B.Tech Final Project

BTP Presentation

Akshay Soni (148)& Tanvi Sharma

(196)Supervisor : Prof.

VijaykumarChakka

OFDM

MulticarrierCommunication

Basics

Diagram

OFDM ChannelEstimation

2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

BTP Presentation

Akshay Soni (148) & Tanvi Sharma (196)Supervisor : Prof. Vijaykumar Chakka

DA-IICTEvaluation Committee : 2

May 4, 2010

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Multicarrier Communication - (1)

The basic idea of wideband communication systems isrelaible and very high-rate data transfer over ISI freechannels.

For an ISI free channel, the symbol time Ts has to besignificantly larger than the channel delay spread τ .

High data rates means Ts is much less than τ , resultingin severe ISI.

Multicarrier modulation divides the high data ratetransmission into K lower rate substreams, each havingdata rate 1/K times the original.

Symbol time increases by the same factor K and foreach substream KTs >> τ holds, hence nullifying theeffect of ISI.

Data transmission occurs over K parallel subcarriersmaintaining the overall high data rate.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Multicarrier Communication - (2)

Essentially, a high data rate signal of rate R bps andwith a passband bandwidth B is broken into K parallelsubstreams, each with rate R/K and passbandbandwidth B/K.

In the time domain, the symbol duration on eachsubcarrier has increased to T = KTs, so letting K growlarger ensures that the symbol duration exceeds thechannel delayspread, T >> τ , which is a requirementfor ISI-free communication.

In the frequency domain, the subcarriers havebandwidth B/K << Bc, which ensures flat fading, thefrequency-domain equivalent to ISI-free communication.

Typically, subcarriers are orthogonal to each otherpreventing the effects of ICI and such modulation istermed as OFDM.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (1)

Adding Cyclic Prefix (CP) as shown in following figure,creates a signal that appears to be x[n]L soy[n] = x[n] ⊗ h[n].

X0 X1 X2 X3 ... XL-v-1 XL-v XL-v+1 … XL-1 XL-v XL-v+1 … XL-1

copy and paste last v symbols.

cyclic prefix OFDM data symbols

Figure: Addition of Cyclic Prefix to OFDM Symbols

Circular convolution in time domain is equivalent tomultiplication in DFT domain.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (2)

OFDM pursues transmission over multiple complexexponential functions, which are orthogonal [1].Exponential functions are choosen because

They are eigenfunctions to an LTI system.

y(t) =∫ ∞

−∞h(τ)ej2πf(t−τ)dτ

= ej2πft

∫ ∞

−∞h(τ)e−j2πfτ)dτ

= H(f) ej2πft

They are orthogonal to each other for any two differentfrequencies.

< ej2πf1t, ej2πf2t > =∫ ∞

−∞ej2πf1t(ej2πf2t)∗dt

=∫ ∞

−∞ej2πf1te−j2πf2tdt

= δ(f2 − f1) = 0 forf1 �= f2

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (3)

The orthogonality property holds over an infinite timeduration. But in OFDM, we use finite time duration T .

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (3)

The orthogonality property holds over an infinite timeduration. But in OFDM, we use finite time duration T .The transmitted complex baseband signal u(t) is

u(t) =K−1∑n=0

B[n]pn(t) (1)

where B[n] is the symbol transmitted andpn(t) = ej2πfntI[0,T ] is the modulating signal, fn is the

frequency of the nth subcarrier and IA is the indicatorfunction of set A.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (3)

The orthogonality property holds over an infinite timeduration. But in OFDM, we use finite time duration T .The transmitted complex baseband signal u(t) is

u(t) =K−1∑n=0

B[n]pn(t) (1)

where B[n] is the symbol transmitted andpn(t) = ej2πfntI[0,T ] is the modulating signal, fn is the

frequency of the nth subcarrier and IA is the indicatorfunction of set A.Now Pn(f) = Tsinc((f − fn)T ) i.e. Pn(f) = 0 if|f − fn| = k/T , where k is integer. Therefore, theorthogonality still holds over finite interval T ifsubcarriers are spaced apart by multiple of 1/T∫ T

0ej2πfnte−j2πfmtdt =

ej2π(fn−fm)t−1

j2π(fn − fm)= 0

for (fn − fm)T = nonzero integer.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (4)

Choosing frequencies of different subcarriers asfn = n/T giving (1) as

u(t) =K−1∑n=0

B[n]pn(t) =K−1∑n=1

B[n]ej2πnt/T I[0,T ] (2)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (4)


u(t) =K−1∑n=0

B[n]pn(t) =K−1∑n=1


If we sample (2) at a rate 1/Ts where Ts = T/N we get

u(kTs) = b(k) =K−1∑n=0

B[n]ej2πnk/N (3)

where k represents the kth subcarrier.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (4)


u(t) =K−1∑n=0

B[n]pn(t) =K−1∑n=1



u(kTs) = b(k) =K−1∑n=0

B[n]ej2πnk/N (3)


(3) is nothing but IFFT of symbol sequence B[n].

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (4)


u(t) =K−1∑n=0

B[n]pn(t) =K−1∑n=1



u(kTs) = b(k) =K−1∑n=0

B[n]ej2πnk/N (3)


(3) is nothing but IFFT of symbol sequence B[n].B[n] can again be generated from b(k) using therelation

B[n] =1K

K−1∑k=0

b[k]e−j2πnk/N (4)

which can be efficiently done using FFT operation.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (5)

Now using cyclic prefix, output of the channel can bewritten as circular convolution giving y = h ⊗ x.

Circular convolution in matrix form can be written as

y = Cx + n (5)

where

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

h0 0 · 0 hL−1 hL−2 · h1

h1 h0 0 · 0 hL−1 · h2

......

......

......

......

hL−1 hL−2 hL−3 · h0 0 · 00 hL−1 hL−2 · h1 h0 · 0...

......

......

......

...0 0 · 0 hL−1 hL−2 · h0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦is a circulant matrix i.e. rows are cyclic shifts of eachother.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Basics - (6)

The matrix C can be eigen decomposed as

C = VHΛV (6)

where V is a unitary matrix i.e. VVH = I, I is identitymatrix, Λ is a diagonal matrix that contains eigenvalues of C.

Using (6), we can write (5) as

y = V−1ΛVx + n (since VH = V−1)

Y = ΛX + N (7)

where Y = Vy, X = Vx and N = Vn.

Cyclic prefix of length v is discarded from the beginninggiving output of length K.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Block Diagram

K

point IFFT

P/SAdd CP

h[n]Delete

CP S/PK

point FFT

FEQ+

Frequency Domain

Time Domain

X x

n

y Y X

A circular channel: y = h x +n*

Figure: Block Diagram Representation for OFDM

At the receiver, output is Yl = HlXl +Nl for subcarrier l.

Each subcarrier can then be equalized via an FEQ by simplydividing by the complex channel gain H [i] for that subcarrier.This results in

Yl/Hl = Xl = Xl +Nl/Hl (8)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

OFDM Channel Estimation

In OFDM, working in transform domain becomes mucheasier.

We use recursive channel estimation algorithms intransform domain to estimate H which are utilized in (8)to obtain the input symbol estimates.

In OFDM, 2D-MMSE channel estimation in frequency andtime domain is optimum, if noise is additive.

However, 2D-MMSE algortihm has computational complexityof O(N3), where N is order of the filter. Also, it requiresexact channel correlation between the data and pilot symboltransmission [2].

2D-RLS algorithm does not require accurate channelstatistics and converges in several OFDM symbol time only[3].

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-RLS Channel Estimation - (1)

At transmission time n, the recieved signal on the kth

subcarrier can be expressed as

Y (n, k) = H(n, k)X(n, k) +N(n, k) (9)

where X(n, k) represents transmitted signal on kth

subcarrier at time n and N(n, k) represents FFT ofadditive complex Gaussian noise with zero mean andvariance σ2, which is uncorrelated for different n or k.

Each frame consists of M OFDM symbols. For the firstframe, initial ‘L’ OFDM symbols are preambles and restare data symbols.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


The output Y (n− L, k) for the first preambleX(n − L, k) after removal of CP and taking K-pointFFT (where K is number of subcarriers), is used for thefirst least square (LS) estimate H(n− L, k) of thechannel as

H(n − L, k) =Y (n − L, k)

X(n − L, k)(10)

Using similar approach, other LS channel estimatesH(n− (L − 1), k), H(n− (L − 2), k) ... H(n− 1, k) areobtained. The ‘L’ LS estimates are stored in LK × 1 inputvector as

P(n) = [H(n − 1, 1)..H(n − 1, K)...H(n − L, 1)..H(n − L, K)]T (11)

and a K × 1 reference vector Href (n) is constructed as

Href (n) = [H(n − 1, 1) H(n − 1, 2) ... H(n − 1, K)]T (12)

where [.]T represents transpose. Back

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Given vectors Href (n) and P(n), the channelestimation in general form is defined as [3]

H(n) = GH(n)P(n) (13)

where H(n) is estimation of H(n), G(n) is LK ×Kweight coefficient matrix and (.)H represents Hermitiantranspose. So the channel estimation error is

e(n) = Href (n) − H(n) (14)

The weighted least square cost function to beminimized is defined as

ε(n) =n∑

i=1

λn−i ‖e(i)‖2 + δλn ‖G(n)‖2F (15)

where λ is the exponential weighting factor Back .

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Minimizing gradient vector of the cost function (15)with respect to GH(n) by equating it to zero, gives[

n∑i=1

λn−iP(i)PH(i) + δλnI

]G(n) =

n∑i=1

λn−iP(i)HH

ref (i)

(16)

where I is LK × LK identity matrix.

Then (16) can be reformulated as

Φ(n)G(n) = Ψ(n) (17)

Equations for Φ(n) and Ψ(n) in iterative form are

Φ(n) = λΦ(n− 1) + P(n)PH(n) (18)

Ψ(n) = λΨ(n − 1) + P(n)HH

ref (n) (19)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Assuming Φ(n) to be non-singular, (17) can berewritten as

G(n) = Φ−1(n)Ψ(n) (20)

For simplicity of description, we further defineLK × LK matrix Q(n) as

Q(n) = Φ−1(n) (21)

Using matrix inversion lemma [4] on (18), we obtain

Q(n) = λ−1Q(n− 1) − λ−1k(n)PH(n)Q(n− 1) (22)

where k(n) is the LK × 1 gain vector

k(n) =λ−1Q(n− 1)P(n)

1 + λ−1PH(n)Q(n− 1)P(n)(23)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Cross multiplying and rearranging (23) gives

k(n) = Q(n)P(n) (24)

Substituting (21), (22) and (24) into (20), defines G(n)as

G(n) = G(n− 1) + k(n)ξH(n) (25)

where ξ(n) is the priori estimation error given as

ξ(n) = Href (n) − GH(n− 1)P(n) (26)

Once G(n) is obtained from (25), new channel estimatecan be calculated using (13) which can be used in (8).

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


2D-RLS algorithm sometimes diverges and becomesunstable when the inverse of input auto-correlationmatrix looses the property of positive definitenes orHermitian symmetry.

To improve the numerical stability of 2D-RLSalgorithm, QR decomposition based algorithms may beused which guarantees the property of positivedefinitenes of input auto-correlation matrix.

Further, to reduce the computations to compute theinverse of the input auto-correlation matrix, InverseQR-2D-RLS adaptive algorithm can be used.

This algorithm directly operates on the inverse of inputauto-correlation matrix, thus saving large number ofcomputations.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Channel Estimation - (1)

IQR-2D-RLS adaptive algorithm propagates the inversesquare root of input auto-correlation matrix instead ofpropagating the inverse of input auto-correlation matrix.

IQR-2D-RLS algorithm guarantees the property ofpositive definiteness and is numerically more stable than2D-RLS algorithm [5].

Inverse of input auto-correlation matrix Q using (22) is

Q(n) = λ−1Q(n− 1) − λ−1Q(n− 1)P(n)λ−1PH(n)Q(n− 1)r(n)

(27)

where r(n) = 1 + λ−1PH(n)Q(n− 1)P(n).

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


There are four distinct matrix terms that constitute theright hand side of (27), which can be written asfollowing 2-by-2 block matrix A(n) as

A(n) =⎡⎢⎢⎣1 + λ−1PH(n)Q(n− 1)P(n)... λ−1PH(n)Q(n− 1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

λ−1Q(n− 1)P(n)... λ−1Q(n− 1)

⎤⎥⎥⎦Now, since Q(n− 1) = Q1/2(n− 1)QH/2(n− 1) andrecognising that A(n) is a nonnegative-definite matrix,we may use Cholesky factorization [4] to express A(n)as follows

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


A(n) =

⎡⎢⎣1... λ−1/2PH(n)Q1/2(n− 1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0... λ−1/2Q1/2(n− 1)

⎤⎥⎦

×

⎡⎢⎢⎣ 1... 0T

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

λ−1/2QH/2(n− 1)P(n)... λ−1/2QH/2(n− 1)

⎤⎥⎥⎦(28)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Using matrix factorization lemma [4] on the firstproduct term of (28) gives⎡⎢⎢⎣1

... λ−1/2PH(n)Q1/2(n − 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0... λ−1/2Q1/2(n− 1)

⎤⎥⎥⎦Θ(n) =

⎡⎢⎢⎣ r1/2(n)... 0

. . . . . . . . . . . . . . . . . . . . . . .

k(n)r1/2(n)... Q1/2(n)

⎤⎥⎥⎦(29)

The unitary matrix Θ(n) is determined by using eitherGivens Rotations or Householder Transformations [6].

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Stability Analysis

The relation between Q(n) and Q1/2(n) is defined by

Q(n) = Q1/2(n)QH/2(n) (30)

where the matrix QH/2(n) is Hermitian transpose ofQ1/2(n).The nonnegative definite character of Q(n) as acorrelation matrix is preserved by virtue of the fact thatthe product of any square matrix and its Hermitiantranspose is always a nonnegative definite matrix [6][7].

The condition number of Q1/2(n) equals the squareroot of the condition number of Q(n).

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Computer Simulations

Number of subcarriers K = 64Length of cyclic prefix CP = 16Rayleigh fading channel with exponential delay profile isused.

Maximum Doppler shift of 100 Hz is taken.

BPSK modulation is utilized.

Number of OFDM symbols in a frame M = 5.Number of preambles in first OFDM symbol L = 2.δ = 0.1 and λ = 0.5.At time instant n = 0, G(0) = 0 and Q1/2(0) = δ−1/2I.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Computational Complexity

Table: Operation Count Per Iteration for L = 2 and K sub-carriers

2D-RLS IQR-Givens IQR-Householders

20K2 + 6K + 2 20K2 + 6K + 3 18K2 + 3K + 1

It is observed from above Table that operation count for2D-RLS algorithm and IQR-2D-RLS algorithm usingGivens Rotations are similar.

But fewer operations per iteration are required forHouseholder Transformations than Givens Rotations.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS BER Performance

2 4 6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

10−1

100

SNR (in dB)

BE

R

IQR−2D−RLS Householder Transformation2D−RLSIQR−2D−RLS Givens Rotations

Figure: BER Performance of 2D-RLS and IQR-2D-RLS Algorithms

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS NMSE Performance

2 4 6 8 10 20 40 70 10010

−4

10−3

10−2

10−1

100

Iteration

NM

SE

IQR−2D−RLS Householder Transformation2D−RLSIQR−2D−RLS Givens Rotations

Figure: NMSE Performance of 2D-RLS and IQR-2D-RLS atSNR 10 dB

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Stability Performance

0 20 40 60 80 100 120 140 160 180 20010

0

105

1010

1015

1020

2D−RLSIQR−2D−RLS (Householders Transformation)IQR−2D−RLS (Givens Rotation)

Figure: Condition Number Result for 2D-RLS andIQR-2D-RLS Algorithms

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

IQR-2D-RLS Conclusion

Due to smaller condition number, the matrix Q inIQR-2D-RLS algorithm is close to non-singularity andhence proposed algorithm is numerically more stablethan 2D-RLS algorithm.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



Also, both the algorithms have computationalcomplexity of O(N2). MATLAB simulations show thatIQR-2D-RLS and 2D-RLS algorithms have similar BERperformance.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




NMSE performance shows that convergence rate ofIQR-2D-RLS algorithm is slightly less than the 2D-RLSalgorithm.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




NMSE performance shows that convergence rate ofIQR-2D-RLS algorithm is slightly less than the 2D-RLSalgorithm.

A. Soni, T. Sharma and V.K. Chakka, “Inverse QR2D-RLS Adaptive Channel Estimation for OFDMSystems,” in IEICE Transactions on CommunicationLetters, (submitted), 2010.

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OFDM


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2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS Channel Estimation - (1)

2D-RLS and IQR-2D-RLS algorithms havecomputational complexity of O(N2).Set Membership Feasibility (SMF) adaptive channelestimation algorithms allow trade-off between the speedof convergence and overall computational complexity.

In SMF algorithms, the adaptive filter coefficients areupdated only when the estimation error is greater thana prescribed threshold.

2D-SM-NLMS algorithm by doing trade-off inconvergence speed, reduces the computationalcomplexity to O(N) [8][9].

The BER performance of the proposed algorithm is notcompromised and is similar to the conventional 2D-RLSalgorithm.

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OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


P and Href are constructed similar to (11) and (12)See .

The adaptive channel estimation equation is given by(13) and priori estimation error is given by (26) See .

The objective of the SMF is to design G(n) such thatthe magnitude of estimation output error is upperbounded by a prescribed threshold γ.

In summary, any filter parameter leading to a magnitudeof the output estimation error smaller than adeterministic threshold is an acceptable solution.

Bounded error constraint results in a set of filters ratherthan a single estimate.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


‘Feasibility Set’ Θ [10] is defined as

Θ =⋃

(P,Href )

{G :∣∣∣Href − GHP

∣∣∣ ≤ γ}

(31)

Given a set of pairs (P and Href ), υ(n) is defined asthe set containing all matrices G(n) such that theassociated output error at time instant n is upperbounded in magnitude by γ.

υ(n) ={G :∣∣∣Href − GHP

∣∣∣ ≤ γ}

(32)

The set υ(n) is the‘constraint set’ whose boundaries arehyperplanes.

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OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Since each pair (P, Href ) has an associated constraintset, the intersection sets over all the available timeinstants is called the ‘exact membership set’ ψ(n),formally defined as

ψ(n) =n⋂

i=0

υ(n) (33)

The set ψ(n) represents a polygon in the parameterspace whose location is one of the main objectives ofthe SMF.

The selective updating of the SMF brings aboutopportunities for power and computational savings.

The goal of SMF adaptive filtering is to adaptively findan estimate that belongs to the feasibility set.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Figure: Exact Membership Set out of Constraint Set

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Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


2D-SM-NLMS performs a test to verify if previousestimate lies outside constraint set υ(n), i.e.,∣∣∣Href (n) − GH(n− 1)P(n)

∣∣∣ > γ.

If yes then the new G(n) is calculated to the closestboundary of υ(n).

Figure: Coefficient matrix updating for 2D-SM-NLMS algorithm

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


The normalized LMS algorithm is given [10] as

G(n) = G(n− 1) +µ(n)

γ + PH(n)P(n)P(n)ξH(n) (34)

where µ(n) is the variable step size and γ (0 < γ < 1)is added to avoid large step size when PH(n)P(n)becomes small.

Considering the equation for posterior error given by

e(n) = Href (n) − GH(n)P(n) (35)

and putting Eq.(34) into Eq.(35), following relation isobtained

µ(n) = 1 − γ

|e(n)| if |e(n)| > γ

= 0 otherwise (36)

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS Computer Simulations

Number of subcarriers K = 64Length of cyclic prefix CP = 16Rayleigh fading channel with exponential delay profile isused.

Maximum Doppler shift of 100 Hz is taken.

BPSK modulation is utilized.

Number of OFDM symbols in a frame M = 5.Number of preambles in first OFDM symbol L = 2.δ = 0.1 and λ = 0.5.At time instant n = 0, G(0) = 0.

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OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS Computational Complexity

Table: Operation Count Per Iteration for L = 2 and Ksub-carriers at SNR 10dB

2D-RLS 2D-SM-NLMS

20K2 + 6K + 2 8K + 11

Table: Total Operation Count for Convergence for L = 2 and64 sub-carriers at SNR 10dB

2D-RLS 493884

2D-SM-NLMS (γ =√

5σ2n) 4788

2D-SM-NLMS (γ =√

0.5σ2n) 6384

2D-SM-NLMS (γ =√

0.05σ2n) 7980

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS BER Performance

2 4 6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

10−1

100

SNR (in dB)

BE

R

2D−RLS2D−SM−NLMS2D−SM−NLMS2D−SM−NLMS

γ =√

5σ2n

γ =√

0.05σ2n

γ =√

0.5σ2n

Figure: BER performance of 2D-RLS and 2D-SM-NLMSalgorithms

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS NMSE Performance

0 5 10 15 20 25 30 35 40 45 5010

−5

10−4

10−3

10−2

10−1

100

Iterations

NM

SE

2D−RLS2D−SM−NLMS2D−SM−NLMS2D−SM−NLMS

γ =√

5σ2n

γ =√

0.05σ2n

γ =√

0.5σ2n

Figure: NMSE performance of 2D-RLS and 2D-SM-NLMSalgorithms at SNR 10dB

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

2D-SM-NLMS Conclusion

The adaptive-filter coefficients are updated only whenthe error is more than the predefined threshold.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



SMF algorithms slightly compromises on theconvergence speed which is acceptable in stationaryenvironments.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




Improvement in the computational complexity (O(N))than the 2D-RLS adaptive channel estimation algorithm(O(N2)).

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




Improvement in the computational complexity (O(N))than the 2D-RLS adaptive channel estimation algorithm(O(N2)).BER performance is not effected by decrease incomputational complexity.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




Improvement in the computational complexity (O(N))than the 2D-RLS adaptive channel estimation algorithm(O(N2)).BER performance is not effected by decrease incomputational complexity.

T. Sharma, A. Soni and V.K. Chakka,“Two-Dimensional Set Membership Normalized LeastMean Square Adaptive Channel Estimation for OFDMSystems,” in IEEE INDICON, December 2009, pp.125 − 128.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

MIMO Two-way Relaying - (1)

OFDM based two-way relaying case where both thenodes have multiple antennas has been discussed [11].

The relay station (RS) is a MIMO station which cantransmit as well as receive signals in two different timeslots (half-duplex relaying).

Frequency Selec�ve Environment

M Antennas at Node-1

M Antennas at Node-2

Minimum 2M Antennas at MIMO-RS

Figure: Two-way MIMO Relaying Model

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VijaykumarChakka

OFDM


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2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work


Amplify and Forward relaying scheme is used at RSwhich provides very less complexity as compared toDecode and Forward relaying scheme [12].

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



Two-way relaying was introduced by Rankov andWittneben [13]

Nodes S1 and S2 transmit simultaneously to RS whichreceives the superposition of both the signals.On second channel resource, RS retransmits thissuperposition after application of spatial filtering.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



Two-way relaying was introduced by Rankov andWittneben [13]

Nodes S1 and S2 transmit simultaneously to RS whichreceives the superposition of both the signals.On second channel resource, RS retransmits thissuperposition after application of spatial filtering.

Unger and Klein [14] discussed the two-hop relayingcase consisting of nodes and RS with multiple antennasgiving the configuration of MIMO.

Nodes S1 and S2 have same number of antennas andRS has minimum of twice the number of antennas thatnodes have.The transceive filter at the RS is responsible for thetransmit and receive processing and hence CSI at thenodes becomes unnecessary.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)

The two nodes S1 and S2 are equipped with Mantennas and the RS should be having minimum 2Mantennas [14].

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OFDM


Basics

Diagram


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IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)


The modulated data symbols at the nodes Sk i.e.

X(k) =[X

(k)1 ,X

(k)2 , ...

], k = 1, 2 are grouped into

blocks of length N .

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VijaykumarChakka

OFDM


Basics

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2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)



X(k) =[X

(k)1 ,X

(k)2 , ...



After IFFT at both nodes, data is given as

x(k) =[x

(k)1 , x

(k)2 , ...

], k = 1, 2.

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OFDM


Basics

Diagram


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IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)



X(k) =[X

(k)1 ,X

(k)2 , ...




x(k) =[x

(k)1 , x

(k)2 , ...

], k = 1, 2.

Cyclic Prefix (CP) is added to each block to reduce ISI.

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VijaykumarChakka

OFDM


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Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)



X(k) =[X

(k)1 ,X

(k)2 , ...




x(k) =[x

(k)1 , x

(k)2 , ...

], k = 1, 2.


Data from node S1 is to be transmitted to node S2 andvice versa via RS.

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VijaykumarChakka

OFDM


Basics

Diagram


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IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)



X(k) =[X

(k)1 ,X

(k)2 , ...




x(k) =[x

(k)1 , x

(k)2 , ...

], k = 1, 2.



The transmit covariance matrices at node Sk are givenby Rx(k) = E

{x(k)x(k)H

}, k = 1, 2.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (1)



X(k) =[X

(k)1 ,X

(k)2 , ...




x(k) =[x

(k)1 , x

(k)2 , ...

], k = 1, 2.



The transmit covariance matrices at node Sk are givenby Rx(k) = E

{x(k)x(k)H

}, k = 1, 2.

The overall 2M × 1 data vector is x =[x(1)T , x(2)T

]Twith covariance as Rx = E

{xxH}.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (2)

The channel matrix from node Sk, k = 1, 2 to RS is

H(k) =

⎡⎢⎣ hk1,1 . . . hk

1,M...

. . ....

hkMRS ,1 . . . hk

MRS ,M

⎤⎥⎦ (37)

where hki,j , i = 1, ...,MRS and j = 1, ...,M is

(hki,j)1 + ...+ (hk

i,j)L, ‘L’ is number of channel taps.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (2)


H(k) =

⎡⎢⎣ hk1,1 . . . hk

1,M...

. . ....

hkMRS ,1 . . . hk

MRS ,M

⎤⎥⎦ (37)


(hki,j)1 + ...+ (hk

i,j)L, ‘L’ is number of channel taps.The overall 2M × 2M channel matrix isH =

[H(1),H(2)

].

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (2)


H(k) =

⎡⎢⎣ hk1,1 . . . hk

1,M...

. . ....

hkMRS ,1 . . . hk

MRS ,M

⎤⎥⎦ (37)


(hki,j)1 + ...+ (hk


[H(1),H(2)

].

Since spatial filtering is to be applied at the RS, onlythe scalar transmit filters Q(1) = q(1)IM andQ(2) = q(2)IM are applied at the nodes where IM is theidentity matrix of M-dimension.

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (2)


H(k) =

⎡⎢⎣ hk1,1 . . . hk

1,M...

. . ....

hkMRS ,1 . . . hk

MRS ,M

⎤⎥⎦ (37)


(hki,j)1 + ...+ (hk


[H(1),H(2)

].

Since spatial filtering is to be applied at the RS, onlythe scalar transmit filters Q(1) = q(1)IM andQ(2) = q(2)IM are applied at the nodes where IM is theidentity matrix of M-dimension.The scalar transmit filters are required to meet thetransmit energy constraints given by

E

{∥∥∥q(k)x(k)∥∥∥2

2

}≤ E(k), k = 1, 2 (38)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (3)

The overall transmit matrix is defined as

Q =

[Q(1) IMIM Q(2)

](39)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (3)


Q =

[Q(1) IMIM Q(2)

](39)

The received vector yRS at the RS is given as

yRS = HQx + nRS (40)

where nRS is an additive white gaussian noise vectorwith covariance matrix RnRS

= E{nRSnH

RS

}.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (3)


Q =

[Q(1) IMIM Q(2)

](39)

The received vector yRS at the RS is given as

yRS = HQx + nRS (40)

where nRS is an additive white gaussian noise vectorwith covariance matrix RnRS

= E{nRSnH

RS

}.

At RS, the received vector yRS is spatially filtered by alinear transceive filter G leading to the RS transmitvector

xRS = GyRS = GHQx + GnRS (41)

which is required to fulfil the energy constraints at RS,given by

E{‖xRS‖2

2

}≤ ERS (42)

where ERS is the maximum transmit energy at RS.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (4)

Since spatial filtering is done only at RS, the nodes areassumed to have scalar receive filter P = pI2M .

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (4)


The channel matrix from RS to nodes is taken as thetranspose of the channel matrix from nodes to RS [15].

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (4)



The estimate for data vector x(2) at node S1 is termedas x1 and estimate for x(1) at node S2 is termed as x2

giving the overall estimated data vector x =[xT1 , x

T2

]Tas

x = p(HT GHQx + HT GnRS + nR

)(43)

where nR is an additive white gaussian noise vectorwith covariance matrix RnR = E

{nRnH

R

}.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

System Model - (4)



The estimate for data vector x(2) at node S1 is termedas x1 and estimate for x(1) at node S2 is termed as x2

giving the overall estimated data vector x =[xT1 , x

T2

]Tas

x = p(HT GHQx + HT GnRS + nR

)(43)

where nR is an additive white gaussian noise vectorwith covariance matrix RnR = E

{nRnH

R

}.

Finally, FFT of the estimated data vector x is taken toobtain estimated QPSK symbols X.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Linear Transceive Filter

The spatial filtering at the RS in MIMO two-wayrelaying is done by utilizing linear transceive filter Gdefined as

G = GT GπGR (44)

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VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



G = GT GπGR (44)

RS mapping filter, Gπ is given by

Gπ =[φM

√1 − βIM√

βIM φM

](45)

where φM is a null matrix of M-dimension and theparameter β with 0 ≤ β ≤ 1 is a weight factor [14].

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



G = GT GπGR (44)

RS mapping filter, Gπ is given by

Gπ =[φM

√1 − βIM√

βIM φM

](45)

where φM is a null matrix of M-dimension and theparameter β with 0 ≤ β ≤ 1 is a weight factor [14].

The linear transmit filter (GT ) and linear receive filter(GR) have been implemented using ZF and MMSEcriteria.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Linear ZF Transceive Filter - (1)

ZF Receive filter GR,ZF :With the transmit power constraint in Eq.(38) and ZFconstraint, ZF optimization can be formulated [15]. Butsince it is not convex, the Karush-Kuhn-Tucker (KKT)conditions [16] are utilized to obtain the ZF receive filter

GR,ZF =(QHHHR−1

nRSHQ)−1

QHHHR−1nRS

(46)

with scalar transmit coefficients

q(k) =

√E(k)

tr {Rx(k)} , k = 1, 2 (47)

where Rx(k) is the covariance matrix of x(k) and tr {.}denotes the sum of diagonal elements of a matrix.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Linear ZF Transceive Filter - (2)

ZF Transmit filter GT,ZF :With the RS transmit power constraint in Eq.(42) andusing the KKT conditions, ZF transmit filter is obtainedas

GT,ZF =1pZF

H∗(HT H∗

)−1

(48)

with the scalar receive filter

pZF =

√√√√ tr{(

HT H∗)−1GπRxRS

GHπ

}ERS

RS estimation vector in Eq.(41) is obtained when RSreceived vector yRS is multiplied by the receive filter

matrix GR and is given as xRS =[x(1)TRS , x

(2)TRS

]Twith

estimate of x(1) as x(1)RS and estimate of x(2) as x

(2)RS .

The covariance matrix is given by

RxRS= E{xRS xH

RS

}= GR

(HQRxQ

HHH + RnRS

)GH

R

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Linear MMSE Transceive Filter - (1)

MMSE Receive filter GR,MMSE:With the transmit power constraint in (38) and MMSEconstraint, MMSE optimization can be formulated. Butsince it is not convex, the KKT conditions are utilizedto obtain the MMSE receive filter

GR,MMSE = RxQHHH

(HQRxQ

HHH + RnRS

)−1

(49)

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Linear MMSE Transceive Filter - (2)

MMSE Transmit filter GT,MMSE:With the RS transmit power constraint in (42) andMMSE constraint, MMSE optimization can beformulated. But since it is not convex, the KKTconditions are utilized to obtain the MMSE transmitfilter

GT,MMSE =1

pMMSE

(H∗HT +

tr {RnR} I

ERS

)−1

H∗ (50)

with the scalar receive filter

pMMSE =

√√√√√√ tr

{(H∗HT +

tr{RnR}I

ERS

)−2

H∗RxRSHT

}ERS

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

MIMO Two-way Relaying Simulations

Number of antennas at nodes M = 1.Number of antennas at RS 2M = 2.QPSK modulation is utilized.

In frequency selective environment, L = 2.Weight factor β = 0.5 to make the RS estimationvectors equally probable.

Number of subcarriers N = 16.Cyclic prefix length CP = 4.Maximum Doppler shift of 100 Hz is considered.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Frequency Flat Using ZF

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

SNR(1) in dB

BE

R

SNR(2) = 10 dB

SNR(2) = 20 dB

SNR(2) = 30 dB

SNR(2) = 40 dB

Figure: BER performance in frequency flat environment using ZF

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Frequency Flat Using MMSE

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

SNR(1) in dB

BE

R

SNR(2) = 10dB

SNR(2) = 20dB

SNR(2) = 30dB

SNR(2) = 40dB

Figure: BER performance in frequency flat environment usingMMSE

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Frequency Selective Using ZF

0 5 10 15 20 25 30 3510

−3

10−2

10−1

100

SNR(1) in dB

BE

R

SNR(2) = 10dB with OFDM




SNR(2) = 40 dB without OFDM

Figure: BER performance in frequency selective environmentusing ZF

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Frequency Selective Using MMSE

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

SNR(1) in dB

BE

R





Figure: BER performance in frequency selective environmentusing MMSE

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

MIMO Two-way Relaying Conclusion

MIMO two-way relaying scenario is considered forfrequency selective environment using OFDM where twonodes S1 and S2 communicate via an intermediateMIMO RS.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work



For two-way relaying, the utilization of spatial filteringonly at RS eliminates the CSI signaling overhead atnodes S1 and S2.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work




The BER performance of MMSE transceive filteroutperforms the ZF transceive filter in both frequencyflat and frequency selective environments.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work





OFDM mitigates the effects of the frequency selectivechannel and plays a critical role in reducing the BER.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work





OFDM mitigates the effects of the frequency selectivechannel and plays a critical role in reducing the BER.

A. Soni, T. Sharma, M. Chandwani and V. Chakka,“MIMO Two-Way Relaying in Frequency SelectiveEnvironment using OFDM,” in First UK-IndiaInternational Workshop on Cognitive Wireless Systems,December 2009.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Future Work

Equalization in MIMO two-way realying using 2D-RLSand IQR-2D-RLS, which we have already developed.

Utilizing lattice structure of recursive algorithms tofurther reduce computational complexity.

Use of two cascaded one-dimensional filters for channelequalization instead of one two-dimensional filter toreduce the computational complexity.

Design the matrices for MIMO two-way relay to reducethe computational complexity of the process.

Use of space-time block codes in MIMO two-way relaysto get the benefit of spatial diversity as well.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

Prof. L.N. Pillutla, “Lecture Slides: Introduction toOFDM,” in DA-IICT, 2009.

S.K. Kim and Y.H. Lee, “3-Dimensional MMSE ChannelEstimation in Multi-Antenna OFDM Systems,” in TheThird International Conference on DigitalCommunications, 2008, pp. 6–10.

X. Hou, S. Li, C. Yin and G. Yue, “Two-DimensionalRecursive Least Square Adaptive Channel Estimation forOFDM Systems,” in International Conference onWireless Communications, Networking and MobileComputing, 2005, pp. 232–236.

S. Haykin, Adaptive Filter Theory, edition 4. PrenticeHall, 2001.

A. Soni, T. Sharma and V.K. Chakka, “Inverse QR2D-RLS Adaptive Channel Estimation for OFDMSystems,” in IEICE Transactions on CommunicationLetters, submitted, 2010.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

A.H. Sayed, Adaptive Filters. John Wiley & Sons, 2008.

J.A. Apolinario Jr., QRD-RLS Adaptive Filtering.Springer, 2009.

T. Sharma, A. Soni and V.K. Chakka,“Two-Dimensional Se Membership Normalized LeastMean Square Adaptive Channel Estimation for OFDMSystems,” in IEEE INDICON, December 2009, pp.125–128.

A. Soni, T. Sharma and V. Chakka, “Two-DimensionalSet Membership Affine Projection Adaptive ChannelEstimation for OFDM Systems,” in National Conferenceon ICT: Theory, Applications and Practices, March 2010.

P.S.R. Diniz, Adaptive Filtering-Algorithms andPractical Implementation, 3rd ed. Springer, 2008.

A. Soni, T. Sharma, M. Chandwani and V. Chakka,“MIMO Two-Way Relaying in Frequency SelectiveEnvironment using OFDM,” in First UK-India

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

International Workshop on Cognitive Wireless Systems,December 2009.

S. Berger, T. Unger, M. Kuhn, A. Klein and A.Wittneben, “Recent Advances in Amplify-and-ForwardTwo-Hop Relaying,” in IEEE CommunicationsMagazine, July 2009.

B. Rankov and A. Wittneben, “Achievable Rate Regionsfor the Two-way Relay Channel,” in IEEE InternationalSymposium on Information Theory (ISIT), July 2006.

T. Unger and A. Klein, “On the Performance ofTwo-Way Relaying with Multiple-Antenna RelayStations,” in 16th IST Mobile and WirelessCommunications Summit, July 2007.

T.Unger and A.Klein, “Linear Transceive Filters forRelay Stations with Multiple Antennas in the Two-WayRelay Channel,” in IEEE Transactions on SignalProcessing, December 2008.

BTP Presentation



VijaykumarChakka

OFDM


Basics

Diagram


2D-RLS

IQR-2D-RLS

Stability

Simulations

Conclusion

2D-SM-NLMS

Simulations

Conclusion

MIMO Relay

System Model

Spatial Filter

ZF Fiter

MMSE Fiter

Simulations

Conclusion

Future Work

S. Boyd and L. Vandenberghe, Convex Optimization, 1sted. Cambridge University Press, 2004.

B.Tech Final Project

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