Nonlinear Communications: Achievable Rates, Estimation, and Decoding

Encryption encoding and secrecy codes
Channel encoding
Channel modelling and achievable rates
Channel estimation and symbol detection

.
Nonlinear communications:
achievable rates, encryption, estimation and
decoding
.

N. Kalouptsidis

Dept. of Informatics & Telecommunications, University of Athens

Second Greek Signal Processing Jam

Coworkers: B. Babadi, A. Katsiotis, N. Kolokotronis, G. Mileounis, I.
Sason, V. Tarokh, K. Xenoulis

N. Kalouptsidis SP JAM 2012 Nonlinear communications: achievable rates, encryption, estimation and decoding 1 / 46

Channel encoding

. Outline


Channel encoding




Channel encoding

Requirements:
MIMO and Nonlinearities

Time varying channel
Reliability

Data integrity and
conﬁdentiality
Complexity


Channel encoding

Requirements: Approach:
MIMO and Nonlinearities Sieve structures and finite
memory
Time varying channel Adaptive methods
Reliability Capacity approaching codes

Data integrity and Secrecy codes, encryption
confidentiality
Complexity Simplifications (EM, relaxation,
sparse models and sparsity
aware schemes)


Channel encoding

. Secrecy Codes

Encryption v
m Encoder

Encryption key
k symmetric key Public key
cryptography
Channel
cryptography
Decryption key

Decryption
ˆ
m Decoder
Eavesdropper
y ˆ
y

Symmetric key cryptography: a common secret key is shared by
encoder/decoder
Public key cryptography: Each user has a public key and private
key. Sender encrypts with the public key of receiver. The receiver
decrypts with its own private key

Channel encoding

. Information theoretic secrecy
Symmetric key cryptography:
(2nR , 2nRk , n) randomized encoder: generates codewords
v(m, k) ∼ P (v|m, k) for each message-key pair
(m, k) ∈ [1 : 2nR ] × [1 : 2nRk ]
Decoder: assigns a message m(y, k) to each received vector y and
ˆ
key k
Decoding rule: joint input-output typicality
Performance characteristics:
Probability of error for the secrecy code
n
Pe = P [m(y, k) = m]
ˆ
n 1
Information leakage rate Rl = n I (M ; Y )


Channel encoding

. Information theoretic secrecy

A rate R is achievable at key rate Rk if there is a sequence of
secrecy codes with Pe → 0 and Rl → 0.
n n

Secrecy capacity for the DMC channel C(Rk ): supremum of
achievable rates at key rate Rk
.
Theorem.
.
{ }
CRk = min Rk , max I(V ; Y )
. P (v)

Secure communication is limited by the key rate until saturated by
the channel capacity


Channel encoding

. The wiretap channel
y ˆ
m
Decoder
v Channel
m Encoder
P (y, y |v)
ˆ ˜
y
Eavesdropper

( )
n 1
Information leakage rate: Rl = n I M ; Y ˜
If the channel to the eavesdropper is a physically degraded version
of the channel to the receiver

P (y, y |v) = P (y|v)P (˜|y)
˜ y

then the secrecy capacity is:
( )
Cs = max I(V ; Y ) − I(V ; Y )
˜
P (v)


Channel encoding

Encryption u Channel v
m
Encoder Encoder

Encryption key
k symmetric key Public key
cryptography cryptography Channel

Decryption key

ˆ
m Decryption Channel
Eavesdropper
Decoder ˆ
u Decoder y ˜
y


Channel encoding

Encryption Encoder: one way function u = E(m, k)
Channel Encoder: (typically linear) adds redundancy to
combat the channel noise.
Channel Decoder: u = maxu P (u|y, k)
ˆ
Decryption Decoder: m = E −1 (ˆ, k)
ˆ u
Both must computationally tractable
Eavesdropper (symmetric key):

max P (k|˜),
y max P (m|˜)
y
k m

must be computationally hard


Channel encoding

. Public key cryptography

Encryption encoder: u = G(m, kpub )
Decryption decoder: m = G−1 (ˆ, ksec )
ˆ u
Eavesdropper:

max P (ksec |˜, kpub )
y
ksec
max P (m|˜, kpub )
y
m


Channel encoding

. McEliece public key encryption scheme

Key generation
k, n, t ﬁxed common integers
choose k × n matrix G which can correct t errors and for
which an eﬃcient decoding algorithm is known (RS codes +
BM decoding, LDPC + sum product)
Draw k × k non-singular S.
Draw n × n permutation P .
ˆ
Compute G = SGP
ˆ
Public key (G, t)
Private key (S, G, P )


Channel encoding

Encryption
ˆ
Use the key of the intended recipient k = (G, t)
Represent the message m as a binary vector of length k
Draw a random binary vector z of weight t
ˆ
Compute u = E(m, k) = mG + z
Decryption
Compute uP −1 = mSG + zP −1 . Note w(zP −1 ) = w(z)
because P permutation.
Use the decoding algorithm to determine mS
Compute mSS −1 = m


Channel encoding

. Fast decoding of regular LDPC codes
(n, k) linear block code with parity check matrix H
Codewords: n dimensional binary vectors: vH T = 0
Syndrome s = rH T , r received vector
error e = r − v satisﬁes

s = eH T
Number of errors: = e 0 << n
Minimum distance decoding:

min e 0 subject to s = eH t

min e 1 subject to s = eH t

“Kalouptsidis, Kolokotronis. Fast decoding of regular LDPC codes using greedy approximation algorithms.”

Channel encoding

The error determines an epsilon sparse representation of the
syndrome vector in the dictionary generated by the columns of H.
The proposed algorithm is motivated by Matching Pursuit but
operates mostly over finite fields
Basic idea: Select columns of H mostly correlated with residual:

⊕
v
ˆ
sv = hλi ∈ F2
m

i=1
Performance guarantees for regular (γ, ρ) LDPC codes
H is sparse
Each column contains γ ones
Each row contains ρ ones
The minimum distance of the code satisfies
dmin ≥ γ + 1
The code can correct ≤ |γ/2|

Channel encoding

input: parity–check matrix H, received word r, maximum
number ν of iterations
initialization: Λ = ∅, i = 0
1. s = rH t mod 2 syndrome
2. While (s = 0) ∧ (i < ν)
3. λ ∈ arg max{ s, hω : ω ∈ Λ}
/ choose randomly
4. s = s ⊕ hλ
5. Λ = Λ ∪ {λ}
6. i=i+1
7. End
output: residual s, error locations Λ


Channel encoding

.
Theorem.
.
Let C be a (γ, ρ)–regular LDPC (n, k) code. The proposed
algorithm is capable of correcting all error patterns e satisfying
γ
≤
2
where
. = e 1.


Channel encoding


Channel encoding

. NL AWGN

Multi Input Multi Output nonlinear channel with
Additive White Gaussian Noise

y(t) =D[v](t) + ξ(t)
ξ(t) : i.i.d ∼ N (0, Q) , Q>0

Channel Operator D: shift invariant, causal, BIBO stable with
fading memory
Then D can be approximated by a ﬁnite memory architecture


Channel encoding

. NL AWGN

v1 (t) Shift Register 1
.
.
··· .

v2 (t) Shift Register 2 . yi (t)
.
. hi
···
.
. .
.
. .
.
.
.
vT (t) Shift Register T

···

Canonical ﬁnite memory form


Channel encoding

Modeling options: nonparametric, semi–nonparametric,
parametric
Focus of this presentation: parametric forms
Each function hi (·) is a polynomial in several variables
More generally, hi (·) is approximated by a member of a sieve
family

Examples of linear sieves: Tensor products of Fourier series,
splines, wavelets
Consequence: Models linear in the parameters

Examples of nonlinear sieves: Neural Networks, Radial Basis


Channel encoding

. Polynomial AWGN
Multi–index i = (i1 , . . . , i ) ir ∈ N and xi = xi1 xi2 · · · xik
Then ∑
h(xi1 , xi2 , . . . , xik ) = hi xi
i∈I
Each output:
∑∑∑
L
(j,k)
yj (t) = hi vk1 (t − i1 )vk2 (t − i2 ) · · · vk (t − i )
=1 k∈K i∈I
∑T ∑q (j,k)
Examples: Linear Systems yj (t) = k=1 i=0 hi vk (t − i)
Quadratic
∑∑
T q
(j,k)
∑ ∑ ∑ ∑
T T q q
(j,k ,k2 )
yj (t) = hi uk (t−i)+ hi1 i21 vk1 (t−i1 )vk2 (t−i2 )
k=1 i=0 k1 =1 k2 =1 i1 =0 i2 =0

(j,k)
Sparsity: Most of the coeﬃcients hi in each hj (·) are zero.

Channel encoding

. Achievable rates for SISO polynomial channels
SISO polynomial channel:

yt = D[v]t + ξt
∑∑
L q ∑
q
D[v]t = h0 + ... hj (i1 , . . . , ij )vt−i1 · · · vt−ij
j=1 i1 =0 ij =0

ξt ∼ N (0, σ 2 )

Let vm denote the transmitted codeword and y the received
vector. A maximum likelihood error occurs if

P (y|vm ) ≥ P (y|vm ), for some m = m


Channel encoding

Since noise is Gaussian, this is equivalent to

Dvm − Dvm + ξ 2
2 ≤ ξ 2
2

Chernoﬀ’s bound and Gallager’s upper bound imply for a speciﬁc
(N, R) code C
∑ ( )
Dvm − Dvm 2
Pe (m|C) ≤ exp −ρ 2
,0 ≤ ρ ≤ 1
8σ 2
m =m

Using a random coding argument, the average error probability
over an ensemble of codes C we obtain
( ) [ ρ ∑N ]
N R− ρ2 N Dv (Q)
Pe ≤ e 4σ E e 8σ2 i=1 Zi , 0 ≤ ρ ≤ 1


Channel encoding

Ft the minimal σ-algebra generated by V1 , V1 , . . . , Vt , Vt
˜ ˜
(F0 {∅, Ω}). (Zt , Ft ) martingale diﬀerence sequence:
   
∑t+q ∑
t+q
Zt −E  wj Ft  + E 
2
wj Ft−1  , wj [Dv]j − [D˜]j
2
v
j=t j=t

1 ∑( [ ] )
n
Output covariance: Dv (Q) E ([Dv]j )2 − (E[Dv]j )2
n
j=1

Martingales enable the development of several concentration
inequalities. For instance Bennett’s inequality:
[ ( )] ( )N
ρ ∑
ρd ρd
γ2 e 8σ2 + e−γ2 8σ2
N
E exp Zi ≤
8σ 2 1 + γ2
i=1


Channel encoding

Suppose
[ ] µ2
max |Zi | ≤ d, max E Zi2 |Fi−1 ≤ µ2 , γ2
vi ,˜i ,j≤i
v vj ,˜j ,j≤i−1
v d2
Then, the bound on average error probability becomes:
( [ ])
P e ≤ exp −N R2 (σ 2 ) − R
 ( 1+γ )

 ( ) d 2
γ2 e 8σ2 −1
 D γ2 2Dv (Q) 1 2Dv (Q)
2 KL 1+γ2 + d(1+γ2 ) 1+γ2 , d < 1+γ2
R2 (σ ) = max d
8σ 2
Q 
1+γ2 e
 Dv (Q)

d
8σ 2 +e
−γ2 d2
− ln γ2 e 1+γ2
8σ
4σ 2 , otherwise

DKL the binary Kullback-Leibler divergence
DKL (p||q) = p log p/q + (1 − p) log(1 − p)/(1 − q)

“Xenoulis,Kalouptsidis,Sason. New achievable rates for nonlinear Volterra channels
via martingale inequalities.”

Channel encoding

Example: Discrete Memoryless binary input AWGN channel
(input: u ∈ {−A, A} with Q(u = A) = α, SNR σ2 ) A

( )
R2 (SNR) = ln 2 − ln 1 + e− 2
SNR
in nats per channel use
0.7
Achievable rates in nats per channel use

0.6

0.5

0.4

0.3
Capacity
0.2
R2 SNR

0.1

0 1 2 3 4 5 6 7 8 9 10
SNR


Channel encoding

. Sparse joint channel state and parameter estimation

Joint state and parameter estimation
Blind estimation via EM and smoothing algorithms


Channel encoding

. Joint detection and estimation
Alternating state estimation and training based parameter
estimation
Channel parameter: θ = [h, Q]
Channel input output form: y t = h(xt ) + ξt
xt = [vt , . . . , vt−q ]

Maximum Likelihood
max log P (y 1:n |x1:n ; θ)
v∈C,θ
= max max log P (y 1:n |x1:n ; θ)
θ v∈C

= max max log P (y 1:n |x1:n ; θ)
θ x1:n ∈S


Channel encoding

Stage 1: State Estimation
Parameter estimate at step i : θ (i) = [h(i) , Q(i) ]
State estimate: v = arg max log P (y 1:n |x1:n ; θ (i) )
ˆ
x1:n ∈S

Convolutional codes of memory < q lead to a Hidden Markov
Process (HMP) (y 1:n , x1:n )
The HMP framework implies
∑
n
ˆ
xi = arg max log P (y t |xt )
x1:n ∈S
t=1

Optimization can be carried out by dynamic programming and the
Viterbi algorithm.

Several relaxations over the real numbers are available for special
cases (decoding by linear programming, semideﬁnite relaxation)

Channel encoding

Example: Binary Input Memoryless channel

∑
n ∑
log P (y t |xt ) = vt γt
t=1
P (y t |1)
γt =
P (y t |0)

Relax the constraints by replacing the convex hull of the codebook
by the intersection of the convex hulls of the parity check
equations.
The problem is converted to a linear programming problem
Decoding by the interior point algorithm


Channel encoding

STEP 2: Parameter Estimation
Given the transmitted message estimate, the decoder updates
channel parameters by the rule
(i+1)
θ (i+1) = arg max P (y 1:n |x1:n ; θ)
θ


Channel encoding

The solutions are given as follows:
. Table look up
1
∑n
t=1 yt δ(xt , x)
h(x) = ∑n
ˆ
t δ(xt , x)
∑(
n )( )H
ˆ = 1
Q ˆ ˆ
y(t) − h(x) y(t) − h(x)
n
t=1
(∑n ) ∑n
.
2 h linear: H ˆ
t=1 xt xt h = t=1 yt xt
(∑n )
.
3 h polynomial: ˆ ∑n yt φ(xt )
H h=
t=1 φ(xt )φ(xt ) t=1
where φ(xt ) = [xt , ⊗2 xt , . . . , ⊗L xt ] with
xt = [v1 (t), . . . , v1 (t − q), · · · , vT (t), . . . , vT (t − q)]


Channel encoding

Adding a sparsity term in the likelihood

θ (i+1) = arg max P (y 1:n |x1:n ; θ) + γ h 1
θ

a convex program results that can be solved by CS methods.


Channel encoding

. Greedy algorithms: CoSaMP/SP
The main ingredients of the CoSaMP/SP algorithms are outlined
below:
.
1 locate the largest components of the proxy

.
2 form a union of two sets of indices

.
3 estimation via LS on the merged set

.
4 prune the LS estimates to s largest components

.
5 updates the error residual

The proposed algorithm modiﬁes the identiﬁcation, estimation and
error residual step. In order to:
sequentially track system variations
reduce the computational complexity
while maintaining the superior performance of CoSaMP/SP

Channel encoding

. The SpAdOMP Algorithm
Algorithm description Complexity
h(0) = 0, w(0) = 0, p(0) = 0
r(0) = y(0)
0<λ≤1
0 < µ < 2λ−1
max
For n := 1, 2, . . . do
1: p(n) = λp(n − 1) + v ∗ (n − 1)r(n − 1) q
2: Ω = supp(p2s (n)) q
3: Λ = Ω ∪ supp(h(n − 1)) s
4: ε(n) = y(n) − v T (n)w|Λ (n − 1)
|Λ s
5: w|Λ (n) = w|Λ (n − 1) + µv ∗ (n)ε(n)
|Λ s
6: Λs = max(|w|Λ (n)|, s) s
7: h|Λs (n) = w|Λs (n), h|Λc (n) = 0
s
8: r(n) = y(n) − v T (n)h(n) s
end For O(q)
“Mileounis,Babadi,Kalouptsidis,Tarokh. An Adaptive Greedy Algorithm With Application to Nonlinear
Communications.”

Channel encoding

. Steady-State MSE of SpAdOMP
.
Theorem.
.
The SpAdOMP algorithm produces an s-sparse approximation
h(n) that satisfies the following steady-state bound

h − h(n) 2 C1 (n) ξ(n) 2 + C2 (n) v |Λ (n) 2 |eo (n)|,

where
eo (n) is the estimation error of the optimum Wiener filter
C1 (n), C2 (n) are constants independent of h
.
The first term is analogous to the SS error of the
CoSaMP/SP algorithm
The second term is induced by performing a single LMS
iteration (instead of using the LS estimate)

Channel encoding

. Simulations on sparse ARMA channels
ARMA channel: yn = a1 yn−6 + a2 yn−48 + vn + b1 vn−13 + b2 vn−34 + ξn ,
and 500 samples from CN (0, 1/5).
0 0.2
LMS
−5 0.1 LOG−LMS
SpAdOMP
−10 0
NMSE (dB)

NMSE (dB)
−15 −0.1

−20 −0.2

−25 −0.3

LMS
−30 −0.4
LOG−LMS
SpAdOMP
−35 −0.5
0 200 400 600 800 1000 250 300 350 400 450 500
Iterations Iterations

a. Learning curve (SNR=23dB) b. Time evolution of (a1 )
( )
NMSE = 10 log10 E{ h(n) − h 22 }/E{ h 22 }
SpAdOMP converges very fast
SpAdOMP achieves an average gain of nearly 19dB

Channel encoding

. Blind estimation via EM and smoothing algorithms

Transmitted sequence unknown
Likelihood maximization is intractable
The Expectation Maximization (EM) method and the
underlying iterative algorithm provides an option that
inherently addresses symbol detection
Augmented likelihood function formed by the state and
received sequence. Given Q(θ, θ ) the expectation step forms

Q(θ, θ ) = Eθ {log P (x1:n , y 1:n ; θ)|y 1:n }

Expectation over the state sequence given the received
sequence.


Channel encoding

Marginal output likelihood

Ln (θ) = log P (y 1:n ; θ)

Jensens inequality implies

Ln (θ) − Ln (θ ) ≥ Q(θ, θ ) − Q(θ, θ)

This suggests the second step
Let θ (i) denote an estimate at step i. Then

θ (i+1) = arg max Q(θ i , θ) and Ln (θ (i+1) ) ≥ Ln (θ i )
θ

For Gaussian noise, P (y(t)|x(t)) is log concave, the minimizer of
Q is unique and the sequence θ i converges to a stationary point of
the likelihood

Channel encoding

The EM method leads to the following estimates
∑n
ˆ (i+1) (x) = ∑ P (xt = x; θ |y 1:n )yt
(i)
t=1
h n
t=1 P (xt = x; θ |y 1:n )
(i)

∑ |M | ( )( )H
q+1
n ∑
ˆ (i+1) = 1
Q ˆ ˆ
P (xt = x; θ (i) |y 1:n ) y t − h(xl ) y t − h(xl )
n t=1
l=1

Determination of the smoothing probabilities P (xt |y 1:n ) by the
forward backward recursions (Chang and Hancock)
P (xt , y 1:n ) = α(xt , y 1:t )β(y t+1:n |xt )
∑
M
α(xt , y 1:t ) = b(yt |xt ) α(xt−1 , y 1:t−1 )αxt−1 xt
xt−1 =1

∑
M
β(y t+1:n |xt ) = αxt xt+1 β(y t+2:n |xt+1 )b(yt+1 |xt+1 )
xt+1 =1


Channel encoding

. The Sparse BW Algorithm

Algorithmic description
For := 0, 1, . . . , do
1: {r ( ) , R( ) }:=Run {Forward/Backward recursions} {symbol detector}
sgn(r i ) [ ( ) ]
( )
( +1)
2: hi = ( )
|r i | − γ
2 {channel estimator}
Ri,i +

2 ( +1)
∑
n 2
3: σq =n 1
yn − x( +1)T h( +1)
n {noise variance estima
n=1
end For

∑
n
( ) ∑
n
( )
∗ ˆ ˆ
r( ) = yi E{xi |y n ; h }, R( ) = E{xi xH |y n ; h }
i
i=1 i=1

“Mileounis,Kalouptsidis,Babadi,Tarokh. Blind identiﬁcation of sparse channels
and symbol detection via the EM algorithm.”

Channel encoding

Adaptive Channel Coding Based on Flexible Trellis
. (Convolutional) Codes
Popular adaptive coding schemes → variable rate punctured
convolutional codes
IEEE 802.22 standard for cognitive WRAN uses a rate 1/2
convolutional code and a set of puncturing matrices that lead
to rates 2/3, 3/4 and 5/6.

Flexible Convolutional Codes

They can vary both their rate and the decoding complexity →
eﬃcient management of the system resources.
Constructed by combining the techniques of path pruning and
puncturing.
Varying quantities associated with the complexity proﬁle of
the trellis diagram→ Varying decoding complexity.

Channel encoding

. Flexible Convolutional Codes
Consider an (n, 1, m) mother convolutional code.
Let ut be the information bit and ut the input bit of the
ˆ
mother encoder at time instant t.
every Tpr time units the single input bit of the encoder is not
an information bit, rather it is computed as a linear
combination of bits of the current state
St = {ˆt−1 , · · · , ut−m }.
u ˆ
{
ut1 (Tpr −1)+t2 , if t2 = 0
ut =
ˆ ∑d ˆ
i=1 ci ut1 Tpr −i , if t2 = 0
ˆ
⌊ ⌋
ˆ
where t1 = Tt , t2 = t mod Tpr , t = 1, 2, . . . , and d is the
pr
∑m
degree of the polynomial c(X) = i=1 ci X i .
“Katsiotis,Rizomiliotis,Kalouptsidis. Flexible Convolutional Codes: Variable Rate and Complexity.”

Channel encoding

. Flexible Convolutional Codes

The ﬁnal step involves periodic puncturing of the encoded bits
with period Tpu = pTpr , in order to adjust the rate.

The complexity proﬁle of the resulting trellis depends solely on
ˆ
the parameters m, Tpr , d and the puncturing matrix.

Large families of high-performance codes of various rates and
values of decoding complexity are constructed.


Channel encoding

. Extending the Constructions
Flexible Turbo Codes

Extending the analysis in the case where recursive mother
encoders are used.
The goal is to construct flexible parallel concatenated
powerful coding schemes.
Preliminary results indicate that varying the complexity profile
of the trellis can be more efficient than simply varying the
number of decoding iterations.

Flexible Secret Codes

Embedding secret keys in the procedures of pruning and
puncturing can result to robust and flexible secret encoders.

Channel encoding

. Publications
G. Mileounis, N. Kalouptsidis, A sparsity driven approach to cumulant
based identiﬁcation, in IEEE Proc. SPAWC 2012, Turkey.
K. Xenoulis, N. Kalouptsidis, I. Sason, New achievable rates for nonlinear
Volterra channels via martingale inequalities, in IEEE proc. ISIT 2012.
A. Katsiotis, P. Rizomiliotis, and N. Kalouptsidis, ”Flexible Convolutional
Codes: Variable Rate and Complexity,” IEEE Trans. Commun., vol. 60,
no. 3, pp. 608-613, March 2012.
N. Kalouptsidis, G. Mileounis, B.Babadi, and V. Tarokh, ”Adaptive
Algorithms for Sparse System Identiﬁcation,” Signal Process., vol. 91, no.
8, pp. 1910-1919, Aug. 2011.
K. Xenoulis and N. Kalouptsidis, ”Tight performance bounds for
permutation invariant binary linear block codes over symmetric channels,
IEEE Trans. Inf. Theory, vol. 57, pp. 6015-6024, Sep. 2011.
K. Limniotis, N. Kolokotronis, and N. Kalouptsidis, ”Constructing
Boolean functions in odd number of variables with maximum algebraic
immunity,” in proc. 2011 IEEE ISIT, pp. 2662-2666, 2011.

Channel encoding

. Publications (contd.)
N. Kalouptsidis and N. Kolokotronis, ”Fast decoding of regular LDPC
codes using greedy approximation algorithms,” in proc. 2011 IEEE ISIT,
pp. 2011-2015, 2011.
A. Katsiotis and N. Kalouptsidis, ”On (n, n-1) punctured convolutional
codes and their trellis modules,” IEEE Trans. Commun., vol. 59, pp.
1213-1217, 2011.
K. Xenoulis and N. Kalouptsidis, ”Achievable rates for nonlinear Volterra
channels,” IEEE Trans. Inform. Theory, vol. 57, pp. 1237-1248, 2011.
A. Katsiotis, P. Rizomiliotis, and N. Kalouptsidis, ”New constructions of
high-performance low-complexity convolutional codes,” IEEE Trans.
Commun., vol. 58, pp.1950-1961, 2010.
G. Mileounis, B. Babadi, N. Kalouptsidis, and V. Tarokh, ”An Adaptive
Greedy Algorithm with Application to Nonlinear Communications,” IEEE
Trans. Signal Proc., vol. 58, No. 6, June 2010.
B. Babadi, N. Kalouptsidis, and V. Tarokh, ”SPARLS: The Sparse RLS
Algorithm,” IEEE Trans. Signal Proc., vol. 58, no. 8, August 2010.

Channel encoding

. Publications (contd.)
N. Kolokotronis, K. Limniotis, and N. Kalouptsidis, ”Best aﬃne and
quadratic approximations of particular classes of Boolean functions, IEEE
Trans. Inform. Theory, vol. 55, pp. 5211-5222, 2009.
T. Etzion, N. Kalouptsidis, N. Kolokotronis, K. Limniotis and K. G.
Paterson, ”Properties of the error linear complexity spectrum,” IEEE
Trans. Inform. Theory, pp. 4681-4686, vol. 55, 2009.
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