Worked on implementation and comparision of dierent channel estimation techniques for OFDM systems, alongwith extension to MIMO-OFDM.

1. BTP Presentation
Akshay Soni (148)
& Tanvi Sharma
(196)
Supervisor : Prof.
Vijaykumar
Chakka
BTP Presentation
OFDM
Multicarrier
Communication
Basics
Akshay 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 Presentation
Multicarrier 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 Presentation
Multicarrier 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
OFDM 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 Presentation
2D-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 Presentation
2D-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 Presentation
2D-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 Presentation
2D-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 Presentation
2D-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 Presentation
2D-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 Presentation
2D-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Relay
Figure: BER Performance of 2D-RLS and IQR-2D-RLS Algorithms System Model
Spatial Filter
ZF Fiter
MMSE Fiter
Simulations
Conclusion
Future Work

32. BTP Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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 Presentation
IQR-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