[86]

CHAPTER 4 ESTIMATION OF FREQUENCY OFFSET

ynchronization is one of the major concerns of OFDM system. Synchronization includes

both timing and frequency offset estimation and correction. The timing synchronization

analysis has been done in previous chapter. This chapter analyzes the effect of CFO and different methods

of CFO estimation. One new method of CFO estimation has also been proposed in this chapter and

compared with other existing method.

4.1 INTRODUCTION

The performance of any digital communication system (Wired or Wireless with SC or MC) very

much depends on the synchronization scheme used in that system. Improper synchronization creates

amplitude and phase distortion at the output of demodulator. Synchronization provides the correct starting

time of a symbol and way to estimate the carrier frequency offset. The three main parameters required by

most receivers are the carrier frequency, the carrier phase and the symbol timing of the received signal

[21]. The carrier frequency of the received signal may be different from that of the nominal value of the

transmitter carrier frequency. The discrepancy between transmitter carrier frequency and received carrier

frequency is called as CFO. This discrepancy may be resulted due to the mismatching of transmitter and

receiver local oscillators and or due to the Doppler Effect when the transmitter is in motion relative to the

receiver. This is called Doppler shift [119, 127]. In reality, it takes a finite amount of time for the

information-bearing electromagnetic wave to travel from the transmitter to the receiver. This transmission

delay introduces a mismatch between the symbol timing at the transmitter and that at the receiver. Recall

that it is required to sample the output of the matched filter at an exact time to optimize the error

performance. The symbol timing (or equivalently, the transmission delay) at the receiver has to be known

in order to eliminate the performance degradation due to the timing mismatch. The carrier phase of the

received signal is the sum of three major components, namely, the random phase of the transmitter

oscillator, the channel phase response and the phase due to the transmission delay.

The process of estimating these parameters is called synchronization. The process of estimating

the carrier phase is known as carrier phase synchronization, which can be accomplished by a phase locked

loop (PLL) circuit. The process of estimating the transmission delay is known as symbol timing

S

[87]

synchronization, which, can be accomplished by a delay-locked loop (DLL) circuit in SC system. The

analysis of symbol timing synchronization in OFDM system has already been discussed in last chapter.

In OFDM system, there is a stringent requirement of carrier frequency synchronization. The

whole concept of OFDM and all other advantages are depending on the orthogonality of sub-carriers. If

the orthogonality between sub-carriers lost due to CFO, then the performance of OFDM system degrades.

CFO causes a number of impairments including attenuation and rotation of each of the subcarriers and ICI

between subcarriers. The effect of CFO in the performance of OFDM system in multipath fading channel

is discussed in following section.

4.2 FREQUENCY OFFSET ANALYSIS

The effect of frequency offset has been analyzed in this section. To analyze the effect of carrier

frequency offset, re-writing the received signal (2.4.2.1) –

푟(푖,푛) = exp푗 2 휋 푛 휖

푁 ℎ (푛) 푥(푖,푛 − 휏 ) + 푤(푖,푛) ,

푓표푟 푛 = 0, 1, … ,푁 − 1 (4.2.1)

where ′ε′ is the carrier-frequency offset normalized by the sub-carrier spacing (1/푇 = 1/푁푇 ), ℎ (푛) is

the impulse response of frequency selective multipath fading channel with path gains {ℎ (푛) ∶ 푙 =

0,1, … , 퐿 − 1} , 휏 is the path delay of 푙 path, and 푤(푖,푛) is a zero-mean, complex value Gaussian noise

process with variance 휎 . By considering a perfect timing synchronization (i.e., timing offset δ = 0), the

output of FFT block, after CP removal, can be written as –

푌(푖, 푝) = 푟(푖,푛) 푒 / ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.2)

Substituting the value of 푟(푖,푛) –

푌(푖,푝) = 푒 / ℎ (푛) 푥(푖,푛 − 휏 ) + 푤(푖,푛) 푒 / ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.3)

[88]

Separating the AWGN term –

푌(푖,푝) = 푒 / ℎ (푛) 푥(푖,푛 − 휏 ) 푒 / + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.4)

where,

푊(푖,푝) = 푤(푖,푛) 푒 / , 푓표푟 푝 = 0, 1, … ,푁 − 1

(4.2.5)

Now, put the value of 푥(푖,푛 − 휏 ) in (4.2.4)

푌(푖,푝) = 푒 / ℎ (푛) 1푁

푋(푖,푘) 푒 푒 /

+ 푊(푖,푝) , 푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.6)

After interchanging the summation, it becomes -

푌(푖, 푝) =1푁

푋(푖,푘) ℎ (푛)e

( )

1 × 푒 ( ) ( / ) + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.7)

Replacing the inner summation term by 퐻(푘) –

푌(푖,푝) =1푁

푋(푖,푘) 퐻(푘) 1 × 푒 ( ) ( / ) + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.8)

[89]

푌(푖,푝) =1푁

푋(푖,푘) 퐻(푘) 푆(푘 − 푝 + 휖) + 푊(푖,푝),

푓표푟 푝 = 0,1, … ,푁 − 1 (4.2.9)

where, 푆(푘 − 푝 + 휖) is an ICI coefficient, and is defined as –

푆(푘 − 푝 + 휖) = 1 × 푒 ( ) ( / ) = 푒 ( ) 푆푖푛 휋 (푘 − 푝 + 휀)

푆푖푛( 휋 (푘 − 푝 + 휀)/ 푁)

(4.2.10)

Now re-writing the above expression (4.2.9) by breaking it into three terms –

푌(푖, 푝) =1푁

푋(푖, 푝) 퐻(푝) 푆(휀) + 1푁

푋(푖,푘) 퐻(푘) 푆(푘 − 푝 + 휖),

+ 푊(푖,푝) ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (4.2.11)

where,

푆(휀) = 1 × 푒

= 푒 푆푖푛( 휋 휀)

푆푖푛( 휋 휀/ 푁)

= 푁, 푓표푟 휀 = 0

푁표푛 푍푒푟표, 푓표푟 휀 ≠ 0

(4.2.12)

Form (4.2.12), it is clearly understood that ICI depends on the CFO (‘ε’), and it can be compensated if

correctly estimated. Therefore, accurate frequency estimation method is required at the receiver to reduce

the ICI and to improve the BER performance of OFDM system. Several CFO estimators are available in

literature and discussed in following section.

[90]

4.3 METHODS OF CARRIER FREQUENCY OFFSET ESTIMATION

In this section, the techniques of different DA methods of frequency offset estimation are

discussed in detail. In DA methods [2, 35, 36, 71, 79, 107, 125 and 142], the frequency offset is estimated

in two steps, the FFO estimation and the IFO estimation.

The first algorithm for frequency offset estimation using preamble was given by P. H. Moose in

1994 [92]. For better accuracy and wide acquisition range than Moose’s estimator, another technique was

proposed by Schmidl & Cox proposed in the year 1997 [125]. They used two training symbols with

differential encoding at identical subcarrier positions for CFO estimation. The first symbol which consists

of two identical halves was used for FFO estimation whereas IFO was estimated in frequency domain by

using second symbol. The estimation range of Schmidl & Cox was ± N/2 times the subcarrier spacing, for

N-point FFT.

To reduce the overhead due to second symbol in Schmidl & Cox method, several other methods

have been proposed in the literature. Few of them, which are considered to be best amongst all available

ones, are discussed here.

4.3.1 Schmidl and Cox Method

Schmidl & Cox [125] designed two training symbol. First training symbol has two identical halves

as described above in previous section. The first preamble is used for both timing offset estimation and

fractional frequency offset estimation. Whereas, the second preamble is used for determining IFO only.

The second block contains a differentially encoded pseudo-noise sequence PN1 on even subcarriers and

another pseudo-noise sequence PN2 on odd subcarriers. The basic concept behind frequency offset

estimation is that the two identical halves will remain identical after passing through channel, except the

phase difference ′휑 between them due to the frequency offset ′∆푓′. The phase difference ′휑′ between

two halves is obtained by calculating their correlation and given as-

휑 = 휋 푇 ∆푓 = 휋 푁 푇 ∆푓 , (4.3.1.1)

where, 푇 (= 푁푇 ) is the useful duration of one OFDM symbol and 푇 is the sampling interval. This phase

difference ′휑′ can be estimated as -

휑 = 푎푛푔푙푒 푃 & (푑) (4.3.1.2)

[91]

where, 푃 & (푑) is defined as the correlation between the two halves of training symbol. If |휑| is less then

π, then the frequency offset estimate is given by –

∆푓 =

, (4.3.1.3)

Otherwise, the actual frequency offset would be estimated by –

∆푓 =

+

, (4.3.1.4)

where, 푍 is an integer. The first term is FFO (∆푓 ) and the second term is IFO (∆푓 ), which will be

estimated using second training symbol by following method. After FFO estimation, both training symbols

are frequency corrected by 휑 휋 푁 푇⁄ (by multiplying the samples by exp −푗 2 휋 푡 (휑 휋 푁 푇⁄ ) ). Then

after, the FFT of both symbols are taken for estimating integer frequency offset. For IFO estimation, the

second training symbol block contains a differentially encoded pseudo-noise sequence on even subcarriers

and another pseudo-noise sequence on odd subcarriers. Let the FFT’s of received first and second training

symbols is 퐹 (푘) and 퐹 (푘) and the differentially modulated even frequencies of second training symbol

be 푈(푘). Let 푋 be the set of indices for the even frequency components, 푋 = {−푊 ,−푊 +

2, … ,−4,−2, 2, 4, … , 푊 − 2, 푊 } . The sliding correlation between the FFTs and 푈(푘) is given as-

퐵(푙) =퐹∗(푘 + 2푙) .푈∗(푘)∈ . 퐹 (푘 + 2푙)

2( |퐹 (푘)|∈ ),

(4.3.1.5)

where, 푊 is the total number of even frequencies with PN sequence. The value of ′푙 at which 퐵(푙)

attains maximum value is used to calculate IFO.

4.3.2 Best Linear Unbiased Estimator (BLUE)

Morelli et.al [79] introduced the best linear unbiased estimator (BLUE) as an extension of Schmidl

& Cox method. Morelli’s method consists of one training symbol with 퓜 identical parts. The Morelli’s

algorithm can estimate integral and fractional part of frequency offset simultaneously. This method

basically exploits the correlations of the samples from the matched filter. The estimated carrier frequency

offset 휀̂ is given as-

[92]

휀̂ = 1

2휋/ℳ푤(푚) 휗(푚),

(4.3.2.1)

where, 퐻 is a design parameter, less than or equal to ℳ− 1 and 푤(푚) is the 푚 component of

푤 = 퐶 1 1 퐶 1⁄ (4.3.2.2)

In this equation, 퐶 is the covariance matrix of 휗 ≜ [휗(1),휗(2), … ,휗(퐻)] and 1≜ [1,2, … ,1]푇 is an H-

dimensional column vector of all ones. The angles {휗(푚),푚 = 1,2, … ,퐻 } is defined as -

휗(푚) ≜ [푎푟푔{푅(푚)} − 푎푟푔{푅(푚 − 1)}] , 푓표푟 1 ≤ 푚 ≤ 퐻 (4.3.2.3)

where, [푥] denotes modulo-2π operation (it reduces x to the interval [-π, π], [. ] is represents the

transpose of matrix [. ] and 푎푟푔{푅(푚)} is the argument of 푅(푚) (output of Matched filter). The 푅(푚) is

given as -

푅(푚) =1

푁 − 푚 푀 푥(푘) . 푥∗(푘 − 푚 푀 ) , 푓표푟 0 ≤ 푚 ≤ 퐻

where, 푀 = 푁/ℳ is the number of samples in each section of training symbol. (4.3.2.4)

4.3.3 Chirp Training Symbol Based Estimator

In the year 2007, Wei et.al proposed a new method which uses chirp signal as training symbol

[71]. The structure of training symbol blocks of Wei et.al method contains two CAZAC sequences or chirp

signal. First sequence is used for FFO estimation and both sequences are used for IFO estimation. The

preamble of Wei et.al method has following form –

푃푟푒푎푚푏푙푒 = [푥(푛, 푟 ) 푥(푛, 푟 ) ] (4.3.3.1)

where, 푥(푛, 푟 ) and 푥(푛, 푟 ) is defined as first and second CAZAC sequences respectively. The CAZAC

sequence in time domain can be written as –

푥(푛, 푟) = exp(푗 휋 푟 푛 / 푁), 푓표푟 푛 = 0, 1, 2, … ,푁 − 1 (4.3.3.2)

where, 푟 is a parameter of CAZAC sequence.

[93]

The FFO estimation is same as given by Schmidl & Cox (using autocorrelation) [125]. The IFO estimation

is based on the fact that DFT of a CAZAC sequence is also a CAZAC sequence. For IFO estimation, this

method uses cross correlation between known training sequence at the receiver and partially (FFO)

compensated training sequence. After FFO estimation, the received signal are corrected by multiplying its

sample with the term 푒 / , where, 휀̂ is the estimated normalized fractional frequency offset. If

푟(푛) is the received signal, then the DFT of 푟(푛) after FFO correction is given as -

푌(푘) = 1√푁

푟(푛)푒 / 푒

(4.3.3.3)

After that, the cross correlation between DFT of known training sequence and the DFT of partially

compensated received signal 푌(푘) is obtained as –

푅 휀 − 휀̂ , 푟, 휏 = 1푁

푋 (푘 − 휏) 푌∗(푘)

(4.3.3.4)

where, 푋 (푘) is the DFT of training sequence 푥(푛, 푟). After that, the normalised integer fractional

frequency offset 휀̃ will be estimated by using following steps:

Step 1: Find the peak location of the two correlations respectively.

퐿 = arg max 푅 휀 − 휀̂ , 푟 , 휏 (4.3.3.5)

퐿 = arg max 푅 휀 − 휀̂ , 푟 , 휏 (4.3.3.6)

Step 2: If 퐿 < , go to step 3, else go to step 4. Step 3: While 퐿 ≥ {퐿 = (퐿 + 푟 ) }; (4.3.3.7) While (퐿 < 퐿 ) {퐿 = (퐿 + 푟 )} . Go to step 5 (4.3.3.8) Step 4: While (퐿 < 퐿 ) and (퐿 > 푟 ) {퐿 = (퐿 + 푟 )} . (4.3.3.9) Step 5: 휀̃ = (퐿 ) ; (4.3.3.10) If 휀̃ > , 휀̃ = 휀̃ − 푁 푎푠푠푢푚푖푛푔 |휀| < . (4.3.3.11)

[94]

4.4 PROPOSED METHOD OF CARRIER FREQUENCY OFFSET ESTIMATION

The proposed carrier frequency offset estimation method uses chirp signal as a training sequence

and exploits the modulation property of FRFT. Once the training sequence is successfully retrieved using

timing estimation procedure described in Chapter-3, it can be further exploited for frequency offset

estimation .The frequency estimation method is divided into two parts – i.e., the FFO and IFO estimation

respectively. Therefore, the net frequency offset (∆f) can be obtained as –

∆푓 = ∆푓 + ∆푓 , (4.4.1)

where, ∆푓 and ∆푓 is a fractional and integer frequency offset respectively. Since the two halves of the

proposed preamble is identical, the FFO (∆푓 ) estimation method applied in proposed scheme is same as

given by Schmidl & Cox. After FFO estimation, the FRFT of first half of received training sequence is

taken. Thereafter, the IFO estimation is carried out by searching the peak position in the FRFT of the first

half of received training sequence (chirp signal). Details of both FFO and IFO estimation method are

described below. The IFO estimation is based on the modulation property of FRFT; therefore the

modulation property of FRFT is described in the next section before describing the proposed IFO and FFO

estimation method.

4.4.1 Modulation Property

The term “modulation” has extensively used in the study of any type of communication system.

Modulation is a process that causes a shift in the spectrum of the modulating signal. Therefore, the signals

that occupy the same range of frequencies can be separated using modulation technique. Hence,

modulation has been considered as a fundamental requisite of carrier based communication systems.

The basic purpose of modulation in carrier based communication system is the translation of

frequency form one range to another (i.e., baseband to some higher frequency range). This translation of

frequency helps in fulfilling the basic requirement of communication system like multiplexing, suitability

of antenna height for design constraint, and narrow-banding [16]. Multiple signals (having same baseband

frequencies) can be transmitted simultaneously using frequency division multiplexing after modulating

them using different carrier signal. This frequency translation also converts a wideband signal to a

narrowband. The concept of frequency translation which is extensively helpful in telecommunication

system are basically comes from the modulation property of transform, specially, Fourier transform.

[95]

The modulation property of Fourier transform (FT) states that “the phase change encountered by

the signal in time-domain results into corresponding shift of the spectrum in frequency domain”. In other

words, “multiplication of the signal by a factor 푒 shifts the spectrum of that signal by 푤 = 푣”. If

퐺(푤) represents the Fourier transform of the signal ‘g(푡)’ then modulation property can be expressed as –

푒−푗 푣 푡푔(푡) ↔ 퐺(푤 − 푣) (4.4.1.1)

In the propose method, a chirp signal, whose frequency components are time-dependant, is used as a

training sequence for frequency offset estimation. Due to this time-variant nature of frequency of chirp

signal, FT fails to localize the chirp signal in frequency domain. The FRFT has already been established as

a superior tool for locating the chirp signal in fractional Fourier domain [40]. A simulation exercise has

been performed for the validation of the superiority of FRFT over FT in the case of chirp signal. The chirp

signal ′푔(푡)′ can be defined as-

푔(푡) = 푒−푗 2 휋 (푎 푡2+푏 푡) 푟푒푐푡 , (4.4.1.2)

where, 푟푒푐푡 =1, |푡| ≤ 휏

0, 표푡ℎ푒푟푤푖푠푒,

where, ′휏′ is the duration of chirp, ′2 푎 휏′ is the bandwidth of chirp, and ′푏′ is the center frequency of chirp

signal. For the simulation, the required values of different parameters are taken as follows -

푎 = 8, 푏 = 55, and 휏 = 1.

The plot of chirp signal, its FT and its FRFT are shown in Figure-4.4.1.1. The optimum value of

angle parameter ′훾′ for calculating the FRFT is obtained by expression (3.4.2.2) as 훾 = 3.13165. As

visible from the Figure- 4.4.1.1 (b), the lack of one sharp peak in the FT of the chirp signal makes it

difficult at receiver to detect the shift of spectrum under noisy condition, when chirp is used as a training

sequence for estimating frequency offset. Subsequently, this drawback can be eliminated if FRFT is used

instead of FT, due to presence of a sharp peak in the FRFT domain, as shown in Figure-4.4.1.1 (c).

Therefore, the modulation property of FRFT has been exploited in the proposed scheme.

[96]

(a)

(b)

(c) Figure – 4.4.1.1: (a) Time domain plot of chirp signal, (b) FT of chirp signal, and (c) FRFT of

chirp signal.

[97]

The modulation property of FRFT is defined in [7], as –

ℑ 푥(푡)푒 = 푋 (푢 − 푣 푆푖푛(훾)) 푒 ( ) ( ) ( ) , (4.4.1.3)

where, 푋 (푢) is the FRFT of a signal 푥(푡) at angle ′훾′. The modulation property of FRFT reveals that

“introduction of phase (i.e. multiplication of signal by factor 푒 ) in the time-domain has an analogous

effect of shift in the position of the peak of FRFT of phase shifted signal by ′푣 푆푖푛(훾) in the fractional

Fourier domain”. Using this property, by observing the shift in the peak position ′푍 of the FRFT of

received signal, the phase shift encountered in time-domain can be estimated as –

푣 =( )

, (4.4.1.4)

where, 푍 is the shift in the peak position, determine by the simulation.

To illustrate this aspect of FRFT, another simulation is undertaken in which the quantity ‘푣’ is considered

as 30 with all other parameter kept same as above. The obtained plot of FRFT of chirp signal and of phase

shifted chirp signal is shown in Figure-4.4.1.2.

Figure – 4.4.1.2: FRFT of chirp and phase shifted chirp signal

[98]

Based on this simulation result, the phase shift introduced in the time-domain signal can be estimated by

measuring the difference between location of peaks of FRFT of original and phase shifted signal in

fractional Fourier domain. The outcome of this analysis is presented in Table-4.1 as given below–

Table – 4.1: Analysis of simulation results

Assumed Value of ′푣′ Location of Peaks of FRFT of Estimated Value of ′푣′

Original Signal Phase-Shifted Signal

30 3.429 5.303 30.013

It is clearly visible from the Table-4.1, that the phase-shift introduced can be measured successfully from

the location of peaks of FRFT of phases shifted signal (within the error of 0.0466 %).

4.4.2 Proposed FFO Estimation

The two halves of training symbol are identical at transmitter side but they will differ at receiver

side by a phase difference ′휃′ which is introduced by the frequency offset ′∆푓′ and due to multipath fading

channel. Let the received training sequence obtained after the timing estimation is g(n) and the two halves

of this received training sequence are represented by g (n) and g (n), where -

푔 (푛) = 푔 (푛) 푒 ∆ (4.4.2.1)

The phase difference ′휃 between g (n) and g (n) is obtained by calculating their correlation and given as

-

휃 = 2 휋 푇 ∆푓 = 2 휋 푁 푇 ∆푓, (4.4.2.2)

where, 푇 (= 푁푇 ) is the useful duration of one OFDM symbol and 푇 is the sampling interval. This phase

difference ′휃′ can be estimated as –

휃 = 푎푛푔푙푒 푃 (푑) , (4.4.2.3)

where, 푃 (푑) is defined as the correlation between the two halves of g(n), represented as-

[99]

푃 (푑) = 푔∗(푑 + 푚) 푔 (푑 + 푚) ,

(4.4.2.4)

where, (.)* is complex conjugate of (.). If |휃| is less then π, then the frequency offset estimate will be –

∆푓 =휃

2 휋 푁 푇 ,

(4.4.2.5)

Otherwise, the actual frequency offset would be estimated by –

∆푓 =휃

2 휋 푁 푇+

푍푁 푇

,

(4.4.2.6)

where, Z is an integer. The first term is FFO ∆푓 and the second term is IFO ∆푓 , which can be

obtained by following the procedure described in the next sub-section.

4.4.3 Proposed Algorithm of IFO Estimation

In IFO estimation, the DFRFT of first half of g(n) is taken and its peak position is determined

which will be utilized for the estimation of Z. The proposed algorithm for estimating the IFO is –

Step 1: Take the DFRFT of first half of training symbol.

퐺 (푢) = ℑ [푔 (푛)], 푓표푟 푛 = 0, 1, 2, … ,푁 − 1 (4.4.3.1)

Step 2: Find the location of peak of 퐆후(퐮) .

푍 = arg푚푎푥 퐺 (푢) (4.4.3.2)

Step 3: Determine the value of 퐙.

푍 =

⎩⎪⎨

⎪⎧

|푍 − (푁/2)|− 1, ∀ (휑/휋) ≥ 0

|푍 − (푁/2)|, 표푡ℎ푒푟푤푖푠푒

(4.4.3.3)

[100]

Figure – 4.4.3.1: Average estimate of frequency offset estimation using proposed method

The estimation range of proposed estimator is ± N/2 times the subcarrier spacing, for N sub-carriers

OFDM system, as shown in Figure- 4.4.3.1. In this plot, an OFDM system with 64 sub-carriers has been

taken.

4.5 PERFORMANCE EVALUATION

The performance of proposed method is presented and compared with Schmidl & Cox [125],

Morelli et.al [79] and Wei et.al [71]. For this comparison, an OFDM system with 256 sub-carriers and 16

cyclic prefix with QPSK modulation is considered. An ISI channel model [55 and 127] consisting of L=8

paths with path delays of 푚 = 0, 1, … , 퐿 − 1 samples and an exponential power delay profile having

average power of 푒 / is used, wherein each path undergoes independent Rayleigh fading. The mean

square error has been taken as performance evaluation parameter.

Figure- 4.5.1 shows the comparison of MSE of frequency offset estimation with all estimators

considered in multipath fading channel. It is clearly visible that the proposed estimator performs better than

all other discussed algorithms.

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Figure – 4.5.1: MSE of frequency offset estimation of different methods with normalized frequency

offset 2.35

With the proposed algorithm of CFO estimation, the performance is improved but at the cost of

increased computational complexity. The computational complexity of various estimators is compared

with proposed algorithm, as shown in Table-4.2.

Table – 4.2: Approximate computational complexity of various methods for CFO estimation

Methods Complex multiplication Complex Addition

Schmidl & Cox [125] 푁 2⁄ + 2 푁 푙표푔 푁 + 푁 (푁 2⁄ − 1)(푁 + 1) + 2 푁 푙표푔 푁

Morelli et.al [79] 3푁푄 8⁄ − 푁 4⁄ 3NQ 8⁄ − N 4⁄ −ℳ 2⁄

Wei et.al [71] 2 푁 푙표푔 푁 + 5푁/2 5 푁 2⁄ − 3 + 2 푁 푙표푔 푁

Proposed 푁 + 푁 푙표푔 푁 푁 − 1 + 푁 푙표푔 푁

* ℳ is the number of identical parts in training symbol of method given by Morelli et.al [79]

[102]

The complexity of proposed method is higher than Morelli’s method [79] but less than Schmidl &

Cox method [125] and Wei et.al method [71], as shown in Table-4.2.

4.6 SUMMARY

A new method is proposed for frequency offset estimation for OFDM system. The proposed

method is based on the localization of chirp signal in the FRFT domain. The MSE of frequency offset

estimation is found very low with proposed method in comparison to other discussed methods, in multipath

fading channel. The betterment in MSE is obtained in the proposed method at the cost of increased

computational complexity.

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