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# Effects analysis of imperfections on multi-antenna transceiving systems

- Huiyong Li
^{1}, - Qi An
^{1}Email author, - Kexin Jia
^{1}, - Hao Wu
^{1}and - Zishu He
^{1}

**2012**:81

https://doi.org/10.1186/1687-1499-2012-81

© Li et al; licensee Springer. 2012

**Received:**28 September 2011**Accepted:**5 March 2012**Published:**5 March 2012

## Abstract

Multi-antenna transceiving systems are widely used in high speed wireless communications due to its higher efficiency comparing to its traditional single-antenna counterparts. In this article, the performance effects of some inevitable imperfections are elaborated on multi-antenna transceiving systems. The simulation results are also displayed. An effective method for phase noise elimination is presented and validated using simulation examples.

## Keywords

- Phase Noise
- Phase Error
- Symbol Error Rate
- Error Vector Magnitude
- Amplitude Error

## 1. Multi-antenna technology scheme

With the growing development of wireless communications in the current society, the conflict between the ever-increasing service requirements of wireless communications and the resource limitation of radio spectrum becomes increasingly apparent [1, 2]. A key technology to solve this problem is multi-antenna techniques, which can enhance the frequency efficiency and link reliability drastically. Compared to the single-antenna communication systems, array gain (i.e., beam forming gain), diversity gain, multiplexing gain, and interference suppression can be obtained by using the multi-antenna communication systems.

In recent communication systems, two of the primary multi-antenna techniques are multi-input multi-output (MIMO) and transceiving beamforming.

MIMO [3, 4], which is developed based on the smart antenna technology, has its distinct superiority. In the MIMO systems, the wireless channels are independent, or weakly correlated. Due to its abundant multi-paths, MIMO systems could enhance the system capability greatly [5].

The signal processing flow on transceiving beamforming is simple and apt to hardware applications because it does not involve so much prior knowledge and any complicated algorithm. However, the channels' consistency is so rigorous in demand. Thanks to the narrow beam, the performance will be decrease sharply if any direction error exists.

According to these disadvantages, some inevitable imperfections will be discussed from four different aspects. In the following section, lots of closed-form expressions are derived and verified by the simulation results subsequently.

## 2. Effects analysis of imperfect factors

*N*×

*N*transceiving beam formation system, there are also lots of imperfect elements even if the beams between transmitting and receiving ends are completely in the right direction. Some of these unsatisfactory factors such as narrowband or wideband amplitude and phase error, the amplitude and phase error in in-phase (I) channel, quadrature (Q) channel, and phase noise, are focused in this article. Each of them is added into the ideal model and discussed, respectively (Figure 1).

### 2.1. Narrowband amplitude and phase error

*s*(

*t*), and the output of the

*m*

_{ th }(

*m*= 1,...,

*N*) transmitting channel with error by ${g}_{m}{e}^{j{\varphi}_{m}}$. Then the output of every transmitting antenna can be written as ${g}_{m}{e}^{j{\varphi}_{m}}\cdot s\left(t\right)$, where

*g*

_{ m }= 1-

*σ*

_{ m }

*, σ*

_{ m }is random and exponentially distributed, and

*ϕ*

_{ m }is uniformly distributed in the range of [0,

*θ*]. The

*n*

_{ th }(

*n*= 1,...,

*N*) channel noise is denoted by

*n*

_{ n }(

*t*). If the amplitude and phase error exists in each signal path at both the transmitting and receiving ends, which are expressed as ${g}_{m}{e}^{j{\varphi}_{m}}$ and ${h}_{n}{e}^{j{\phi}_{n}}$ respectively, the total signal received will be

### 2.2 Wideband amplitude and phase error

*h*

_{ mn }represents the amplitude fading and phase difference between the

*m*

_{ th }transmitting antenna and the

*n*th receiving antenna (Figure 2).

*h*

_{ mn }varies with time and can be modeled as [7]

*τ*is the counter-shifting,

*ϕ*is the relative phase, $k=\frac{2\pi}{\lambda}$ is the wave number and

*d*

_{ mn }is the distance between the

*m*th transmitting antenna and the

*n*th receiving end. The Fourier transform of the channel matrix is

The channel gain is *G* = 10 · log_{10}*P*.

### 2.3. Amplitude and phase error in in-phase and quadrature channels

*T*

_{ m }(

*t*) = 2[

*k*

_{ m }cos(

*ωt*), sin(

*ωt*+

*φ*

_{ m })] and

*R*

_{ n }(

*t*) = [

*l*

_{ n }cos (

*ωt*), sin(

*ωt*+

*γ*

_{ n })]. The

*k*

_{ m }and

*l*

_{ n }(

*m*,

*n*= 1,...,

*N*) represent the amplitude errors and they are both in the range of [1-

*ξ*,1+

*ξ*], where

*ξ*is a tiny number. The phase errors

*φ*and

*γ*are assumed to be much less than $\frac{\pi}{2}$. After the low-pass filtering, the received signal of the

*n*th path is

*H*

_{ n }represents the

*n*th channel matrix of I/Q channel and is defined as

The noise form in (8) is n_{
nr
}(*t*) = *LPF*〈R_{
n
}(*t*)*n*_{
n
} (*t*)〉.

*N*channels is

### 2.4. The model and correction of phase noise

*k*th symbol by

*e*

^{ jφk }. The discrete form of phase noise can be written as

*φ*obeys the normal distribution and is random generated, that is, Δ

*φ*~

*N*(0,

*σ*

^{2}), ${\sigma}^{2}=\frac{4\pi {f}_{\Delta \mathsf{\text{3dB}}}}{{F}_{s}}$,

*f*

_{Δ3dB}is the 3 dB line width,

*F*

_{ s }is the sampling frequency,

*α*is a constant near but less than 1. The relational expression displayed in (9) indicates that the current phase noise is related with its previous moment, which is different to the two other counterparts. Assumed that

*s*(

*t*) is the output signal in each transmitting path and all the transmitted signals can be received by every receiving antenna. The sampled received signal in the

*n*th path can be written as

*N*denotes the number of antennas which is the same for both the transmitting and receiving ends. The sum of received signals is equivalent to

*N*

^{2}times the transmitting signal in the ideal case. Then the error can be expressed as

Thanks to the especial character, a few approaches are derived to eliminate the error *e*(*n*). Gitlin [12] modeled the phase noise as sinusoidal signal and presented self-adaptation and compensation method named finite impulse response-adaptive line enhancer (FIR-ALE). However, the ability of compensation is limited. Another method, the infinite impulse response-adaptive line enhancer (IIR-ALE) [13] has obvious effect provided many prior information being known. extended Kalman filter (EKF) [14] is used to estimate phase noise by linearizing first order digital phase locked loop (DPLL), in which, however, the non-convergent condition happens sometimes.

As an improvement approach to FIR-ALE, the input of LMS algorithm is a more accurate input, which is closer to the real phase offset caused by phase noise. It means a clearer aim and a better compensation ability.

### 2.5 Simulation results

*f*

_{ s }= 4 MHz. Rooted raised cosine (RRC) filters are adopted in both the transmitting and receiving ends. The roll-off factor is

*α*= 0.02. The rate of sampling period and symbol period is 2, and the order of RRC is 16. The effects of different factors on the transceiving performance are shown in Figures 4, 5, and 6. As is shown in Figure 4, symbol error rate (SER) gets a sharply increase when the narrowband phase error starts rising. Comparatively, the amplitude error puts a much less influence on it. Figure 5 displays the effect of wideband amplitude and phase error. When the channel fading is small (i.e., b1 = 0.01), the situation is very similar to the ideal case. However, just a little increase of the fading can lead to the rapid rise of SER much less a large value near 1. The amplitude and phase error caused in I/Q channels are measured by another criteria error vector magnitude (EVM) as it is shown in Figure 6. Yet the amplitude error and phase error both have a drastic influence on the transceiver's performance.

## 3. Conclusion

Aiming at analyzing the transceiving beam forming systems, some imperfect factors are considered and analyzed theoretically. Phase noise is presented as a more inevitable factor. An effective compensation method is presented and verified by simulation results.

## Declarations

### Acknowledgements

The authors would like to thank the support of Fundamental Research Funds for the Central Universities (ZYGX2010J015).

## Authors’ Affiliations

## References

- Paulraj AJ, Nabar RU, Gore DA:
*Introduction to space-time wireless communications [M].*Cambridge University Press, Cambridge; 2002.Google Scholar - Liu X, Yang LT, Sohn KHigh-Speed Inter-view Frame Mode Decision Procedure for Multi-view Video Coding, Future Generation Computer Systems (Elsevier), May 2011, doi:10.1016/j.future.2011.05.013Google Scholar
- Gesbert D, Shafi M: From theory to practice: an overview of MIMO space-time coded wireless systems.
*IEEE J Sel AreasCommun*2003, 21(3):281-302. 10.1109/JSAC.2003.809458View ArticleGoogle Scholar - Goldsmith A, Jafar SA, Jindal N, Vishwanath S: Capacity limits of MIMO channels.
*IEEE J Sel Areas Commun*2003, 21(5):684-702. 10.1109/JSAC.2003.810294View ArticleMATHGoogle Scholar - He Q, Blum RS, Haimovich AM: Non-coherent MIMO radar for location and velocity estimation: more antennas means better performance.
*IEEE Trans Signal Process*2010, 58(7):3661-3680.MathSciNetView ArticleGoogle Scholar - Ingason T, Liu H, Coldrey M, Wolfgang A, Hansryd J: Impact of frequency selective channels on a line-of-sight MIMO microwave radio link.
*Vehicular Technology Conference (VTC 2010-Spring)*2010, 1-5.View ArticleGoogle Scholar - Georgiadis A: Gain, phase imbalance, and phase noise effects on error vector magnitude.
*IEEE Trans Veh Technol*2004, 53(2):443-449. 10.1109/TVT.2004.823477MathSciNetView ArticleGoogle Scholar - Chen Z, Dai FF: Effects of LO phase and amplitude imbalances and phase noise on M-QAM transceiver performance.
*IEEE Trans Indust Electron*2010, 57(5):1505-1517.View ArticleGoogle Scholar - Kasdin NJeremy: Discrete simulation of colored noise and stochastic processes and 1/f
^{α}power law noise generation.*Proc IEEE*1995, 83(5):802-827. 10.1109/5.381848View ArticleGoogle Scholar - Berger S, Wittneben A: Comparison of channel estimation protocols for coherent AF relaying networking in the presence of additive noise and LO phase noise.
*EURASIP J Wirel Commun Netw*2010., 2010:Google Scholar - Georgiadis A: Gain, phase imbalance, and phase noise effects on error vector magnitude.
*IEEE Trans Veh Technol*2004, 53(2):443-449. 10.1109/TVT.2004.823477MathSciNetView ArticleGoogle Scholar - Wu S, Liu P, Bar-Ness Y: Phase noise estimation and mitigation for OFDM systems.
*IEEE Trans Wirel Commun*2006, 5(12):3616-3625.View ArticleGoogle Scholar - Gholami MR, Nader-Esfahani S, Eftekhar AA: A new method of phase noise compensation in OFDM. IEEE International Conference on Communications (ICC '03) 3443-3446. Vol. 5, Issue Date 11-15 May 2003Google Scholar
- Farhang-Boroujeny B: Pre-equaliser cancellation of sinusoidal phase jitter.
*IEE Proc Commun*1995, 142(4):216-220. 10.1049/ip-com:19952054View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.