Carrier Frequency Synchronization in the Downlinkof 3GPP LTE

Qi Wang, Christian Mehlfuhrer, and Markus Rupp

Institute of Communications and Radio-Frequency EngineeringVienna University of Technology

Gusshausstrasse 25/389, A-1040 Vienna, AustriaEmail: {qwang,chmehl,mrupp}@nt.tuwien.ac.at

Web: http://www.nt.tuwien.ac.at/rapid-prototyping

Abstract—In this paper, we investigate carrier frequency syn-chronization in the downlink of 3GPP Long Term Evolution(LTE). A complete carrier frequency offset estimation and com-pensation scheme based on standardized synchronization signalsand reference symbols is presented. The estimation performancein terms of mean square error is derived analytically andcompared to simulation results. The impact of estimation erroron the system performance is shown in terms of uncoded biterror ratio and physical layer coded throughput. Comparedto perfect synchronization, the presented maximum likelihoodestimator shows hardly any performance loss, even when themost sophisticated MIMO schemes of LTE are employed.

I. INTRODUCTION

Long Term Evolution (LTE) [1] has been standardized as thesuccessor of the Universal Mobile Telecommunication System(UMTS). The downlink transmission utilizes Orthogonal Fre-quency Division Multiple Access (OFDMA) as the physicallayer technique which enables high data rate transmissionin frequency selective fading scenarios. However, one of thedrawbacks of OFDMA is its vulnerability to Carrier FrequencyOffset (CFO). Given a carrier frequency of 2.5 GHz, a typicalfrequency drift of 10ppm (10 × 10−6) of the local oscillatorresults in an offset of 25 kHz. In LTE, which employs a fixedsubcarrier spacing of 15 kHz, such an offset corresponds to1.67 subcarrier spacings.

Synchronization for general OFDM systems has been in-vestigated for example in [2–8]. The basic idea in these publi-cations is to split the CFO into a Fractional Frequency Offset(FFO), an Integer Frequency Offset (IFO), and a ResidualFrequency Offset (RFO), which can be estimated individually.In [2–4], a Cyclic Prefix (CP) based FFO estimator has beenpresented and the Cramer-Rao lower bound of this estimatorhas been derived in [8]. In order to expand the estimationrange, an IFO estimator based on the maximum likelihoodprinciple has been derived in [5]. The impact of RFO as wellas RFO compensation schemes were investigated in [4, 6, 7].Specifically for LTE, a synchronization concept was presentedin [9]. However, the performance was only evaluated in termsof simulated Mean Squared Error (MSE).

In this work, we present an entire chain of carrier frequencyoffset estimation blocks based on the standardized synchro-nization signals and reference symbols of LTE. Furthermore,

we derive analytic expressions for the MSE of the RFO andthe FFO estimators. The mathematical analysis is comple-mented by simulations that are carried out using a standardcompliant MATLAB-based downlink physical layer simulatorfor LTE [10]. In addition to the frequency offset MSE, also thesystem performance after compensation is evaluated in termsof uncoded bit error ratio and physical layer coded throughput.All algorithm implementations will be made publicly availablein a future release of our LTE physical layer simulator [10].

The paper is organized as follows. In Section II, thestandardized synchronization and reference signals of LTEare described. Section III explains our system model and theimpact of CFO on an OFDM system. Our carrier frequencyoffset estimation scheme and its analytical MSE are presentedin Section IV. Section V shows the physical layer simulationresults. Finally we draw our conclusions in Section VI.

II. SYNCHRONIZATION AND REFERENCE SIGNALS IN LTEAs specified in [11], radio frames in an LTE downlink

transmission have a duration of Tframe = 10ms. Each frameconsists of 10 subframes of equal length. Each subframecontains two consecutive slots of time duration Tslot = 0.5ms.The smallest scheduling unit is called one resource blockwhich is a time-frequency grid of 12 subcarriers over Ns = 7OFDM symbols for the normal CP length and Ns = 6 forthe extended CP length. In the LTE downlink two kinds ofsignals can be utilized for synchronization at the receiver.Firstly, the dedicated synchronization signals, and secondlythe cell-specific reference symbols. Both are briefly explainedbelow.

A. Synchronization SignalsThe LTE standard defines two synchronization signals,

namely the primary (PSCH) and the secondary (SSCH). InFDD mode, these signals are located on the 62 subcarrierssymmetrically arranged around the DC-carrier in the first slotin the sixth and seventh OFDM symbols of the first and thesixth subframes, as shown in Fig. 1.

B. Cell-specific Reference SymbolsCell-specific reference signals utilize 4-QAM modulated

symbol alphabets and are mapped to the resource elements

The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)

1 frame 10ms

1 subframe 1ms

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

slot 1 slot 1

PSCH

SSCH

Fig. 1. Synchronization signals in LTE FDD downlink.

in the frequency domain as shown in Fig. 2. Whenever thereis one antenna port transmitting a reference symbol on oneresource element, all the other antenna ports transmit a ”zero”symbol at this position. Thus, interference of the referencesymbols transmitted from different antennas is avoided andchannel estimation is simplified.

III. CARRIER FREQUENCY OFFSET IN OFDMIn this section, we define an OFDM system model and

show what impact a carrier frequency offset has on the OFDMsignal. The transmitted time-domain OFDM signal is denotedby x(m)

l,n , the channel impulse response by h(m)l,n , the additive

Gaussian noise by v(m)l,n , and the received time-domain signal

by r(m)l,n . Here, l is the OFDM symbol index within one

subframe, n ∈ [1, N +Ng] the time index within one OFDMsymbol and m the receive antenna index. N denotes the FFTsize and Ng the CP length.

Using the above definitions, the transmission with CFO isdescribed in the time-domain as

r(m)l,n =

{x(m)l,n ∗ h

(m)l,n + v

(m)l,n

}· ej2πεCFO(n+l(N+Ng))/N , (1)

where εCFO is the frequency mismatch between transmitter andreceiver carrier frequency normalized to the subcarrier spacing.

Correspondingly, we define in the frequency-domain X(m)l,k

as the transmitted symbol, H(m)l,k as the channel frequency

response, V (m)l,k as the additive Gaussian noise, and R

(m)l,k as

the received symbol on the k-th subcarrier of the l-th OFDMsymbol at the m-th receive antenna. When the discrete Fouriertransform is applied to the received signal r(m)

l,n in Eq. (1), weobtain according to [2]

R(m)l,k = γ ·X(m)

l,k′ H(m)l,k′ + I

(m)l,k′ + V

(m)l,k , (2)

with

γ =sin(πε′)

N sin(πε′/N)· ejπε

′(N−1)/N · ej2πεCFOl(N+Ng)/N ,

(3)

I(m)l,k′ =

∑p 6=k′

X(m)l,p H

(m)l,p ·

sin(π(p+ εCFO − k))N sin(π(p+ εCFO − k)/N)

· ejπ(p+εCFO−k)(N−1)/N · ej2πεCFOl(N+Ng)/N .(4)

antenna port 1

time

freq

uenc

y

time

freq

uenc

y

time

freq

uenc

y

time

freq

uenc

y

antenna port 2

antenna port 3 antenna port 4

reference symbol of antenna port 1

reference symbol of antenna port 2

reference symbol of antenna port 3

reference symbol of antenna port 4

“zero“ symbol

Fig. 2. Cell-specific reference signals in LTE FDD downlink.

where the relations k′ = k − εIFO and ε′ = εCFO − εIFO =εFFO + εRFO are used. The last relation is obtained becausethe total CFO can be split into the FFO, IFO, and the RFO:εCFO = εFFO + εIFO + εRFO. The factor γ is the degradation onthe desired subcarrier caused by the CFO. I(m)

l,k′ contains theinter-carrier interference from the neighboring subcarriers. Theindex k′ implies that data symbols transmitted on subcarrierk − εIFO are received on subcarrier k.

IV. CARRIER FREQUENCY OFFSET ESTIMATION SCHEME

In this section, we develop a three-stage CFO compensationscheme for UMTS LTE. We assume the channel to be slowfading, that is, the variation of the channel impulse responsewithin one subframe of duration 1ms is negligible.

A. FFO Estimation

As shown in Eq. (3) and (4), FFO is the major source ofinter-carrier interference destroying the orthogonality betweensubcarriers. Therefore, it has to be compensated before theFFT operation in the OFDM receiver.

In [3], a Maximum Likelihood (ML) estimator for the FFOhas been derived. In the context of slow fading, this methodcan be extended to subframe base as below:

εFFO = − 1

2πarg

NR∑m=1

Nf∑l=1

Ng∑n=1

r(m)l,n r

(m)∗

l,n+N

. (5)

In this equation, NR denotes the number of receive antennas,Nf the number of OFDM symbols in one subframe, and Ng theCP length. The estimation range (normalized to the subcarrierspacing) of this stage is (−0.5, 0.5) and is determined by thearg{·} operation. In an AWGN channel where sufficientlylarge SNR is assumed, the MSE of this FFO estimator canbe derived as

MSEεFFO =2σ2

vσ2r + σ4

v

8π2NRNgNfσ4r

=2γt + 1

8π2NRNgNfγ2t≈

≈ 1

4π2NRNgNfγt, (6)

The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)

where γt = σ2r/σ

2v � 1 is the signal-to-noise ratio of the

received signal in time domain. A detailed derivation of Eq. (6)is provided in Appendix I.

It is obvious that the estimation error can be reduced byusing more receive antennas, by extending the CP length, orby taking more OFDM symbols into account.

B. IFO Estimation

After the FFO estimation stage, we assume the fractionalpart of the CFO has been mostly corrected. It can be seenfrom Eq. (2) and Eq. (3) that the integer part of the CFO hastwo effects on the received signal. The first is the FFT indexshift, that is, the symbol transmitted on subcarrier k− εIFO isreceived on subcarrier k. The second effect is a phase rotationproportional to the OFDM index l.

These two effects can be compensated once the IFO isestimated, for example by the ML estimator proposed in [5].The symbols used for estimation can be either pre-definedpilots or PSK modulated data symbols. In the context of LTE,we choose to only apply it to the standardized synchronizationsignals, leading to the following IFO estimator:

εIFO = argmaxi

{<[ej2πiNgN · (7)

·NR∑m=1

∑k∈KSCH

(R

(m)∗SSCH,k+iR

(m)PSCH,k+i

)(X

(m)∗SSCH,kX

(m)PSCH,k

)∗]},

where <{·} returns the real part of the argument and i ∈(−31, 31) corresponds to the set of potential integer offsetsthat can be estimated. This set is determined by the definitionof the synchronization signals in LTE. The set KSCH repre-sents the subcarrier indices that contain the synchronizationsignals. Note that due to the fact that in LTE the synchroniza-tion signals only exist in every fifth subframe, the estimationof εIFO can only be carried out in these subframes.

C. RFO Estimation

The magnitude of the RFO depends on the estimationerror of the FFO. Although its impact on the current OFDMsymbol may not be visible, the estimation error leads to anincreasing phase shift for subsequent OFDM symbols, seeEq. (1). Therefore, it is necessary to improve the estimationby using the cell-specific reference signals to correct the RFOafter the FFT.

Given Eq. (2) and Eq. (3), we assume εFFO and εIFO havebeen corrected to the extent that the interference is smallenough to be neglected. When two symbols on subcarrier kin OFDM symbols l and l + L are observed, there is

W(m)l,k = R

(m)l,k R

(m)∗

l+L,k ·(X

(m)l,k X

(m)∗

l+L,k

)∗= (8)

= e−j2πεRFOL(N+Ng)/N · |X(m)l,k |

2|X(m)l+L,k|

2|H(m)l,k |

2 + V(m)l,k .

The RFO εRFO is only contained in the argument of the signalterm. This holds under the assumption that the channel doesnot change during L OFDM symbols. When applied to thecell-specific reference signals in LTE, L = Ns is the number of

TABLE ISIMULATOR PARAMETERS FOR SIMULATIONS WITH CFO

Parameter ValueBandwidth 5MHz

Transmission setting 1× 1, 2× 2, 4× 2Number of users 1

CQI 9Symbol alphabet 16QAM

Coding rate 616/1024 = 0.602Transmission mode Open loop spatial multiplexing

Maximum HARQ retransmission 0Channel model AWGN1/ITU PedB [12]

OFDM symbol timing PerfectChannel estimation Perfect

Receiver type Soft sphere decoderNumber of subframes 1000

Introduced CFO 3.1415 subcarrier spacings

OFDM symbols in one slot. The corresponding RFO estimatorcan be expressed as

εRFO =− 1

2π

N

Ns(N +Ng)· (9)

· arg

∑m

∑(k,l)∈KP

R(m)l,k R

(m)∗

l+Ns,k·(X

(m)l,k X

(m)∗

l+Ns,k

)∗ .

The estimation is performed on subframe basis with the setKP corresponding to the joint set of subcarrier indices andOFDM symbol numbers on which the reference symbols arelocated.

Similar to the FFO estimation, the estimation range forεRFO is (−N/(2Ns(N + Ng)), N/(2Ns(N + Ng))). Specif-ically, for the 5 MHz mode with normal CP length, εRFO ∈(−0.067, 0.067). The derivation of the theoretical MSE issimilar to that of the FFO estimator and results in

MSEεRFO ≈N2

4π2N2s (N +Ng)2NRKγf

, (10)

where γf = σ2s/σ

2v is the post-FFT signal-to-noise ratio. In

LTE, the factor K is the total number of reference symbolsin one slot. Specifically for the 5 MHz mode we have K =100 when one transmit antenna port is used and K = 200when two transmit antenna ports are used. Since the RFOestimator relies on pairs of reference symbols located on thesame subcarrier, the reference signals of antenna ports threeand four have no contribution.

V. SIMULATION RESULTS

In this section, we present simulation results and evaluatethe performance of the CFO compensation scheme. Simula-tions are carried out using a standard compliant LTE linklevel simulator that was developed at the Vienna Universityof Technology [13]. According to Eq. (6),(10), the frequencyoffset estimation error does not depend on the magnitude of theCFO. Therefore, we introduced a constant CFO and evaluatethe system performance comparing to a perfectly synchronized

1The channel matrix of the 4×2 AWGN channel is defined as [1 1 1 1; 1 −1 − 1 1].

The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)

0 5 10 15 20 25 3010-9

10-8

10-7

10-6

10-5

10-4

10-3

SNR [dB]

Mea

n Sq

uare

d E

rror

FFO estimatorperformanceFFO estimatorperformance

RFO estimatorperformanceRFO estimatorperformance

theoretical 1x1 AWGN

1x1 PedB

1x1 AWGN

Fig. 3. Calculated and simulated Mean Squared Error for FFO and RFO in1× 1 transmission.

transmission. According to [14], sampling frequencies on bothends are assumed to be perfectly synchronized. The othersimulation parameters are listed in Table I. Using the bootstrapalgorithm [15] we calculated the 95% confidence intervalsfor all simulated curves. These intervals are indicated by thevertical bars in the simulated curves.

A. Estimation Performance

The estimation performance is presented in Fig. 3, Fig. 4and Fig. 5 in terms of MSE for the FFO/RFO and errorprobability for the IFO. In the simulated 1000 subframestransmission, FFO and IFO estimations are carried out in everyfifth subframe, leading to a relatively large confidence intervalcompared to the RFO estimation curves.

In Fig. 3 and Fig. 4, it is shown that for AWGN transmis-sion, the simulated curves agree quite well with the calculatedtheoretical MSE. For ITU Pedestrian B (PedB) [12] channelswith time dispersion, the MSE of the FFO estimator saturatesaround 10−5 since the beginning of the CP is corrupted bythe delayed version of the previous OFDM symbol. As longas the FFO estimation error does not exceed the estimationrange of the RFO estimator, this does not affect the overallestimation performance.

For the IFO estimation, errors only occur for the SISOtransmission in PedB channel (Fig. 5). The error in the IFOestimation results in a subcarrier mismatch which fails theRFO estimation where correct reference symbol extraction isrequired (Fig 3, SNR < 10 dB).

B. System Performance

In order to investigate the impact of the CFO on the systemperformance when no CFO compensation is employed, weintroduced 100 logarithmically spaced CFOs between 10−4

and 10−1 subcarrier spacings and observe the variation of thephysical layer coded throughput for a series of CQI values(Fig. 6 and Fig. 7). The simulated throughput curves dropsharply when the CFOs reach certain levels. The higher CQIs

0 5 10 15 20 25 3010-9

10-8

10-7

10-6

10-5

10-4

10-3

SNR [dB]

Mea

n Sq

uare

d E

rror

FFO estimatorperformanceFFO estimatorperformance

RFO estimatorperformanceRFO estimatorperformance

theoretical 2x2 AWGN

2x2 PedB

2x2 AWGN

Fig. 4. Calculated and simulated Mean Squared Error for FFO and RFO in4× 2 transmission.

0 2 4 6 8 10SNR[dB]

inte

ger

part

err

or p

roba

bilit

y10

10

10

10

0

-1

-2

-3

Fig. 5. Error probability of the IFO estimation.

which have larger modulation scheme and higher coding ratealso require more accurate frequency synchronization.

The uncoded Bit Error Ratio (BER) curves for simulationsin AWGN and PedB channels are shown in Fig. 8 and Fig. 9.When comparing to the perfect synchronization cases, both1 × 1 SISO and 4 × 2 MIMO transmission show hardly anyloss. The same trend is observed for the coded throughputcurves in Fig. 10.

Note that CQI 15, which has the most demanding require-ments on CFO compensation, requires the CFO to be lessthan about 10−3 (see Fig. 6) which corresponds to 10−6 ofMSE. This requirement is fullfilled by our CFO compensationscheme at SNR 30 dB (see Fig. 3), explaining the negligibleloss in the simulated BER and throughput curves.

VI. CONCLUSION

In this paper, a CFO compensation scheme for 3GPP LTEis presented and evaluated. Simulation results show that thepresented three stage scheme is sufficient to compensate evenrelatively large CFOs in slow fading scenarios. Comparedto perfect synchronization, the performance loss is hardlynoticeable. Since the synchronization and reference signalsin the standard are not dedicated for carrier frequency syn-

The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)

10-4 10-3 10-2 10-10

5

10

15

20

25

carrier frequency offset [subcarrier spacing]

thro

ugh

put

[Mbit/s

]

CQI=15CQI=15

CQI=10CQI=10CQI=9CQI=9

CQI=7CQI=7

CQI=1CQI=1

CQI=6CQI=6

AWGN channel100 subframes

SNR=30dBSISO

AWGN channel100 subframes

SNR=30dBSISO

Fig. 6. Throughput of SISO transmission over an AWGN channel whenCFO is introduced.

10-4 10-3 10-2 10-10

5

10

15

20

carrier frequency offset [subcarrier spacing]

thro

ughp

ut [M

bit/

s]

CQI=13CQI=13

CQI=10CQI=10

CQI=9CQI=9

CQI=7CQI=7CQI=6CQI=6

CQI=1CQI=1

PedB channel100 subframes

SNR=30dBSISO

PedB channel100 subframes

SNR=30dBSISO

Fig. 7. Throughput of SISO transmission over a PedB channel when CFOis introduced.

chronization but mainly used for cell search and channelestimation, it is not necessary to apply more sophisticatedcompensation schemes to reduce the resource overhead. Weplan further investigation in high mobility scenarios with fastfading channels.

ACKNOWLEDGMENTS

Funding for this research was provided by the fFORTE WIT- Women in Technology Program of the Vienna University ofTechnology. This program is co-financed by the Vienna Uni-versity of Technology, the Ministry for Science and Researchand the fFORTE Initiative of the Austrian Government. Also,this work has been co-funded by the Christian Doppler Lab-oratory for Wireless Technologies for Sustainable Mobility.

0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

SNR [dB]

unco

ded

BE

R

15.4 15.6 15.8 16 16.2 16.4

10-2.7

10-2.6

1x1, perfect sync

4x2, perfect sync

1x1, offset compensated

4x2, offset compensated

4x2, AWGN

4x2, AWGN

1x1, AWGN

1x1, AWGN

Fig. 8. Uncoded BER of CQI=9 SISO transmission in AWGN channel.

0 5 10 15 20 25 3010-4

10-3

10-2

10-1

100

SNR [dB]

unco

ded

BE

R

15.4 15.6 15.8 16 16.2

10-1.4

10-1.3

1x1, perfect sync

4x2, perfect sync

1x1, offset compensated

4x2, offset compensated

4x2, PedB

4x2, PedB

1x1, PedB

1x1, PedB1x1, PedB1x1, PedB

4x2, PedB4x2, PedB

Fig. 9. Uncoded BER of CQI=9 SISO transmission in PedB channel.

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

SNR [dB]

code

d th

roug

hput

[M

bit/

s]

11 11.5 12 12.5

5

5.5

6

6.5

7

1x1, AWGN1x1, AWGN

1x1,

Ped

B

1x1,

Ped

B

1x1,

PedB

1x1,

PedB

4x2,

AW

GN

4x2,

AW

GN

4x2,

Ped

B4x

2, P

edB

perfect syncperfect sync

offset compensatedoffset compensated

Fig. 10. Coded Throughput of CQI=9 SISO/MIMO transmission over AWGNand PedB channel.

The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)

APPENDIX I. DERIVATION OF THE THEORETICAL MSE OFTHE FFO ESTIMATOR

In order to derive the theoretical MSE of the FFO estimator,we rewrite Eq. (5) as

εFFO = − 1

2πarctan

={∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+N

}<{∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+N

} ,(11)

where <{·} and ={·} are the real and imaginary operators.The estimation error can be written as

εFFO − εFFO = (12)

=1

2πarctan

={∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+Nej2πεFFO

}<{∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+Nej2πεFFO

} .When the error is small, Eq. (12) can be approximated to

εFFO − εFFO ∼= (13)

∼=1

2π

={∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+Nej2πεFFO

}<{∑NR

m=1

∑Nf

l=1

∑Ng

n=1 r(m)l,n r

(m)∗

l,n+Nej2πεFFO

} .In order to find the real and imaginary terms, we rewriteEq. (1) for simplicity

r(m)l,n =

(r(m)l,n + v

(m)l,n

)ej2πεFFO(n+l(N+Ng))/N , (14)

with r(m)l,n = x

(m)l,n ∗ h

(m)l,n denoting the received signal without

CFO. Thus, we have

r(m)l,n r

(m)∗

l,n+Nej2πεFFO = (15)

=(r(m)l,n + v

(m)l,n

)(r(m)l,n+N + v

(m)l,n+N

)∗ej2π(εFFO−εFFO)

=(r(m)l,n r

(m)∗l,n+N + r

(m)l,n v

(m)∗l,n+N + v

(m)l,n r

(m)∗l,n+N + v

(m)l,n v

(m)∗l,n+N

)· ej2π(εFFO−εFFO)︸ ︷︷ ︸≈1 for small error

.

When relatively large SNR is assumed, there is

<{r(m)l,n r

(m)∗l,n+N

}�<

{r(m)l,n v

(m)∗l,n+N + v

(m)l,n r

(m)∗l,n+N

+v(m)l,n v

(m)∗l,n+N

}. (16)

Thus, the denominator in Eq. (13) becomes

<

NR∑m=1

Nf∑l=1

Ng∑n=1

r(m)l,n r

(m)∗

l,n+Nej2πεFFO

=

NR∑m=1

Nf∑l=1

Ng∑n=1

r(m)l,n r

(m)∗l,n+N = NRNfNgσ

2r , (17)

where σ2r is the average signal power of the received signal

in time domain.

Based on the fact that E{v(m)l,n } = 0, E{|v(m)

l,n |2} = σ2v ,

the variance of the nominator in Eq. (13) can be found bystraightforward calculation:

E{|={·}|2} = NRNfNg

(σ2vσ

2r +

1

2σ4v

)(18)

Therefore, the MSE is given by

E{|εFFO − εFFO|2} =1

4π2

E{|={·}|2}(NRNfNgσ2

r)2=

2σ2vσ

2r + σ4

v

8π2NRNgNfσ4r

=2γt + 1

8π2NRNgNfγ2t≈ 1

4π2NRNgNfγt, (19)

where γt = σ2r/σ

2v � 1 is the signal-to-noise ratio of the

received signal in time domain.

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The 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'10)