& advanced tx methods • equalization is required in inter-symbol interference (isi) channels...

S-72.3281 Advanced Transmission Methods

Vector Signal Model

Olav Tirkkonen

TKK, Department of Communications and Networking

Baseband Receiver

• Here, concentrate on equalization in signal space• after Rx pulse shaping, sampling and A/D conversion

Vector Signal Model 2

Equalization & Advanced Tx Methods

• Equalization is required in Inter-Symbol Interference (ISI)channels

• modern advanced transmission methods for ISI channels aredesigned to allow almost perfect equalization• Orthogonal Frequency Domain Multiplexing (OFDM)

• Single carrier FDMA with cyclic prefix

• To understand these, essential to understand equalization insignal space• as opposed to pre-sampling equalization based on frequency domain

filtering

• Also, to understand Rx-aspects of modern multiantenna (MIMO)techniques, essential to understand signal space equalization


Vector Signal Model I• There are Nr received signals, y is a Nr × 1 vector

• There are Nt transmitted symbols, x is a Nt × 1 vector• the symbols are independent, and normalized to have power 1:

E{xxH

}= I

• The identity matrix is denoted by I

• The noise n is additive white Gaussian, a Nr × 1 vector• noise covariance is E

{nnH

}= N0 I

• PDF of noise component : p(n) = 1πN0

e−|n|2/N0

• joint noise PDF: p(n) = 1(πN0)Nr e

−|n|2/N0 where |n|2 = nHn

• channel H is a Nr × Nt matrix, includes:• physical channel

• pulse shaping filtering at Tx and Rx

• Tx power level


Vector Signal Model, Example

• three received signals, two transmitted symbols

• Vector signal model⎡⎣ y1

y2

y3

⎤⎦ =

⎡⎣ h11 h12

h21 h22

h31 h32

⎤⎦[ x1

x2

]+

⎡⎣ n1

n2

n3

⎤⎦

consists of the three equations:

y1 = h11x1 + h12x2 + n1

y2 = h21x1 + h22x2 + n2

y3 = h31x1 + h32x2 + n3


Vector Signal Model II

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• The task of the receiver is to estimate the transmitted symbolsfrom this set of equations.

• Based on the same signal model receivers for a multitude ofcases may be understood• Multple input, multiple output channels (MIMO)

• multiuser detection• multiple transmitters, multiple receive antennas (“virtual MIMO”)

• CDMA multiuser detection

• equalization (ISI)


Signal Model for MIMO

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• signal model represents reception in one symbol period

• Nt transmit antennas

• Nr receive antennas

• hmn is channel between Tx antenna n and Rx antenna m

• Multiple symbols transmitted simultaneously:• symbol xn transmitted from Tx antenna n

• intentional non-orthogonality

• interference between simultaneously transmitted symbols


Multiuser Detection, multiple Rx Antennas

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• “virtual MIMO”

• signal model represents reception in one symbol period

• Nt users synchronously transmitting on the same channel

• Nr receive antennas

• hmn is channel between user n and Rx antenna m

• Multiple symbols received simultaneously: xn from user n• intentional non-orthogonality (e.g. CDMA)

• interference between users


Multiuser Detection, CDMA

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• signal model represents reception over multiple chip periods• Nr received chips, Nr = spreading factor

• one spread symbol transmitted per user

• Nt users synchronously transmitting on the same channel

• hmn is channel times spreading code of user n in received chip m

• Multiple symbols received simultaneously: xn from user n• intentional non-orthogonality

• interference between users


Channel Convolution and ISI I

• The received signal is a sum (convolution) of multiple transmittedsymbols


Channel Convolution and ISI II• sampled receive signal, sampling at times kT ′ + d

yk = y(kT ′ + Δ) =

−∞∑m=0

xm h(kT ′ − mT + d) + nr(kT ′ + d)

• sum of contribution from echoes of (in principle) all previous symbols

• convolved with channel impulse response h(t) (including pulse chaping)

• Symbol spaced sampling: One sample per transmitted symbol

yk = hkkxk +∑m �=k

hkm xm︸︷︷︸ISI

+ nk

• if hkm �= hδkm, there is Inter-Symbol-Interference (ISI)

• nk is AWGN


Channel Convolution and ISI III• Tapped Delay Line (TDL):

• channel taps are constant hkm = hk−m

• causality realized as hm = 0 for m < 0

• there are L non-zero channel taps {hm}L−1m=0:

• Signal model becomes: yk = h0 xk +∑L−1

m=1 hm xk−m + nk


Signal Model for Equalization: Finite Block

• For finite block of Nr Rx symbols, vector signal model can be used

• the channel matrix is a Toeplitz matrix.

• Example: block of Nr = 5 Rx symbols, L = 3 channel taps

⎡⎢⎢⎢⎢⎣

y1

y2

y3

y4

y5

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

h2 h1 h0 0 0 0 00 h2 h1 h0 0 0 00 0 h2 h1 h0 0 00 0 0 h2 h1 h0 00 0 0 0 h2 h1 h0

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

x−1

x0

x1

x2

x3

x4

x5

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎣

n1

n2

n3

n4

n5

⎤⎥⎥⎥⎥⎦

• E.g.: y5 = h0x5 + h1x4 + h2x3︸︷︷︸ISI

+n5


Signal Model for Equalization

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• signal model represents reception for sequence of symbol periods

• Nt symbols in a transmission block that is equalized

• Nr samples at the receiver

• hmn is channel between transmitted symbol n and receivedsample m

• Multiple symbols received simultaneously• inter-symbol interference

• non-orthogonality caused by the channel (and possibly Rx/Tx filters)


S-72.3281 Advanced Transmission Methods

Linear Receivers for Interference Channels

Olav Tirkkonen

TKK, Department of Communications and Networking

Receiver in Vector Signal Space

• Estimate transmitted signal x from received samples y usingsignal model

yNr×1

= HNr×Nt

xNt×1

+ nNr×1

• H is assumed to be known (by estimation).

• Linear receivers• use linear algebra to construct symbol estimates xm

• bit decisions (soft or hard) made by quantizing xm

• non-linear receivers• use discreteness of x to construct the symbol estimates

• iterative receivers: iterated (linear algebra + decisions)

• (approximative) Maximum Likelihood Sequence Estimators

Linear Receivers for Interference Channels 2

MAP Receiver• Optimum non-linear receivers consider all possible transmitted

symbol vectors• M alternatives for x: {xm}M

m=1

• Maximum A Posteriori (MAP) receiver• optimum receiver

• finds most probable transmitted symbol vector for given received signalvector:

x = arg maxm

P (xm|y)

• “A posteriori” = afterwards


ML receiver• Maximum Likelihood (ML) receiver

• finds most likely transmitted symbols xm

• likelihood function = probability density of received signal given transmittedsignal

• transmitted symbol vector for which the likelihood function is highest, giventhe transmitted symbol vector:

x = arg maxm

p(y|xm)

• Relation of ML and MAP• Bayes’ rule: P (xm|y) = p(y|xm)P (xm)

p(y)

• If all signals equiprobable: P (xm) = 1/M

arg maxm

P (xm|y) = arg maxm

p(y|xm)

ML and MAP are equivalent


Vector Likelihood Function• Vector signal model:

y = Hx + n

• Gaussian noise:

p(n) =1

(πN0)Nre−|n|2/N0

• Received signal conditioned on channel, transmitted signal andnoise:

p(y|H,x,n) = δ2Nr (y − Hx − n)

• completely determined by vector signal model

• Likelihood function:

p(y|H,x) =

∫CNr

dn p(y|H,x,n) p(n) =1

(πN0)Nre−|y−Hx|2/N0


ML Detection Metric• ML detection metric:

• logarithm of likelihood function:

MML = |y − Hx |2

• ML decision:arg min

mMML(xm)

• exhaustive search over all possible xm

• often prohibitively complex

• e.g. decoding a vector of four 64-QAM symbols:• there are 644 = 224 ≈ 16 ∗ 106 possible transmitted vectors

• ML (or MAP) searches over all of these

• linear receivers are a viable, often the only alternative


Linear Receivers• Set of linear equations y = Hx + n

1. Solve esimate x from this set using linear algebra

2. decide symbol based on x

• Symbol estimate in two decoupled steps:1. continuous symbol estimate: apply linear filter (matrix) F to y

xL = Fy

2. discrete symbol estimate: decide transmitted constellation points x from xL

• Denote rows of filter by fHk , i.e. FH = [f1 f2 . . . fNt]

• Decision metric decouples

MF = |x − Fy|2 =∑

n

|xn − fHn y|2

• xn is the constellation point closest to fHn y


When is a Linear Receiver Optimal?

1. If H singular, x cannot be solved from y = Hx + n even if nknown⇒ linear filter may be optimal only if H non-singular

2. A receiver is optimal if decision metric is equivalent to ML metric:

aMF = MML

• proportionality constant a may depend on H,y but not on x

⇒ a |x − Fy|2 =∣∣H (

x − H−1y)∣∣2

⇒ a (x − Fy)H (x − Fy) =(x − H−1y

)HHHH

(x − H−1y

)• should hold for all possible x,y,H

⇒ HHH = aI and F = H−1

• Thus linear receiver is the optimum receiver only ifthe channel is orthogonal, H proportional to a unitary matrix.• Orthogonal signaling

• if channel considered in correct coordinates it is visible that there is no ISI


White Noise Approximated Matched Filter

• A Matched Filter based on a White Noise approximation (WN-MF)is one of the simplest linear receivers

• Combine coherently samples relating to symbol of interest

F =(diag

(HHH

))−1HH

fn =(hH

nhn

)−1hn

• hn is n:th column of H

• The inverses are just a scaling of the decision surfaces

• Example: 2 × 2 channel[y1

y2

]=

[h11 h12

h21 h22

] [x1

x2

]+

[n1

n2

]• WN-MF estimate is

x1 =h∗

11y1 + h∗21y2(

|h11|2 + |h21|2) = x1 +

(h∗11h12 + h∗

21h22) x2(|h11|2 + |h21|2

) + filtered noise


Maximum Ratio Combining• A true Matched Filter is a Maximum Ratio Combining (MRC)

receiver• reliability scaling according to noise + interference power

• Definition of Maximum Ratio Combining:• diversity branches combined coherently

• each received signal that carries information about a given transmitted symbol is called adiversity branch

• combining weights selected to maxmize post-combining SINR

• assuming noise + interference that corrupts branches uncorrelated

• Optimum MRC weights will be solved below

• Usually in text books, white vs. coloured noise aspects ofMatched Filter receivers are not stressed• see Benedetto-Biglieri, exercise 2.26, p. 102


MRC Weight Derivation I (EC)• EC = Extra Curricular, not demanded for exam

• Rewrite signal model for receiving xk

y = hk xk +∑j �=k

hj xj + n

︸︷︷︸≡i

• the covariance of noise + interference is

E{i iH}

=∑j �=k

hj hHj + N0I

• for MRC i is approximated as uncorrelated interference i:

E{im i∗l

}=

⎛⎝∑

j �=k

|hmj|2 + N0

⎞⎠ δml


MRC Weight Derivation II (EC)• The filter is

fk = Ak hk

where Ak = diag[ak1 ak2 . . . akNt] is a diagonal matrix of realreliability weights

• signal power after filtering is

Sk =∣∣hH

k Akhk

∣∣2 =

(∑m

akm |hmk|2)2

• the (Approximative) noise plus interference power is

Ik = E{|hH

k Aki|2}

= hHk Ak E

{i iH}

Akhk

=∑m

a2km|hmk|2

⎛⎝∑

j �=k

|hmj|2 + N0

⎞⎠


MRC Weight Derivation III (EC)• The set of reliability weights may be scaled with any number

without changing SINR• (Extracurricular) Derivation of optimum weights

• Choose scale so that signal power Sk = μ2

• minimize interference + noise power subject to constraint∑m akm |hmk|2 = μ

• Lagrangian optimization (λ is a Lagrange multiplier):

L = Ik + 2λ

(μ −

∑m

akm |hmk|2)

• Find minima of L

dLdakm

= 2akm|hmk|2⎛⎝∑

j �=k

|hmj|2 + N0

⎞⎠− 2λ|hmk|2 = 0

⇒ akm =λ∑

j �=k |hmj|2 + N0


MRC Weights, Summary• MRC estimate of xk is

xk = fHk y

• the m:th diversity branch contributing to xk is

ym = hmkxk +∑j �=k

hmjxj + nm︸︷︷︸interference+noise

• the combining weight of the m:th branch is

fmk =λ hmk∑

j �=k |hmj|2 + N0

• divided by interference + noise power per branch, assuming non-correlated

• λ is chosen so that we get an unscaled estimate:

λ = 1/∑m

|hmk|2∑j �=k |hmj|2 + N0


MRC Example

• 2 × 2 WN-MF example above

• diversity branches

y1 = h11 x1 + h12 x2 + n1

y2 = h21 x1 + h22 x2 + n2

• WN-MF• coherent combining with weights λh11 and λh21

• SIR for symbol x1 is (omitting N0)

SIR1 =

∣∣∣∣∣ |h11|2 + |h21|2h∗

11h12 + h∗21h22

∣∣∣∣∣2

• this is not MRC optimum


MRC Example II

• when receiving x1, interference powers in y1 and y2 are

E{i1 i∗1

}= |h12|2 + N0

E{i2 i∗2

}= |h22|2 + N0

• MRC reliability weights are

a11 = λ

a12 = λ|h12|2 + N0

|h22|2 + N0≡ λa

• MRC coherent combining weights λh11 and λa h21:

SIR1 =

∣∣∣∣∣ |h11|2 + a |h21|2h∗

11h12 + a h∗21h22

∣∣∣∣∣2


MRC Example III

• for example if channel is near orthogonal

h11 = h22 = 10, h12 = h21 = 1

• plain WN-MF gives

SIRMF1 =

(101

20

)2

≈ 25

• MRC gives

SIRMRC1 =

(100 + a

10 + 10a

)2

=

(100.01

10.1

)2

≈ 100

• NOTE: MRC does not maximize SINR• MRC: best SINR assuming uncorrelated noise + interference


Zero Forcing• also known as de-correlating receiver

• Why not solve x directly form y = Hx + n, forgetting the noise?

x = H−1y

• If H is singular, use Moore-Penrose pseudo-inverse:

x =(HHH

)−1HH y

• note: if H non-singular, we have(HHH

)−1HH = H−1

(HH)−1

HH = H−1

• ZF symbol estimate

x = x + coloured noise

• all ISI has been forced to zero

• noise is coloured (if channel not orthogonal)


Minimum Mean Square Error Estimator

• The Linear Minimum Mean Square Error (MMSE) Estimatorsuppresses interference, but takes existence of the noise intoaccount

• MMSE indeed minimizes mean square error of the symbolestimate

• MMSE is ZF regularized by noise term:

FMMSE = HH(HHH + N0I

)−1=(HHH + N0I

)−1HH

• For small N0, MMSE becomes ZF

• For large N0, MMSE becomes WN-MF (up to scaling)

• ZF and MMSE are ML if channel is orthogonal


MMSE Derivation I (EC)

• To understand Minimum Mean Square Error (MMSE) Estimation,some covariance matrices needed

E{yxH

}= E

{(Hx + n)xH

}= H E

{xxH

}= H

E{yyH

}= E

{(Hx + n)

(xHHH + nH

)}= H E

{xxH

}HH + E

{nnH

}= HHH + N0I

• The Mean Square Error

E = E{|x − Fy|2} = Tr E

{(x − Fy)

(xH − yHFH

)}= Tr

[E{xxH

}− 2Re[FE

{yxH

}]+ F E

{yyH

}FH]

= Tr[I − 2Re [FH] + F

(HHH + N0I

)FH]


MMSE Derivation II (EC)

• Find extrema of

E = Tr[I − 2Re [FH] + F

(HHH + N0I

)FH]

• differentiate w.r.t. the elements of F:dEdF

= −2HH + 2F(HHH + N0I

)• Solve for extremum filter matrix:

FMMSE = HH(HHH + N0I

)−1

• this filter minimizes the MSE


MMSE Derivation III (EC)

• To see difference of MMSE and ZF, another form of filter needed

• use matrix inversion lemma

V(A−1 + VVH

)−1=(I + VHAV

)−1VHA

• MMSE for coloured nose• noise covariance E

{nnH

}= C

• MMSE filter becomes

FMMSE = HH(HHH + C

)−1=(HHC−1H + I

)−1HHC−1


Linear Receivers, Summary• White-Noise approximated Matched Filter (WN-MF)

• treats each symbol separately

• for each symbol, there is a number of diversity branches• diversity branches combined coherently

• weighted by channel power

• Maximum Ratio Combining (MRC)• treats each symbol separately• diversity branches combined coherently

• weighted by channel power divided by expected interference + noise power

• assuming that noise and interference non-correlated

• Zero Forcing (ZF)• treats symbols jointly• solves symbols from signal model forgetting the noise

• inverts the channel, i.e. forces interference to zero

• Minimum Mean Square Error Estimator (MMSE)• treats symbols jointly

• solves symbols from signal model taking noise power into account


SINR analysis of LinearReceivers


SINR for Linear Receiver• For performance analysis, post-processing SINR after linear

receiver may be calculated• signal power corrupted by possible residual ISI, and possibly coloured noise

• analysis valid for any linear receiver: FH = [f1 f2 . . . fNt]

• filter output for symbol k is

zk = fHk H x + fH

k n

= fHk hk xk︸︷︷︸

wanted signal

+∑j �=k

fHk hj xj + fH

k n

︸︷︷︸noise and interference

• Channel matrix is H =[h1 h2 . . . hNt

]• symbol estimate is xk = zk/f

Hk hk


SINR for Linear Receiver II• assuming Gaussian interference and noise, performance is

characterized by the ratio of average signal power and averagenoise + interference power

• signal power after filtering is

Sk =∣∣fH

k hk

∣∣2• the noise plus interference power is

Ik =∑j �=k

∣∣fHk hj

∣∣2 + N0 fHk fk

• post-processing SINR is

SINRk =Sk

Ik


MF, ZF, MMSE & Channel Covariance

• WN-MF, ZF and MMSE can be written in the form: F = LHH

• LH =[l1 l2 . . . lNt

]is a channel inversion matrix

• filtered signal isz = LRx + LHHn ,

• the channel covariance matrix is

R = HHH

• diagonal elements: coherently combined (WN-MF) channels of symbol xk

rkk =(HHH

)kk

= hHk hk

• off-diagonal elements: ISI between xk and xj after MF

rkj =(HHH

)kj

= hHk hj


Signal and Interference Power: MF, ZF, MMSE

• Everything can be understood in terms of inversion matrix andchannel covariance

• Concentrate on symbol xk

• signal power after filtering

Sk =∣∣lHk HHhk

∣∣2 = |(LR)kk|2

• noise plus interference power is

Ik =∑j �=k

∣∣lHk HHhj

∣∣2 + N0 lHk R lk

=∑j �=k

∣∣∣(LR)kj

∣∣∣2 + N0

(LRLH

)kk


Post-processing SINR

• Denote G = LR

• post-processing SINR:

SINRk =|gkk|2∑

j �=k |gkj|2 + N0 (LRLH)kk

• first term in denominator comes from residualpost-processing self-interference

• second term is possibly enhanced and coloured noise


SINR in Orthogonal Channel

• H proportional to a unitary matrix

• channel covariance proportional to identity, R = rI• rkk = r is the gain of the MRC combined channels

• optimum inversion matrix proportional to identity, L = l I

• SINR becomes

SINRk =(lr)2

N0l2 r=

r

N0,

• no residual self-interference

• no noise enhancement


SINR for (WN-)Matched Filter

• inversion matrix inverts just the coherently combined powers.• linear scaling, no effect on SINR, omitted here:

L = I

• SINR becomes

SINRk =r2kk∑

j �=k |rkj|2 + N0rkk

=rkk∑

j �=k |rkj|2 /rkk + N0

• no self-interference is suppressed

• noise is not enhanced


SINR for Zero Forcing

• inversion matrix inverts the channel covariance,

L = R−1

• G = I

• SINR becomes

SINRk =1

N0 (R−1)kk

• self-interference vanishes completely

• noise is enhanced

• for an orthogonal channel we reproduce the result above


Noise Enhancment by ZF, Example I

• Example: 2 × 2 channel[y1

y2

]=

[h11 h12

h21 h22

] [x1

x2

]+

[n1

n2

]• The channel covariance matrix is

R =

[ |h11|2 + |h21|2 h∗11h12 + h∗

21h22

h∗12h11 + h∗

22h21 |h12|2 + |h22|2]

=

[r11 r12

r∗12 r22

]


Noise Enhancment by ZF, Example II

• The determinant of R is

detR = r11r22 − |r12|2

• the inverse of R is

R−1 =1

detR

[r22 r∗12

r12 r11

]• the ZF SINR for symbol x1 is

SINR1 =1

N0 (R−1)kk

=detR

N0 r22=

1

N0

(r11 − |r12|2

r22

)≤ r11

N0

• equality only if r12 = 0, i.e. orthogonal channel

• comparing to contribution of N0 to SINR for MF, noise is enhanced by ZFreceiver


SINR for MMSE• inversion matrix inverts channel covariance + noise term,

L = (R + N0I)−1

• expression for SINR non-transparent• both noise enhancement and some residual ISI

• In limit N0 → 0, Zero Forcing result reproduced• In limit N0 → ∞, we have

L → 1

N0I

• in expression for SINR, 1/N0 factors cancel• MF result reproduced

• For orthogonal channel, we have

L = (r + N0)−1I

G =r

r + N0I

• result for MF and ZF reproduced


Performance Analysis of Linear Filters

• With the SINR values calculated above, performance of a detectorcan be analyzed• up to accuracy of approximation interference + noise as Gaussian

• For example, if QPSK is used, the BER of a symbol with SINRk

isBERk ≈ Q

(√SINRk

)where

Q(x) = 12 erfc

(x/

√2)

• The average performance can be estimated by averaging theBERs


S-72.3281Advanced Transmission Methods

Part3: Linear Receiver PerformanceExample:Multiuser Detection for UL CDMA

Olav TirkkonenTKK, Department of Communications and Networking

2

UL CDMA

F There are U users simultaneously transmittingF each user is using a spreading code cu of length SF “chips”

I SF is the spreading factorI The spreading code is interpreted as a column vector

cu = [c1u c2u . . . cSF,u]T

I elements of spreading code have norm 1I usually cju {1, 1} or cju {1, 1, j , j }

F the user is spreading the transmission of each symbol xu over SF chipsI example: SF = 4, one-tap channel hu, transmitted symbol xuI received signal from the transmission of user u:

yu = hu

c1uc2uc3uc4u

xu + n

F There are U users simultaneously transmittingF each user is using a spreading code cu of length SF “chips”

I SF is the spreading factorI The spreading code is interpreted as a column vector

cu = [c1u c2u . . . cSF,u]T

I elements of spreading code have norm 1I usually cju {1, 1} or cju {1, 1, j , j }

F the user is spreading the transmission of each symbol xu over SF chipsI example: SF = 4, one-tap channel hu, transmitted symbol xuI received signal from the transmission of user u:

yu = hu

c1uc2uc3uc4u

xu + n

3

UL CDMA, Power control

F Power Control (PC) is required in CDMA UL due tonear-far e ectI if no PC, signal from a user close to base stationdrowns signal of a far-away user belowdynamic range of A/D converter

F Fast PCI attempts to follow fast fadingI instantaneous received signal power of di erentusers equal

F Slow PCI attempts to follow slow fadingI mitigate shadowing and path lossI average received power of di erent users equal

F Power Control (PC) is required in CDMA UL due tonear-far e ectI if no PC, signal from a user close to base stationdrowns signal of a far-away user belowdynamic range of A/D converter

F Fast PCI attempts to follow fast fadingI instantaneous received signal power of di erentusers equal

F Slow PCI attempts to follow slow fadingI mitigate shadowing and path lossI average received power of di erent users equal

4

UL CDMA, simplification for MUD analysis

F in WCDMA, UL is asynchronousI timing of spreading codes of di erent usersis not synchronized

I new symbol starts in di erent chip for di erent usersF the spreading sequences of di erent usersare not orthogonalI spreading codes are pseudo-random sequencesI good cross-correlation and auto-correlationproperties

F to simplify analysis of e ect of inter-userinterference on UL CDMA with and withoutMultiuser Detection (MUD), we assumesynchronous UL with non-orthogonal spreading codes

F in WCDMA, UL is asynchronousI timing of spreading codes of di erent usersis not synchronized

I new symbol starts in di erent chip for di erent usersF the spreading sequences of di erent usersare not orthogonalI spreading codes are pseudo-random sequencesI good cross-correlation and auto-correlationproperties

F to simplify analysis of e ect of inter-userinterference on UL CDMA with and withoutMultiuser Detection (MUD), we assumesynchronous UL with non-orthogonal spreading codes

5

UL CDMA MUD, signal modelF signal model

y =

h1 0 · · · 00 h2 · · · 0

. . .

0 0 · · · hU

£c1 c2 · · · cU

¤

| {z }H

x1x2...xU

+ n

F elements of covariance matrix ruv = huhv cHu cv

I interference between users u and v if spreading codesnot orthogonal

F when elements of spreading code normalized to 1:

cHu cu = SF

F coherently combined channel gain for user u is ruu = SF |hu|2

F signal model

y =

h1 0 · · · 00 h2 · · · 0

. . .

0 0 · · · hU

£c1 c2 · · · cU

¤

| {z{ }}H

x1x2...xU

+ n

F elements of covariance matrix ruv = huhv cHu cv

I interference between users u and v if spreading codesnot orthogonal

F when elements of spreading code normalized to 1:

cHu cu = SF

F coherently combined channel gain for user u is ruu = SF |hu|2

6

CDMA SINR for Matched Filter receiver

F MRC (=WN-MF) for CDMA is the well-known RAKE receiverF Matched filter SINR for user u is

SINRu =ruuP

v 6=u |rvu|2 /ruu +N0=

SF |hu|2Pv 6=u |hv |2 |cHv cu|2 /SF +N0

I we see the processing gain = SF against noise and interference¨ wanted signal combines coherently¨ noise and interference non-coherently

F for random sequences E©cHu cv

ª= SF for u 6= v

I can be used to approximate SINR when many interferers

F with perfect PC

SINRu =SF

U 1 + N0/|hu|2I with increasing load, SINR decreasesI with “full load”, U = SF , SINR 0 dB

F MRC (=WN-MF) for CDMA is the well-known RAKE receiverF Matched filter SINR for user u is

SINRu =ruuP

v 6=66 u |rvu|2 /ruu +N0=

SF |hu|2Pv 6=66 u |hv |2 |cHv cu|2 /SF +N0

I we see the processing gain = SF against noise and interference¨ wanted signal combines coherently¨ noise and interference non-coherently

F for random sequences E©cHu cv

ª= SF for u 6=66 v

I can be used to approximate SINR when many interferers

F with perfect PC

SINRu =SF

U 1 + N0/|hu|2I with increasing load, SINR decreasesI with “full load”, U = SF , SINR 0 dB

7

UL CDMA MUD Performance Plots

F next pages: performance plotsI synchronous CDMA, SF = 16

¨ processing gain 10 log10(16) = 12.04 dB

I random complex spreading codesI di erent number of users from U = 1 to U = 16I MF, ZF and MMSE receiversI slow and fast PC

F next pages: performance plotsI synchronous CDMA, SF = 16

¨ processing gain 10 log10(16) = 12.04 dB

I random complex spreading codesI di erent number of users from U = 1 to U = 16I MF, ZF and MMSE receiversI slow and fast PC

8

Typical plot

Number of usersMF is interference

limited

At high SNR, ZF approaches MMSE

At low SNR, MF approaches

MMSE, ZF worse than MF

Average received SNR of the users

Average BER of the users

9

One & two users, slow PC

Single user: orthogonal system. All detectors same performance

Adding a user makes MF interference limited. practically no effect on

ZF& MMSE

10

Half and full load (8 users, 16users), slow PC

Half load: MF error floor rises,

small effect on ZF & MMSE, ZF at low SNR suboptimal

Full load: MF error floor rises, ZF very suboptimal,

4-6 dB loss to MMSE

11

One & two users, fast PC

Single user: orthogonal system. Fast PC

removed fading: AWGN performance in Rx

SNR. Processing gain compared to AWGN

~12 dB

Adding a user makes MF interference limited. practically no effect on

ZF& MMSE

12

Half and full load (8 users, 16users), fast PC

Half load: MF error floor rises,

small effect on MMSE, ZF > 1dB suboptimal

Full load: MF error floor rises,

ZF 5-6 dB suboptimal

13

Observations from UL CDMA MUD

ZF performs worst when system is most interference limited

this is a consequence of noise enhancementthe more interference to cancel, the more noise enhancement

when load is nearly full, channel covariance R has small eigenvalues

these lead to noise enhancementregulating with N0 I in MMSE makes R better conditioned

less noise enhancement

UL CDMA is desinged to operate with high load, close to ”pole capacity”at high load, ZF performs badlyfor MMSE, accurate estimate of N0, and R, required


Part4: Transversal Filters for ISI Channels


2

Toeplitz Matrix for ISI Channel

F ISI channel:

yk = h0 xk +

L 1Xm=1

hm xk m + nk

F vector formy = Hx+ n

F with Toeplitz channel, example

y1y2y3y4y5

=


x 1

x0x1x2x3x4x5

+

n1n2n3n4n5

F ISI channel:

yk = h0 xk +

L 1Xm=1

hm xk m + nk

F vector formy = Hx+ n

F with Toeplitz channel, example

y1y2y3y4y5

=


x 1

x0x1x2x3x4x5

+

n1n2n3n4n5

3

Linear Equalizers for ISI: Transversal Filter

F construct a transversal filterI a set of taps operatng on a number of consecutive symbols,giving a symbol estimate¨ fT is 1×Nr vector, fHTy = xD

I the estimated symbol may be any of the symbolsappearing in the received signals, given by delay D¨ in example above, it may be any of the x 1, . . . , x5¨ the equlizer performance is di erent for di erent delay D¨ the best delay may be sought for (Krauss & al.)¨ often the middle symbol is taken, x2 in example above

F construct a transversal filterI a set of taps operatng on a number of consecutive symbols,giving a symbol estimate¨ fTff is 1×NrNN vector, fHTff y = xD

I the estimated symbol may be any of the symbolsappearing in the received signals, given by delay D¨ in example above, it may be any of the x 1, . . . , x5¨ the equlizer performance is di erent for di erent delay D¨ the best delay may be sought for (Krauss & al.)¨ often the middle symbol is taken, x2 in example above

4

Transversal Filter II

z-1: symbol delay operationc: transversal filter tap coefficients

5

Transversal Filter III

F problem: H is Nr × (Nr + L 1)I more symbols in channel model than samplesI y = Hx + n is underdetermined

I channel covariance R is singularI may be cured by oversampling (or multiple Rx antennas)I alternatively this may be simply overlooked

¨ with MMSE, the matrix to be inverted is non-singular¨ poor performance for symbol at both ends¨ performance of symbols in the middle is almost optimum¨ with MRC, this is not a problem

F Example:I 4-tap channel, taps [0.2, 0.9, 0.8, 0.2]I SNR=10dB

I 7-tap transversal filter, 7 samples, 7+4-1=10 symbols in filterI MMSE SINR for the symbols are:

[ 12.72, 0.35, 1.64, 3.44, 3.95, 3.98, 3.87, 3.47, 3.42, 12.72]dB

I MRC SINR is -0.11 dB

F problem: H is NrNN × (NrNN + L 1)I more symbols in channel model than samplesI y = Hx + n is underdetermined

I channel covariance R is singularI may be cured by oversampling (or multiple Rx antennas)I alternatively this may be simply overlooked

¨ with MMSE, the matrix to be inverted is non-singular¨ poor performance for symbol at both ends¨ performance of symbols in the middle is almost optimum¨ with MRC, this is not a problem

F Example:I 4-tap channel, taps [0.2, 0.9, 0.8, 0.2]I SNR=10dB

I 7-tap transversal filter, 7 samples, 7+4-1=10 symbols in filterI MMSE SINR for the symbols are:

[ 12.72, 0.35, 1.64, 3.44, 3.95, 3.98, 3.87, 3.47, 3.42, 12.72]dB

I MRC SINR is -0.11 dB

6

Constructing Transversal Filter

F construct MMSE filter for vector channel modelusing Toeplitz matrix

F either take filter column corresponding tocentral symbol D = bNt/2c,

F or optimize delay D

fT =³H¡HHH+N0I

¢ 1´D

F construct MMSE filter for vector channel modelusing Toeplitz matrix

F either take filter column corresponding tocentral symbol D = bNtNN /2c,

F or optimize delay D

fTff =³H¡HHH+N0I

¢ 1´D

7

ISI equalization & Transversal Filter performance

3-tap channeltap amplitudes [0.41, 0.82, 0.41]

sum of tap powers normalized to 1random phases for each tapthis set of taps includes some very difficult channels to equalize

the previous and next symbols may conspire to remove the center tap altogether, 0.41 x0 + 0.41 x2 may be – 0.82 x1

4-tap channel tap amplitudes [0.15, 0.75, 0.6, 0.23]

BER for QPSK modulation calculatedaveraged over 1000 realizations of the tap phasesusing Gaussian approximation and correpsonding post-processing SINR to estimate transversal filter performance

8

MRC vs WN-MF, 3-tap channel

true MRC several dBs better at high SNRdifference in BER not too significant

both suffer from error floor due to non-canceled interference

9

MMSE Transversal Filter, different lengths, 3-tap channel

performance improvement almost saturates at filter length 7 taps

10

MMSE filter length, 4-tap channel

performance improvement almost saturates at 9 tapsusually 2L+1 taps is sufficient filter length

11

Performance of Different Transversal Filters

Same general characteristics as in UL CDMA example:

ZF, MMSE, MRC

MFB: Matched Filter Bound, performance if all ISI were

removed Performance gap between linear equalizer and no ISI

12

Summary: Transversal Filters

A transversal filter estimates one symbol from Nrconsecutive samplesFilter taps solved in time domain

e.g. one column of MMSE filter matrix2L+1 taps typically enough

L taps to collect the energy of the symbol of interestremaining taps to have sufficient independent samples to subtract interference

the delay of the filterwhich symbol is estimated from the set of samplesoptimum delay depends on the Power Delay Profilesimple solution: take middle symbol

In WCDMA terminals, DL receivers based on chip-level transversal filtering in time domain are going to be used


Part5: Frequency Domain Equalization, Block Transmission, and Cyclic Prefix;OFDM and Single-carrier FDMA


2

Discrete Fourier Transform (DFT)

F DFT transforms a sequence of N complex numbers {tk}Nk=1 to anothersequence of complex numbers {fk}Nk=1 by the equations

fk =1

N

NXn=1

tne2 j (k 1)(n 1)/N

F Inverse DFT (IDFT) transforms inverts this dependence:

tn =1

N

NXk=1

fke2 j (k 1)(n 1)/N

F DFT transforms a set of time domain samples into the frequency domainF IDFT transforms a set of frequency domain samples into the time domain

F DFT transforms a sequence of N complex numbers {tk}Nk=1 to anothersequence of complex numbers {fk}Nk=1 by the equations

fk =1

N

NXn=1

tne2 j (k 1)(n 1)/N

F Inverse DFT (IDFT) transforms inverts this dependence:

tn =1

N

NXk=1

fke2 j (k 1)(n 1)/N

F DFT transforms a set of time domain samples into the frequency domainF IDFT transforms a set of frequency domain samples into the time domain

3

Discrete Fourier Transform, Matrix Representation

F the DFT can be expressed in matrix formF define an N × N DFT matrix M with elements:

mkn =1

Ne 2 j (k 1)(n 1)/N

I the DFT is fk =PN

n=1 tnmkn

I the IDFT is tn =PN

k=1 tnmkn

F construct a vector of the time domain sequence: t =£t1 t2 . . . tN

¤TF construct a vector of the frequency domain sequence f =

£f1 f2 . . . fN

¤TF the DFT can be expressed as

f =Mt

F the inverse DFT can be expressed as

t =MHf

F the DFT matrix is unitary: MHM = I

F the DFT can be expressed in matrix formF define an N × N DFT matrix M with elements:

mkn =1

Ne 2 j (k 1)(n 1)/N

I the DFT is fkff =PN

n=1 tnmkn

I the IDFT is tn =PN

k=1 tnmkn

F construct a vector of the time domain sequence: t =£t1 t2 . . . tN

¤TF construct a vector of the frequency domain sequence f =

£f1 f2ff . . . fNff

¤TF the DFT can be expressed as

f =Mt

F the inverse DFT can be expressed as

t =MHf

F the DFT matrix is unitary: MHM = I

4

DFT Matrix Examples

F for N = 2, 4, 8 we have

M2 =1

2

1 11 1

¸M4 =

1

2

1 1 1 11 j 1 j1 1 1 11 j 1 j

M8 =1

2 2

1 1 1 1 1 1 1 11 j 3 1 5 j 7

1 j 1 j 1 j 1 j1 3 j 1 7 j 5

1 1 1 1 1 1 1 11 5 j 7 1 j 3

1 j 1 j 1 j 1 j1 7 j 5 1 3 j

where = e j /4

F in a DFT of size N, in principle N 2 multiplications have to be doneF Fast Fourier Transform (FFT) is an implementation of DFT of certainsizes where redundant calculations are removedI for N power of 2, FFT can be performed with N log2N multiplications

F for N = 2, 4, 8 we have

M2 =1

2

1 11 1

¸M4 =

1

2

1 1 1 11 j 1 j1 1 1 11 j 1 j

M8 =1

2 2

1 1 1 1 1 1 1 11 j 3 1 5 j 7

1 j 1 j 1 j 1 j1 3 j 1 7 j 5

1 1 1 1 1 1 1 11 5 j 7 1 j 3

1 j 1 j 1 j 1 j1 7 j 5 1 3 j

where = e j /4

F in a DFT of size N, in principle N 2 multiplications have to be doneF Fast Fourier Transform (FFT) is an implementation of DFT of certainsizes where redundant calculations are removedI for N power of 2, FFT can be performed with N log2N multiplications

5

Circulant matrices, DiagonalizationF columns of circulant matrix C are rotated versions of the first column c1:

C =

a d c bb a d cc b a dd c b a

F the product of two circulant matrices is a circulant matrixF define N × N DFT matrix M with elements:

mkl = e2 j (k 1)(l 1)/N

I taking the DFT of a vector corresponds to multiplying the vector withM

F a circulant matrix can be diagonalized with DFT:

MCMH = diag£d1 d2 d3 d4

¤D

F The diagonal elements dn are Fourier transforms of the first column, thevector of diagonal elements is

d = NMc1

F columns of circulant matrix C are rotated versions of the first column c1:

C =

a d c bb a d cc b a dd c b a

F the product of two circulant matrices is a circulant matrixF define N × N DFT matrix M with elements:

mkl = e2 j (k 1)(l 1)/N

I taking the DFT of a vector corresponds to multiplying the vector withM

F a circulant matrix can be diagonalized with DFT:

MCMH = diag£d1 d2 d3 d4

¤D

F The diagonal elements dn are Fourier transforms of the first column, thevector of diagonal elements is

d = NMc1

6

Circulant matrices, Inversion

F a circulant matrix is very simple to invert:

C 1 =MHD 1M

F Fourier transforms, and inverting diagonal matrixF typically we are not iterested in the overall scale, it is su cient to knowthe adjoint matrix Aadj = detA ·A 1

I the adjoint fulfils Aadj A = detA · IF adjoint can be constructed without divisions, which are very heavy com-putationallyI for generic matrix, constructing adjoint takes O(N3) multiplicationsI for circulant matrix C, significantly less:

¨ determinant: detC = detD =QNn=1 dn

¨ adjoint: detC ·C 1 =MH¡detD ·D 1

¢M =MHDadj M

¨ adjoint of diagonal matrix: 3N 6 multiplications¨ if N = 2m, FFT takes N log2N multiplications

adjoint of circulant matix: O(N log2N) multiplications

F significant complexity reduction in matrix inversion of circulant matrix

F a circulant matrix is very simple to invert:

C 1 =MHD 1M

F Fourier transforms, and inverting diagonal matrixF typically we are not iterested in the overall scale, it is su cient to knowthe adjoint matrix Aadj = detA ·A 1

I the adjoint fulfils Aadj A = detA · IF adjoint can be constructed without divisions, which are very heavy com-putationallyI for generic matrix, constructing adjoint takes O(N3) multiplicationsI for circulant matrix C, significantly less:

¨ determinant: detC = detD =QNyy

n=1 dn¨ adjoint: detC ·C 1 =MH

¡detD ·D 1

¢M =MHDadj M

¨ adjoint of diagonal matrix: 3N 6 multiplications¨ if N = 2m, FFT takes N log2N multiplications

adjoint of circulant matix: O(N log2N) multiplications

F significant complexity reduction in matrix inversion of circulant matrix

7

Frequency Domain Equalization (FDE)

F The channel covariance of a Toeplitz matrix is almost circulantF Example: 3-tap channel, 5 Rx samples:

H =


R =

|h2|2 h2h1 r2 0 0 0 0h1h2 |h1|2 + |h2|2 r1 r2 0 0 0r2 r1 r0 r1 r2 0 00 r2 r1 r0 r1 r2 00 0 r2 r1 r0 r1 r20 0 0 r2 r1 |h0|2 + |h1|2 h1h00 0 0 0 r2 h0h1 |h0|2

r0 = |h0|2 + |h1|2 + |h2|2r1 = h1h0 + h2h1

r2 = h2h0

F The channel covariance of a ToepTT litz matrix is almost circulantnnF Example: 3-tap channel, 5 Rx samRR ples:

H =


R =

|h2|2 h2h1 r2 0 0 0 0h1h2 |h1|2 + |h2|2 r1 r2 0 0 0r2 r1 r0 r1 r2 0 00 r2 r1 r0 r1 r2 00 0 r2 r1 r0 r1 r20 0 0 r2 r1 |h0|2 + |h1|2 h1h00 0 0 0 r2 h0h1 |h0|2

r0 = |h0|2 + |h1|2 + |h2|2r1 = h1h0 + h2h1

r2 = h22h0

8

FDE II

F if the channel covariance is forced to circulant, it can be inverted withDFT

I replace R with

eR =

r0 r1 r2 0 0 r2 r1r1 r0 r1 r2 0 0 r2r2 r1 r0 r1 r2 0 00 r2 r1 r0 r1 r2 00 0 r2 r1 r0 r1 r2r2 0 0 r2 r1 r0 r1r1 r2 0 0 r2 r1 r0

I calculate e =MeRMH, the eigenvalue matrix of the approximative channelcovariance eR

I invert channel + noise covariance, calculate approximative MMSE weights:

³HHH+N0I

´ 1

HH MH³e +N0I´ 1

MHH

F if the channel covariance is forced to circulant, it can be inverted withDFT

I replace R with

eR =

r0 r1 r2 0 0 r2 r1r1 r0 r1 r2 0 0 r2r2 r1 r0 r1 r2 0 00 r2 r1 r0 r1 r2 00 0 r2 r1 r0 r1 r2r2 0 0 r2 r1 r0 r1r1 r2 0 0 r2 r1 r0

I calculate e =MeRMH, the eigenvalue matrix of the approximative channelcovariance eR

I invert channel + noise covariance, calculate approximative MMSE weights:

³HHH+N0I

´ 1

HH MH³e +N0I´ 1

MHH

9

Block Transmission and Guard Interval

F transversal filter equalizes continuous single carrier transmissionI decision made on incomplete info of transmitted symbolsI some energy related to symbols in the equalization windowalways left outside the window

I all information available for equalization cannot be usedexcept for infinite length equalizer

F this can be improved by designing a block transmissionI a block of consecutive symbols is transmitted, preceded bya guard interval¨ length of GI at least L 1

I in the guard interval the ISI from the previous blockis allowed to vanish, no new information transmitted¨ transmission rate lower for a block transmissionthan for continuous transmission

F transversal filter equalizes continuous single carrier transmissionI decision made on incomplete info of transmitted symbolsI some energy related to symbols in the equalization windowalways left outside the window

I all information available for equalization cannot be usedexcept for infinite length equalizer

F this can be improved by designing a block transmissionI a block of consecutive symbols is transmitted, preceded bya guard interval¨ length of GI at least L 1

I in the guard interval the ISI from the previous blockis allowed to vanish, no new information transmitted¨ transmission rate lower for a block transmissionthan for continuous transmission

10

Block Transmission and GI: Example I

F guard interval filled with zeros

y1y2y3y4y5

=


00x1x2x3x4x5

+

n1n2n3n4n5

F samples y1, . . . , y5, have only contributions from x1, . . . , x5.F no Inter-Block InterferenceF symbols x1, . . . , x2 equalized together from samples y1, . . . , y5F lower rate due to no information in GI

F guard interval filled with zeros

y1y2y3y4y5

=


00x1x2x3x4x5

+

n1n2n3n4n5

F samples y1, . . . , y5, have only contributions from x1, . . . , x5.F no Inter-Block InterferenceF symbols x1, . . . , x2 equalized together from samples y1, . . . , y5F lower rate due to no information in GI

11

Block Transmission and GI: Example II

F received power in following GI may be used to gather power for reception:

y1y2y3y4y5y6y7

=

h0 0 0 0 0h1 h0 0 0 0h2 h1 h0 0 00 h2 h1 h0 00 0 h2 h1 h00 0 0 h2 h10 0 0 0 h2

x1x2x3x4x5

+

n1n2n3n4n5n6n7

F the equalizer may now detect all x1, . . . x5 from this blockF this requires calculating a full TDE filter for all symbols in the blockF FDE does not workF after equalization, symbols in the block see a di erent channel

I symbols close to block end su er from less ISI

F received power in following GI may be used to gather power for reception:

y1y2y3y4y5y6y7

=

h0 0 0 0 0h1 h0 0 0 0h2 h1 h0 0 00 h2 h1 h0 00 0 h2 h1 h00 0 0 h2 h10 0 0 0 h2

x1x2x3x4x5

+

n1n2n3n4n5n6n7

F the equalizer may now detect all x1, . . . x5 from this blockF this requires calculating a full TDE filter for all symbols in the blockF FDE does not workF after equalization, symbols in the block see a di erent channel

I symbols close to block end su er from less ISI

12

Cyclic Prefix IF A very special block transmission: fill GI with Cyclic Prefix (CP):F The CP is a copy of a few last transmitted symbols:

y1y2y3y4y5

=


x4x5x1x2x3x4x5

+

n1n2n3n4n5

F some transmitted symbols shown multiple times in vector signal modelabove

F Remove this redundancy by dropping the CP:

y1y2y3y4y5

=

h0 0 0 h2 h1h1 h0 0 0 h2h2 h1 h0 0 00 h2 h1 h0 00 0 h2 h1 h0

x1x2x3x4x5

+

n1n2n3n4n5

F Channel matrix is circulant! Vector signal model:

y = Hcircx + n

FF A very special block transmission: fill GI with Cyclic Prefix (CP):F The CP is a copy of a fewff last transmitted symbols:

y1y2y3y4y5

=


x4x5x1x2x3x4x5

+

n1n2n3n4n5

F some transmitted symbols shown multiple times in vector signal modelabove

F Remove this redundancy by dropping the CP:

y1y2y3y4y5

=


x1x2x3x4x5

+

n1n2n3n4n5

F Channel matrix is circulant! VecVV tor signal model:

y = Hcircx + n

13

Cyclic Prefix II

F note that a circulant channel matrix appears only if the length of CPL 1, where L is length of tapped delay line.

F The circulant channel matrix consists of rotated versions of the tappeddelay line vector extended to the length N

hT =£h0 h1 . . . hL 1 0 . . . 0

¤TF channel matrix (and channel covariance) can be exactly diagonalized withDFT:

MHcircMH =Hdiag = diag [hF]

I elements of diagonalized channel matrix are the frequency domain channelcoe cients

hF = NMhT

F note that a circulant channel matrix appears only if the length of CPL 1, where L is length of tapped delay line.

F The circulant channel matrix consists of rotated versions of the tappeddelay line vector extended to the length N

hT =£h0 h1 . . . hL 1 0 . . . 0

¤TF channel matrix (and channel covariance) can be exactly diagonalized withDFT:

MHcircMH =Hdiag = diag [hF]

I elements of diagonalized channel matrix are the frequency domain channelcoe cients

hF = NMhT

14

Cyclic Prefix: FDE

1. Perform DFT at receiver:

My = MHcircMHM| {z }=I

x +Mn

= HdiagMx + n0

F note that M is unitary, no noise colouring2. Single tap equalization of the channels. For example, ZF:

H 1diagMy =Mx +H

1diagn

0

F here, the noise components are coloured according to inverse channel power3. Inverse FFT to decouple transmitted symbols:

x =MHH 1diagMy = x+M

HH 1diagn

0

F the DFT matrix mixes the noise components, all symbols have same SINR

F note delicate issue: in FDE discussion above, complexity gains came fromsimplified tap solving (inversion of cirulant matrix)

F here, the whole equalization process can be performed with FFTs (andinversion of diagonal matrix)

F Single-carrier transmisison with CP is UL modulation of LTE

1. Performff DFT at receiver:

My = MHcircMHM|M{z{ }M}=I

x +Mn

= HdiagMx + n0

F note that M is unitary, no noise colouring2. Single tap equalization of the channels. For examFF ple, ZF:

H 1diagMy =Mx +H

1diagn

0

F here, the noise components are coloured according to inverse channel power3. Inverse FFT to decouple transmitted symbols:

x =MHH 1diagMy = x+M

HH 1diagn

0

F the DFT matrix mixes the noise components, all symbols have same SINR

F note delicate issue: in FDE discussion above, complexity gains came fromffsimplified tap solving (inversion of cirulant matrix)

F here, the whole equalization process can be performeff d with FFTs (andinversion of diagonal matrix)

F Single-carrier transmisison with CP is UL modulation of LTE

15

OFDM IF Move the last Inverse FFT above from the receiver to the transmitter:

I time domain transmitted symbol vector x is a Inverse Fourier Transformof a frequency domain symbol vector:

x =MHxF

I the information is packed into constellation points (QPSK, M-QAM) ofthe elements in xF

F this e ectively diagonalizes the channel at the transmitterI note that Hcirc =MHHdiagM

y1y2y3y4y5

=


MH

xF,1xF,2xF,3xF,4xF,5

+

n1n2n3n4n5

= MHHdiagxF + n

F simple DFT (or FFT) processing at Rx makes the channel completelydiagonal:

My =HdiagxF +Mn

FF Move the last Inverse FFT above fromff the receiver to the transmitter:I time domain transmitted symbol vector x is a Inverse FourFF ier TransTT formffof a freff quency domain symbol vector:

x =MHxF

I the informaff tion is packed into constellation points (QPSK, M-QAM) ofthe elements in xF

F this e ectively diagonalizes the channel at the transmitterI note that Hcirc =MHHdiagM

y1y2y3y4y5

=


MH

xF,1xF,2xF,3xF,4xF,5

+

n1n2n3n4n5

= MHHdiagxF + n

F simple DFT (or FFT) processing at Rx makes the channel completelydiagonal:

My =HdiagxF +Mn

16

OFDM II

F This is Orthogonal Frequency Domain Multiplexing (OFDM)I requires no information of the channel at transmitterI makes channel orthogonalI removes all ISII transmission moved to the frequency domain

F multicarrier transmission

I there are N narrowband subcarriersI the narrowband channel observed on this subcarrier is Fourier transformof TDL

I subcarrier bandwidth W/NI a symbol is transmitted on a subcarrierI a subcarrier corresps to Fourier waveform in time domain

F OFDM provides perfect removing of ISI with simple receiver

F This is Orthogonal Frequency Domain Multiplexing (OFDM)I requires no information of the channel at transmitterI makes channel orthogonalI removes all ISII transmission moved to the frequency domain

F multicarrier transmission

I there are N narrowband subcarriersI the narrowband channel observed on this subcarrier is Fourier transformof TDL

I subcarrier bandwidth W/NI a symbol is transmitted on a subcarrierI a subcarrier corresps to Fourier waveform in time domain

F OFDM provides perfect removing of ISI with simple receiver

17

16-tap MMSE, TD continuous & block transmissions

FDE: frequency domain tap solving for continuous transmissionperformance saturates at high SNR due to intentional error in channel inversion

TDE: time domain equalization for continuous transmission CP: block transmission with cyclic prefix: FDE0-GI: block transmission with silent guard interval. Block-TDE

note: block transmissions have lower rate

price of FDE

}}

Continuouos transmission: loss due to fixed D and limited information of interference

{Gain from using all Rx power

Continuous transmission

Blocktransmission

18

Single tap equalization: single carrier vs. OFDM

MRC error floor due to interference

OFDM:severe fading,

no diversity

OFDM:no ISI• OFDM: ”single tap equalizer” is FFT receiver

• single carrier: single tap equalizer is MRC detector (RAKE)

19

Single tap equalization: single carrier vs. OFDM II

uncoded BER: OFDM (much) worse than single carrier block transmission

block transmissions provide perfectly equalized multipath diversity all symbols are received over the wide band, components from allcannel taps

OFDM has no ISInarrowband transmission, no multipath diversityfrequency selective fading

recall that here the multiapath channel is not truly fadingamplitudes are fixed

With diversity (though channel code etc) OFDM performance similar or slightly better than single carrier block transmission

slightly better with linear receivers: better channel estimation, no errors in equalizationhere, channel estimation errors were not modeled

20

Summary: FDE

with large equalizers, solving for transversal filter taps in time domain is computationally challengingapproximative filter taps may be solved in frequency domain, using Fourier transform

approximation error decreases with length of filter, increases with length of channelat high SNR; approximation error dominates performance

error floor due to intentional equalization inaccuracy

21

Summary Block transmissions & OFDM

blocks of consecutive transmission symbols may be isolated from each other by inserting a Guard Interval between blocks

no inter-block interferencelower transmission rate due to GI

equalization becomes a block-by-block operationIf Guard Interval filled with Cyclic Prefix, FDE may be used without approximation

significant equalization complexity reduction when FFT applicableIf CP is used, and pre-equalization with FFT is used at transmitter, we have constructed a multicarrier OFDM transmission

no ISIeach symbol sees a narrowband channel

& advanced tx methods • equalization is required in inter-symbol interference (isi) channels...

Documents