A Study of Channel Estimation in Fast Fading Environments

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 4, ISSUE 08, AUGUST 2015 ISSN 2277-8616

196IJSTR©2015www.ijstr.org

A Study Of Channel Estimation In Fast FadingEnvironments

Fatima Abdeen Farah Edris, Dr. Ashraf Gasim Elsid, Dr. Mohamed. H. M. Nerma

Abstract: This paper will be a study of two types of channel estimation techniques (LS & MMSE) that based on comb pilot insertion arrangementin fast fading channels environment which is consist of two types Rayleigh & Rician channel models, and a comparison between the twotechniques will be done depending on different parameter and implement these into a simulation program which will be here MATLAB.

Index Terms: wireless channel, Fast fading, Rayleigh, Rician, comb pilot insertion, channel estimation, OFDM, LS, MMSE.————————————————————

1. INTRODUCTION

In wireless telecommunication environment, the signals thattransfer through it are very susceptible to changing inamplitude and phase due to many factors such as multipathloss which is leading to phenomenon called fading. Fadingrefers to the fluctuations in signals strength when received atthe receiver and Multipath fading is a common phenomenonthat occurs in wireless signal transmission [1]. Wireless

environments change quickly due to reflection, diffraction andscattering and thus introduce a complexity on the channelresponse. Due to the channel response that changed duringthe transmission and the received signal that distorted by thechannel characteristics, the channel effect must be estimatedand compensated in the receiver in order to recover thetransmitted bits. Two main types of fading channel models areused here _in this paper_ Rayleigh and Rician channelmodels, and for channel estimation technique LS and MMSEwill be used. Finally, implement of two channel estimationtechniques in a simulation program, which will be the MATLABprogram and the results will be shown later.

2. Overview of wireless communication

The performance of wireless communication systems is mainlygoverned by the wireless channel environment. As opposed tothe typically static and predictable characteristics of a wiredchannel, the wireless channel is rather dynamic and

unpredictable, which makes an exact analysis of the wirelesscommunication system often difficult [2]. One of the mostdisturbing aspects in the wireless communication is fading,which is present when there are several multipath componentsand these components arrive at the receiver at slightlydifferent times[3]. Two models of fast fading _Rayleigh &Rician_ channels will be studied and reviewed in order tounderstand the fading environments and its effect on thereceived channel response at the receiver.

3. Introduction to fadingFading refers to the distortion that a carrier-modulatedtelecommunication signal experiences over certainpropagation media. In wireless systems, fading is due tomultipath propagation and is sometimes referred to asmultipath induced fading [4]. It’s also known as the fluctuationsin signals strength when received at the receiver [1]. Tounderstand fading, it is essential to understand multipath. In

wireless telecommunications, multipath is the propagationphenomenon that results in radio signals reaching thereceiving antenna by two or more paths, however, causes ofmultipath include atmospheric ducting, ionosphere reflectionand refraction, and reflection from terrestrial objects, such asmountains and buildings. The effects of multipath includeconstructive and destructive interference, and phase shifting othe signal. This distortion of signals caused by multipath isknown as fading. In other words it can be said that in the reaworld, multipath occurs when there is more than one pathavailable for radio signal propagation. The phenomenon oreflection, diffraction and scattering all give rise to additionaradio propagation paths beyond the direct optical LOS (Line oSight) path between the radio transmitter and receiver. Fadingcan be classified into two main types named as fasfading/small-scale fading, and slow fading/large-scale fadingLarge-scale fading occurs as the mobile moves through alarge distance, for example, a distance of the order of cell sizeand small-scale fading refers to rapid variation of signal levelsdue to the constructive and destructive interference of multiplesignal paths (multi-paths) when the mobile station moves shordistances [2].

Fast fading refers to the rapid fluctuations in the amplitudephase, or multipath delays of the received signal, due to theinterference between multiple versions of the same

______________________

Fatima Abdeen FarahEdris is currently pursuingmasters degree program in Telecommunication &

Networking engineering in Future University, Sudan E-

mail: [email protected]. Supervisor: Dr. Ashraf Gasim Elsid Dean of faculty of

telecommunication in future university. External supervisor: D. Mohamed. H. M. Nerma,

faculty of electronics in Sudan university.

Fig. 1: illustrates fading channel classification andin this paper will consider the fast fading’s types

which is consist of Rayleigh and Rician channels.





transmitted signal arriving at the receiver at slightly differenttimes. So, the time between the reception of the first version ofthe signal and the last echoed signal is called delay spread .The multipath propagation of the transmitted signal, whichcauses fast fading, is because of the three propagationmechanisms described as Reflection, diffraction andscattering. There are various models to describe statisticalbehavior of this phenomenon (Fading); the two common

models that

been used for fast fading are Rayleigh and Ricefading channel models [2].

A. Rayleigh fading channel model Rayleigh fading occurs when there are multiple indirect pathsbetween the transmitter and the receiver; which scatter, diffractor reflect the signal. Rayleigh fading is the specialized modelfor stochastic fading when there is no line of sight signal, andis sometimes considered as a special case of the moregeneralized concept of Rician fading. In Rayleigh fading, theamplitude gain is characterized by a Rayleigh distribution, itcan be a useful model in heavily built-up city centers wherethere is no line of sight between the transmitter and receiverand many buildings and other objects attenuate, reflect, refract

and diffract the signal. [5]. in which the fading coefficients areassumed to be zero-mean, complex Gaussian distributed. Itoccurs when no LOS path exists in between transmitter andreceiver, but only have indirect path than the resultant signalreceived at the receiver will be the sum of all the reflected andscattered waves [2]. Probability density function for Rayleighfading is:

f xx =x

σ2e− x2

2σ2 (1)

B. Rician fading channel modelWhen there is a strong dominate component is present, thenthe model must be follow the Rician distribution. The model ofRician fading is similar to the model of Rayleigh fading, unlessthat in Rician fading a strong dominant component is present

[4]. It occurs when one of the paths, typically a line of sightsignal, is much stronger than the others [5]. The pdf of Ricianfading will be expressed as:

f xx x

σ2e−x2+c2

2σ2 I0 xc

σ2 (2)

Where I0() is the modified Bessel function of the first kind withorder zero and c is the strength of the direct component.In Rician fading, the amplitude gain is characterized by aRician distribution. The Rician distribution is often described interms of the Rician factor K, defined as the ratio between thedeterministic signal power (from the direct path) and thediffuse signal power (from the indirect paths). K is usuallyexpressed in decibels as [6]:

k dB = 10 log10(

c2

2σ2) (3)

4. CHANNEL ESTIMATION During the past few years, the developments in digitalcommunication are rapidly increasing to meet the everincreasing demand of higher data rates. Orthogonal frequencydivision multiplexing (OFDM) is a special case of multi-carriertransmission and it can accommodate high data raterequirement of multimedia based wireless systems. It has anedge over other frequency multiplexing techniques by usingmore densely packed carriers, thus achieving higher datarates using similar channels [7]. It is also a key technique forwireless communication because of its robustness for narrow

band interference, frequency selective fading and spectraefficiency [8]. The orthogonality of OFDM allows eachsubcarrier component of the received signal to be expressedas the product of the transmitted signal and channel frequencyresponse at the subcarrier. Thus, the transmitted signal can berecovered by estimating the channel response just at eachsubcarrier [2]. So the characteristics of wireless signachanges as it travels from the transmitter antenna to the

receiver antenna. These characteristics depend upon thedistance between the two antennas, the paths taken by thesignal and the environment around the path. In general, thepower profile of the received signal can be obtained byconvolving the power profile of the transmitted signal with theimpulse response of the channel. Convolution in time domainis equivalent to multiplication in the frequency domain [5]Therefore, the transmitted signal x, after propagation throughthe channel becomes y as in equation:

y = x ⊗ h + z (4) Where h is the channel response, z is noise and ⊗ is thecircular convolution. Note that x, y, h, and z in the aboveequation are all functions of the signal frequency f. The objectslocated around the path of the wireless signal reflect the

signal. Some of these reflected waves are also received at thereceiver. Since each of these reflected signals takes adifferent path, it has a different amplitude and phase [5]. Theestimation of channel is an important factor in OFDM systemand it mostly done by three ways; training based channeestimation _it’s types will be discussed in details later_, blind

channel estimation and semi blind channel estimation.

Training-based (Pilot based) channel estimationknown symbols are transmitted specifically to aid thereceiver’s channel estimation algorithms. Heretraining symbols or pilot tone that are known a priorto the receiver, are multiplexed along with the datastream for channel estimation.

Blind channel estimation: the receiver mus

determine the channel without the aid of knownsymbols. It is carried out by evaluating the statisticainformation of the channel and certain properties ofthe transmitted signals. Although higher-bandwidthefficiency can be obtained in blind techniques due tothe lack of training overhead, the convergence speedand estimation accuracy are significantlycompromised. Blind Channel Estimation has itsadvantage in that it has no overhead loss. It is onlyapplicable to slowly time varying channels due to itsneed for a long data record.

Semi blind channel estimation: is hybridcombination between blind and training techniqueutilizing pilots and other natural constraints to perform

channel estimation [5].

Pilot based channel estimationPilot-based approaches are used widely to estimate thechannel properties and correct the received signal. It providescoherent data detection to decrease error [2]. Receiveperforms channel estimation based on received pilot symbolsand it is used for synchronization and continuity. The channeestimation in this case is mostly done by inserting pilotsymbols into all of the subcarriers of an OFDM symbol orinserting pilot symbols uniformly into some of the sub-carriersof each OFDM symbol [7]. Pilot-based method channe





estimation can be performed by either block type pilots or bycomb type pilots as will shown below.

a) Block type pilotsIn this type, the OFDM symbols of the channel estimation aretransmitted periodically in time-domain and all subcarriers areused as pilots. It is particularly suitable for slow-fading radiochannels.

b) Comb type pilotIn this case, the pilot will be distributed uniformly within theOFDM symbol, and it suitable for fast fading channel which itis the study target field for this paper.

Where sf is represent the period of pilot symbols in frequencyand σmax is the inverse of the maximum delay spread which isuse to determine the coherence bandwidth. Since channelestimation is an integral part of OFDM systems, it is critical tounderstand the basis of channel estimation techniques forOFDM systems so that the most appropriate method are

based on least square (LS) and minimum mean-square erro(MMSE) will be discussed in this paper.

A. LS channel estimation algorithmThe goal of the channel least square estimator (LS) is tominimize the square distance between the received signal andthe original signal.

J

H

=

Y

−XH

2

= Y − XHHY − XH (5)

Y − YH XH − HHXHY + HHXHXH Where (. )H is the conjugate transpose operator. By setting the

derivative of the function with respect to H H to zero∂J(H)∂HH= −2(YH X)∗ + 2(XHXH)∗

Now, the LS channel estimation as:

HLs = YXH X−1XH Y = Y X−1 (6)

Where Y= XH

HLs = [k], k= 0,1,2,…,N-1.

Also HLs can be written for each subcarrier as:

HLs = Y

X = Y

k

Xk T

(7)

Where (. )T is the transpose operator and k = 0,1,2,… , N − 1.The mean-square error (MSE) of the LS channel estimate isgiven as:

MSELS = E{H − HLs HH − HLs }

= E{H − YX−1HH − YX−1} = EX−1ZHX−1Z (8)

E = { ZHXXH−1Z}

=σz

2

σ²x

There, MSE is inversely proportional to the SNR σz2/σ²x, which

implies that it may be subject to noise enhancement

especially when the channel is in a deep null. Due to itssimplicity, however, the LS method has been widely used forchannel estimation. Note that this simple LS estimator doesnot exploit the correlation of channel across subcarriers infrequency and across the OFDM symbols in time. Withouusing any knowledge of the statistics of the channel, the LSestimator can be calculated with very low complexity, but ithas a high mean-square error since it does not take intoaccount of the effect of noise on the signal [2]. This algorithmis very simple, because it is not considering any channestatistical parameters it is prone to the noise. So theperformance is worse [9].

B. MMSE channel estimation algorithm

The MMSE channel estimation method finds a better (linear)estimate in terms of W in such a way that the MSE in Equation(9) is minimized. The orthogonality principle states that theestimation error vector e = H − H is orthogonal to H, such that:

Fig. 2 illustrate the block type pilot with =1

Where is represent the period of pilot symbols intime and is the Doppler frequency.

Fig. 3 illustrate the comb type pilot with =1

Where is represent the period of pilot symbols in

frequency and is the inverse of the maximumdelay spread which is use to determine the coherence

bandwidth.





JH = Ee2 = E H − H2 (9)

E{eHH} = E = {(H − H )HH}

= E{(H

−WH

)H

H}

= EHHH

− EW{ HHH

} (10) RHH − WRH H = 0

Where RAB is the cross-correlation matrix of N X N matrices A

and B (i.e., RAB = E[ABH]), and H is the LS channel estimategiven as:

H = X−1Y = HX−1Z (11) Solving Equation (10) for W yields:

W = RHHRH H−1 (12)

Where RH H is the autocorrelation matrix of H given as:

RH H = EHHH = E{

X−1Y

X−1Y

H }

= E{H+X−1

ZH + X−1

ZH

} = EHHH+X−1ZHH+HZHX−1H + X−1ZZHX−1H (13) = E{HHH}+ E {X−1ZZHX−1H}

= E{HHH} +σz

2

σ²x

I

And RHH is the cross-correlation matrix between the truechannel vector and temporary channel estimate vector in thefrequency domain. Using Equation (13), the MMSE channelestimate follows as:

H = WH = RHHRH H−1 H RHH +σz

2

σ²xI−1

(14)

MMSE estimator employs the second-order statistics of the

channel conditions to minimize the mean-square error. Toestimate the channel for data symbols, the pilot subcarriersmust be interpolated. Popular interpolation methods includelinear interpolation, second-order polynomial interpolation [2].

5. MODEL DESCRIPTION

The data symbols is first grouped and mapped according tothe modulation technique [10]. Then it will be converted fromserial to parallel for pilot inserting. For comb type pilot

subcarrier arrangement, the Kp pilot signals Xp(m), m =0, 1, 2…, Kp are uniformly inserted into X(k). That is, the total Nsubcarriers are divided into Kp groups, each with L= N/Kpadjacent subcarriers. In each group, the first subcarrier is usedto transmit pilot signal. The OFDM signal modulated on the kthsubcarrier is shown in two following equations, where:

X(k)= X(mL+l)

X(k) = Xpm, where l = 0

inf.Data, where l = 1,2,… , L − 1 (15)

Where Xp(m) is the mth pilot carrier value [11]. After insertingpilots uniformly between the information data sequence, IDFTblock _which is a kind of FFT used for digital data_ is used totransform the data sequence of length N fX(k) frequencydomain signal into time domain signal fx(n) with the following

equation:x(n)= IDFT{X(k)} n= 0,1,2,….,N-1

= X(k)N−1k=0 e

2πkn

N (16)where N is the DFT length [12]. Following IDFT block, guardtime, which is chosen to be larger than the expected delayspread is inserted to prevent inter-symbol interference.This guard time includes the cyclic prefix which is a part ofOFDM symbol in order to eliminate inter-carrier interference(ICI). The resultant OFDM symbol is given as follows:

xf (n) = xN + n, n = −Ng , −Ng + 1, … ,−1

xn, n = 0,1 … , N − 1 (17)

where Ng is the length of the guard interval [13][14]. Thetransmitted signal xf (n) will pass through the Rayleigh and

Rician fading channels with noise. In Rayleigh channel modethe receiver receives a number of reflected and scatteredwaves. Because of the varying path lengths, the phases arerandom, and consequently, the instantaneous received powebecomes a random variable. In the case of un modulatedcarrier, the transmitted signal at frequency ωc reaches thereceiver via a number of paths, the ith path having anamplitude ai, and a phase ∅i. If assumed that there is no direcpath or line-of sight (LOS) component, the received signal s(t)can be expressed as:

st = aicos (ωct + ∅i)Ni=1 (18)

where N is the number of paths. The phase ∅i depends on thevarying path lengths, changing by 2π when the path length

Fig. 4: MMSE channel estimation algorithm





changes by a wavelength. Therefore, the phases are uniformlydistributed over [0, 2π]. The envelope in this case has aRayleigh density function given by equation (1) and the pdf ofis governed by the scale parameter σ, the greater the scaleparameter the wider distribution. In Rician channel model, theRician distribution is observed when, in addition to themultipath components, there is an exist of direct path or line-of-sight component between the transmitter and the receiver.

In the presence of such a path, the transmitted signal can bewritten as: st = ai cosωct + ωdi t+∅i + c cos (ωct +N−1

i=1

ωdt) (19)Where the constant c is the strength of the direct component,ωd is the Doppler shift along the LOS path, and ωdi are theDoppler shifts along the indirect paths given by equation (18).The envelope in this case has a Rician probability densityfunction given by equation (2) [6]. The distribution based onRician is calculated according to the values using thecorresponding scale parameter function in (2). In thereceiver part, first the received signal must be converted fromserial to parallel in order to remove the cyclic prefix and guardinterval. The cyclic prefix would be of all zero samplestransmitted in front of each OFDM symbol. It does not contain

any useful data; it would be discarded at the receiver. Thelength of the guard interval should be longer than the timespan of the channel, such that the OFDM symbol itself will notbe distorted. Thus, by eliminating the guard interval, theimpacts of intersymbol interference can be removed [9] andthe signal is then will be equal to equation (4).Where z(n) isthe noise, h(n) is the channel impulse response and ⊗ iscircular convolution. yf n for − Ng ≤ n ≤ N − 1 is equal to

yn = yf n + Ng n = 0,1,2,… . . , N − 1 (20)

Where y(n) is the received signal after CP removal and theny(n) [14] is sent to DFT block for defining Y(k) the followingoperation:

Y(k)= DFT{y(n)} k=0,1,2,…,N_1

=

1

N y(n)N

−1

k=0 e

2πkn

N (21)Assuming there is no ISI, the following equation will show therelation of the resulting Y(k) to H(k) with the following equation:

Y(k) = X(k)H(k) + Z(k) (22)Where X is a matrix of size K x K with the elements of thetransmitted signals on its diagonal, Y is the received vector ofsize K x l, H is a channel frequency response of size K x 1 andZ is noise with zero-mean and variance. The noise Z isassumed to be uncorrelated with the channel H [24].

Y ≜ Y[0]⋮Y[N − 1]

= X[0] ⋯ 0⋮ ⋱ ⋮0 ⋯ X[N − 1]

H[0]⋮H[N − 1]

+

Z[0]

⋮Z[N − 1] (23)

Next block will be the channel estimation block, which _in thispaper_ will be using the LS and MMSE channel estimationalgorithms. The received pilot signal vector, Yp =[Yp(0), Yp(1), … Yp(Np − 1)]Tcan be expressed as given bythe following:

Yp=XpHp + Zp (24)

Where:

Xp=Xp ⋯ 0⋮ ⋱ ⋮0 ⋯ Xp(Np − 1)

(25)

Hp (k) is the frequency response of the channel at pilot sub-

carriers and defined as Hp=[Hp(0), Hp(1),

… Hp(Np

−1)]T ,

and Zp=[Zp(0), Zp(1), … Zp(Np − 1)]T is the noise vector in

pilot subcarriers where [. ]T is a transpose operator [2]. Theequations that used for both LS and MMSE is reported indetails in the previously in equations (5) to (14). In channeestimation based on comb type pilot insertion, an interpolationtechnique is necessary in order to estimate channel at datasub-carriers by using the channel information at pilot subcarriers. The channel estimation at the data-carrier using linea

interpolation is given by:Hek = He(mL + l) Hpm + 1 − Hpm l

L+ Hp(m) (26)

The second-order interpolation results to be better than thelinear interpolation. The channel estimated by second-ordeinterpolation is given by:

Hek = He(mL + l) c1Hpm − 1 + c0Hpmc−1Hp(m + 1) (27)

Where

c1 =α(α−1)

2

c0 = α − 1α + 1, α =l

N

c−1 =α(α+1)

2

(28)

After estimate the channel at pilot sub-carriers by mean ofchannel estimation and data sub-carriers by using the channeinformation at pilot sub-carriers by mean of interpolation, thedata now is ready to be un modulated after transfer it fromparallel to serial in order to show the output signal at the end[8].

6. SIMULATION DESIGN

The MATLAB environment provides an accurate simulation othe application in real world, the more details and parametersare defined the more accurate the simulation will be, thusproviding strong results back to the conclusion. In this thesisthe attempt is to show comparison between LS & MMSE interm of MSE average and SNR using different modulations

schemes.A. Implementing channel estimation in MATLABMATLAB Program will perform the LS and MMSE channelestimation, respectively. It will calculate MSE & SNR for eachand compare it in one plot. Next table will show equations thatused to calculate average MSE.

TABL E 1 Average MSE for the LS&MMSE estimators.

Estimator Average MSE

LSβ

SNR

MMSE1

N

β

SNR λk,N

λk,N +β

SNR

N−1

k=0

The simulation uses these parameters that mentionedabove will be repeated in order of different modulationtechniques using 64 subcarriers which is supporting Wi-Fi and 512 subcarriers which is supporting Wi-MAX.





B. LS & MMSE ResultsIn figures 6, 7, 8, and 9, the comparison is done in order ofcalculating MSE average and SNR for both channel estimationalgorithms. It shows that the MMSE has low MES & SNR andthus doing better performance than LS whatever themodulation used.

7. CONCLUSION

In this thesis, a full review of fading channels model (Rayleigh& Rice), LS estimator and MMSE estimator is given. Rayleighfading model is happens when no LOS path exists in between

transmitter and receiver, but only have indirect path than theresultant signal received at the receiver will be the sum of althe reflected and scattered waves. Rician fading model will beappeared once the receiver receives one strong componentwhich will be a line of sigh signal. MMSE estimator has abetter performance than LS estimator in order of average MSEand SNR but it has a computational complexity so it use onlyin the low SNR environments. However, MMSE estimator wilbe used in systems that require precise measurement andtime sensitive environments such as high speedcommunications systems. On modulation side, the resultsshow that the M-QAM has a better performance for bothchannel estimation algorithms than M-PSK modulation.

Fig .6: the two techniques of channel estimation is usedfor 64 subcarriers, with cyclic prefix equal to 1/16 whichwill be expected to be larger than maximum expected

Fig. 7: also the two techniques of channel estimation isused for 64 subcarriers, with cyclic prefix equal to 1/16

and using 16 QAM modulation.

Fig. 8: shows the two techniques will be used for 512subcarriers with cyclic prefix 1/16 and using BPSK

modulation scheme.

Fig. 9: also done in 512 subcarriers with the same CPused for all previous simulations and using 16 QAM

modulation scheme.





8. Future RecommendationsIn channel estimation, instead of use training-based channelestimation, the DFT-based channel estimation technique mustbe used because it was been derived to improve theperformance of LS or MMSE channel estimation by eliminatingthe effect of noise outside the maximum channel delay.Technically, the performance of LS can be improved by usingtechnique called Least Mean Square LMS. The LMS estimator

uses one tap LMS adaptive filter at each pilot frequency. Thefirst value is found directly through LS and the following valuesare calculated based on the previous estimation and thecurrent channel output. For MMSE, although it has betterperformance the LS but its performance can be improved alsoby implementing Linear Minimum Mean Square LMMSE whichhas the best performance since it has the accurate channelstatistical parameters and at high SNR, the performance ofthis algorithm is good.

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A Study of Channel Estimation in Fast Fading Environments