n(t) has zero mean, and

∣s =

− ∣s2

= [, + − ,∣,]2 =

2

.

(22)

With Cov [

∣s

] =

− ( −

)∣s

= [

]and some manipulations, one can have

[ ] = 02

0

()() =

0/2 =

0 ∕= .. (23)




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pDecision regions and criterion


Thus, conditioned on the transmitted constellation s, the ’sare uncorrelated and, since they are Gaussian and also

independent. Moreover,

2

= 0/2.

We have shown that, conditioned on the transmittedconstellation s, is a Gauss-distributed random variable thatis independent of , ∕= and has mean , and variance 0/2.

Thus, the conditional distribution of r is given by

(r∣s sent) =

=1

( ∣) =1

( 0)/2

exp

⎡⎣− 1

0

=1

( − ,)2

⎤⎦ .

(24)




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Decision regions and criterion


It is also straightforward to show that [ ()∣s] = 0 forany , 0 ≤ < . Thus, since conditioned on s and ()are Gaussian and uncorrelated, they are independent.

Also, since the transmitted signal is independent of the noise,, is independent of the process ().

We now discuss the receiver design criterion and show it is notaffected by discarding ().

The goal of the receiver design is to minimize the probability

of error in detecting the transmitted message givenreceived signal ().

To minimize = (

∕= ∣()) = 1 − (

= ∣()), we

maximize (

= ∣()).


Digital modulation techniquesSignal Space AnalysisReceiver Structure and Sufficient Statistics

E P b bili A l i d h U i B d

General resultsProofs of sufficient statistics for optimal detectionD i i i d i i

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Therefore, the receiver output

given received signal ()

should correspond to the message that maximizes ( sent∣()).

Since there is a one-to-one mapping between messages andsignal constellation points, this is equivalent to maximizing ( sent

∣()).



E P b bilit A l i d th U i B d

General resultsProofs of sufficient statistics for optimal detectionD i i i d it i

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Recalling that () is completely described by = (1,..., )and (), we have

(s sent

∣()) = ((,1,...,, ) sent

∣(1,..., , ()))

= ((,1,...,, ) sent, (1,..., ), ())

((1,..., ), ())

= ((,1,...,, ) sent, (1,..., )) (())

((1,..., )) (())

= ((,1,...,, ) sent∣(1,..., )) . (25)

where the third equality follows from the fact that the () isindependent of both (1,..., ) and of (,1,...,, ).



Error Probabilit Anal sis and the Union Bo nd

General resultsProofs of sufficient statistics for optimal detectionDecision regions and criterion

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This analysis shows that (1,..., ) is a sufficient statistic for

() in detecting , in the sense that the probability of erroris minimized by using only this sufficient statistic to estimatethe transmitted signal and discarding the remainder noise.

Since r is a sufficient statistic for the received signal (), wecall r the received vector associated with ().



Error Probability Analysis and the Union Bound


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Decision regions

As aforementioned, the optimal receiver minimizes errorprobability by selecting the detector output

that maximizes

the probability of correct detection

1 − = ( sent∣r received).In other words, given a received vector r, the optimal receiverselects

= corresponding to the constellation s that

satisfies (s∣r) > (s

∣r) ,

∀

∕= (26)

where (s∣r) ≜ (s sent∣r received) for the sake of notational simplicity.





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Decision regions(cont.)

Thus, the decision regions ( 1,..., ) corresponding to(s1, ...,s ) are the subsets of the signal space ℛ anddefined by

= (r : (s∣r) > (s ∣r) , ∀ ∕= ) . (27)

Once the signal space has been partitioned by decisionregions, for a received vector r ∈ , the optimal receiveroutputs the message estimate =

The receiver processing consists of ) computing the receivedvector r from (), ) finding which decision region contains r, and ) outputting the corresponding message .





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An example on decision regions

This process is illustrated in the below figure, that shows atwo-dimensional signal space with four decision regions 1,..., 4 corresponding to four constellations s1, ..., s4.The received vector r lies in region 1, so the receiver willoutput the message

1as the best message estimate given

received vector r.

φ (t)1

φ (t)2

s

s

s

s1

2

3

4

1

ZZ3

Z2

Z4

x





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o obab ty a ys s a d t e U o ou dPassband modulation

ec s o eg o s a d c te o

Decision criterion

Using Bayes rule, one can have

(s∣r) = (r∣s) (s)

(r). (28)

To minimize error probability, the receiver output = corresponds to the constellation point s that maximizes (s∣r), i.e., the detected transmitted constellation point scan be determined by

s = arg maxs

(r∣s) (s) (r)

= arg maxs

(r∣s) (s) , = 1,...,

(29)where the second equality follows from the fact that (r) isnot a function of s





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y yPassband modulation

g

Decision criterion (cont.)

Assuming that transmitted messages are equally likely(i.e., (s) = 1/ ), (29) becomes

s = arg maxs

(r∣s) , = 1,...,. (30)

Let define the likelihood function associated with the receiveras

(s) = (r∣s) . (31)

Given a received vector r, a maximum likelihood (ML)receiver outputs = corresponding to the constellationpoint s that maximizes (s).





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Passband modulation


Since the log function is increasing in its argument,maximizing s is equivalent to maximizing the log likelihoodfunction, defined as (s) = log (s). Using (24) for (s) = log (s) yields

(s) = − 1

0

=1

( − ,)2

+constant = − 1

0∥r− s∥2+constant.

(32)

Based on (30), the detected transmitted constellation point s can

be determined by the ML criterion as

s = arg mins

∥r− s∥2 , = 1,...,. (33)





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Passband modulation


Under the aforementioned assumption of equiprobable

(transmitted) messages , the ML structure minimizes theprobability of detection error at the receiver.

Under the aforementioned assumption of equiprobable(transmitted) messages , the ML structure minimizes theprobability of detection error at the receiver.



Error Probability Analysis and the Union BoundP b d d l i

Error probabilityThe union bound on error probability

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Passband modulation

Error probability of ML detection

With ( sent) = 1/ , the error probability of the ML receiver:

s1

s

s

s

s

s

s

s

2

3

4

5

6

7

8

0

1Z

1

dmin

r=s +n

=

=1

(r /∈ ∣ sent) ( sent)

= 1

=1

(r /∈ ∣ sent)

= 1 − 1

=1

(r ∈ ∣ sent)

= 1 − 1

=1

(r∣) r

= 1 − 1

=1

(r = s + n∣s) r



Error Probability Analysis and the Union BoundP b d d l ti


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Passband modulation

The union bound on error probability

As observed, (34) gives an exact solution to the errorprobability but it is impossible to solve this error probability inclosed-form. Therefore, the union bound on error probabilityis investigated.

Let , denote the event that ∥r − s∥ < ∥r − s∥ given thatthe constellation point s was sent.

If the event , occurs, then the constellation point will bedecoded in error since the transmitted constellation point s is

not the closest constellation point to the received vector r.However, event , does not necessarily imply that s will bedecoded instead of s, since there may be anotherconstellation point s with ∥r− s∥ < ∥r − s∥ < ∥r − s∥.





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Passband modulation

The union bound on error probability (cont.)

The constellation point is decoded correctly if ∥r − s∥ < ∥r − s∥ ∀ ∕= . Therefore,

( sent) =

⎛⎜⎜⎝

=1∕=

,

⎞⎟⎟⎠ ≤

=1∕=

(,) . (35)

where the inequality follows from the union bound onprobability.More specifically, (,) can be determined by

(,) = (∥s − r∥ < ∥s − r∥ ∣ sent)

= (∥s − (s + n)∥ < ∥s − (s + n)∥)

= (∥n + s − s∥ < ∥n∥)

= 2

∥n

∥ ∥s

−s

∥cos <

−∥s

−s

∥2





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Passband modulation


Since ⟨n, s − s⟩ = ∥n∥ ∥s − s∥ cos , one will have

= ∥n∥ cos = ⟨n,s−s⟩∥s−s∥ is a Gaussian random variable with

zero-mean and variance 0/2

As a result, (,) can be simplified to

(,) =

< −∥s − s∥

2

=

>

,2

= ∞,/2

1√ 0

exp−2

0 = ,√

2 0 .





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Passband modulation


Substituting (36) into (35), one can have

( sent) ≤ =1∕=

,√2 0

. (36)

where the function, (), is defined as the probability thata Gaussian random variable with zero-mean and variance of 1 is bigger than , i.e.,

() = ( > ) = ∞

1√2

exp−2

2 . (37)

Summing (36) over all possible messages yields the union bound

=

=1

() ( sent) ≤ 1

=1

=1∕=

,√2 0

. (38)





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Passband modulation


Note that the function cannot be solved in closed-form. Itcan be obtained from the complementary error function as

() = 12erfc

√ 2

.

One can upper bound () with the closed-form expression

() ≤ 1

√

2exp

−2/2

. (39)

and this bound is quite tight for ≫ 0.

Let define the minimum distance of the constellation as = min, ,, one can simplify (41) with looser bound

≤ ( − 1)

√

2 0

. (40)




Passband modulation


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Using (39) for the function yields a closed-form bound

≤

− 1

√

exp−2

4 0 . (41)

Note that for binary modulation ( = 2), there is only oneway to make an error and is the distance between thetwo signal constellation points, so the bound is exact

= √

2 0

. (42)


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Passband modulation




Bit error rate (cont.)

With such a mapping, assuming that mistaking a signalconstellation for a constellation other than its nearestneighbors has a very low probability, we can make theapproximation

≈ 2

. (43)

The most common form of mapping with the property iscalled Gray coding, which will be discussed in more detail.

Signal space concepts are applicable to any modulation wherebits are encoded as one of several possible analog signals,including the amplitude, phase, and frequency modulations asdiscussed later.




Passband modulation

General principlesAmplitude and phase modulationPulse amplitude modulation (MPAM)Phase shift keying (MPSK)Quadrature amplitude modulation (MQAM)

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General principles

The basic principle of passband digital modulation is toencode an information bit stream into a carrier signal which isthen transmitted over a communications channel.

Demodulation is the process of extracting this information bit

stream from the received signal. Corruption of the transmittedsignal by the channel can lead to bit errors in thedemodulation process.

The goal of modulation is to send bits at a high data rate

while minimizing the probability of data corruption.In general, modulated carrier signals encode information in theamplitude (), frequency (), or phase () of a carriersignal.




Passband modulation


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General principles (cont.)

Thus, the modulated signal can be represented as

() = ()cos[2 ( + ()) + () + 0] = ()cos[2 + () + 0]

where () = 2 () + () and 0 is the phase offset of the

carrier. This representation combines frequency and phasemodulation into angle modulation.

One can rewrite the right-hand side of (44) in terms of its in-phaseand quadrature components as:

() = ()cos ()cos[2 ] − ()sin ()sin[2 ]= ()cos[2 ] − ()sin[2 ] (44)

where () = ()cos () is called the in-phase component of ()and () = ()sin () is called its quadrature component.




Passband modulation


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General principles (cont.)

We can write () in its complex baseband representation as

() = Re () 2 (45)

where () = () + ().

This representation is useful since receivers typically processthe in-phase and quadrature signal components separately.




Passband modulation


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Amplitude and phase modulation

In amplitude and phase modulation the information bit streamis encoded in the amplitude and/or phase of the transmittedsignal.

Specifically, over a time interval of , = log2 bits are

encoded into the amplitude and/or phase of the transmittedsignal (), 0 ≤ < .

The transmitted signal over this period() = ()cos[2 ] − ()sin[2 ] can be written interms of its signal space representation as

() = ,11() + ,22() (46)

where basis functions 1() = ()cos(2 + 0) and2() = −()sin(2 + 0), where () is a shaping pulse.




Passband modulation


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Amplitude and phase modulation (cont.)

To send the th message over the time interval [ , ( + 1) ),we set () = ,1() and () = ,2(). These in-phaseand quadrature signal components are baseband signals withspectral characteristics determined by the pulse shape ().

In particular, their bandwidth equals the bandwidth of g(t),and the transmitted signal () is a passband signal withcenter frequency fc and passband bandwidth 2.

In practice we take = / where depends on the

pulse shape: for rectangular pulses = .5 and for raisedcosine pulses .5 ≤ ≤ 1.

Thus, for rectangular pulses the bandwidth of () is .5/ and the bandwidth of () is 1/ .




Passband modulation


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Since the pulse shape () is fixed, the signal constellation foramplitude and phase modulation is defined based on theconstellation point: (,1, ,2) ∈ ℝ2, = 1,..., .

The complex baseband representation of () is

() = Re

() 0 2

(47)

where:

() = () + () = (,1 + ,2) ().

The constellation point s = (,1, ,2) is called the symbolassociated with the log2 bits and is called the symbol time and the bit rate for thismodulation is bits per symbol or = log2 / bits persecond.





General principles

Amplitude and phase modulationPulse amplitude modulation (MPAM)Phase shift keying (MPSK)Quadrature amplitude modulation (MQAM)

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There are three main types of amplitude/phase modulation:

Pulse Amplitude Modulation (MPAM): information encoded inamplitude only.Phase Shift Keying (MPSK): information encoded in phase

only.Quadrature Amplitude Modulation (MQAM): informationencoded in both amplitude and phase.

The number of bits per symbol = log2 , signalconstellation (,1, ,2)

∈ℝ2, = 1,..., , and the choice of

shaping pulse () determines the digital modulation design.The pulse shape () is designed to improve spectral efficiencyand combat inter-symbol-interference (ISI).





General principles


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Amplitude and phase modulation over a given symbol periodcan be generated using the modulator structure shown in thenext Figure.

Note that the basis functions in this figure have an arbitrary

phase 0 associated with the transmit oscillator.Demodulation over each symbol period is performed using thedemodulation structure of Figure xx +1, which is equivalentto the structure of for 1() = ()cos(2 + ) and2() =

−()sin(2 + ).

Typically the receiver includes some additional circuitry forcarrier phase recovery that matches the carrier phase at thereceiver to the carrier phase 0 at the transmitter, which iscalled coherent detection.





General principles


( )

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If − 0 = △ ∕= 0 then the in-phase branch will have anunwanted term associated with the quadrature branch andvice versa, i.e., 1 = ,1 cos(△) + ,2 sin(△) + 1 and2 = ,1 sin(△) + ,2 cos(△) + 2 can result in significantperformance degradation.

The receiver structure also assumes that the sampling functionevery seconds is synchronized to the start of the symbolperiod, which is called synchronization or timing recovery.

Receiver synchronization and carrier phase recovery are

complex receiver operations that can be highly challenging inwireless environments.

We will assume perfect carrier recovery in our discussion of MPAM, MPSK and MQAM, and therefore set = 0 = 0 fortheir analysis.





General principles


P l li d d l i (MPAM)

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Pulse amplitude modulation (MPAM)

We will start by looking at the simplest form of linearmodulation, one-dimensional MPAM, which has noquadrature component (,2 = 0).

For MPAM all of the information is encoded into the signal

amplitude . The transmitted signal over one symbol time isgiven by

() = Re

() 2

= ()cos(2 ), 0 ≤ ≤ ≫ 1/ ,

(48)

where = (2 − 1 − ), = 1, 2,..., defines the signalconstellation, parameterized by the distance which istypically a function of the signal energy, and () is theshaping pulse.





General principles


P l li d d l i ( )

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Pulse amplitude modulation (cont.)

π2

ShapingFilter g(t)

ShapingFilter g(t)

s(t)

In−Phase branch

Quadrature Branch

i1

i2s

s i1s g(t)

s g(t)i2

c−sin(2 f t+ )

cos(2 f t+ )cπ φ

0

cos(2 f t+ )cπ φ

0

π φ0

Figure 2: Amplitude and phase modulator.Mobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 62




General principles


P l lit d d l ti ( t )

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The minimum distance between constellation points is = , ∣ − ∣= 2. The amplitude of thetransmitted signal takes on M different values, which impliesthat each pulse conveys 2 = bits per symbol time .

Over each symbol period the MPAM signal associated withthe th constellation has energy

=

0

2 () =

0

2

2()cos2(2 ) = 2 . (49)

It is noted that the energy is not the same for each signal(), = 1,..., .





General principles



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Assuming equally likely symbols, the average energy is

=1

=1

2 . (50)

i

1

m̂=m

Find i: x Zi

i

T

TIn−Phase branch

π/2

g(T−t)

g(T−t)

cos (2 f t+ )φ

i1 1

2 i2 2

Quadrature branch

s

s

π c

−sin (2 f t+ )cπ φ

r(t)=s (t)+n(t)

r =s +n

r =s +n





General principles



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The constellation mapping is usually done by Gray encoding,where the messages associated with signal amplitudes that areadjacent to each other differ by one bit value, as illustrated inthe below figure.With this encoding method, if noise causes the demodulation

process to mistake one symbol for an adjacent one (the mostlikely type of error), this results in only a single bit error in thesequence of K bits. Gray codes can be designed for MPSK andsquare MQAM constellations, but not rectangular MQAM.

M=4, K=2

00 01 11 10

M=8, K=3

000 001 011 010 110 111 101 100

2d

2dMobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 65




General principles


Pulse amplitude modulation (cont )

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An example: For () = 2/ , 0 ≤ < a rectangularpulse shape, find the average energy of 4PAM modulation.Solution: For 4PAM, the values are = {3, −,, 3}.Hence, the average is

=2

4 (9 + 1 + 1 + 9) = 52. (51)

The decision regions , = 1,..., associated with thepulse amplitude = (2 − 1 − ) for = 4 and = 8as shown in the next figure. Mathematically, for any , these

decision regions are defined by

=

⎧⎨(−∞, + ) = 1,

[ − , + ) 2 ≤ ≤ − 1

[ −

,∞

) =

(52)





General principles



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One can see that MPAM has only a single basis function1() = () cos(2 ).

Thus, the coherent demodulator for MPAM reduces to thedemodulator shown in the next figure, where themulti-threshold device maps to a decision region andoutputs the corresponding bit sequence

=

={

1

,..., }

.





General principles



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2d

2d

A1

A A A2 3 4

Z1

Z Z2

Z3 4

A A A A

Z Z Z ZZ1

Z Z

A1

A2 3

Z2 3 4 5 6 7 8

A A87654

Figure 5: Decision Regions for MPAM.

is (t)+n(t)s

xg (T −t) 0X

cos(2 f t)cπ

^

s

T

Multithreshold Device

2d

4d

−2d

−4d

−(M−2)d

(M−2)d

m=m =b b ...bi 1 2 K

} Zi





General principles


Phase shift keying (MPSK)

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Phase shift keying (MPSK)

For MPSK modulation, all of the information is encoded inthe phase of the transmitted signal.

Thus, the transmitted signal over one symbol time is given by

() = Re () 2(−1)/ 2 , 0 ≤ ≤ (53)

= ()cos

2 +

2( − 1)

= ()cos 2(

−1)

cos(2 )

− ()sin

2( − 1)

sin(2 ).





General principles


Phase shift keying (cont )

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Phase shift keying (cont.)

The constellation points or symbols (,1, ,2) are given by

,1 = cos2(−1)

and ,2 = sin

2(−1)

for

= 1,..., . =2(−1) , = 1, 2,..., = 2 are the

different phases in the signal constellation points that conveythe information bits.

The minimum distance between constellation points is = 2 sin(/ ), where is typically a function of thesignal energy.

2PSK is often referred to as binary PSK or BPSK, while 4PSKis often called quadrature phase shift keying (QPSK), and isthe same as MQAM with = 4 which is defined below.





General principles



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All possible transmitted signals () have equal energy:

=

02 () = 2. (54)

Note that for () =

2/ , 0 ≤ ≤ , i.e., a rectangularpulse, this signal has constant envelope, unlike the otheramplitude modulation techniques MPAM and MQAM.

However, rectangular pulses are spectrally-inefficient, andmore efficient pulse shapes make MPSK nonconstantenvelope.





General principles



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Analogous to MPAM, MPSK constellation mapping is usuallydone by Gray encoding, where the messages associated withsignal phases that are adjacent to each other differ by one bitvalue, as illustrated in the below Figure.With this encoding method, mistaking a symbol for an

adjacent one causes only a single bit error.

=4, K=2

0011

01

10

M=8, K=3

000

001

011

110

100

010

110

101

si1

si2

si1

si2

Figure 7: Gray Encoding for MPSK.Mobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 72


Signal Space AnalysisReceiver Structure and Sufficient StatisticsError Probability Analysis and the Union Bound

Passband modulation

General principles


Phase shift keying: Decision region

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Phase shift keying: Decision region

The decision regions , = 1,..., , associated with MPSKfor = 8 are shown in the next figure.

If we represent r = ∈ ℝ2 in polar coordinates then thesedecision regions for any are defined by

=

: 2( − .5)/ ≤ ≤ 2( + .5)/

. (55)

For the special case of BPSK, the decision regions simplify to 1 = (r : r > 0) and 2 = (r : r

≤0).

Moreover BPSK has only a single basis function1() = () cos(2 ) and, since there is only a single bittransmitted per symbol time , the bit duration = .




Passband modulation

General principles


Phase shift keying: Decision region (cont.)

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Thus, the coherent demodulator for BPSK reduces to thedemodulator shown in the next figure, where the thresholddevice maps to the positive or negative half of the real line,

and outputs the corresponding bit value.We have assumed in this figure that the messagecorresponding to a bit value of 1, 1 = 1, is mapped toconstellation point 1 = and the message corresponding to

a bit value of 0, 2 = 0, is mapped to the constellation point2 = −.




Passband modulation

General principles



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ase s t ey g ec s o eg o (co t )

Z1

Z2

Z3

Z4

Z1

Z2

Z

Z

Z

Z

Z

Z

34

5

6

7

8

Figure 8: Decision Regions for MPSK.

is (t)+n(t)g (T −t)

b 0X

cos(2 f t)cπ

^Tb

Threshold Device

2

m=1 or 0

m=1^

m̂=0

}

}r 1

Z :r>0

Z :r<0

Figure 9: Coherent Demodulator for BPSK.Mobile communications - Chapter 3: Physical-layer transmissions Section 3.1: Digital modulations 75



Passband modulation

General principles


Quadrature amplitude modulation (MQAM)

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Q p ( Q )

For MQAM, the information bits are encoded in both theamplitude and phase of the transmitted signal.

Thus, whereas both MPAM and MPSK have one degree of freedom in which to encode the information bits (amplitude orphase), MQAM has two degrees of freedom.

As a result, MQAM is more spectrally-efficient than MPAMand MPSK, in that it can encode the most number of bits per

symbol for a given average energy.




Passband modulation

General principles


Quadrature amplitude modulation (cont.)

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Q p ( )

The transmitted signal is given by() = Re

() 2

(56)

= cos()() cos(2 ) − sin()()sin(2 ).

where 0≤

≤

.

The energy in () is

=

0

2 () = 2 . (57)

that is the same as for MPAM.

The distance between any pair of symbols in the signalconstellation is

, = ∥s − s∥ =

(,1 − ,1)2 + (,2 − ,2)2. (58)




Passband modulation

General principles


Quadrature amplitude modulation (cont.)

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p ( )

For square signal constellations, where ,1 and ,2 takevalues on (2 − 1 − ), = 1, 2,..., = 2, the minimumdistance between signal points reduces to = 2, the

same as for MPAM.In fact, MQAM with square constellations of size 2 isequivalent to MPAM modulation with constellations of size on each of the in-phase and quadrature signal components.

Common square constellations are 4QAM and 16QAM, whichare shown in the below figure.




Passband modulation

General principles


Quadrature amplitude modulation: constellation and

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decision regions

4−QAM 16−QAM

Figure 10: 4QAM and 16QAMConstellations.

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

Z10

Z11

Z12

Z13

Z14

Z15

Z16

Figure 11: Decision Regions forMQAM withM = 16.




Passband modulation

General principles


Quadrature amplitude modulation: constellation and

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decision regions

These square constellations have = 22 = 2 constellationpoints, which are used to send 2 bits/symbol, or bits perdimension.

It can be shown that the average power of a square signalconstellation with bits per dimension, , is proportional to4/3, and it follows that the average power for one more bitper dimension +1 ≈ 4 .

Thus, for square constellations it takes approximately 6 dBmore power to send an additional 1 bit/dimension or 2bits/symbol while maintaining the same minimum distancebetween constellation points.


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section 3 1 digital mod tech v7

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