System Model - Advanced Modulation/Demodulation Schemes For Wireless Communications

real-isti ommuni ation systems.

Se ondly, we show that all the multi arrier modulation formats an be

derived from su h adis rete-time model, with ageneral prototype lter

(rather than with the standard re tangular lter) and a general time

and frequen y spa ing between the information symbols; inotherword,

we apply pulseshaping te hnique to all the s hemes, so extending the

DCT-OFDM,DST-OFDM andDTT-OFDMs hemesalreadypresentin

literature.

Finally,inspiredbytheWilsonbase[87,Se tion4.2℄,thatturnsouttobe

a leverwayto design well-lo alized orthogonalGaborframeswithhigh

time-frequen ye ien ythatavoidthelimitingfa torduetothe

Balian-Lowtheorem[87℄,[88℄,wederiveapra ti almulti arriermodulationwith

ex ellent performan e on doubly-sele tive hannels.

For all the above mentioned modulation s hemes, we rst assume a

re t-angular prototype pulse, as is usually done in the literature [74,81℄, then we

designasuitableprototypepulse,welllo alizedinboththetimeandfrequen y

domains, by meansof thete hnique proposedin[89℄ or in[90℄.

w(t)

S/P

y _−Q ^m

y _Q ^m

kT

MODULAT OR CHANNEL DEMODULAT OR

x(t) r(t)

y _−Q+1 ^m x i

u e _−Q+1 (t)

h(t, τ ) u e _−Q (t)

u e Q (t)

v e ^∗ _−Q (−t)

v e _Q ^∗ (−t) v e ^∗ _−Q+1 (−t)

Figure5.1: Ideal ontinuous-time lter-bank system.

transmitted symbols

{x i }

^are^zero-mean independent random variables (r.v.s) belonging to a given omplex onstellation. We dene

σ _x ² , E

|x n | ²

. The

transmitted signal, whi h is obtained bymodulating a set of ontinuous-time

lters

{e u _n (t)}

^,^reads

x(t) = X

m∈Z

X Q n=−Q

x ^m _n u e _n (t − mT ).

^(5.1)

Filters

{e u _n (t)}

^will^be^hosenêqual ^toâ^base^pulse ^(eitherâre tangular pulse oranappropriatelydesignedpulse,a ordingtotheapproa hdes ribedin[85℄)

withasuitablemultiplexing inthefrequen ydomain,asintheuniform

lter-bank approa h [91, Chapter 9℄. Hen e, index

n

îs ûsually ^referred ^to âs ^the

frequen y (or sub arrier) index. We will dene the spe tral e ien y

η

ex-pressed insymb/s/Hz, asthe amount of ode symbols that an be loaded on

atime-frequen y region hara terized byaunitary bandwidthandtimewidth.

All the onsidered modulation formats will be designed so that they a hieve

thesame spe trale ien y,inorder to arry out afair omparison.

Aftertransmissionoveranoisydoubly-sele tive hannel,there eivedsignal

r(t)

^is

r(t) = Z _+∞

−∞ h(t, τ ) x(t − τ) dτ + w(t)

^(5.2)

wherethezero-meanGaussianrandompro ess

h(t, τ )

^denotes^thetime-varying hannelimpulseresponse(CIR).Awide-sensestationaryun orrelated

s atter-ing(WSSUS) hannel is assumed,and

h(t, τ ) = 0

^if

τ < 0

^or

τ > τ max

^,^being

τ _max

^the ^maximal^ex
ess ^delay ^of^the^hannel.

τ _max

^givesân îndi
ation âbout

the hannel spreading in the time domain. A similar parameter, denoted as

ν max

^(maximal ^Doppler ^spread), ^quanties ^the ^hannel ^spreading ⁱⁿ ^the

fre-quen y domain [92℄.

w(t)

îs â ômplex ^white ^Gaussian ^pro
ess ^with ^power

spe traldensity

2N ₀

Atthe re eiver,thesignalispro essedbyabankof

2Q + 1

^lters.^The

r

^-th

sampleat the output ofthe

k

^-th ^lter

{ev _k ^∗ (−t)}

^is

y _k ^r , y k (rT ) =

Z _+∞

−∞ r(τ ) ev k ^∗ (τ − rT ) , k = −Q, . . . , Q .

^(5.3)

In order to obtain a pra ti al implementation modelfor the mentioned

lter-bank s heme, we resort to a dis rete-time model. In parti ular, ea h impulse

responselter

e u n (t)

^(and ^similarly^for

e v _k (t)

⁾ ^isapproximately representedby

e

u _n (t) ≃ X

i∈Z

e u _n (i) ⊓

t T _s − i

(5.4)

where

e u _n (i) , e u _n (iT _s )

T _s , T/N

⁽

N

^is^the oversamplingfa tor)and

⊓(t)

^is^a

re tangular pulse dened as

⊓ (α) ,

( 1

^if

0 ≤ α < 1 0

^otherwise

.

(5.5)

The dis rete-time representations of the lters, a ording to (5.4) , is

su- iently a urate provided that a large enough oversampling fa tor

N

^is

em-ployed. Under this assumption, an equivalent dis rete-time representation of

thetrans eiverisshowninFig.5.2, where thetwo squarelters

⊓ (·)

^play^the

role ofdigital-to-analog and analog-to-digital onverters, respe tively.

S/P

y _−Q ^m y _−Q+1 ^m MODULAT OR CHANNEL DEMODULAT OR

h(t, τ ) x(t)

y Q ^m

r(t) u e _−Q (i)

lT s

v e Q ^∗ (i)

⊓ ^{− t} T s

x i

w(t) u e _−Q+1 (i)

u e Q (i)

v e ^∗ _−Q+1 (i) v e ^∗ _−Q (i)

⊓ T ^t s

Figure5.2: Pra ti al oversampled dis rete-time lter-bank system.

The dis rete-time representation of thetransmittedsignal(5.1) is

x(t) = r 1

T _s X

m∈Z

X Q n=−Q

x ^m _n X

i

e u _n (i) ⊓

t − mT T _s − i

(5.6)

where the onstant

p 1/T s

îs ân ênergy normalization term. The

dis rete-time re eived samples an be obtained byusing (5.6) and (5.2) in (5.3),thus

obtaining

y _k ^r = r 1

T _s X

l∈Z

e v ^∗ _k (l)

Z _∞

−∞ r(t) ⊓

t − rT T _s − l

dt

= X

m∈Z

X Q n=−Q

H _k,n ^r,m x ^m _n + n ^r _k

^with

k = −Q, . . . , Q r ∈ N

(5.7)

where

H _k,n ^r,m , X

l∈Z

e

v ^∗ _k (l − rN) X

i∈Z

e

u _n (i − mN) h l,l−i

^(5.8)

with

h _l,i , 1 T _s

Z Z _∞

−∞

h(t + lT _s , τ ) ⊓

t − τ T _s + i

⊓

t T _s

dτ dt

^(5.9)

and

n ^r _k , r 1

T s

X

l∈Z

v e _k ^∗ (l − rN) Z _∞

−∞ w(t) ⊓

t T s − l

dt.

^(5.10)

CHANNEL

MODULATOR DEMODULATOR

x ^m _n s(i) p(l) y _k ^r

Figure 5.3: System model de omposed into three blo ks: the modulator, the

hanneland the demodulator.

From (5.9),thedis rete-time CIR

h _l,i

^has^supp^ort^over

0 ≤ i ≤ L

^,^where

L = ⌊τ max /T _s ⌋ + 1

and

⌊t⌋

^means^round

t

^down ^to ^the ^nearest^integer.

Finally,weprovideanalternative systemdes ription,aimingatthesystem

separation into three dierent blo ks: the transmitter, the hannel, and the

re eiver. Repla ing (5.8) in (5.7),with a few mathemati al manipulations we

obtain the following expressionfor thestatisti s

y ^r _k y _k ^r = X

l∈Z

e v _k ^∗ (l − rN)

 

 X

i∈Z



 X

m∈Z

X Q n=−Q

x ^m _n u e _n (i − mN)



 h _l,l−i

 

 + n ^r _k .

^(5.11)

Hen e we seethat our systemis omposed of thefollowing three stages,

rep-resentedinFig 5.3.

Modulator : omputes thedis rete-time signal

s(i) s(i) , X

m∈Z

X Q n=−Q

x ^m _n u e _n (i − mN)

^(5.12)

whi h is the sum of the output of a bank of lters whose input are

the information symbols

{x ^m n }

^.^The ^signal

s(i)

^is^then ^forwarded ^to ^the

hannel.

Channel : omputes the output dis rete-time signal

p(l)

^by ^the onvolution ofthe inputsignal

s(i)

^and^the ^hannel ^oe
ients

h _l,i

p(l) , X

i∈Z

s(i)h _l,l−i .

^(5.13)

From(5.9),the hannel oe ient

h _l,i

^is^given^by^the ontributionofthe time-varying hannel impulse response and the two re tangular lters

(thedigital-to-analog and analog-to-digital onverters).

Demodulator : derives the su ient statisti s

y ^r _k

^, ^by ^ltering ^the ^re
eived

signal

p(l)

^by^the ^bank ^of^lter

{v k (l)}

y _k ^r = X

l∈Z

e

v _k ^∗ (l − rN)p(l) + n ^r k .

^(5.14)

Obviously the thermal noise is generated by the hannel, but it is

on-venient for our purposetosee su hnoise asaGaussian randomvariable

n ^r _k

^introdu
ed ^by^thedemodulator.

Theabovegeneral modelisvalidfor allmulti arrier s hemes,whi harebased

howeveron dierentmodulatorand demodulator stages.Hen e, itwillbe

suf- ienttoprovideforthevariouss hemes,thedierentexpressionforthesignal

s(i)

ⁱⁿ^(5.12) ^and ^for ^the^statisti

y ^r _k

ⁱⁿ^(5.14).

5.2.1 Uniform Filter-Bank

As,alreadyanti ipated inSe tion 5.2,all multi arrier modulations hemeswe

will onsider, derivefromtheuniformlter-banks system(see[91℄, hapter9):

frequen yresponsesoflters

{e u _n (i)}

^and^lters

{ev n (i)}

^are^obtained^by^shifting

thefrequen y responseof thesame real lter

u(t)

^, ^denoted^as^prototype ^lter,

bya spa ing given by

F _n = nF

^,^where

F

^is ^the ^spa
ing ^between ^the ^v^arious

sub arriers.

Sin e

T

îs ^the ÔFDM ^symbol ^period ând

F

^the ^arrier separation, ea h oded symbols

x ^m _n

^an ^be ^asso
iated ^to ^the ^point

(T m , F n ) = (mT, nF )

^of ^a

bidimensional grid inthe time-frequen y plane [85℄.Hen e, dening

ρ , F T

theinverseof

ρ

ân^be^seenâsâ^measureôf^the^spe
trâl ê
ien
y

η

⁽ⁱⁿ^terms^of

datasymbolsperse ondsperHertz),sin ehigher

ρ ⁻¹

^values^lead^to^a^redu
ed

spa e-frequen y distan e between symbols (i.e., a redu ed distan e between

two adja ent gridpoints).

Nel documento Advanced Modulation/Demodulation Schemes For Wireless Communications (pagine 194-200)

System Model

w(t)

S/P

y −Q m

y Q m

kT

MODULAT OR CHANNEL DEMODULAT OR

x(t) r(t)

y −Q+1 m x i

u e −Q+1 (t)

h(t, τ ) u e −Q (t)

u e Q (t)

v e ∗ −Q (−t)

v e Q ∗ (−t) v e ∗ −Q+1 (−t)

{x i }

σ x 2 , E 

|x n | 2

{e u n (t)}

x(t) = X

m∈Z

X Q n=−Q

x m n u e n (t − mT ).

{e u n (t)}

n

η

r(t)

r(t) = Z +∞

−∞ h(t, τ ) x(t − τ) dτ + w(t)

h(t, τ )

h(t, τ ) = 0

τ < 0

τ > τ max

τ max

τ max

ν max

w(t)

2N 0

2Q + 1

r

k

{ev k ∗ (−t)}

y k r , y k (rT ) =

Z +∞

−∞ r(τ ) ev k ∗ (τ − rT ) , k = −Q, . . . , Q .

e u n (t)

e v k (t)

e

u n (t) ≃ X

i∈Z

e u n (i) ⊓

 t T s − i



e u n (i) , e u n (iT s )

T s , T/N

N

⊓(t)

⊓ (α) ,

( 1

0 ≤ α < 1 0

.

N

⊓ (·)

S/P

y −Q m y −Q+1 m MODULAT OR CHANNEL DEMODULAT OR

h(t, τ ) x(t)

y Q m

r(t) u e −Q (i)

lT s

v e Q ∗ (i)

⊓  − t T s

x i 

w(t) u e −Q+1 (i)

u e Q (i)

v e ∗ −Q+1 (i) v e ∗ −Q (i)

⊓  T t s



x(t) = r 1

T s X

m∈Z

X Q n=−Q

y _−Q ^m

y _Q ^m

y _−Q+1 ^m x i

u e _−Q+1 (t)

h(t, τ ) u e _−Q (t)

v e ^∗ _−Q (−t)

v e _Q ^∗ (−t) v e ^∗ _−Q+1 (−t)

σ _x ² , E

|x n | ²

{e u _n (t)}

x ^m _n u e _n (t − mT ).

{e u _n (t)}

r(t) = Z _+∞

τ _max

τ _max

2N ₀

{ev _k ^∗ (−t)}

y _k ^r , y k (rT ) =

Z _+∞

−∞ r(τ ) ev k ^∗ (τ − rT ) , k = −Q, . . . , Q .

e v _k (t)

u _n (t) ≃ X

e u _n (i) ⊓

t T _s − i

e u _n (i) , e u _n (iT _s )

T _s , T/N

y _−Q ^m y _−Q+1 ^m MODULAT OR CHANNEL DEMODULAT OR

y Q ^m

r(t) u e _−Q (i)

v e Q ^∗ (i)

⊓ ^{− t} T s

x i

w(t) u e _−Q+1 (i)

v e ^∗ _−Q+1 (i) v e ^∗ _−Q (i)

⊓ T ^t s

T _s X

x ^m _n X

e u _n (i) ⊓

t − mT T _s − i

y _k ^r = r 1

T _s X

e v ^∗ _k (l)

Z _∞

t − rT T _s − l

dt

H _k,n ^r,m x ^m _n + n ^r _k

H _k,n ^r,m , X

v ^∗ _k (l − rN) X

u _n (i − mN) h l,l−i

h _l,i , 1 T _s

Z Z _∞

h(t + lT _s , τ ) ⊓

t − τ T _s + i

t T _s

n ^r _k , r 1

v e _k ^∗ (l − rN) Z _∞

t T s − l

x ^m _n s(i) p(l) y _k ^r

h _l,i

L = ⌊τ max /T _s ⌋ + 1

y ^r _k y _k ^r = X

e v _k ^∗ (l − rN)

x ^m _n u e _n (i − mN)

 h _l,l−i

 + n ^r _k .