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- 1. BTP Presentation Akshay Soni (148) & Tanvi Sharma (196) Supervisor : Prof. Vijaykumar Chakka BTP Presentation OFDM Multicarrier Communication BasicsAkshay Soni (148) & Tanvi Sharma (196) Diagram OFDM Channel Supervisor : Prof. Vijaykumar Chakka Estimation 2D-RLS IQR-2D-RLS DA-IICT Stability Simulations Evaluation Committee : 2 Conclusion 2D-SM-NLMS Simulations May 4, 2010 Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 2. BTP PresentationMulticarrier Communication - (1) Akshay Soni (148) & Tanvi Sharma The basic idea of wideband communication systems is (196) Supervisor : Prof. relaible and very high-rate data transfer over ISI free Vijaykumar Chakka channels. OFDM For an ISI free channel, the symbol time Ts has to be Multicarrier Communication signiﬁcantly larger than the channel delay spread τ . Basics Diagram High data rates means Ts is much less than τ , resulting OFDM Channel Estimation in severe ISI. 2D-RLS IQR-2D-RLS Multicarrier modulation divides the high data rate Stability Simulations transmission into K lower rate substreams, each having Conclusion data rate 1/K times the original. 2D-SM-NLMS Simulations Conclusion Symbol time increases by the same factor K and for MIMO Relay each substream KTs >> τ holds, hence nullifying the System Model Spatial Filter eﬀect of ISI. ZF Fiter MMSE Fiter Simulations Data transmission occurs over K parallel subcarriers Conclusion maintaining the overall high data rate. Future Work
- 3. BTP PresentationMulticarrier Communication - (2) Akshay Soni (148) & Tanvi Sharma Essentially, a high data rate signal of rate R bps and (196) Supervisor : Prof. with a passband bandwidth B is broken into K parallel Vijaykumar Chakka substreams, each with rate R/K and passband bandwidth B/K. OFDM Multicarrier Communication In the time domain, the symbol duration on each Basics subcarrier has increased to T = KTs , so letting K grow Diagram OFDM Channel larger ensures that the symbol duration exceeds the Estimation 2D-RLS channel delayspread, T >> τ , which is a requirement IQR-2D-RLS Stability for ISI-free communication. Simulations Conclusion In the frequency domain, the subcarriers have 2D-SM-NLMS Simulations bandwidth B/K << Bc , which ensures ﬂat fading, the Conclusion MIMO Relay frequency-domain equivalent to ISI-free communication. System Model Spatial Filter Typically, subcarriers are orthogonal to each other ZF Fiter MMSE Fiter preventing the eﬀects of ICI and such modulation is Simulations Conclusion termed as OFDM. Future Work
- 4. BTP PresentationOFDM Basics - (1) Akshay Soni (148) & Tanvi Sharma Adding Cyclic Preﬁx (CP) as shown in following ﬁgure, (196) Supervisor : Prof. creates a signal that appears to be x[n]L so Vijaykumar y[n] = x[n] ⊗ h[n]. Chakka cyclic preﬁx OFDM data symbols OFDM Multicarrier XL-v XL-v+1 … XL-1 X0 X1 X2 X3 ... XL-v-1 XL-v XL-v+1 … XL-1 Communication Basics Diagram copy and paste last v symbols. OFDM Channel Estimation 2D-RLS Figure: Addition of Cyclic Preﬁx to OFDM Symbols IQR-2D-RLS Stability Simulations Conclusion Circular convolution in time domain is equivalent to 2D-SM-NLMS Simulations multiplication in DFT domain. Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 5. BTP PresentationOFDM Basics - (2) Akshay Soni (148) & Tanvi Sharma OFDM pursues transmission over multiple complex (196) Supervisor : Prof. exponential functions, which are orthogonal [1]. Vijaykumar Chakka Exponential functions are choosen because They are eigenfunctions to an LTI system. OFDM Multicarrier ∞ Communication j2πf (t−τ ) y(t) = h(τ )e dτ Basics Diagram −∞ ∞ OFDM Channel j2πf t −j2πf τ ) Estimation =e h(τ )e dτ 2D-RLS −∞ IQR-2D-RLS Stability j2πf t = H(f ) e Simulations Conclusion 2D-SM-NLMS They are orthogonal to each other for any two diﬀerent Simulations Conclusion frequencies. MIMO Relay ∞ System Model j2πf1 t j2πf2 t j2πf1 t j2πf2 t ∗ <e ,e >= e (e ) dt Spatial Filter ZF Fiter −∞ MMSE Fiter ∞ Simulations = ej2πf1 t e−j2πf2 t dt Conclusion −∞ Future Work = δ(f2 − f1 ) = 0 f orf1 = f2
- 6. BTP PresentationOFDM Basics - (3) Akshay Soni (148) & Tanvi Sharma The orthogonality property holds over an inﬁnite time (196) Supervisor : Prof. duration. But in OFDM, we use ﬁnite time duration T . Vijaykumar Chakka OFDM Multicarrier Communication Basics Diagram OFDM Channel Estimation 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
- 7. BTP PresentationOFDM Basics - (3) Akshay Soni (148) & Tanvi Sharma The orthogonality property holds over an inﬁnite time (196) Supervisor : Prof. duration. But in OFDM, we use ﬁnite time duration T . Vijaykumar Chakka The transmitted complex baseband signal u(t) is K−1 OFDM Multicarrier u(t) = B[n]pn (t) (1) Communication Basics n=0 Diagram OFDM Channel where B[n] is the symbol transmitted and Estimation pn (t) = ej2πfn t I[0,T ] is the modulating signal, fn is the 2D-RLS IQR-2D-RLS frequency of the nth subcarrier and IA is the indicator Stability Simulations Conclusion function of set A. 2D-SM-NLMS Simulations Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 8. BTP PresentationOFDM Basics - (3) Akshay Soni (148) & Tanvi Sharma The orthogonality property holds over an inﬁnite time (196) Supervisor : Prof. duration. But in OFDM, we use ﬁnite time duration T . Vijaykumar Chakka The transmitted complex baseband signal u(t) is K−1 OFDM Multicarrier u(t) = B[n]pn (t) (1) Communication Basics n=0 Diagram OFDM Channel where B[n] is the symbol transmitted and Estimation pn (t) = ej2πfn t I[0,T ] is the modulating signal, fn is the 2D-RLS IQR-2D-RLS frequency of the nth subcarrier and IA is the indicator Stability Simulations Conclusion function of set A. 2D-SM-NLMS Now Pn (f ) = T sinc((f − fn )T ) i.e. Pn (f ) = 0 if Simulations Conclusion |f − fn | = k/T , where k is integer. Therefore, the MIMO Relay System Model orthogonality still holds over ﬁnite interval T if Spatial Filter subcarriers are spaced apart by multiple of 1/T ZF Fiter MMSE Fiter Simulations T ej2π(fn −fm )t−1 Conclusion ej2πfn t e−j2πfm t dt = =0 Future Work 0 j2π(fn − fm ) for (fn − fm )T = nonzero integer.
- 9. BTP PresentationOFDM Basics - (4) Akshay Soni (148) & Tanvi Sharma Choosing frequencies of diﬀerent subcarriers as (196) fn = n/T giving (1) as Supervisor : Prof. Vijaykumar Chakka K−1 K−1 u(t) = B[n]pn (t) = B[n]ej2πnt/T I[0,T ] (2) OFDM Multicarrier n=0 n=1 Communication Basics Diagram OFDM Channel Estimation 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
- 10. BTP PresentationOFDM Basics - (4) Akshay Soni (148) & Tanvi Sharma Choosing frequencies of diﬀerent subcarriers as (196) fn = n/T giving (1) as Supervisor : Prof. Vijaykumar Chakka K−1 K−1 u(t) = B[n]pn (t) = B[n]ej2πnt/T I[0,T ] (2) OFDM Multicarrier n=0 n=1 Communication Basics Diagram If we sample (2) at a rate 1/Ts where Ts = T /N we get OFDM Channel Estimation K−1 2D-RLS B[n]ej2πnk/N IQR-2D-RLS u(kTs ) = b(k) = (3) Stability Simulations n=0 Conclusion 2D-SM-NLMS where k represents the kth subcarrier. Simulations Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 11. BTP PresentationOFDM Basics - (4) Akshay Soni (148) & Tanvi Sharma Choosing frequencies of diﬀerent subcarriers as (196) fn = n/T giving (1) as Supervisor : Prof. Vijaykumar Chakka K−1 K−1 u(t) = B[n]pn (t) = B[n]ej2πnt/T I[0,T ] (2) OFDM Multicarrier n=0 n=1 Communication Basics Diagram If we sample (2) at a rate 1/Ts where Ts = T /N we get OFDM Channel Estimation K−1 2D-RLS B[n]ej2πnk/N IQR-2D-RLS u(kTs ) = b(k) = (3) Stability Simulations n=0 Conclusion 2D-SM-NLMS where k represents the kth subcarrier. Simulations Conclusion (3) is nothing but IFFT of symbol sequence B[n]. MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 12. BTP PresentationOFDM Basics - (4) Akshay Soni (148) & Tanvi Sharma Choosing frequencies of diﬀerent subcarriers as (196) fn = n/T giving (1) as Supervisor : Prof. Vijaykumar Chakka K−1 K−1 u(t) = B[n]pn (t) = B[n]ej2πnt/T I[0,T ] (2) OFDM Multicarrier n=0 n=1 Communication Basics Diagram If we sample (2) at a rate 1/Ts where Ts = T /N we get OFDM Channel Estimation K−1 2D-RLS B[n]ej2πnk/N IQR-2D-RLS u(kTs ) = b(k) = (3) Stability Simulations n=0 Conclusion 2D-SM-NLMS where k represents the kth subcarrier. Simulations Conclusion (3) is nothing but IFFT of symbol sequence B[n]. MIMO Relay System Model B[n] can again be generated from b(k) using the Spatial Filter ZF Fiter relation MMSE Fiter K−1 1 Simulations B[n] = b[k]e−j2πnk/N (4) Conclusion K Future Work k=0 which can be eﬃciently done using FFT operation.
- 13. BTP PresentationOFDM Basics - (5) Akshay Soni (148) & Tanvi Sharma Now using cyclic preﬁx, output of the channel can be (196) written as circular convolution giving y = h ⊗ x. Supervisor : Prof. Vijaykumar Chakka Circular convolution in matrix form can be written as OFDM y = Cx + n (5) Multicarrier Communication Basics Diagram where OFDM Channel ⎡ h ⎤ Estimation 0 0 · 0 hL−1 hL−2 · h1 2D-RLS IQR-2D-RLS ⎢ h1 h0 0 · 0 hL−1 · h2 ⎥ Stability ⎢ . . . . . . . .⎥ Simulations ⎢ . . . . . . . .⎥ Conclusion ⎢ . . . . . . . .⎥ 2D-SM-NLMS C = ⎢hL−1 hL−2 hL−3 · 0 · 0⎥ Simulations ⎢ h0 ⎥ Conclusion ⎢ 0 hL−1 hL−2 · h1 h0 · 0⎥ ⎢ . ⎥ MIMO Relay ⎣ . . . . . . . . . . . . . .⎦ . System Model . . . . . . . . Spatial Filter ZF Fiter 0 0 · 0 hL−1 hL−2 · h0 MMSE Fiter Simulations Conclusion is a circulant matrix i.e. rows are cyclic shifts of each Future Work other.
- 14. BTP PresentationOFDM Basics - (6) Akshay Soni (148) & Tanvi Sharma The matrix C can be eigen decomposed as (196) Supervisor : Prof. Vijaykumar C = VH ΛV (6) Chakka where V is a unitary matrix i.e. VVH = I, I is identity OFDM Multicarrier Communication matrix, Λ is a diagonal matrix that contains eigen Basics Diagram values of C. OFDM Channel Estimation Using (6), we can write (5) as 2D-RLS IQR-2D-RLS y = V−1 ΛVx + n (since VH = V−1 ) Stability Simulations Conclusion 2D-SM-NLMS ˜ ˜ ˜ Y = ΛX + N (7) Simulations Conclusion ˜ ˜ ˜ where Y = Vy, X = Vx and N = Vn. MIMO Relay System Model Spatial Filter Cyclic preﬁx of length v is discarded from the beginning ZF Fiter MMSE Fiter giving output of length K. Simulations Conclusion Future Work
- 15. BTP PresentationOFDM Block Diagram Akshay Soni (148) & Tanvi Sharma Time Domain (196) Supervisor : Prof. n Vijaykumar K K ˆ X x Add Delete y Y FEQ X Chakka P/S h[n] + CP S/P point CP point IFFT FFT OFDM A circular channel: y = h * x +n Multicarrier Communication Basics Frequency Domain Diagram OFDM Channel Figure: Block Diagram Representation for OFDM Estimation 2D-RLS IQR-2D-RLS Stability At the receiver, output is Yl = Hl Xl + Nl for subcarrier l. Simulations Conclusion Each subcarrier can then be equalized via an FEQ by simply 2D-SM-NLMS Simulations dividing by the complex channel gain H[i] for that subcarrier. Conclusion This results in MIMO Relay System Model Spatial Filter ˆ Yl /Hl = Xl = Xl + Nl /Hl (8) ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 16. BTP PresentationOFDM Channel Estimation Akshay Soni (148) & Tanvi Sharma In OFDM, working in transform domain becomes much (196) Supervisor : Prof. easier. Vijaykumar Chakka We use recursive channel estimation algorithms in ˆ transform domain to estimate H which are utilized in (8) OFDM Multicarrier to obtain the input symbol estimates. Communication Basics In OFDM, 2D-MMSE channel estimation in frequency and Diagram time domain is optimum, if noise is additive. OFDM Channel Estimation However, 2D-MMSE algortihm has computational complexity 2D-RLS IQR-2D-RLS of O(N 3 ), where N is order of the ﬁlter. Also, it requires Stability Simulations exact channel correlation between the data and pilot symbol Conclusion 2D-SM-NLMS transmission [2]. Simulations Conclusion 2D-RLS algorithm does not require accurate channel MIMO Relay statistics and converges in several OFDM symbol time only System Model Spatial Filter [3]. ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 17. BTP Presentation2D-RLS Channel Estimation - (1) Akshay Soni (148) At transmission time n, the recieved signal on the kth & Tanvi Sharma (196) Supervisor : Prof. subcarrier can be expressed as Vijaykumar Chakka Y (n, k) = H(n, k)X(n, k) + N (n, k) (9) OFDM Multicarrier where X(n, k) represents transmitted signal on kth Communication Basics Diagram subcarrier at time n and N (n, k) represents FFT of OFDM Channel additive complex Gaussian noise with zero mean and Estimation 2D-RLS variance σ 2 , which is uncorrelated for diﬀerent n or k. IQR-2D-RLS Stability Simulations Each frame consists of M OFDM symbols. For the ﬁrst Conclusion 2D-SM-NLMS frame, initial ‘L’ OFDM symbols are preambles and rest Simulations Conclusion are data symbols. MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 18. BTP Presentation2D-RLS Channel Estimation - (2) Akshay Soni (148) & Tanvi Sharma (196) Supervisor : Prof. The output Y (n − L, k) for the ﬁrst preamble Vijaykumar Chakka X(n − L, k) after removal of CP and taking K-point OFDM FFT (where K is number of subcarriers), is used for the Multicarrier ﬁrst least square (LS) estimate H(n − L, k) of the Communication Basics Diagram channel as Y (n − L, k) OFDM Channel H(n − L, k) = (10) Estimation X(n − L, k) 2D-RLS IQR-2D-RLS Using similar approach, other LS channel estimates Stability H(n − (L − 1), k), H(n − (L − 2), k) ... H(n − 1, k) are Simulations Conclusion obtained. The ‘L’ LS estimates are stored in LK × 1 input 2D-SM-NLMS Simulations vector as Conclusion MIMO Relay T P(n) = [H(n − 1, 1)..H(n − 1, K)...H(n − L, 1)..H(n − L, K)] (11) System Model Spatial Filter and a K × 1 reference vector Href (n) is constructed as ZF Fiter MMSE Fiter Simulations T Conclusion Href (n) = [H(n − 1, 1) H(n − 1, 2) ... H(n − 1, K)] (12) Future Work where [.]T represents transpose. Back
- 19. BTP Presentation2D-RLS Channel Estimation - (3) Akshay Soni (148) & Tanvi Sharma (196) Supervisor : Prof. Given vectors Href (n) and P(n), the channel Vijaykumar Chakka estimation in general form is deﬁned as [3] OFDM H(n) = GH (n)P(n) Multicarrier (13) Communication Basics Diagram where H(n) is estimation of H(n), G(n) is LK × K OFDM Channel weight coeﬃcient matrix and (.)H represents Hermitian Estimation 2D-RLS IQR-2D-RLS transpose. So the channel estimation error is Stability Simulations Conclusion e(n) = Href (n) − H(n) (14) 2D-SM-NLMS Simulations Conclusion The weighted least square cost function to be MIMO Relay System Model minimized is deﬁned as Spatial Filter ZF Fiter n MMSE Fiter 2 2 (n) = λn−i e(i) + δλn G(n) Simulations F (15) Conclusion i=1 Future Work where λ is the exponential weighting factor Back .
- 20. BTP Presentation2D-RLS Channel Estimation - (4) Akshay Soni (148) & Tanvi Sharma Minimizing gradient vector of the cost function (15) (196) with respect to GH (n) by equating it to zero, gives Supervisor : Prof. Vijaykumar Chakka n n H λn−i P(i)PH (i) + δλn I G(n) = λn−i P(i)Href (i) OFDM Multicarrier Communication i=1 i=1 Basics (16) Diagram where I is LK × LK identity matrix. OFDM Channel Estimation 2D-RLS Then (16) can be reformulated as IQR-2D-RLS Stability Simulations Φ(n)G(n) = Ψ(n) (17) Conclusion 2D-SM-NLMS Simulations Equations for Φ(n) and Ψ(n) in iterative form are Conclusion MIMO Relay H System Model Φ(n) = λΦ(n − 1) + P(n)P (n) (18) Spatial Filter ZF Fiter MMSE Fiter H Ψ(n) = λΨ(n − 1) + P(n)Href (n) Simulations (19) Conclusion Future Work
- 21. BTP Presentation2D-RLS Channel Estimation - (5) Akshay Soni (148) & Tanvi Sharma Assuming Φ(n) to be non-singular, (17) can be (196) Supervisor : Prof. rewritten as Vijaykumar G(n) = Φ−1 (n)Ψ(n) (20) Chakka OFDM For simplicity of description, we further deﬁne Multicarrier LK × LK matrix Q(n) as Communication Basics Diagram Q(n) = Φ−1 (n) (21) OFDM Channel Estimation 2D-RLS IQR-2D-RLS Using matrix inversion lemma [4] on (18), we obtain Stability Simulations Conclusion Q(n) = λ−1 Q(n − 1) − λ−1 k(n)PH (n)Q(n − 1) (22) 2D-SM-NLMS Simulations Conclusion where k(n) is the LK × 1 gain vector MIMO Relay System Model Spatial Filter −1 λ Q(n − 1)P(n) ZF Fiter k(n) = (23) MMSE Fiter 1 + λ−1 PH (n)Q(n − 1)P(n) Simulations Conclusion Future Work
- 22. BTP Presentation2D-RLS Channel Estimation - (6) Akshay Soni (148) & Tanvi Sharma Cross multiplying and rearranging (23) gives (196) Supervisor : Prof. Vijaykumar k(n) = Q(n)P(n) (24) Chakka OFDM Substituting (21), (22) and (24) into (20), deﬁnes G(n) Multicarrier Communication as Basics G(n) = G(n − 1) + k(n)ξ H (n) Diagram (25) OFDM Channel Estimation where ξ(n) is the priori estimation error given as 2D-RLS IQR-2D-RLS Stability ξ(n) = Href (n) − GH (n − 1)P(n) (26) Simulations Conclusion 2D-SM-NLMS Simulations Once G(n) is obtained from (25), new channel estimate Conclusion can be calculated using (13) which can be used in (8). MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 23. BTP Presentation2D-RLS Channel Estimation - (7) Akshay Soni (148) & Tanvi Sharma 2D-RLS algorithm sometimes diverges and becomes (196) Supervisor : Prof. unstable when the inverse of input auto-correlation Vijaykumar Chakka matrix looses the property of positive deﬁnitenes or Hermitian symmetry. OFDM Multicarrier Communication To improve the numerical stability of 2D-RLS Basics Diagram algorithm, QR decomposition based algorithms may be OFDM Channel used which guarantees the property of positive Estimation 2D-RLS deﬁnitenes of input auto-correlation matrix. IQR-2D-RLS Stability Further, to reduce the computations to compute the Simulations Conclusion inverse of the input auto-correlation matrix, Inverse 2D-SM-NLMS Simulations QR-2D-RLS adaptive algorithm can be used. Conclusion MIMO Relay This algorithm directly operates on the inverse of input System Model Spatial Filter auto-correlation matrix, thus saving large number of ZF Fiter MMSE Fiter computations. Simulations Conclusion Future Work
- 24. BTP PresentationIQR-2D-RLS Channel Estimation - (1) Akshay Soni (148) & Tanvi Sharma IQR-2D-RLS adaptive algorithm propagates the inverse (196) Supervisor : Prof. square root of input auto-correlation matrix instead of Vijaykumar Chakka propagating the inverse of input auto-correlation matrix. OFDM IQR-2D-RLS algorithm guarantees the property of Multicarrier Communication positive deﬁniteness and is numerically more stable than Basics Diagram 2D-RLS algorithm [5]. OFDM Channel Estimation Inverse of input auto-correlation matrix Q using (22) is 2D-RLS IQR-2D-RLS H λ−1 Q(n − 1)P(n)λ−1 P (n)Q(n − 1) Stability Q(n) = λ−1 Q(n − 1) − Simulations Conclusion r(n) 2D-SM-NLMS (27) Simulations Conclusion where r(n) = 1 + λ−1 PH (n)Q(n − 1)P(n). MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 25. BTP PresentationIQR-2D-RLS Channel Estimation - (2) Akshay Soni (148) & Tanvi Sharma There are four distinct matrix terms that constitute the (196) Supervisor : Prof. right hand side of (27), which can be written as Vijaykumar Chakka following 2-by-2 block matrix A(n) as OFDM A(n) = Multicarrier Communication ⎡ ⎤ Basics −1 PH (n)Q(n − 1)P(n) . λ−1 PH (n)Q(n − 1) Diagram . ⎢1 + λ . ⎥ OFDM Channel ⎢. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎥ Estimation ⎣ ⎦ 2D-RLS −1 Q(n − 1)P(n) . . −1 Q(n − 1) IQR-2D-RLS λ . λ Stability Simulations Conclusion 2D-SM-NLMS Now, since Q(n − 1) = Q1/2 (n − 1)QH/2 (n − 1) and Simulations Conclusion recognising that A(n) is a nonnegative-deﬁnite matrix, MIMO Relay we may use Cholesky factorization [4] to express A(n) System Model Spatial Filter ZF Fiter as follows MMSE Fiter Simulations Conclusion Future Work
- 26. BTP PresentationIQR-2D-RLS Channel Estimation - (3) Akshay Soni (148) & Tanvi Sharma (196) Supervisor : Prof. ⎡. ⎤ Vijaykumar 1 . . λ−1/2 PH (n)Q1/2 (n − 1) Chakka ⎢ ⎥ A(n) = ⎣. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎦ OFDM . Multicarrier 0 . λ−1/2 Q1/2 (n − 1) Communication . ⎡ ⎤ Basics Diagram . . T ⎢ 1 . 0 ⎥ OFDM Channel Estimation × ⎢. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎥ ⎣ ⎦ 2D-RLS IQR-2D-RLS λ −1/2 QH/2 (n − 1)P(n) . λ−1/2 QH/2 (n − 1) . . Stability Simulations Conclusion (28) 2D-SM-NLMS Simulations Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 27. BTP PresentationIQR-2D-RLS Channel Estimation - (4) Akshay Soni (148) & Tanvi Sharma Using matrix factorization lemma [4] on the ﬁrst (196) Supervisor : Prof. product term of (28) gives Vijaykumar Chakka ⎡ ⎤ . −1/2 H . λ 1/2 ⎢1 . P (n)Q (n − 1)⎥ OFDM Multicarrier ⎢. . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎥ Θ(n) = Communication ⎣ ⎦ Basics . . −1/2 Q1/2 (n − 1) Diagram 0 . λ OFDM Channel ⎡ ⎤ (29) Estimation 1/2 (n) . . 2D-RLS ⎢ r . 0 ⎥ IQR-2D-RLS ⎢ . . . . . . . . . . . . . . . . . . . . . . .⎥ Stability ⎣ ⎦ Simulations Conclusion k(n)r 1/2 (n) . Q1/2 (n) . . 2D-SM-NLMS Simulations Conclusion MIMO Relay The unitary matrix Θ(n) is determined by using either System Model Spatial Filter Givens Rotations or Householder Transformations [6]. ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 28. BTP PresentationIQR-2D-RLS Stability Analysis Akshay Soni (148) The relation between Q(n) and Q1/2 (n) is deﬁned by & Tanvi Sharma (196) Supervisor : Prof. Vijaykumar Q(n) = Q1/2 (n)QH/2 (n) (30) Chakka OFDM where the matrix QH/2 (n) is Hermitian transpose of Multicarrier Communication Q1/2 (n). Basics Diagram The nonnegative deﬁnite character of Q(n) as a OFDM Channel Estimation correlation matrix is preserved by virtue of the fact that 2D-RLS IQR-2D-RLS the product of any square matrix and its Hermitian Stability Simulations transpose is always a nonnegative deﬁnite matrix [6][7]. Conclusion 2D-SM-NLMS The condition number of Q1/2 (n) equals the square Simulations Conclusion root of the condition number of Q(n). MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 29. BTP PresentationIQR-2D-RLS Computer Simulations Akshay Soni (148) & Tanvi Sharma Number of subcarriers K = 64 (196) Supervisor : Prof. Length of cyclic preﬁx CP = 16 Vijaykumar Chakka Rayleigh fading channel with exponential delay proﬁle is OFDM used. Multicarrier Communication Maximum Doppler shift of 100 Hz is taken. Basics Diagram BPSK modulation is utilized. OFDM Channel Estimation Number of OFDM symbols in a frame M = 5. 2D-RLS IQR-2D-RLS Stability Number of preambles in ﬁrst OFDM symbol L = 2. Simulations Conclusion δ = 0.1 and λ = 0.5. 2D-SM-NLMS Simulations At time instant n = 0, G(0) = 0 and Q1/2 (0) = δ−1/2 I. Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 30. BTP Presentation IQR-2D-RLS Computational Complexity Akshay Soni (148) & Tanvi Sharma (196)Table: Operation Count Per Iteration for L = 2 and K sub-carriers Supervisor : Prof. Vijaykumar Chakka 2D-RLS IQR-Givens IQR-Householders 20K 2 + 6K + 2 20K 2 + 6K + 3 18K 2 + 3K + 1 OFDM Multicarrier Communication It is observed from above Table that operation count for Basics Diagram 2D-RLS algorithm and IQR-2D-RLS algorithm using OFDM Channel Estimation Givens Rotations are similar. 2D-RLS IQR-2D-RLS But fewer operations per iteration are required for Stability Simulations Householder Transformations than Givens Rotations. Conclusion 2D-SM-NLMS Simulations Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 31. BTP Presentation IQR-2D-RLS BER Performance Akshay Soni (148) & Tanvi Sharma 0 10 (196) Supervisor : Prof. Vijaykumar −1 10 Chakka OFDM −2 10 Multicarrier Communication BER Basics Diagram −3 10 OFDM Channel Estimation −4 2D-RLS 10 IQR-2D-RLS IQR−2D−RLS Householder Transformation Stability 2D−RLS IQR−2D−RLS Givens Rotations Simulations −5 10 Conclusion 2 4 6 8 10 12 14 16 18 20 2D-SM-NLMS SNR (in dB) Simulations Conclusion MIMO RelayFigure: BER Performance of 2D-RLS and IQR-2D-RLS Algorithms System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 32. BTP PresentationIQR-2D-RLS NMSE Performance Akshay Soni (148) & Tanvi Sharma 0 10 (196) IQR−2D−RLS Householder Transformation Supervisor : Prof. 2D−RLS IQR−2D−RLS Givens Rotations Vijaykumar Chakka −1 10 OFDM Multicarrier Communication NMSE −2 10 Basics Diagram OFDM Channel Estimation −3 10 2D-RLS IQR-2D-RLS Stability Simulations −4 10 Conclusion 2 4 6 8 10 20 40 70 100 2D-SM-NLMS Iteration Simulations Conclusion MIMO Relay Figure: NMSE Performance of 2D-RLS and IQR-2D-RLS at System Model SNR 10 dB Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 33. BTP PresentationIQR-2D-RLS Stability Performance Akshay Soni (148) & Tanvi Sharma 20 10 (196) Supervisor : Prof. Vijaykumar Chakka 15 10 OFDM Multicarrier Communication 10 10 Basics Diagram OFDM Channel 5 Estimation 10 2D-RLS IQR-2D-RLS 2D−RLS IQR−2D−RLS (Householders Transformation) Stability IQR−2D−RLS (Givens Rotation) Simulations 0 10 Conclusion 0 20 40 60 80 100 120 140 160 180 200 2D-SM-NLMS Simulations Conclusion Figure: Condition Number Result for 2D-RLS and MIMO Relay System Model IQR-2D-RLS Algorithms Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work
- 34. BTP PresentationIQR-2D-RLS Conclusion Akshay Soni (148) & Tanvi Sharma Due to smaller condition number, the matrix Q in (196) Supervisor : Prof. IQR-2D-RLS algorithm is close to non-singularity and Vijaykumar Chakka hence proposed algorithm is numerically more stable than 2D-RLS algorithm. OFDM Multicarrier Communication Basics Diagram OFDM Channel Estimation 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
- 35. BTP PresentationIQR-2D-RLS Conclusion Akshay Soni (148) & Tanvi Sharma Due to smaller condition number, the matrix Q in (196) Supervisor : Prof. IQR-2D-RLS algorithm is close to non-singularity and Vijaykumar Chakka hence proposed algorithm is numerically more stable than 2D-RLS algorithm. OFDM Multicarrier Communication Also, both the algorithms have computational Basics complexity of O(N 2 ). MATLAB simulations show that Diagram OFDM Channel IQR-2D-RLS and 2D-RLS algorithms have similar BER Estimation 2D-RLS performance. 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
- 36. BTP PresentationIQR-2D-RLS Conclusion Akshay Soni (148) & Tanvi Sharma Due to smaller condition number, the matrix Q in (196) Supervisor : Prof. IQR-2D-RLS algorithm is close to non-singularity and Vijaykumar Chakka hence proposed algorithm is numerically more stable than 2D-RLS algorithm. OFDM Multicarrier Communication Also, both the algorithms have computational Basics complexity of O(N 2 ). MATLAB simulations show that Diagram OFDM Channel IQR-2D-RLS and 2D-RLS algorithms have similar BER Estimation 2D-RLS performance. IQR-2D-RLS Stability NMSE performance shows that convergence rate of Simulations Conclusion IQR-2D-RLS algorithm is slightly less than the 2D-RLS 2D-SM-NLMS Simulations algorithm. Conclusion MIMO Relay System Model Spatial Filter ZF Fiter MMSE Fiter Simulations Conclusion Future Work

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