Effective Analysis of Timing Jitter and I/Q Imbalance in OFDM Systems

N.Thirumala Reddy1 ,CH.Nagaraju2

1M.Tech Student, Annamacharya Institute of Technology &Sciences, Rajampet

2Associate Professor, Department of ECE, Annamacharya Institute of Technology &Sciences, Rajampet

1tirumal4a2@gmail.com,2ch_nag2000@yahoo.com

ABSTRACT

In the high data rate orthogonal frequency division multiplexing (OFDM) systems has the problem of intercarrier interference (ICI) due to Timing jitter and I/Q imbalance, this will lead the system performance as high bit error rate. We propose a new algorithm to analyze the interaction between timing jitter and I/Q imbalance which leads the extra ICI terms in their interaction. The analysis indicates that intercarrier interference (ICI) has equal real and imaginary components and is independent of received subcarrier index. Furthermore it is shown that the parameters values influence the intercarrier interference (ICI) from the relative contribution on timing jitter and I/Q imbalance, timing jitter taking over for larger jitter values and I/Q imbalance dominating when timing jitter is relatively small. The interaction is negligible for extra ICI in all cases. The experimental results best matches with the analytical value.

Key words: - OFDM, ICI(Inter carrier interference), Timing jitter, I/Q imbalance

I. INTRODUCTION

A widely held technique for transmission of signals over wireless channels is Orthogonal Frequency Division Multiplexing (OFDM). In a several wireless

standards such as digital video broadcasting (DVB-T), digital audio broadcasting (DAB), the IEEE 802.11a [1] local area networks (LAN) standard and the IEEE 802.16a [2] has been using OFDM scheme.

In the optical fiber systems data rates are extremely high, for example the transmission of 121.9 Gbits/s within an optical bandwidth of 22.8GHz has been shown up [3]. For the very high data rates, the OFDM systems demand high speed digital to analog converters (DACs) and analog to digital converters (ADCs) using precise sampling clocks. However the signal edges of the Practical sampling clocks deviate from the ideal position and these deviations are stated as timing jitter.

The performance of the OFDM system is limited by timing jitter this has been analyzed in recent times [4-7]. The timing jitter causes noticeable performance degradation in high frequency band pass sampling receivers and mitigation techniques are proposed in [4]. In [5], an upper bound for the interference caused by timing jitter is derived and the effects of integer oversampling are studied. In [6], a more extensive analysis of timing jitter is presented including the effect of both white and colored timing jitter. In [6], we introduced a timing jitter matrix to describe the rotational and intercarrier interference (ICI) effect of timing jitter in OFDM

systems and in [7] we applied this matrix to show that both fractional oversampling and integer oversampling can be used to reduce the ICI power due to timing jitter.

Fig 1. Simplified OFDM block diagram.

I/Q imbalance occurs when a front-end component doesn’t respect the power balance or the orthogonality between the I and Q branch [9]. While being able to easily cope with the frequency selective nature of a multi-path propagation channel, multi-carrier systems are very sensitive to I/Q imbalance [10]. In order to cope with these impairments, numerous approaches for a digital compensation of the I/Q imbalance have been proposed in the literature, see for example [11].

In this paper organized as section II describes the system model, Section III presents an analysis of the ICI power caused by the combined effect of timing jitter and I/Q imbalance. Simulations are presented in Section IV. Conclusions are drawn in Section V.

II. SYSTEM MODELAs shown in the figure .1,consider a

OFDM system consist of N subcarriers and the OFDM symbol period ,not including cyclic prefix (CP), is T. At the transmitter, in each symbol period complex values representing the constellation points are

used to modulate N subcarriers. The data to be transmitted in each OFDM symbol period is represented by complex vector X of length N. In most OFDM systems the band-edge subcarriers are not used, so some of the elements of X are zero. The complex time domain samples at the output of the transmitter inverse fast Fourier transform (IFFT) are given by

xn=1

√N ∑k=−N

2+1

N /2

Xk exp( j 2πnkN ) .(1)

By using DACs, the real and imaginary parts of the digital baseband signal xn are converted to analog baseband signals given by

xI (t )=ℜ{ 1√N ∑

k =−N

2 +1

N2

Xk exp( j2 πtkT )}(2)

xQ ( t )=ℑ { 1√ N ∑

k=−N2 +1

N2

Xk exp( j2πtkT )}(3)

where x I (t ) and xQ (t ) denote the real and imaginary parts of analog baseband signal and Re {・ } and Im {・ } are the real and imaginary parts of the argument.x I ( t ) and xQ ( t )are subsequently combined by an I/Q mixer, which we assume to be ideal, to give the passband transmitted signal

x p (t )=x I ( t ) cos (2π f c t )−xQ (t )sin ( 2π f c t )¿ ℜ {x (t)exp ( j 2 π f c t)} ,(4)

Where f c is the RF or optical carrier frequency, and x (t )=x I ( t )+ j . xQ ( t ) .

At the receiver, the receiver signal isy p (t )=x p (t ) hp ( t )+ηp (t ) , (5 )

where ηp ( t )is bandpass AWGN and hp ( t ) is bandpass channel impulse response. Note that y p ( t ) , x p ( t ) ,h p (t ) , ηp ( t ) are all real, while all the baseband signals such as X k, xn, Y k, yk, x I , n, xQ, n, y I ,nand yQ, n, baseband

-

channel impulse response, h(t), and baseband AWGN, η(t), are all complex. In the case of perfect matching between the I and Q branches, the quadrature demodulation and low-pass filtering of the received signal y p ( t ) result in baseband I and Q components y I ( t ) and yQ ( t )given byy I ( t )=LPF {cos (2 π f c t ) . y p ( t ) } (6)y p ( t )=ℜ{x (t ) h ( t )+η (t )y I (t )=LPF {−sin (2π f c t ) . y p (t ) } (7)y p ( t )=ℑ{x (t )h ( t )+η (t )

where LPF{ ・ } represents the low-pass filtering. The quadrature down-converted signals are sampled by I and Q branch ADCs and these each introduce timing jitter [12]. We assume the timing jitter in the I branch is the same as timing jitter in the Q branch. The signal samples after the two ADCs are given by

y I ,n ( t )=ℜ { 1√ N ∑

k=−N

2 +1

N2

H k Xk exp( j 2 πtkT

+τ n)+ηn}(8)

Fig. 2. Block diagram of quadrature down-converter with I/Q amplitude and

phase imbalance.

yQ, n (t )=ℑ { 1√ N ∑

k=−N2 +1

N2

H k X k exp ( j 2πtkT

+τ n)+ηn}(9)

where τ n is the discrete timing jitter and H k is the discrete frequency domain channel

response of the kth subcarrier. The resulting complex samples at the input to the FFT are

yn= y I , n+ j . yQ, n .(10)In any practical system, perfect matching between I and Q branches is not possible due to limited accuracy in the implementation of the RF or optical front-end. In this paper, we consider only the I/Q imbalance at the receiver side. I/Q imbalance can be modeled as either symmetrical or asymmetrical. Both models are equivalent representations [15]. We will use the symmetrical model in this paper. In the symmetrical model [14], each arm experiences half of the phase and amplitude imbalance as shown in Fig. 2. Assume that there is a phase imbalance of θ degrees and an amplitude imbalance of δ dB and that θ and δ are frequency independent. In this case, the FFT output is given by [16]

Y l=∝1

√N ∑n=−N

2+1

N /2

yn exp(− j 2 πnlN )+β 1

√N ∑n=−N

2+1

N /2

yn¿ exp(− j 2 πnl

N )(11)

Withα=cos (θ /2 )+ j (ε /2 ) sin (θ/2 )(12)

β=(ε /2)cos (θ/2 )− jsin (θ /2 )(13)Where the superscript * denotes the complex

conjugate and ε2=(10

δ10−1)/ (10

δ10 +1)

III.TIMING JITTER ANALYTICAL AND I/Q IMBALANCE ANALYSIS

In this section, it is indicated that the noise as ICI components are added due to timing jitter in the received signal. We originate the ICI power caused by I/Q imbalance and timing jitter. Altering (11) into the compact matrix form

Y=α WHX+β W H ¿ X m¿ +N ,(14)

where

Y=[Y −N2

+1⋯Y O⋯Y N

2 ]T ,

X m¿ =[X N

2

¿ ⋯ X0¿⋯ X −N

2+1

¿ ]T ,

W =[w−N2 +1 ,−N

2 +1⋯ w−N

2 +1, N2

⋮ ⋱ ⋮wN

2,− N

2+1

⋯ w N2

, N2

],N=[N −N

2+1⋯ NO⋯ N N

2 ]T

The elements of X m¿ are the complex

conjugate of the transmitted signal’s mirror image. The elements of W are given by

w l ,k=1N ∑

n=−N2

+1

N /2

exp (2 πkT (nT

N+τn))exp(− j2 πnl

N ) .(15)

Both I/Q imbalance and timing jitter cause added noise like components in the received signal. From (14)

Y=α HX+α (W −I ) HX β+W H ¿ Xm¿ +N (16)

ICI due to both timing jitter and I/Q imbalance

Substituting (12) into the first component of the right hand side of (16), we obtain

Y=cos(θ2 )HX+ j( ε

2 )sin( θ2 )HX+α (W −I ) HX+β (W −Im ) H ¿ Xm

¿ +β H ¿ X ¿+N (17)

Where I m is the mirror image of I. We are pondering a unity gain flat channel so H k=1. In order to recover the transmitted signal, both sides of (16) are scaled by

cos (θ2 ) to give

Y

cos( θ2 )

=X+ j( ε2 ) tan(θ

2 )X+ α

cos( θ2 )

(W−I ) X+ β

cos( θ2 )

(W −Im ) Xm¿ + N

cos ( θ2 )

, (18 )

Where X is a wanted component From (15) and (17), we obtain

Y l

cos (θ/2)=X l− jtan(θ

2 )X l¿+( ε

2 )X l¿+ j( ε

2 ) tan( θ2 )X l+( wl , k−Il , k ) X l+ j( ε

2 ) tan( θ2 )+( ε

2 ) ( wl , k−Il ,−k ) X−l¿ − jtan( θ

2 ) ( wl , k−Il ,−k ) X−l¿ +

N l

cos (θ2)(19)

In the right hand side of (19), all the components except the X l component are noise and ICI components related to various impairments. There impairments come as timing jitter, I/Q imbalance and AWGN. At the rear of, we look into the consequences of the both I/Q imbalance and timing jitter in a noiseless channel. Due to I/Q imbalance and timing jitter are independent of the subcarrier index, the average ICI power is the same as the average ICI power for each subcarrier. First, consider the contribution to ICI set off by the interaction between I/Q imbalance and jitter. This is given by the 6 th, 7th and 8th components of the right hand side of (19). The ICI power due to these is

P jitter+θ+ε=E {| j( ε2 ) tan ( θ

2 ) ∑k=−N

2+1

N2

(w l ,k−I l , k) X k|2

}+E {|( ε

2 ) ∑k=−N

2+1

N2

( wl , k−I l ,−k ) X−k¿ |

2

}+

E {|− jtan( θ2 ) ∑

k =−N2

+1

N2

(w l ,k−Il ,−k ) X−k¿ |

2

}.(20)

We take up the timing jitter is white, i.e, the correlation between different timing jitter samples is zero. By using the method in [5], which applies a Taylor series expansion, (20) can be simplified to give

P jitter+θ+ε=σs

2σ 2

6 (( ε2 )

2

tan2( θ2 )+( ε

2 )2

+ tan2(θ2 ))( Nπ

T )2

. (21 )

Thus from (19) and (21) the ratio of the total ICI power (taking in all componets) to signal power ratio is given by

γ=Ptotal

σ s2 {(1+( ε

2 )2

tan2( θ2 )+( ε

2 )2

+ tan2( θ2 ))× 1

6π2 σ j

2}+ tan2( θ2 )+( ε

2 )2

+( ε2 )

2

tan2( θ2 ) ,(22)

Where σ j=σ j N /T is the normalized standard deviation(SD) of the timing jitter.

IV. EXPERIMENTAL RESULTS

In this section, the simulation results shows that the impairments caused by IQ imbalance and timing jitter are examination,to conform the derived analytical results. We employ a system with the subcarriers of N=512, a flat channel and 2000 OFDM symbols. The ICI is camputed based on the extending of constallation points [17].

We observe the combined outcome of timing jitter and IQ imbalance. The I/Q imbalance and timing jitter values are extremely system dependent [17]. Fig. 3 exposes the influence on the ICI to signal power ratio, γ , of varying phase imbalance and timing jitter when there is no amplitude imbalance. Fig. 4 shows the analytical and simulation results for changing amplitude imbalance and timing jitter when there is no phase imbalance,at the same time Fig. 5 gives results for a range of combinations of amplitude and timing jitter phase imbalance.Fig. 3 (no amplitude imbalance) shows that for σ j<0.01 , the ICI hangs on strongly on te phase imbalance, but that as timing jitter levels increase,the effect of the timing jitter dominates and increasing the phase imbalance has only a small effect. Fig. 4 (no phase imbalance) shows a similar effect. For σ j<0.01 the ICI depends strongly on the amplitude imbalance, but as timing jitter increases the effect of timing jitter dominates.

The simulation results agree with analytical results given in (22). Fig. 5 shows the average ICI to signal power ratio against the timing jitter with both phase and amplitude imbalance. When θ=1ο and δ = 0.1 dB, there is very little increase in ICI power compared with the plot without I/Q imbalance. However even when θ=12οand δ = 1.5 dB, and for σ j>0.15, the ICI power increase is not significant compared with the plot without I/Q imbalance. This indicates that timing jitter introduces more ICI power than I/Q imbalance.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-45

-40

-35

-30

-25

-20

-15

--tet

--gam

m(db)

performance Analysis at Del=0

sig=0sig=0.01sig=0.03sig=0.06sig=0.1

Fig. .γ versus the phase imbalance with σ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-40

-35

-30

-25

-20

-15

-10

-5

--Normalized sigma

--gam

m(db)

del=0 theta=0del=0.1 theta=1del=0.3 theta=4del=0.9 theta=8del=1.5 theta=12

Fig. γ. versus timing jitter with I/Q imbalance

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-40

-35

-30

-25

-20

-15

-10

--Delta (db)

--gam

m(d

b)

sig=0sig=0.01sig=0.03sig=0.06sig=0.1

Fig. 4. γ versus the amplitude imbalance with σ

CONCLUSION

In this paper, we showed that the impact and influence of the timing jitter and IQ imbalance in the high data rate OFDM systems. The results say that due to the IQ imbalance and the timing jitter introduces the presents the extra ICI at the receiver and this can produce a critical degradation in the OFDM system. Due to the phase and amplitude imbalance in the OFDM system cause ICI besides timing jitter jointly produce the ICI in the system. The parameter values (θ ,σ , δ) of system influence on the ICI from the relative contribution of the timing jitter and IQ imbalance, for the relative larger values of timing jitter domination and IQ imbalance dominating when timing jitter is relatively small.

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