Channel and clipping level estimation for ofdm in io t –based networks a review

T. K Ranjith, M. Sini, Rasheed Sareena Abdul, International Journal of Advance Research, Ideas and Innovations in
Technology.
© 2017, www.IJARIIT.com All Rights Reserved Page | 1444
ISSN: 2454-132X
Impact factor: 4.295
(Volume 3, Issue 6)
Available online at www.ijariit.com
Channel and Clipping Level Estimation for OFDM in
IoT –Based Networks: A Review
Ranjith T. K
S. N. Polytechnic, Kanhangad, Kerala
ranjith.falcon@gmail.com
Sini .M
JDT Polytechnic, Kozhikkode, Kerala
sinimrafi@gmail.com
Sareena Abdul Rasheed
JDT Polytechnic, Kozhikkode, Kerala
ar22sare@gmail.com
Abstract: Internet of Things (IoT) is the idea to connect all devices to the internet. To implement such systems, we need to design
low cost and less complex transmitters and make the receiver side complex. Now days OFDM is mainly used for communication
due to its great advantages. But it faces the main problem such as PAPR due to the non-linear performance of High power
amplifiers. There are so many methods are available to reduce the effect of PAPR in OFDM transmission, among this clipping
is the simplest one. In this paper, we propose two algorithms to find the clipping level as well as the channel estimation. The
efficiency of these algorithms is evaluated by using CLRB calculation.
Keywords: OFDM, PAPR, Clipping, Channel, Estimation.
INTRODUCTION
Orthogonal Frequency Division Multiplexing or OFDM is a modulation technique that is used for many wireless and
telecommunications standards. It is a form of the multicarrier modulation method. An OFDM signal consists of a number of closely
spaced modulated carriers. These carriers are mutually orthogonal. When modulation of any form - voice, data, etc. is applied to a
carrier, then sidebands spread out either side. It is necessary for a receiver to be able to receive the whole signal to be able to
successfully demodulate the data. As a result, when signals are transmitted close to one another they must be spaced so that the
receiver can separate them using a filter and there must be a guard band between them. This is not the case with OFDM. Although
the sidebands from each carrier overlap, they can still be received without the interference that might be expected because they are
orthogonal to each another. This is achieved by having the carrier spacing equal to the reciprocal of the symbol period.
But this system suffers from envelope fluctuations producing a problem called Peak-to-Average Power Ratio (PAPR). This
leads to an unwelcome tradeoff between the linearity of the transmitted signal and the cost of the High Power Amplifier (HPA). A
high PAPR causes the HPA to work in the saturation region which introduces nonlinear distortion at the transmitter output.
PAPR (Peak-to-Average Power Ratio)
When the independently modulated subcarriers are added coherently, the instantaneous power will be more than the average power.
High PAPR degrades the performance of OFDM signals by forcing the amplifier to work in a nonlinear manner, distorting this way
the signal and making the amplifier to consume more power. The PAPR for the continuous-time signal x (t) is the ratio of the
maximum instantaneous power to the average power. For the discrete-time version x[n], PAPR is expressed as,
PAPR(x[n]) = 𝑚𝑎𝑥0≤𝑛≤𝑁
|𝑥(𝑛)|2
𝐸|𝑥(𝑛)|2

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The performance of a PAPR reduction scheme is usually described by three main factors: the complementary cumulative distributive
function (CCDF), bit error rate (BER), and spectral spreading. There are also other factors to be considered such as transmitted
signal power, computational complexity, and bandwidth expansion and data rate loss.
PAPR reduction techniques can be broadly classified into three main categories:
 Signal distortion techniques,
 Multiple signaling and probabilistic techniques and
 Coding techniques
PAPR Reduction Techniques
System Model
Consider an OFDM system with N number of subcarriers, in which S= [𝑠0, 𝑠1, 𝑠2 , … . 𝑠 𝑁−1] 𝑇
is the frequency domain symbol vector
selected from a constellation such as QAM. The time-domain symbols are obtained by taking the Inverse Discrete Fourier Transform
(IDFT) from the frequency-domain symbols as:
𝑥 𝑛 =
1
√𝑁
∑ 𝑠 𝑘 𝑒
−𝑗2𝜋𝑛
𝑁𝑁−1
𝑘=0 , n = 0, 1, 2,…., N-1 (1)
Rewrite this equation in matrix for as:
x = F 𝐻
s (2)

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Where ‘F’ is the N× 𝑁 unitary discrete Fourier transform matrix. The channel is slow fading with L+1 taps (L<<N), denoted as h
= [ℎ0, ℎ1, ℎ2 , … . ℎ 𝐿] 𝑇
, g(. ; A) is the limiter non linearity with the following amplitude modulation and phase modulation
conversion characteristics:
𝐹(𝑟) = {
𝑟, 𝑟 ≤ 𝐴
𝐴, > 𝐴
(3)
∅(𝑟) = 0 (4)
Where r is the magnitude of the limiter input signal, and A is the CA. Combining AM/AM and AM/PM conversion characteristics,
g(u; A) as a function of complex scalar u (and parameterized by A) can be written as,
𝑔(𝑢; 𝐴) = {
𝑢, |𝑢| ≤ 𝐴
𝐴𝑒 𝑗 arg(𝑢)
, |𝑢| > 𝐴
(5)
The output of the limiter is z = g(x; A), in which g(: ;A) is taken element-wise. However, it is difficult to directly work with the
output of the limiter, where the output of the limiter can be represented in a linear fashion by introducing N augmented binary
variables 𝑐 𝑛 for 𝑛 ∈ {0, 1 , … . , 𝑁 − 1} indicating whether the sample at time n has been clipped (𝑐 𝑛 = 1) or not (𝑐 𝑛 = 0), i.e.
𝑐 𝑛 = {
1, 𝑟𝑛 > 𝐴
0, 𝑟𝑛 ≤ 𝐴
(6)
Which lead to,
𝑧 𝑛=(1− 𝑐 𝑛 )𝑥 𝑛+𝐴𝑐 𝑛 𝑒 𝑗∅ 𝑛 n= 0,1,…,N-1, (7)
It can be represented in vector form
𝑧 = (1 − 𝑐) ⊙ 𝑥 + 𝐴𝑐 ⊙ 𝑒 𝑗∅
(8)
To remove the Inter-Symbol Interference (ISI), a Cyclic Prefix (CP) with a length Lcp (≥ L) is pre-added to the time-domain
symbols at the transmitter and is removed at the receiver. After the process of adding and removing CP, the matrix form time-
domain representation of the OFDM transmission can be written as:
u =Hz + w (9)
Where H is an N × N circulant matrix whose first column which represents the circular convolution operator, and ‘w’ is a zero-
mean circularly symmetric complex Gaussian noise vector, Taking the DFT from both sides of (9), we obtain the frequency-domain
representation of the OFDM transmission as:
y = Fu
= √𝑁 𝐷 𝐻 𝐹𝑧 + 𝑤̃ (10)
Where 𝐷 𝐻 is a diagonal matrix with the N-point DFT of h as its diagonal elements, 𝑤̃ is the DFT of time – domain noise vector.
Joint Channel and Clipping Amplitude Estimation
Here propose an alternating optimization algorithm to jointly estimate the channel and CA. To do this, we use frequency-domain
block-type training symbols. Once we have the estimates, we can use them to detect the transmitted symbols in the subsequent
OFDM blocks. The intuition behind this algorithm is that CA is a slowly time-varying parameter (much slower than channel
variations). Moreover, wireless channels are usually slow time varying, so the block-fading channel model, which remains the same
during the transmission of several OFDM blocks is reasonable. Using frequency-domain block-type training symbols, the problem

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of joint Maximum-Likelihood (ML) estimation of channel and CA can be formulated as the following Least- Squares (LS) using
(10):
min
𝐴>0,ℎ
‖𝑦 − √𝑁𝐷 𝐻 𝐹𝑧‖
2
(11)
Where ‖. ‖denotes the Euclidean norm.
A. Clipping Amplitude Estimation Given the Channel
First, solve (11) for A given h. To do so, we first sort the elements in r and construct a new vector denoted as𝑟̇. Note that, 𝑟̇ . = Ps r,
in which Ps is the sorting permutation matrix. Using the sorted vectors, the estimate of CA at the 𝑖 𝑡ℎ
iteration can be written as,
𝐴 = 𝑎𝑟𝑔 min
𝐴>0
‖𝑦 − √𝑁 𝐷 𝐻
𝑖
𝐹𝑃𝑠
𝑇 𝑧̇‖
2̂
(12)
Where
𝑧 = 𝑥 ⊙ (1 − 𝑐̇) + 𝐴𝑒 𝑗∅ ⊙ 𝑐̇̇̇
(13)
In which ˚x = Psx, ˚c = Psc, and φ˚ = Psφ. Note that ˚c is constant within each interval ˚rk−1 ≤ A ≤ ˚rk, k = 0 . . . N −1,
consequently (13) is piecewise affine in A, therefore the minimization problem (12) can be written as
Â = 𝑎𝑟𝑔 min
𝐴>0
𝐽(𝐴) (14)
Where
𝐽(𝐴) = {
𝐽0(𝐴), 0 ≤ 𝐴 ≤ 𝑟0̇
𝐽1(𝐴), 𝑟0̇ ≤ 𝐴 ≤ 𝑟1̇
𝐽 𝑁−1(𝐴), 𝑟𝑁−2 ≤ 𝐴 < 𝑟𝑁−1̇̇
(15)
In which
𝐽 𝑘(𝐴) = ‖𝑦 − √𝑁𝐷 𝐻 𝐹𝑃𝑠
𝑇
𝑍 𝑘
̇ ‖
2
= ‖𝐵(𝐶 𝑘 ⊙ 𝑒 𝑗∅̇ )‖
2
𝐴2
− 2ℜ {[𝑦 − 𝐵(𝑥̇ ⊙ 𝐶 𝑘
̇̅̅̅)]
𝐻
𝐵(𝐶 𝑘 ⊙ 𝑒 𝑗∅
)} + ‖𝑦 − 𝐵(𝑥̇ ⊙ 𝐶 𝑘
̇̅̅̅)‖
2
(16)
As we can see, for each interval, (16) is a convex quadratic function of A. Consequently, for each interval, we can easily
find the minimizer by taking the first derivative of (16) and setting it to zero, and check if the solution lies in that interval
or not. If it lies in that interval we have already found the minimizer, otherwise, the minimum takes place at the interval’s end point
with the smaller cost function. Mathematically, for each interval [˚rk−1,˚rk], the minimizer is
Â 𝑘 = {
Ă 𝑘 , 𝑟𝑘−1 ≤ Ă 𝑘 ≤̇ 𝑟𝑘̇
𝑟𝑘−1,̇ 𝐽 𝑘(𝑟𝑘−1) < 𝐽 𝑘(𝑟𝑘)̇̇
𝑟𝑘,̇ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(17)
Where
Ă 𝑘 =
ℜ{[ 𝑦−𝐵(𝑥⊙𝑐 𝑘̇̅̅̅̅̇ ] 𝐻 𝐵(𝑐 𝑘⊙𝑒 𝑗∅)̇ }
‖𝐵(𝑐 𝑘⊙𝑒 𝑗∅)‖
2 (18)

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To find the global minimizer, we find the minimizer for each interval using (17), and then find the minimizer among those
minimizers. Since J (A) is a continuous function, in each interval, we need just check the point at which the derivative is zero
and the rightmost corner point. Therefore, we have
Algorithm1. Estimation of Clipping Amplitude (A)
1. Inputs:
y, r and exp(j∅)
F and 𝐷 𝐻
𝑐 𝑘̇ for k =0, 1, ….N-1
2. Initialize:
[𝑟,̇ 𝑃𝑠] = sort(r)
𝑒 𝑗∅
= 𝑃𝑠 𝑒 𝑗∅
𝐵 = √𝑁𝐷 𝐻 𝐹𝑃𝑠
𝑇
𝑟̇ − 1 = 0
3. for k =0 to N-1 do
4. Ă 𝑘 =
ℜ{[ 𝑦−𝐵(𝑥⊙𝑐 𝑘̇̅̅̅̅̇ ] 𝐻 𝐵(𝑐 𝑘⊙𝑒 𝑗∅)̇ }
‖𝐵(𝑐 𝑘⊙𝑒 𝑗∅)‖
2
5. if 𝑟𝑘−1 <̇ Ă 𝑘 < 𝑟𝑘̇ then
6. Â 𝑘 = Ă 𝑘
7. 𝐞𝐥𝐬𝐞
8. Â 𝑘 = 𝑟𝑘̇
9. end if
10. 𝑗̂ 𝑘 = 𝐽 𝐾(Â 𝑘 )
11. end for
12. 𝑘̂ = 𝑎𝑟𝑔 min
0≤𝑘≤𝑁−1
𝐽̂ 𝐾
13. Ậ = Â 𝑘
B. Channel Estimation gave Clipping Amplitude
Here, we solve (11) for h given A. We can rewrite (11) as:
min
ℎ
‖y − √N diag(Fz)F̆h‖
2
(19)
Therefore, the LS channel estimate can be computed as:
ĥ =
1
√N
(FH̆ diag(Fz)H
) diag(Fz)F̆)ϯ
FH̆ diag(Fz)H
y (20)
Where (. )+
denotes the Moore–Penrose pseudo–inverse.
C. Initialization of the Alternating Algorithm
Two alternative initialization strategies to the alternating optimization algorithm as follows:
1) Initializing by the Channel: Since the value of A is unknown at the beginning, the channel can be estimated using the un-
clipped version of transmitted time-domain symbols. It is equivalent to putting x instead of z in (20), and using s = Fx.
Therefore, the initializing channel estimate is:
h(0)̂ =
1
√N
(FH̆ diag(s)H
diag(s)F̆)−1
FH̆ diag(s)H
y (21)

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2) Initializing by the Clipping Amplitude: By substituting (20) into the cost function in (19), the resulting LS error for A is:
𝜀(𝐴) = 𝑦 𝐻 (𝐼 − 𝑇)𝑦 (22)
By minimizing (22) over A, the CA can be estimated. However, (22) is a complicated function of A and is not convex. Nevertheless,
one can find its approximate solution by using grid search methods. Therefore, we can find the initializing CA by solving
𝐴(0)̂ = 𝑎𝑟𝑔 min
𝐴>0
𝜀(𝐴) (23)
Due to the non-convexity of 𝜀(A), it is highly possible that even more advanced optimization methods than grid search just find a
local minimum.
D. Alternating Optimization Algorithm
Depending on which initialization is used, we have two alternating algorithms denoted as Algorithms 2 and 3:
E. Convergence
Note that both algorithms of alternating optimization are guaranteed to converge to a local optimum because in every iteration we
can find the unique optimal solution to the optimization problem. In particular, we have the following proposition:
Proposition 1: Both algorithms 2 and 3 converge to a local optimum point of the cost function given in (11) (or (12)). Proof: let us
denote the cost function in (11) by J (h; A), then for Algorithm 2, we have
Ậ(𝑖−1)
= 𝑎𝑟𝑔 min
𝐴>0
𝐽(ĥ(𝑖−1)
, 𝐴) (24)
Algorithm 2 Alternating Optimization with Initializing Channel
1. Inputs:
y, r and exp(j∅)
F
𝑐 𝑘̇ for k =0, 1, ….N-1
2. Initialize:
[𝑟,̇ 𝑃𝑠] = sort(r)
𝑒 𝑗∅
= 𝑃𝑠 𝑒 𝑗∅
𝐷̂ 𝐻
(0)
= 𝑑𝑖𝑎𝑔 (𝐹̆ℎ̂(0)
), ℎ̂(0)
𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦(21)
𝑟̇ − 1 = 0
i=1
3. while(convergence criteria not met ) do
4. B = √𝑁 𝐷̂ 𝐻
(𝑖−1)
𝐹𝑃𝑠
𝑇
5. calculate Ậ(𝑖−1)
𝑎𝑛𝑑 𝑘̂(𝑖−1)
using algorithm 1
6. 𝑧̇ = 𝑥̇ ⊙ (1 − 𝑐̇ 𝑘(𝑖−1)) + Ậ(𝑖−1)
𝑒 𝑗∅
⊙ 𝑐̇ 𝑘(𝑖−1)
7. 𝑉 = 𝑑𝑖𝑎𝑔 (𝐹𝑃𝑠
𝑇
𝑧)𝐹̆̇
8. ℎ̂(𝑖)
= (𝑉 𝐻
𝑉)ϯ
𝑉 𝐻
𝑉
9. 𝐷̂ 𝐻
(𝑖)
= 𝑑𝑖𝑎𝑔(𝐹̆ℎ(𝑖)
)
10. 𝑖 = 𝑖 + 1
11. end while
Therefore
𝐽(ĥ(𝑖−1)
, Ậ(𝑖−1)
) ≤ 𝐽(ĥ(𝑖−1)
, 𝐴), ∀ 𝐴 > 0 (25)
Also, note that,

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ĥ(𝑖)
= arg min
ℎ
𝐽( ℎ, Ậ(𝑖−1)
) (26)
Hence
𝐽( ĥ(𝑖),
, Ậ(𝑖−1)
) ≤ ℎ, Ậ(𝑖−1)
, ∀ℎ ∈ 𝐶 𝐿+1
(27)
Algorithm 3 Alternating Optimization with Initializing CA
1. Inputs:
y, r and exp(j∅)
F
ck̇ for k =0, 1, ….N-1
2. Initialize:
[r,̇ Ps] = sort(r)
ej∅
= Ps ej∅
Â(i)
by solving (23)using a grid search
ṙ − 1 = 0
i=1
3. while(convergence criteria not met ) do
4. V = diag (FPs
T
ż(i−1)
)F̆
5. ĥ(i−1)
= (VH
V)ϯ
VH
y
6. D̂H
(i−1)
= diag(F̆h(i−1)
)
7. B = D̂H
(i−1)
FPs
T
8. calculate Â(i)
and k̂(i)
using algorithm 1
9. i = i + 1
10. end while
Therefore, the objective function is decreasing at each iteration and eventually converges to a local minimum. The same argument
is also valid for Algorithm 3.For Symbol Error Rate (SER) simulations use the iterative detection algorithm introduced in, but with
the estimated channel and CA as follows:
Algorithm 4 Iterative Detection Algorithm
1. Inputs:
y, F, and 𝑁𝑞
ĥ and Ậ calculated by Algorithm 2 (or 3)
2. Initialize
𝐷̂ 𝐻=diag (F̆ ĥ)
k= 1-𝑒−𝐴2
+ Ậ √
𝜋
2
erfc(Ậ)
𝑑(0)
= 0
3. for i=1 to 𝑁𝑞 do
4. 𝑠̂(𝑖)
= 〈
1
𝑘
(𝐷̂ 𝐻
−1
𝑦 − 𝑑(𝑖−1)
)〉
5. 𝑥̂(𝑖)
= 𝐹 𝐻
𝑠̂(𝑖)
6. 𝑑(𝑖)
=F(g(𝑥̂(𝑖)
; Ậ) - 𝑥̂(𝑖)
)

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7. the end of
Cramer-Rao Lower Bound
The Cram´er Rao Lower Bound (CRLB) expresses a lower bound on the achievable variance of unbiased estimators. The ML
estimator asymptotically achieves the CRLB under some regularity conditions. Here, the estimation parameters comprise of
complex-valued channel taps and a real-valued CA. Therefore, to calculate the CRLB, we denote the parameter vector as 𝜃
=[ℎ 𝑇
, ℎ 𝐻
, 𝐴] 𝑇
, which is used to derive the complex Fisher Information Matrix (FIM). The logarithm of the Probability Density
Function (PDF) of the frequency domain observation vector y given in (10) can be written as:
log 𝑝(𝑦: 𝜃) = −
1
𝜎2
‖𝑦 − √𝑁 𝐷 𝐻 𝐹𝑧‖
2
+ constant
Where constant comprises the terms which are independent of the estimation parameter vector 𝜃 the log PDF is a differentiable
function of ‘h’ in the whole parameter space.
𝐶𝐿𝑅𝐵(𝐴) =
1
𝑞 − 2𝑃 𝐻 𝐶1
−1
𝑃
And
𝐶𝐿𝑅𝐵(ℎ) = 𝐶1
−1
+
1
𝑞 − 2𝑃 𝐻 𝐶1
−1
𝑃
𝐶1
−1
𝑃𝑃 𝐻
𝐶1
−𝐻
RESULTS
NMSE performance of the CA estimation using Algorithms
2 and 3, when L + 1 = 7,
N = 128 and CL = 1 dB.
NMSE performance of the channel estimation using
Algorithms 2 and 3, when L+1 = 7
N = 128 and CL = 1dB.
NMSE performance of the CA estimation using Algorithms
2 and 3, when L + 1 = 7,
N = 512 and CL = 3 dB.
NMSE performance of the channel estimation using
Algorithms 2 and 3, when L+1 = 7
N = 512 and CL = 3dB

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SER performance of the iterative detection (Algorithm 4)
vs. SNR, when L + 1 = 7
N =128 and CL = 1 dB.
N = 512 and CL = 1 dB
N = 256 and CL = 3 dB
vs. CL, when L + 1 = 7,
N = 128 and SNR = 20 dB
CONCLUSION
In this, we, studied joint maximum-likelihood estimation of channel and clipping level at the receiver side in future IoT-based
OFDM networks, where there are lots of low-cost low-power nodes transmitting to and receiving from more complex nodes such
as a BTS. In particular, we have proposed two alternating optimization algorithms, in which we have optimally solved a non-smooth
non-convex optimization problem. And also computed the theoretical lower bounds (CRLB) on the performance of these estimators,
and showed that they attain these lower bounds. Next, we have combined the channel and the CA estimates with the iterative
detection method from to perform symbol detection at the receiver. Finally, we have shown by simulations that the performance of
the iterative detection method using the proposed algorithms is almost the same as the one of the cases that the receiver has genie-
aided knowledge of the channel and CA.
REFERENCE
1. Joint Channel and Clipping Level Estimation for OFDM in IoT-based Networks Ehsan Olfat, Student Member, IEEE,
Mats Bengtsson, Senior Member, IEEE,
2. J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang,(2014) “What will 5G
be?” IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp. 1065–1082, June.
3. Y. Rahmatallah and S. Mohan, (2013) “Peak-to-average power ratio reduction in OFDM systems: A survey and
taxonomy,” IEEE Commun. Surveys Tuts., vol. 15, no. 4, pp. 1567–1592, Mar.
4. J. Kim and K. Konstdntinou, (2001) “Digital predistortion of wideband signals based on power amplifier model with
memory,” Electronics Letters, vol. 37, no. 23, pp. 1417–1418, Nov.
5.L. Ding, G. T. Zhou, D. R. Morgan, Z. Ma, J. S. Kenney, J. Kim, , and C. R. Giardina, (2004) “A robust digital base
band pre distorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, no. 1, pp. 159–165, Jan.
6.F. H. Gregorio, S. Werner, J. Cousseau, J. Figueroa, and R. Wichman, (2011) “Receiver-side nonlinearities mitigation
using an extended iterative decision-based technique,” Signal Processing, vol. 91, no. 8, pp. 2042– 2056, Mar.
10.F. Gregorio, T. I. Laakso, and J. E. Cousseau,(2006)“Receiver cancellation of nonlinear power amplifier distortion in
SDMA-OFDM systems,” in Proc. ICASSP, vol. 4, pp. 325–328.

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