Growth of Wireless and Mobile Communication

 Growth of Wireless and Mobile Communication
 
 
 
 
 
 
 
 
 
 
 
 
 
1.0 INTROUCTION
 

Figure1.1: Graph showing number of mobile phone users.
 
Several new technologies have been explored and deployed in this regard to make effective use of the limited resources. One way to improve capacity is by using the concept of cellular technology, which involves dividing a large coverage zone into small hexagonal cells. Therefore, a single high power transmitter is replaced with many low power transmitters. Each cell is allocated a set of frequency channels that are different from those allocated to the neighboring cells. However the same set of frequencies can be reused by another cell as long as they are separated well enough not to cause interference. Since each of these cells reuses the frequency spectrum, a significant increase in capacity can be achieved. However, increasing the number of cells to accommodate growing subscriber needs is neither an economical nor an effective option. Within the cells in a cellular network a further increase in capacity is achieved by efficiently sharing the frequency channels using multiple access techniques. The first generation cellular networks used analog frequency division multiple access (FDMA) technique. FDMA allocates different carrier frequencies to different users. The second-generation cellular systems were based on digital modulation techniques and they used time division multiple access (TDMA) scheme. In the TDMA scheme, different users using the same frequency channel are allocated different time slots. Third generation cellular system uses another type of access technique referred to as the code division multiple access (CDMA) technique. It is based on the spread spectrum technology where individual users are identified by the use of signature codes and they occupy the same bandwidth.

Figure1.2: Graph showing number of mobile subscriber in the world
So far these technologies have brought about tremendous increase in wireless network capacity to meet the increasing demand for wireless services. However personal wireless communications is getting more and more popular and is continuing to grow at an exponential rate. Therefore new technologies particularly antennas in mobile communications are needed to accommodate future capacity needs. Antennas employed at the base station, radiates suitable beams to serve different users.
1.2 Antenna
An antenna (or aerial) is a transducer designed to transmit or receive electromagnetic waves. In other words, antennas convert electromagnetic waves into electrical currents and vice versa. They are used with waves in the radio part of the electromagnetic spectrum, that is, radio waves, and are a necessary part of all radio equipment. Antennas are used in systems such as radio and television broadcasting, point-to-point radio communication, wireless LAN, cell phones, radar, and spacecraft communication. Antennas are most commonly employed in air or outer space, but can also be operated under water or even through soil and rock at certain frequencies for short distances.
Physically, an antenna is an arrangement of one or more conductors, usually called elements in this context. In transmission, an alternating current is created in the elements by applying a voltage at the antenna terminals, causing the elements to radiate an electromagnetic field. In reception, the inverse occurs: an electromagnetic field from another source induces an alternating current in the elements and a corresponding voltage at the antenna's terminals. Some receiving antennas (such as parabolic and horn types) incorporate shaped reflective surfaces to collect EM waves from free space and direct or focus them onto the actual conductive elements.
 
1.3 Smart Antenna
A smart antenna is a digital wireless communications antenna system that takes advantage of diversity effect at the source (transmitter), the destination (receiver), or both. Diversity effect involves the transmission and or reception of multiple radio frequency (RF) waves to increase data speed and reduce the error rate.
 
Smart antennas have promised to provide significant increases in system capacity and performance in wireless communication systems. In turn, this leads to increased revenue for the telecommunications companies and also a reduction in dropped and blocked calls. Other benefits include greater coverage, meaning less base stations are needed to cover the same area compared to conventional antennas. For these reasons, smart antennas have gained greater interest over the recent years.
 
Smart antennas were first used in RADAR applications in the form of phased array. Research on application of smart antennas has paved the way for their use in commercial wireless systems. Smart antennas are currently used in wireless communication systems to provide interference reduction and enhance user capacity, data rates. Current applications of the smart antennas are predominantly at the cellular base stations due to area and processing power requirements.
 
However, recent propagation measurements for smart antennas and the development of faster and low-power processors have enabled the use of this technology at the access points in a WLAN system in the form of dual diversity reception. Mobile terminal based smart antennas are still in the research stage for large network and ad-hoc network scenarios where they have been touted to improve average network capacity. In future one can expect smart antenna technology to be present at base stations and mobile terminals. The goal of this analysis is to present in a simple yet comprehensive manner the smart antenna technology to the non-specialists.
 
1.4 Outline of a Smart Antenna
 
The definition of a Smart antenna is an antenna array system that is aided by a processing system that processes the signals received by the array or transmitted by the array using suitable array algorithms to improve wireless system performance. An antenna array consists of a set of distributed antenna elements (dipoles, monopoles or directional antenna elements) arranged in certain geometry (e.g., linear, circular or rectangular grid) where the spacing between the elements can vary. The signals collected by individual elements are coherently combined in a manner that increases the desired signal strength and reduces the interference from other signals.
 
Hence a smart antenna can be viewed as a combination of “regular or conventional” antenna elements whose transmit or received signals are processed using “smart” algorithms. In the Figure 1.3 Generic implementation of Smart Antenna System, shows a generic implementation smart antenna system.

Figure 1.3 Generic implementation of Smart Antenna System
 
The antenna arrays have input or output as RF signals in the analog domain. These signals are passed to or from the RF analog front end which usually consists of low noise amplifiers, mixers and analog filters. In the receive mode, the RF signals are converted to digital domain by analog to digital converters (ADCs) and in transmit mode, the base band digital signals are converted to RF using digital to analog converters (DACs). The down-conversion from RF to base band or up-conversion from base band to RF can involve the use of IF signals. The base band signals received from each antenna is then combined using the “smart” algorithms in a digital processing section. Each antenna element hence has a RF chain going from the antenna element to RF front end to digital conversion for receiver and vice-versa for transmitter. The digital processing section can be implemented on a microprocessor or a DSP or FPGA.
 
Hence the “smart” algorithm implementation usually is a software code unless implemented in an ASIC or FPGA. An example of the array processing in the digital domain is shown in Figure 1.4.

Figure 1.4: Block diagram representation of antenna array processing.
 
The diagram illustrates the operation of an M-element antenna array system in the receive mode. The signals collected by the antenna elements are down converted, sampled and digitized to generate the beam former inputs (x1, x2,.. xM). These signals contain both the desired signal and the interfering signals and these are appropriately scaled by complex gain vectors, also known as weight vectors (w1, w2… wM) and combined to generate the array output y as:
y (t)= w (t) x( t)                                                                                     ( 1 . 1 )
The array output is then compared with some reference signal in the ‘Generate Error Signal’ block to generate an error signal which is then adaptively minimized by an adaptive algorithm. This adaptation process involves changing the weight vector according to some minimization criteria. For example, for stochastic gradient based Least Mean Square (LMS) algorithm, the weight update equation has the following form:
w (k + 1) = w (k) + μ e( k) *x( k)                                                          ( 1 . 2 )
Where w (k), e (k) and x (k) are the weight vector, error signal and input signal vector at the k-th instant and ‘*’ denotes complex conjugate operation. In most cases, the weight vector is updated during some training sequence when some known or pilot symbols are transmitted and at the end of the training sequence, the array output is fed to the demodulator and subsequently to the upper layers of the system.
 
 
IMPLEMENTATION OF A SMART ANTENNA SYSTEM
2.1 Execution technique of a Smart Antenna System
The wireless spectrum is limited and during the last decade it has become a precious resource. Achieving the capacities needed for future wireless systems without increasing the required spectrum will only be accomplished by the design and implementation of advanced communications techniques such as multi-antenna systems. These systems are realized by time-consuming and computationally complex algorithms.
 
Multi-antenna systems consist of two or more antenna elements either at the transmitter, the receiver, or both. Here the two different groups of multi-antenna systems, smart antenna systems and multiple input – multiple output systems (MIMO), will be discussed.
 
2.2 Smart / Multi-antenna System
 
A smart antenna is a digital wireless communications antenna system with multiple antenna elements at the source (transmitter), the destination (receiver), or both, where signals from the different antenna elements are combined or created in an intelligent way by an algorithm. The smart antenna system can be utilized in a number of ways. It can be used to increase the capacity and the coverage (beam forming) in a mobile communication system. It can also be used for improving the link quality, user position estimation, and to decrease the delay dispersion. There are a few techniques that are used as an approach to this system.
 
2.2.1 SISO
 
Conventional wireless communications, a single antenna is used at the source, and another single antenna is used at the destination as shown in Figure 2.1. This communication system is referred to as a single input – single output (SISO) system.
 
Assume that a transmitter with a single antenna element transmits Omni directional, meaning that the signal or wave front is transmitted in all directions, and that the receiver antenna listens for signals coming from all directions. Sending signals by transmitting energy in all directions is not energy efficient.

Figure 2.1 A conventional SISO communications system.
 
A better way is to only transmit in the direction of the receiver. In the same manner it is more efficient to only listen in the direction of the transmitter and not in all directions at the same time. This will increase energy efficiency and will also lead to a reduction in interference between different transmitters and thereby increase the efficiency in an interference limited system.    
 
Another drawback with SISO systems is that they are vulnerable to multipath effects. When the electromagnetic wave front travels towards the receiver, its propagation path can be obstructed by objects. In an outdoor environment this can for instance be caused by objects such as hills, buildings, trees, cars, etc., while in an indoor scenario the signal can be obstructed by doors, walls, people, furniture, etc. The wave fronts will then be reflected and scattered by these objects, thus creating multiple paths to the receiver (figure 2.2).

Figure 2.2 Scattered and reflected signals due to obstruction, causing multipath effects.
 
The wave front, arriving in scattered portions at different time instances, can cause problems resulting in intermittent reception [2, 6]. In digital communications this can cause an increase in the number of errors resulting in a reduction in data rate. The use of smart antennas can reduce the deterioration of the transmitted wave front caused by multipath wave propagation by automatically changing the directionality of its radiation patterns in response to its signal environment.
 
 
2.2.2 SIMO
 
There are mainly two different categories of smart antenna systems [6]. Single input – multiple output system (SIMO). In a SIMO system, one antenna is used at the transmitter, and two or more antennas are used at the receiver as shown in Figure 2.3.

Figure 2.3 A single input – multiple output system.
 
2.2.3 MISO
 
Multiple inputs – single output (MISO). In a MISO system, two or more antennas are used at the transmitter, and one antenna is used at the receiver as shown in Figure 2.4.

Figure 2.4 A multiple input – single output system.
 
By applying the techniques shown in Figures 2.3-2.4 we can transmit in a specific direction or listen in a specific direction. Figure 2.5 shows the same scenario as in figure 2.2 but with a smart antenna as a receiver. The smart antenna system detects the three multi-paths and creates “listening” beams for those directions. Subsequently, all other signals are suppressed.
 
In this way the signals coming from the directions of the listening beams can be combined at the receiver, thus increasing the signal-to-noise ratio and lowering the bit error rate. The concept of using smart antennas to transmit and receive data more intelligently has existed for many years. Simple smart antenna techniques, like the switched beam technology, where the antenna systems from multiple fixed beams with heightened sensitivity in particular directions, have been used in commercial applications for some time . These antenna systems detect signal strength, choose from one of several predetermined, fixed beams, and switch from one beam to another as the mobile device moves throughout the beam pattern.
 
Smart antenna technology represents the most advanced approach to date taking advantage of its ability to effectively locate and track various types of signals to minimize interference and maximize signal reception.

 
Figure 2.5 A SIMO system where the multiple antennas at the receiver create beams that listen in the directions of the multipath.
 
One sophisticated utilization of smart antenna technology is spatial division multiple access (SDMA). In this technique, single mobile terminals are located and tracked by adaptively steering transmission signals toward users and away from interferers (figure 2.6). In this way a high level of interference suppression is achieved, making possible more efficient reuse of the frequency spectrum.
 
Smart antenna technology can, with some modification, be integrated into all major access methods such as frequency division multiple access (FDMA), time division multiple access (TDMA), code division multiple access (CDMA), etc. and has widespread applications in several different areas such as digital television (DTV), body area networks (BAN), personal area networks (PAN), wireless local area networks (WLAN), metropolitan area networks (MAN), and mobile communications. However, the technique requires sophisticated algorithms and computationally heavy algorithms to operate in real-time.

Figure 2.6 Smart antenna techniques can be used in satellite transmission to cover small hotspots, or in cellular systems to track individual mobiles.
 
2.2.4 MIMO
 
MIMO systems are characterized by having multiple antennas at both the transmitter and the receiver as shown in figure 2.7. The number of antenna elements does not have to be the same at the transmitter and the receiver.

 
Figure 2.7 A multiple input – multiple output system.
 
A MIMO system is mainly used for three different purposes; beam forming, diversity, and spatial multiplexing. Both beam forming and diversity can be use in the same way as in the case of the smart antenna system [6]. By applying a MIMO beam forming system to the scenario in figure 2.2, the signal can be transmitted in one or more favorable directions. Figure 2.8 shows how the signal is transmitted in two beams from the transmitter and received via two beams formed by the receiver antenna.

 
Figure 2.8 A MIMO system using beam forming to transmit the signal in specific directions and creating beams to listen for signals coming from those directions.
 
In this way transmission energy is saved, since less energy is transmitted in other directions than those of the receiver. Another way of using a MIMO system that has attracted lots of interest in recent years is spatial multiplexing. Spatial multiplexing offers an improvement of the capacity by simultaneously transmitting multiple data-streams.
 
This is done by multiplexing a data-stream into several parallel data-streams that are sent from separate antenna elements as shown in figure 2.9.

 
Figure 2.9 A data-stream is multiplexed onto different antenna elements in a MIMO System.
 
Data transmitted from the multiple antenna elements will be mixed when traveling throughout the propagation channel. Each individual antenna element in the receiver will detect a combination of the transmitted data.
 
The received data must then be resolved by signal processing algorithms before it can be combined into a single data stream again. In this way MIMO can exploit the phenomena of multipath propagation to increase throughput, or reduce bit error rates, rather than suffer from it.
 
MIMO will be incorporated into the new IEEE 802.1 1n standard for local-area wireless networks, which will improve the coverage and data rate significantly. The IEEE 802.1 1n standard is still being discussed, but data throughput is estimated to reach a theoretical 540 Mbit/s. The data rate requirement at the physical layer may be even higher, prompting for new high-speed hardware solutions. Although a few manufacturers have released consumer products with so called pre-n hardware, exploiting rudimentary diversity by using 2 to 4 antenna elements, the widespread usage of MIMO will not be a reality before the standard is set. MIMO has also been added to the latest draft version of Mobile WiMAX (IEEE 802.1 6e).
 
To be able to fully take advantage of the emerging standards, new high throughput hardware architectures must be developed. A first step in the development process is to analyze the multi-antenna algorithms and to identify common algorithmic features.
 
2.3 Classification of Smart Antennas
 
Based on the signal processing technique followed at the baseband output of the antenna array smart antennas can be grouped into four basic types based on:
i)  Beam forming
ii) Diversity combining
iii) Space time equalization
iv) Multiple input multiple output (MIMO) processing.
 
 
2.3.1 Beam forming
 
Through beam forming, a smart antenna algorithm can receive predominantly from a desired direction (direction of the desired source) compared to some undesired directions (direction of interfering sources). This implies that the digital processing has the ability to shape the radiation pattern for both reception and transmission [3] and to adaptively steer beams in the direction of the desired signals and put nulls in the direction of the interfering signals. This enables low co-channel interference and large antenna gain to the desired signal. Figure 2.10 shows the formation of transmit beams to desired users. Figure 2.11 shows a receiver beam former which puts a null in the direction of interferer by choosing appropriate weights using adaptive beam forming algorithm.

 
Figure 2.10 User beam forming transmission
 

Figure 2.11 Receiver beam forming reducing co-channel interference
 
Beam forming systems can be implemented in two ways; fixed beam forming systems or fully adaptive systems. A fixed beam forming system has a beam forming network (BFN) followed by RF switches which operate in the RF/analog domain. The switches are controlled by a control logic which selects a particular beam. Here the processing required is minimal as the control logic has to choose one of the predetermined set of weights to select a beam. In adaptive beam forming, the antenna gains or weights are chosen adaptively through running array algorithms in the digital domain.
 
2.3.2 Diversity combining
 
A major limiting factor in wireless communication is multipath fading where the amplitude of the received signal fluctuates over time. The occurrence of a deep fade where the signal amplitude becomes very small can impair the communications link for a conventional or a single antenna system. When multiple antennas are used it becomes less likely that two or more antennas undergo deep fades at the same time. This diversity in the received signal, for the same transmitted information, is exploited by smart antenna processing schemes. Many simple algorithms, such as maximal ratio combining, equal gain combining, and selection diversity have been developed to take advantage of using antenna arrays to exploit diversity reception in wireless systems. These algorithms weight the received signal similar to beam forming but based on a different criterion used in the algorithm.
 
2.3.3 Space-time equalization
 
The preceding two techniques usually assume that the signal of interest is a narrowband signal compared to the coherence bandwidth of the channel and is thus subjected to flat fading across the bandwidth of the signal. Multipath fading in wireless communication can also introduce frequency distortion to the received signal. By introducing temporal processing in each antenna element to remove the effect of frequency distortion and doing a spatial combining described above results in mitigating channel induced frequency selective fading and providing antenna gain. Such schemes are called space-time adaptive processing (STAP) or equalization.
 
2.3.4 Multiple Input Multiple Output (MIMO)
 
As the name suggests this scheme requires array processing at the transmitter and receiver. There are two different types of MIMO schemes: one uses spatial multiplexing to enhance data rate for a given bandwidth (thus, the spectral efficiency) and the other uses space time coding using diversity combining techniques to combat fading. In the multiplexing scheme, data is serial to parallel converted and transmitted simultaneously over multiple antenna elements. The receiver also uses multiple antenna elements to receive the signal and applies a maximum likelihood (ML) algorithm to retrieve the simultaneously transmitted symbols. One key assumption in this case is that the propagation environment has to provide rich scattering; in other words, the propagation channel has to include a large number of scattering objects that will generate independent fading at the antenna elements. In the case of space-time coding, symbols to be transmitted are coded over multiple antennas and symbol time durations in such a way that the receiver can easily regenerate the transmitted signals by doing a linear processing on received signal.
 
The space-time codes rely on the orthogonality present in the coded symbols for proper detection, and additionally they require the fading to be independent between the antenna elements for best performance results. Figure 2.12 shows the setup of a MIMO system with nT and nR antennas at the transmitter and receiver respectively.

Figure 2.12 MIMO system set-up
 
2.4 Advantages and disadvantages of smart antennas
 
Primarily smart antennas were used at base stations in a cellular network to improve user capacity. Capacity here refers to the number of subscribers that can be simultaneously serviced in a system. Usage of Omni-directional antennas causes co-channel interference when two users use the same band of frequency that eventually limits the user capacity in a system. Since smart antennas can focus their beams towards desired user reducing interference to other users using the same frequency band, the user capacity in a system can be improved using spatial division multiple access (SDMA). Figure 2.15 [7] shows this advantage of SDMA compared to the Omni-directional case, which can reduce co-channel interference using beam forming.

 
Figure 2.15 Omni-directional and Smart antennas based cellular system.
 
Other advantages as seen from various types of smart antennas studied in section 2 include robustness against multipath fading and noise which improves reliability of received signal; reduced power consumption for handsets; low probability of interception and detection; enhanced location estimates and enhanced range of reception. Recent studies on use of smart antennas in mobile terminals have also shown to improve network capacity in ad-hoc networks.
 
One of the major existing disadvantages of smart antennas is in their design and implementation in hardware. Multiple RF chains can increase the cost and make the transceiver bulkier. Most of the baseband processing requires coherent signals. This means that all the mixer LOs and ADC clocks need to be derived from same sources. This can present significant design challenges. The phase characteristics of RF components can change over time. These changes are relatively static and hence need calibration procedures to account for phase differences. Most of the devices such as mixers amplifiers and ADCS used are non-linear devices.
 
Using smart antennas can increase the number of such components used. This can affect the performance of the array if not checked periodically. Further more since antenna arrays use more than one source of signal the data bandwidth required for digital processing increases linearly with number of antenna elements used. This can limit data rates for different applications. Note that the technological challenges in terms of hardware and processing load can be satisfactorily met by resorting to present-day miniaturized RF components and faster and low power processors.
 
The accommodation of the antenna array itself within a small factor device, however remains a challenge. Access points and base stations can easily host antenna arrays of four or more elements but with existing micro strip or patch antenna technology, up to three elements can be fitted in a handset form-factor. The wrapping of the hand around a handheld device may diminish the performance of a handheld smart antenna system.
 
Antenna arrays
 
For some applications single element antennas are unable to meet the gain or radiation pattern requirements. Combining several single antenna elements in an array can be a possible solution. This article introduces the basic concepts of antenna arrays.
 
3.1 Antenna arrays, radiation pattern and array factor
The antenna elements can be arranged to form a 1 or 2 dimensional antenna array. A number of antenna array specific aspects will be outlined; we used 1­dimensional arrays for simplicity reasons. Antennas exhibit a specific radiation pattern. The overall radiation pattern changes when several antenna elements are combined in an array. This is due to the so called array factor: this factor quantifies the effect of combining radiating elements in an array without the element specific radiation pattern taken into account. The overall radiation pattern of an array is determined by this array factor combined with the radiation pattern of the antenna element. The overall radiation pattern results in a certain directivity and thus gain linked through the efficiency with the directivity. Directivity and gain are equal if the efficiency is 100%.
3.2 Broadside vs. end fire arrays
Arrays can be designed to radiate in either broadside i.e. radiation perpendicular to array orientation (the z axis in fig 3.1) or end fire i.e. radiation in the same direction as the array orientation (the y-axis in fig 3.1). We will focus on broadside arrays and only radiation in the z direction is considered. This allows for easy transformation to 2-dimensional planer arrays with the elements in the xy plane. For linear arrays the radiation patterns given below are a cross section in the yz plane. Actually the 3 dimensional radiation pattern of a linear array is a rotation around the y-axis of the patterns given.
 

 
Figure 3.1: Topology of a linear array.
 
3.3 Defining array factor
The array factor depends on the number of elements, the element spacing, amplitude and phase of the applied signal to each element. The number of elements and the element spacing determines the surface area of the overall radiating structure. The surface area is called aperture. A larger aperture results in a higher gain. The aperture efficiency quantifies how efficient the aperture is used. The influence of these parameters will be further explained with the aid of a linear array of isotropic radiating elements. An isotropic radiating element radiates an equal amount of power in all directions, i.e. it has a directivity of 1(0dB) and a gain of 1(0dB) if the efficiency were 100%. In the outline below the array factor is normalized to the array directivity. This results in more intuitive and realistic radiation pattern plots
.      
3.4 Influence of the number of elements on the array factor
The array directivity increases with the number of elements. Figure3. 2 show the directivity of 3 arrays with 2 (red), 5(green) and 10 (blue) elements. The element spacing is 0.4 times the wavelength (λ) for all the arrays in figure 2. Note the presence of side lobes next to the main lobes: this is typical for arrays. The number of side lobes and the side lobe level increase with the number of elements. It is important to note that due to the array factor definition there are 2 main lobes. There is a main lobe at theta 0° (positive z axis)and a main lobe at theta 180°/­180°(negative z axis).
 
 
 

Figure 3.2: Directivity of a 2(red), 5(green) and 10 (blue) element arrays with 0.4λ element spacin.
 3.5 Influence of the element spacing on the array factor
The element spacing has a large influence on the array factor as well. A larger element spacing results is a higher directivity. However, the element spacing is generally kept smaller than λ/2 to avoid the occurrence of grating lobes. A grating lobe is another unwanted peak value in the radiation pattern of the array.
 

Figure 3.3: Directivity of a 5 elements array with 0.2 (red), 0.3(green) and 0.5(blue) time’s λ element spacing.
Increasing the element spacing towards λ results in an increased directivity and grating lobe effect with a maximum grating lobe amplitude equal to the main lobe magnitude at an element spacing λ as shown in figure (3.4).

 Figure 3.4: Directivity of a 5 elements array with 0.5(red), 0.75(green) and 1(blue) time’s λ elements spacing.
 
An element spacing beyond λ becomes impractical and results in multiple unwanted grating lobes as depicted in figure 3.5.

Figure3.5: Directivity of a 5 elements array with 1(red), 1.5(green) and 2(blue) time’s λ element spacing
3.6 Influence of the radiating element properties on the overall radiation pattern.
 
A number of examples of total radiation patterns are given below in order to give an idea of the effect of the radiating element radiation pattern on the overall array radiation pattern. Figure 3.6 shows the radiation pattern of an isotropic element (red), the array factor and the combined radiation pattern(both green).In this case the overall radiation pattern is the same as the array factor since an isotropic element radiates the same amount of power in all directions.
 
 
Figure 3.6: Directivity of an isotropic source (red) in a 5 elements array (green) with 0.4 λ element spacing.
 
Figure 3.7 shows the radiation pattern of a dipole (red), the same array factor as in figure 6 (green) without dipoles and the overall radiation pattern of the array with dipoles (blue). The overall radiation pattern is clearly different from the array factor i.e. the directivity has increased with the dipole’s directivity and the overall radiation pattern is lightly modified due to the dipoles radiation pattern.
 

 
Figure3.7: Directivity of a dipole in a 5 elements array with 0.4 λ element spacing.
 
Figure 3.8 shows the radiation pattern of a dipole on an infinite ground plane (red), the same array factor as in figure 3.6(green) without dipoles and the overall radiation pattern of the array with dipoles on an infinite ground plane (blue). The dipole has a radiation lobe in the positive z axis only (broadside and because of the ground plane). Note that the overall array does indeed not radiate in directions were the antenna element does not radiate i.e.  no radiation in the negative z direction any more. The overall array has thus a perfect front to back ratio; this makes sense because we have used an infinite ground plane.

Figure 3.8: Directivity of a dipole on infinite ground in a 5 elements array with 0.4 λ elements spacing.
 
 
3.7 Feeding of an array
 
In the previously discussed arrays the element spacing has been kept constant and the elements were fed with the same amplitude and phase. The resulting arrays were linear arrays with uniform spacing, uniform amplitude and equal phase. However, the power does not necessarily have to be distributed with equal amplitude and/or phase. Unequal power and phase distribution to the individual elements can be used to modify the side lobe level, directivity and direction of the main lobe. A range of standard amplitude and phase distributions exists (e.g. uniform cosine pedestal) but this is beyond the scope of this article. The reader should realize that any modification to an array will have some adverse effect on the performance of the array and a careful trade off is required. When the power distribution is optimized to reduce the side lobe level, the efficiency of the array decreases; when the phase distribution is optimized to do beam steering, new side lobes will show up as the main beam is deflected sideways.
 
3.8 Feed network of the array
 
The individual antenna elements in an array are fed using a feed network. The complexity of the feed network depends on the number of elements, the amplitude and/or phase distribution between the elements, the ability to do beam steering. It is important to realize that the fees network is the most complex part of the array.
 
SMART ANTENNA ALGORITHMS
 
4.1 Usage of Algorithms
It is important to point out that the antennas themselves are not “smart”, it is rather the underlying antenna systems that have the intelligence in the form of advanced signal processing algorithms. In order to be able to take full advantage of the multi-antenna techniques, discussed in the previous section, advanced and computationally heavy communications algorithms must be used. There are myriads of different algorithms, which are optimized and specialized for different multi-antenna systems and for different user scenarios. A brief discussion on smart antenna and MIMO methods are given below.
 
4.2 Smart antenna algorithms
 
Smart antennas, in their simplest form, linearly combines antenna signals into a weight vector that is used to control the beam pattern. The weights can be determined in a number of ways using different algorithms. These smart antenna algorithms can crudely be divided into three classes of algorithms, spatial reference, temporal reference, and blind algorithms. The common features of the two first algorithm classes are that they both form beam patterns and they are based on linear weighting and addition of received signals at the antenna elements. The difference between the two classes is in how they calculate the antenna weights. The third class of algorithms uses neither of the features used by spatial and temporal reference algorithms. Instead they exploit the statistical properties of the transmit signal.
 
In Spatial reference algorithms (SR) the antenna weights are chosen based on knowledge of the array structure [6],[7]. These algorithms estimate the direction of arrival (DOA) of both the desired and interfering signals. The DOAs can be determined by applying different methods to the sampled data from the antenna array. The simplest way of extracting the DOAs is to use spatial Fourier transform on the signal vector.
 
This method is limited by its resolution (size of antenna array) and has therefore limited usages. In cases where good resolution is necessary, so called high resolution methods could be used. High-resolution methods are limited only by the modeling errors and noise and not by the size of the antenna array. Common high-resolution algorithms include:

Figure 4.1 Block diagram of proposed adaptive equalization system.
 
The role of the equalizer is to resolve the distortion of the channel while minimizing the effect of additive noise at its output [9]. For an unknown channel 1/H, an equalizer with the transfer function F = H produces an overall channel-equalizer transfer function of F/H = 1. This implies that in the case of no interference being experienced, the output from the equalizer, x(k), will be the original transmitted signal u(k). We can think of F as being an equalizer of 1/H, or an estimator of H.
 
4.4 Standard LMS Algorithm
There are several assumptions that need to be made with regards to the system being considered. Reference can be made to Figure 3.2 while reading the following assumptions. We assume that the unknown channel 1/H is linear, time invariant and able to be modeled as a discrete-time IIR filter with n taps.
 
H= h1,h2, h(n-d)                                                         (3.1)
 
The time invariant, n-tuple, IIR modeled, unknown channel 1/H is a sparse channel with only mn nonzero (active) taps.
The LMS adaptive FIR filter (equalizer) has a tap delay line structure and a length of n.
F(1) = Lk,                                                                    (3.2)
The tap coefficients of F(k) are initially set to zero.
Fi(0) = 0, for i = 0, 1, 2, …, n-1                                   (3.3)
The input signal u(k), and interference signal ni(k) are assumed to be zero mean, bounded and wide-sense stationary processes. They are also assumed to be uncorrelated with each other over time.
In order to calculate the output r(k) from the unknown channel 1/H, the system requires knowledge of the last n –1 values of r. That is, the vector
R(k) = [r(k-1), r(k-2), r(k-3), r(k-n-1)]             (3.4)
is used to calculate the output as
1/(H) r(k) = H(1) -H(2: n) RT (k))                                            (3.5)
Where, H(1) = ho, and H(2:n) = h2, h(n-d)
The output signal then has the interference signal ni(k) added to it to produce the received signal.
y(k) = r(k) + ni(k)                                                                    (3.6)
The received signal y(k) is then added to an array of the last n-1 received signals to form the received signal vector.
Y(k)= [y(k), y(k-1), y(k-2),…., y(k-n-1)]                                                           (3.7)
This vector is then input into the adaptive equalizer F to produce the estimate x(k) of the transmitted signal.
x(k) = F(k-1)Y T(k)                                                                   (3.8)
the estimate x(k) is then compared to the original transmitted signal u(k) to provide an error signal
e(k) = u(k)—x(k)                                                                      (3.9)
Ideally, the error signal e(k) should be equal to the interference signal ni(k).
This would indicate that the LMS adaptive equalizer has successfully estimated H. It can
be shown that the tap weights of the equalizer F are functions of the sampling instant k.
This indicates that the tap weights of the adaptive equalizer are time dependent, since they are continuously being adapted. The LMS algorithm adjusts the tap weights, or coefficients of the FIR equalizer in an attempt to minimize the mean squared error (MSE) (k). However, the MSE requires large amounts of memory, so the instantaneous error e(k) is used to estimate the gradient of the MSE surface [8]. Eventually, the Standard LMS equation the FIR equalizer is given by
F(k)= F(k-1) + Y(k) e(k)                                               (3.10)
Where Y (k) is the received signal vector from (3.7) and p is the step-size parameter.
 

ANALYSIS AND RESULT
5.1 Analysis of Smart Antenna Simulation
 
In this case all adaptive array smart antenna simulations have a sum of 10000 input signals of the training sequence have signed values of 1 or -1 to simulate a transmitter sending binary values. Although there are 10000 sampling instants, the results only show up to 300 intervals due to the extremely high rate of convergence of the system. The step-size parameter μ for the Least Mean Square (LMS) algorithm is set to 0.008 to keep simulation as realistic as possible, especially for those simulations with more than one multipath, each multipath experiences a different gain, which contains both amplitude and phase components. It was found that the amplitude of the gain had the most effect on the system, with the phase having little to no effect at all.
 
The carrier frequency, fc of transmitted training sequences is set to 600MHz, which means the value of the wavelength λ is set to 0.5m. To satisfy an element spacing d of λ/2 then means that d is set to 0.25m. Simulations with only one transmitted signal, the propagation delay from transmission to reaching the first antenna element is set to 100μs, and for those with a second transmitted signal, the second Propagation delay is set at 150μs. Even though only four simulation results are being presented, there were many other simulations that were used to progress to the final simulations.
 
The gain and noise terms were initially left out of the system to ensure that the simulations were achieving the correct result in the ideal environment. Also, in order to reach a simulation of signals with three multipath, a simulation with two multipath was first examined. Such simulations have not been included to avoid repetition.
5.2 Results of One White Signal with One DOA
To ensure that the system worked correctly, the first simulation investigated was the reception of one signal with the one path that indicates the direction of arrival (DOA) at the base station at angle of 30°. A gain with amplitude of 0.5 was introduced to the input signal as it was propagated to the antenna. Figure 4.1 illustrates that the received signal error converges.
 
Figure 5.1 shows that the beam pattern of the system correctly steers the main beam in the direction of 30 with beam strength of two. This is due to the signal experiencing a gain of amplitude 0.5, which reduces the power of the signal by half. To counter this, the beam adjusts its gain to the inverse of the signal power in order to receive a signal similar to the original signal.
 

 
Figure 5.1 Smart antenna simulation received signal error for 1 white signal with 1 DOA.

Figure 5.2 Smart antenna simulation beam pattern for 1 white signal with 1 DOA.
 
5.3 Results of One White Signal with Three DOAs
The next simulation is again for the transmission of one training sequence, but this time with three multipaths that have directions of arrival of 50°, 20° and -20°. Each Multipath arrives at the antenna system with a difference of one sampling period 1/fc, so we can denote the signals arriving at time instant t as u(t), u(t-1), and u(t-2).
 
The corresponding gains introduced to each of the multipath components have amplitudes of 0.5, 0.66, and 1.0. In this instance, three different weight vectors are used in the adaptive antenna system, one for each multipath. This means that three LMS equations are running simultaneously to each produce a main lobe in the direction of a multipath.
 

 
Figure 5.3 Smart antenna simulation received signal error for 1 white signal with 3 DOAs.
 
The received signal error plot shown in Figure 5.3 illustrates the effects of having different gain terms. The smaller the gain amplitude, the longer it takes for the antenna array to adapt and correctly estimate the transmitted signal.
 
In Figure 5.4 it shows that the antenna systems beam pattern. It also demonstrates its ability to steer separate beams in multiple directions and nulls in the directions of interferers.

Figure 5.4 Smart antenna simulation beam pattern for 1 white signal with 3 DOAs
 
5.4 Results of Two White Signals with One DOA Each
The simulation of transmitting two different signals with one DOA each is in effect the same as sending one signal with two multipath separated by at least one sample period. This is because in both situations the two signals are uncorrelated with each other. The 1st signal is exposed to a gain with amplitude 0.5 and the 2nd signal 1.0. Figure 5.5 once again shows that it takes longer for the system to converge when the gain term is smaller. In this case two LMS equations are running simultaneously to determine the weight vectors to produce the two beams in each desired direction.

 
Figure 5.5 Smart antenna simulation received signal error for 2 white signals with 1 DOA each.
The beam pattern in Figure 5.6 shows the two beams from each set of weights is able to correctly identify the DOAs of each signal as being 40° and –30°. The gain of the 1st set of weights in the direction of the 2nd signal is 0.0537 and the gain of the 2nd set of weights in the direction of the 1st signal is 0.0105, demonstrating the smart antenna's ability to distinguish between desired signals and interfering ones.
 

 
Figure 5.6 Smart antenna simulation beam pattern for 2 white signals with 1 DOA each.
 
5.5 Results of Three White Signals with Two DOAs
 
The final smart antenna simulation is the most complex and provided the most unexpected results. In this simulation we transmit three training sequences, first two with two multipath components and last one is one path component. Essentially, this means that the 2^nd and 3^rd multipath is arriving at the base station at the same time but from different directions. As from Figure 5.7, we see three different white noise displayed.

Figure 5.7 Smart antenna simulation received signal error for 3 white signals.
 

 
Figure 5.8 Smart antenna simulation beam pattern for 3 white signals with 2 DOA each.
 
After this finding, it was expected that the main beam would either be directed in the direction of the closest multipath or the one with the greatest gain. However, the beam pattern shown in Figure 5.8 displays the three different beam patterns but first signals have two main lobes and 2^nd signal has a nearly same side lobe in the correct directions of the 2nd and 3rd multipath. The gains of these beams are half what they would normally be and swapped between the multipath components.
 
This was a major result, to prove the ability for weight vectors to steer multiple beams in multiple directions. Whereas before this, it was always under the impression, that each set of weight vectors could only steer one beam in one specific direction.
 
5.6 Smart Antenna Summary
 
The smart antenna simulations confirmed that smart antenna systems have an ability to distinguish between signals of interest and interferers by directing beams in the directions of the desired signals and nulls in the directions of interferers. These interferers can either be other transmitted signals from other mobile or multipath components of the same signal. The major finding of the smart antenna simulations is that adaptive array smart antenna systems are able to deploy multiple main beams in multiple directions if multipath of the same desired signal arrives at the base station at the same time.
 
Reference:
 
[1] L. C. Godara, Application of Antenna Arrays to Mobile Communications, Part I: Performance Improvement, Feasibility, and System Considerations, Proceedings of the IEEE, Vol. 85, No. 7, July 1997, pp. 1029-1060.
 
[2] Wong, K.K., Murch, R.D. & Letaief, K.B. 2001. Optimizing Time and Space MIMO Antenna System for Frequency Selective Fading Channels. IEEE Journal on Selected Areas in Communications, July, pp.1395-1406.
 
[3] Rappaport, T.S. 1996 Wireless Communications: Principles & Practice, Prentice Hall Communications Engineering and Emerging Technology Series.
 
[4] Oeting, J. 1983. Cellular Mobile Radio — An Emerging Technology. IEEE Communications Magazine, November, pp. 10-15.
 
[5] Rosol, G. 1995. Base Station Antennas: Part 1, Part 2, Part 3. Microwaves & RF, August, pp. 117-123, September, pp. 127-131, October, pp. 116-124.
 
[6] Brickhouse, R.A., and Rappaport, T.S. 1997. A Simulation Study of Urban In-Building Frequency Reuse. IEEE Personal Communications Magazine, February, pp. 19-23.
 
[7] Liberti, J. C. & Rappaport, T.S. 1999. Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications, Prentice Hall Communications Engineering and Emerging Technology Series, NJ.
 
[8] NURALINA ZUREENBT ROSLI, 2008. Applying LMS algorithm on Smart Antenna System, Universiti Teknologi Malaysia.
 
[9] R. H. Roy, “An overview of smart antenna technology: The next wave in wireless communications,” in Proc. 1998 IEEE Aerospace Conf., vol. 3, May 1998, pp. 339–345.
 
[10] Haykin, S. 1996. Adaptive Filter Theory, Third Edition, Prentice Hall Inc., pp. 365 — 405. How, L. 2001. Signed LMS Adaptive Filtering with Detection. Undergraduate Thesis, School of Information Technology and Electrical Engineering, University of Queensland, Brisbane.
 
[11] Homer, J., Mareels, I., Bitmead, R., Wahlberg, B., & Gustafsson, F. 1998. LMS Estimation via Structural Detection. IEEE Transactions on Signal Processing, Vol. 46, No. 10, pp. 2651-2663.
 
[12] Pattan, B. 2000. Robust Modulation methods and Smart Antennas in Wireless Communications, Prentice Hall PTR, NJ.
 
[13] Zooghby, A. 2001. Potentials of Smart Antennas in CDMA Systems and Uplink Improvements. IEEE Antennas and Propagation Magazine, pp. 172177.
 
[14] Widrow, B., Mantey, P. E., Griffiths, L. J., and Goode, B. B. 1967. Adaptive Antenna Systems. Proc. of the IEEE, December.
 
[15] Monzingo, R. & Miller, T. 1980. Introduction to Adaptive Arrays, Wiley and Sons, NY.
 
[16] Frost, 0. L., III. 1972. An Algorithm for Linearly Constrained Adaptive Array Processing, Proc. of the IEEE, August.
 
[17] Jing Jiang, R. Michael Buehrer, and William H. Tranter, Antenna Diversity in Multiuser Data Networks, IEEE Trans. Comm., vol. 52, no. 3, pp. 490-497, Mar. 2004.
 
[18] Jeffrey H. Reed, Software Radio: A modern approach to radio engineering, Prentice Hall Communications Engineering and Emerging technology series 2002.
 
[19] S. Choi and J. H. Reed, “Smart Antenna API,” a power point presentation submitted to Technical Committee SDRF, June 15, 2004.
 
APPENDICES
Math lab Code:
1)     For Single Signal And Single DOA:
clc
clear all
n = 8;                              %no of antenna elements
u = sign(randn(n,10000));           %source input
N = length(u);                      %no of input signals                                             
ni = (randn(N,n)+j*randn(N,n))*0.1; %noise input
st = 0.008;                          %step-size
T1 = 100*10^(-6);               %time for signal to arrive at first element
fc = 6*10^8;                         %carrier frequency
c = 3*10^8;                           %speed of light
lam = c/fc;                           %wavelength
d = lam/2;                            %element spacing
DOA = 30;                              %Direction Of Arrival(DOA) for u
sin_DOA= sin(DOA*pi/180);             
r = zeros(1,n);                        %received signal for each element
y = zeros(1,n);                       %received signal plus noise
x = zeros(1,N);                       %received signal estimation
e = zeros(1,N);                        %error
B = zeros(1,n);                         %mean squared error
F= zeros(1,n);                          %initialize SA weight vectors to zero
expA = 0;                                %phase delay due to propagation
expB = zeros(1,n);                       %additional delay at each element
Gain = 0.5*(exp(j*pi/3));                  %Gain
for k = 1:N
    U = Gain*[u(k),u(k),u(k),u(k),u(k),u(k),u(k),u(k)];
    for m =1:n
        expA = exp(-j*2*pi*fc*T1);
        expB(m) = exp(-j*2*pi*(m-1)*d*sin_DOA/lam);
        r(m) = U(m)*expA*expB(m);
        y(m) = r(m)+ni(k,m);
    end
    x(k) = y*F';
    e(k) = u(k)-x(k);
    F = F+st*y*conj(e(k));
   B(k) = e(k)*e(k)';
end
figure(1);clf;
subplot(2,1,1);
semilogy(abs(B),'b');
xlim([0,1500]);
ylim([10^-7,10^1]);
grid on;
title('ONE WHITE SIGNAL');
xlabel('Sample Interval');
ylabel('Recieved signal Error');
angle_min = -90*pi/180;                % Minimum angle of polar graph
angle_max = 90*pi/180;                  %Maximum angle of polar graph
angle_incr = 1*pi/180;                  %angle increasing  
q=0;
F=conj(F);
for angle1=angle_min:angle_incr:angle_max
    q=q+1;
    angle2(q)=2*pi*d*sin(angle1)/lam;
    for t=1:n
        G(t)=exp(j*angle2(q)*(t-1));
    end;
    beam(q)=abs(F*G');
end;
angle_range=angle_min:angle_incr:angle_max;
figure(2);clf;
polar(angle_range, beam,'b');           %polar curve   
view(90,-90);                           % view in Horizontal
 
2)     For Multiple Signal And multiple DOAs:
clc
clear all;
u1=sign(randn(1,10000));           %1st multipath input for 1st signal
u2=[0,u1];                          %2nd multipath input for 1st signal
u3=sign(randn(1,10000));           %3rd multipath input for 1st signal
N=length(u1);                       %number of input signals
n=8;                                %number of antenna inputs
ni=(randn(N,n)+j*randn(N,n))*0.1;    %noise
st=0.008;                            %step-size
fc=6*10^8;                           %carrier frequency
T1=100*10^(-6);             %time for first signal to arrive
T2=T1+1/fc;                  %time for second signal to arrive
T3=T2+1/fc;                  %time for third signal to arrive
c=3*10^8;                           %speed of light
lam=c/fc;                             %wavelength
d=lam/2;                           %element spacing
DOA1=50;                           %direction of arrival at u1
DOA2=20;                             %direction of arrival at u2
DOA3=-20;                           %direction of arrival at u3
sin1=sin(DOA1*pi/180);               %sine of DOA1 in radian
sin2=sin(DOA2*pi/180);                %sine of DOA2 in radian
sin3=sin(DOA3*pi/180);                 %sine of DOA3 in radian
r=zeros(3,n);                          %received signals at each signal
y=zeros(3,n);                         %received signal plus noise
x=zeros(3,N);                          %transmitted signal estimation
e=zeros(3,N);                             %error between
B=zeros(3,n);                            %squared error for each signal
F=zeros(3,n);                           %initialize SA weight vectors to zero
exp1A=0;                         %phase delay due to propagation time of u1
exp2A=0;                         %phase delay due to propagation time of u2
exp3A=0;                        %phase delay due to propagation time of u3
exp1B=zeros(1,n);                %additional delay at each element
exp2B=zeros(1,n);                 %additional delay at each element
exp3B=zeros(1,n);                 %additional delay at each element
Gain1=0.5*(exp(j*pi/3));           %Gain experienced by u1
Gain2=0.66*(exp(j*pi/6));           %Gain experienced by u2
Gain3=1*(exp(j*pi/4));              %Gain experienced by u3
for k=1:N
U1=Gain1*[u1(k),u1(k),u1(k),u1(k),u1(k),u1(k),u1(k),u1(k)];
U2=Gain2*[u2(k),u2(k),u2(k),u2(k),u2(k),u2(k),u2(k),u2(k)];
U3=Gain3*[u3(k),u3(k),u3(k),u3(k),u3(k),u3(k),u3(k),u3(k)];
for m=1:n
exp1A=exp(-j*2*pi*fc*T1);
exp1B(m)=exp(-j*2*pi*(m-1)*d*sin1/lam);
exp2A=exp(-j*2*pi*fc*T2);
exp2B(m)=exp(-j*2*pi*(m-1)*d*sin2/lam);
exp3A=exp(-j*2*pi*fc*T3);
exp3B(m)=exp(-j*2*pi*(m-1)*d*sin3/lam);
r(1,m)=U1(m)*exp1A*exp1B(m)+U2(m)*exp2A*exp2B(m)+U3(m)*exp3A*exp3B(m);
y(1,m)=r(1,m)+ni(k,m);
r(2,m)=U2(m)*exp2A*exp2B(m)+U3(m)*exp3A*exp3B(m)+U1(m)*exp1A*exp1B(m);
y(2,m)=r(2,m)+ni(k,m);
r(3,m)=U3(m)*exp3A*exp3B(m)+U1(m)*exp1A*exp1B(m)+U2(m)*exp2A*exp2B(m);
y(3,m)=r(3,m)+ni(k,m);
end;
x(1,k)=y(1,:)*F(1,:)';          
e(1,k)=u1(k)-x(1,k);                             
F(1,:)=F(1,:)+st*y(1,:)*conj(e(1,k));                              
B(1,k)=e(1,k)*e(1,k)';                                            
x(2,k)=y(2,:)*F(2,:)';                                             
e(2,k)=u2(k)-x(2,k);                                                
F(2,:)=F(2,:)+st*y(2,:)*conj(e(2,k));                              
B(2,k)=e(2,k)*e(2,k)';                                             
x(3,k)=y(3,:)*F(3,:)';                                             
e(3,k)=u3(k)-x(3,k);                                             
F(3,:)=F(3,:)+st*y(3,:)*conj(e(3,k));                              
B(3,k)=e(3,k)*e(3,k)';                                            
end;
figure(1);clf;
subplot(2,3,1:3);
semilogy(abs(B(1,:)),'b');
xlim([0,1000]);
ylim([10^-7,10^1]);
hold on;
semilogy(abs(B(2,:)),'r');
xlim([0,1000]);
ylim([10^-7,10^1]);
hold on;
semilogy(abs(B(3,:)),'g');
hold off;
xlim([0,1000]);
ylim([10^-7,10^1]);
grid on;
title('ONE WHITE SIGNAL USING 3DOA');
xlabel('Sample Interval');
ylabel('Received Signal Error');
legend('DOA1','DOA2','DOA3',4);
subplot(2,3,4)
semilogy(abs(B(1,:)),'b');
xlim([0,1000]);
ylim([10^-7,10^1]);
xlabel('Sample Interval for DOA1');
ylabel('Received Signal Error');
subplot (2,3,5)
semilogy(abs(B(2,:)),'r');
xlim([0,1000]);
ylim([10^-7,10^1]);
xlabel('Sample Interval for DOA2');
ylabel('Received Signal Error');
subplot(2,3,6)
semilogy(abs(B(3,:)),'g');
xlim([0,1000]);
ylim([10^-7,10^1]);
xlabel('Sample Interval for DOA3');
ylabel('Received Signal Error');
angle_min=-90*pi/180;
angle_max=90*pi/180;
angle_incr=1*pi/180;
q=0;
F(1,:)=conj(F(1,:));
F(2,:)=conj(F(2,:));
F(3,:)=conj(F(3,:));
for angle1=angle_min:angle_incr:angle_max
q=q+1;
angle2(q)=2*pi*d*sin(angle1)/lam;
for t=1:n
G(t)=exp(j*angle2(q)*(t-1));
end;
beam1(q)=abs(F(1,:)*G');
beam2(q)=abs(F(2,:)*G');
beam3(q)=abs(F(3,:)*G');
end;
angle_range=angle_min:angle_incr:angle_max;
figure(2);clf;
polar(angle_range,beam1,'b');
hold on;
polar(angle_range,beam2,'r');
hold on;
polar(angle_range,beam3,'g');
hold off;
view(90,-90);
legend('DOA1','DOA2','DOA3',2);