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\title{Performance Analysis of Zero Forcing Equalizer in 2×2 and 3×3 MIMO Wireless Channel}
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             \author[1]{Nezam  Uddin}

             \author[2]{Motiur  Rahman}

             \author[3]{Tanvir  Ahmed}

             \affil[1]{  Khulna University of Engineering and Technology}

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\date{\small \em Received: 6 December 2013 Accepted: 3 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


Wireless transmission is affected by fading and interference effects which can be combated with equalizer. The use of MIMO system promises good improvement in terms of spectral efficiency, link reliability and Signal to Noise Ratio (SNR). The effect of fading and interference always causes an issue for signal recovery in wireless communication. Equalization compensates for Inter-Symbol Interference (ISI) created by multipath within time dispersive channels. This paper analyzes the performance of Zero forcing method for MIMO wireless channels. The simulation results are obtained using MATLAB. The Bit Error Rate (BER) characteristics for the various transmitting and receiving antenna simulates in MATLAB and describes many advantages and disadvantages of the system. The simulation results show that the equalizer based zero forcing receivers is good for noise free channel and is successful in removing ISI.

\end{abstract}


\keywords{MIMO system, zero forcing equalizer, 2×2 MIMO channel, 3×3 MIMO channel, inter symbol interference (ISI), bit error rate (BER).}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
uring the past there has been an explosion in wireless technology. This growth has opened a new dimension to future wireless communication whose ultimate goal is to provide universal personal and multimedia communication without regard to mobility or location with high data rates. To achieve such an objective, the next generation personal communication network will need to be support a wide range of services which will include high quality voice, data, facsimile, still pictures and streaming video. These future services are likely to include applications which require high transmission rates of several Mega bit per seconds (Mbps). The data rate and spectrum efficiency of wireless mobile communications have been improved over the last decade  {\ref [1]}.\par
In mobile communication systems, data transmission at high bit rates is essential for many services such as video, high quality audio and mobile integrated service digital network. When the data is transmitted at high bit rates, over mobile radio channels, the channel impulse response can extend over many symbol periods, which lead to inter-symbol interference (ISI). This paper discuss the performances of equalization techniques by considering 2 transmitting channel) and 3 transmitting and 3 receiving antenna case (resulting in a 3x3 MIMO channel). Assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK. The ultimate goal is to provide universal personal and multimedia communication without regard to mobility or location with high data rates. To achieve such an objective we need a strong equalization technique to compensate ISI. Hence, there is need for the development of novel practical, low complexity equalization techniques and for understanding their potentials and limitations when used in wireless communication system characterized by very high rates, high mobility and the presence of multiple antennas \hyperref[b1]{[2]}. 
\section[{II.}]{II.} 
\section[{System Overview a) MIMO Systems Basics}]{System Overview a) MIMO Systems Basics}\par
Multiple-Input Multiple-Output (MIMO) \hyperref[b4]{[5,}\hyperref[b7]{8]} technology is a wireless technology that uses multiple transmitters and receivers to transfer more data at the same time shown inFig.1. MIMO technology takes advantage of a radio-wave phenomenon called multipath where transmitted information bounces off walls, ceilings, and other objects, reaching the receiving antenna multiple times via different angles and at slightly different times. data streams arriving from different paths and at different times to effectively increase receiver signalcapturing power. Smart antennas use spatial diversity technology, which puts surplus antennas to good use. If there are more antennas than spatial streams, the additional antennas can add receiver diversity and increase range. MIMO (multiple-in, multiple-out) takes advantage of multiplexing to increase wireless bandwidth and range. MIMO algorithms send information out over two or more antennas and the information is received via multiple antennas as well. On normal radio, multiplexing would cause interference, but MIMO uses the additional pathways to transmit more information and then recombines the signal on the receiving end. MIMO systems provide a significant capacity gain over conventional single antenna systems, along with more reliable communication. The benefits of MIMO lead many to believe it is the most promising of emerging wireless technologies.  
\section[{i. 2×2 MIMO Channel}]{i. 2×2 MIMO Channel}\par
In a 2×2 MIMO channel shown in Fig. \hyperref[fig_2]{2}, probable usage of the available 2 transmitting antennas can be as follows:\par
? Consider that we have a transmission sequence, for example\{x 1 , x 2 , x 3 ?.x n \}.\par
? In normal transmission, we will be sending x 1 in the first time slot, x 2 in the second time slot, x 3 and so on.\par
? However, as we now have 2 transmitting antennas, we may group the symbols into groups of two. In the first time slot, send x 1 and x 2 from the first and second antenna. In second time slot, send x 3 and x 4 from the first and second antenna send x 5 and x 6 in the third time slot and so on.\par
? Notice that as we are grouping two symbols and sending them in one time slot, we need only n/2 time slots to complete the transmission. Hence the data rate is doubled.\par
? This forms the simple explanation of a probable MIMO transmission scheme with 2 transmitting antennas and 2 receiving antennas. The two transmitted symbols interfered with each other and we will use zero forcing equalizer to minimize the interference. ? The channel is flat fading. In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication \hyperref[b3]{[4]}.\par
? The channel experienced by each transmitting antenna is independent from the channel experienced by other transmitting antennas.\par
? For i th the transmitting antenna to j th receiving antenna, each transmitted symbol gets multiplied by a randomly varying complex number h j,i . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of hj, i are Gaussian n distributed having mean ? hj,i =0and variance ? 2 h j,i =1/2 .\par
? The channel experienced between each transmitter to the receiving antenna is independent and randomly varying in time.\par
? On the receive antenna, the noise has the Gaussian probability density function with??(??) = 1 ?2???? 2 ?? ? (???µ) 2 2?? 2\par
With µ = 0 and ?? 2 = ??? 2\par
? The channel h j,i is known at the receiver. 
\section[{i. 3×3 MIMO Channel}]{i. 3×3 MIMO Channel}\par
In a 3×3 MIMO channel shown in Fig. \hyperref[fig_3]{3}, probable usage of the available 3 transmitting antennas can be as follows:\par
? Consider that we have a transmission sequence, for example\{x 1 , x 2 , x 3 ?.x n \}.\par
? In normal transmission, we will be sending x 1 in the first time slot, x 2 in the second time slot, x 3 and so on.\par
? However, as we now have 3 transmitting antennas, we may group the symbols into three groups. In the first time slot, send x 1 , x 2 and x 3 from the first, second and third antenna. In second time slot, send x 4 , x 5 and x 6 from the first, second and third antenna, send x 7 , x 8 and x 9 in the third time slot and so on.4. 
\section[{Global Journal of Researches in Engineering}]{Global Journal of Researches in Engineering}\par
( )F Volume XIV\par
Notice that as we are grouping three symbols and sending them in one time slot, we need only n/3 time slots to complete the transmission. Hence the data rate is tripled.\par
? This forms the simple explanation of a probable MIMO transmission scheme with 3 transmitting antennas and 3receiving antennas. The three transmitted symbols interfered with each other and we will use zero forcing equalizer to minimize the interference. Zero Forcing Equalizer \hyperref[b6]{[7]} refers to a form of linear equalization algorithm used in communication systems which applies the inverse of the frequency response of the channel. This form of equalizer was first proposed by Robert Lucky. The Zero-Forcing Equalizer applies the inverse of the channel frequency response to the received signal, to restore the signal after the channel. It has many useful applications. For example, it is studied heavily for IEEE 802.11n (MIMO) where knowing the channel allows recovery of the two or more streams which will be received on top of each other on each antenna. The name Zero Forcing corresponds to bringing down the inter-symbol interference (ISI) to zero in a noise free case. This will be useful when ISI is significant compared to noise \hyperref[b2]{[3]}.\par
For a channel with frequency response (?) the zero forcing equalizer (?) is constructed by (?) =1/F (?). Thus the combination of channel and equalizer gives a flat frequency response and linear phase F (?) C (?) =1.\par
In reality, zero-forcing equalization does not work in most applications, for the following reasons: 1. Even though the channel impulse response has finite length, the impulse response of the equalizer needs to be infinitely long. 2. At some frequencies the received signal may be weak. To compensate, the magnitude of the zero-forcing filter ("gain") grows very large. As a consequence, any noise added after the channel gets boosted by a large factor and destroys the overall signal-to-noise ratio. Furthermore, the channel may have zeroes in its frequency response that cannot be inverted at all. (Gain * 0still equals 0). This second problem is often the more limiting condition. These problems can be addressed by making as mall modification to the denominator of (?):(?) =1/(F (?) +k)where k is related to the channel response and the signal SNR \hyperref[b5]{[6]}.If the channel response (or channel transfer function) for a particular channel is (s) then the input signal is multiplied by the reciprocal of it. This is intended to remove the effect of channel from the received signal, in particular the inter-symbol interference (ISI).\par
The zero-forcing equalizer removes all ISI, and is ideal when the channel is noiseless. However, when the channel is noisy, the zero-forcing equalizer will amplify the noise greatly at frequencies f where the channel response (j2 ??) has a small magnitude (i.e. near zeroes of the channel) in the attempt to invert the channel completely. A more balanced linear equalizer in this case is the minimum mean-square error equalizer, which does not usually eliminate ISI completely but instead minimizes the total power of the noise and ISI components in the output.\par
Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,y 1 = h 1,1 x 1 +h 1,2 x 2 +n 1 =[h 1,1 h 1,2 ]? x 1 x 2 ?+n 1\textbf{(2)}\par
The received signal on the second receive antenna is,y 1 = h 2,1 x 1 +h 2,2 x 2 +n 2 =[h 2,1 h 2,2 ]? x 1 x 2 ?+n 2\textbf{(3)}\par
where y 1 , y 2 are the received symbol on the first and second antenna respectively, h 1 , 1 is the channel from 1 st transmit antenna to 1 st receive antenna, h 1 , 2 is the channel from 2 nd transmit antenna to 1streceive antenna, h 2 , 1 is the channel from 1 st transmit antenna to 2 nd receiveantenna, h 2 , 2 is the channel from 2 nd transmit antenna to 2 nd receive antenna, x 1 , x 2 are the transmitted symbols and n 1 , n 2 is the noise on 1 st , 2 nd receive antennas.\par
We assume that the receiver knows h 1 , 1 , h 1 , 2 , h 2 , 1 andh 1 , 2 . The receiver also knows y 1 and y 2 . The unknowns arex1and x 2 . With two equations and two unknowns we cansolve it.\par
For convenience, the above equation can be represented in matrix notation as follows:? ?? 1 ?? 2 ? = ? h 1,1 h 1,2 h 2,1 h 2,2 ? ? x 1 x 2 ? + ? n 1 n 2 ?\textbf{(4) Equivalently, y= Hx + n (5)}\par
Global Journal of Researches in Engineering ( )F Volume XIV Issue IX Version I\par
To solve for, we know that we need to find a matrix Wwhich satisfies WH + I.\par
The Zero Forcing (ZF) linear detector for meeting this constraint is given by,W = (H H H) -1 H H\textbf{(6)}\par
This matrix is also known as the pseudo inverse for a general m x n matrix. The term,? h 1,1 * h 2,1 * h 1,2 * h 2,2 * ? ? h 1,1 h 1,2 h 2,1 h 2,2 ? = ? ?h 1,1 ? 2 + ?h 2,1 ? 2 h 1,1 * h 1,2 + h 2,1 * h 2,2 h 1,2 * h 1,1 + h 2,2 * h 2,1 ?h 1,2 ? 2 + ?h 2,2 ? 2 ?\textbf{(7)}\par
ii. Zero forcing (ZF) equalizer for 3×3 MIMO channel\par
Let us now try to understand the math for extracting the two symbols which interfered with Each other for 3×3 MIMO channel. In the first time slot, the received signal on the first receive antenna is, (8)   The received signals on the second and third receive antenna are,y1=h1,1x1 +h1,2x2+h1,3x3+n1 = [ h1,1 h1,2 h1,3]? ?? 1 ?? 2 ?? 3 ? + ny2=h2,1x1 +h2,2x2+h2,3x3+n2= [ h2,1 h2,2 h2,3]? ?? 1 ?? 2 ?? 3 ? + n2 (9) y3=h3,1x1 +h3,2x2+h3,3x3+n3=[ h3,1 h3,2 h3,3]? ?? 1 ?? 2 ?? 3 ? +n3 (10)\par
We assume that the receiver knows h 1 , 1 , h 1 , 2 , h 1 , 3 ,h 2 , 1 ,h 2 , 2 ,h 2 , 3 ,h 3 , 1 ,h 3 , 2 , and h 3 , 3 . The receiver also knows y 1 , y 2 and y 3 . The unknowns are x 1 , x 3 and x 3 . With three equations and three unknowns we can solve it.\par
For convenience, the above equation can be represented in matrix notation as follows:? ?? 1 ?? 2 ?? 3 ? = ? h 1,1 h 1,2 h 1,3 h 2,1 h 2,2 h 2,3 h 3,1 h 3,2 h 3,3 ? ? x 1 x 2 x 3 ? Equivalently, y= Hx + n (11)\par
To solve for x, we know that we need to find a matrix which satisfies WH +I.\par
The Zero Forcing (ZF) linear detector for meeting this constraint is given by,W = (H H H) -1 H H\textbf{(12)}\par
iii. BER with ZF equalizer with 2×2and 3×3 MIMO Note that the off diagonal terms in the matrix HHH are not zero. Because the off diagonal terms are not zero, the zero forcing equalizer tries to null out the interfering terms when performing the equalization, i.e. when solving for x 1 the interference from x 2 is tried to be nulled and vice versa While doing so, there can be amplification of noise. Hence Zero Forcing equalizer is not the best possible equalizer to do the job. However, it is simple and reasonably easy to implement. Further, it can be seen that, following zero forcing equalization, the channel for symbol transmitted from each spatial dimension (space is antenna) is a like a 1×1 Rayleigh fading channel. Hence the BER for 2×2 and 3×3MIMO channel in Rayleigh fading with Zero Forcing equalization is same as the BER derived for a 1×1 channel in Rayleigh fading \hyperref[b3]{[4]}.\par
For BPSK modulation in Rayleigh fading channel, the bit error rate is derived as,P b = ? ? E b N 0 ? ? E b N 0 ?+1 (13) III. 
\section[{Result and Discussion}]{Result and Discussion}\par
As expected, the simulated results with a 2×2 MIMO system using BPSK modulation in Rayleigh channel is showing matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel shown in Fig.  {\ref 4}. The ZF equalizer helps us to achieve the data rate gain, but not take advantage of diversity gain (as we have two receiving antennas). We might not be able to achieve the two fold data rate improvement in  
\section[{Conclusions}]{Conclusions}\par
This paper presents a simulation study on the performance analysis of ZF equalizer based MIMO receiver. The simulation result shows the BER characteristics for the ZF equalizer. From the simulation result we can summarize that, ZF equalization in addition of noise gets boosted up and thus spoils the overall signal to noise ratio. Hence it is considered good to a receiver under noise free conditions. The multiple antennas are used to increase data rates through multiplexing or to improve performance through diversity. This technique offers higher capacity to wireless systems and the capacity increases linearly with the number of antennas and link range without additional bandwidth and power requirements.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_2}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-4.png}
\caption{\label{fig_3}Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-5.png}
\caption{\label{fig_4}GlobalFig. 5 :}\end{figure}
 			\footnote{© 2014 Global Journals Inc. (US) Performance Analysis of Zero Forcing Equalizer in 2×2 and 3×3 MIMO Wireless Channel} 			\footnote{where y 1 , y 2 and y 3 are the received symbol on the first, second and third antenna respectively, h 1 , 1 is the channel from 1 st transmit antenna to 1 st receive antenna, h 1 , 2 is the channel from 2 nd transmit antenna to 1 st receive antenna, h 1 , 3 is the channel from 3 rd transmit antenna to 1 st receive antenna, h 2 , 1 is the channel from 1st transmit antenna to 2 nd receive antenna, h 2 , 2 is the channel from 2 nd transmit antenna to 2 nd receive antenna, h 2 , 3 is the channel from 3 rd transmit antenna to 2 nd receive antenna, h 3 , 1 is the channel from 1 st transmit antenna to 3 rd receive antenna, h 3 , 2 is the channel from 2 nd transmit antenna to 3 rd receive antenna, h 3 , 3 is the channel from 3 rd transmit antenna to 3 rd receive antenna, x 1 , x 3 and x 3 are the transmitted symbols and n 1 , n 2 and n 3 are the noise on 1 st , 2 nd and 3 rd receive antennas respectfully.} 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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