The Electromagnetic Radiation from an antenna, particularly dipole antenna, has been studied in great detail. The mathematical framework proposed by Maxwell has stood the test of time and theoretical concepts have been verified through physical measurements. But the behavior of Electromagnetic (EM) waves close to the radiating antenna is not that well understood. This region that extends to about a wavelength from the antenna is called Near Field, as opposed to Far Field, which extends further out. The Near Field is further divided into Reactive Near Field and Radiative Near Field.Continue reading Near Field of an Antenna
There has been a continuous debate about harmful effects of Electromagnetic Radiations ever since they came into existence. Most of the research results suggest that there are no harmful effects, if the rules and regulations are followed. But there is a small body of research that suggests that there might be some harmful effects and more research needs to be carried out. This is particularly important now as 5G Wireless Technology is being rolled out around the world and it uses millimeter waves for which we have limited data. Also, 5G would be using much smaller cells meaning that base stations would be closer to human beings.
We have previously discussed beamforming for single carrier signals. Now we turn our attention to multicarrier signals particularly at mmWave where the bandwidths are two orders of magnitude (100x) higher than at sub 6GHz band. We want to investigate that whether there is any distortion in the array response due to high signal bandwidths at mmWave.
But let us start with the case that we have discussed so far i.e. 1GHz single carrier case and a Uniform Linear Array (ULA). We then add two other carriers at 1.2GHz and 0.80GHz, quite an extreme case, stretching the bandwidth to 400MHz. Antenna spacing is still λ/2=0.15m corresponding to the center frequency of 1GHz.Continue reading Multicarrier Beamforming at mmWave
In the previous two posts we discussed the fundamentals of array processing particularly the concept of beamforming (please check out array processing Part-1 and Part-2). Now we build upon these concepts to introduce some linear estimation techniques that are used in array processing. These are particularly suited to a situation where multiple users are spatially distributed in a cell and they need to be separated based upon their angles of arrival. But first let us introduce the linear model; I am sure you have seen this before.
Here, s is the vector of symbols transmitted by M users, H is the N x M channel matrix, w is the noise vector of length N and x is the observation vector of length N. The channel matrix formed by the channel coefficients is deterministic (as opposed to probabilistic) in nature as it is purely dependent upon the phase shifts that the channel introduces due to varying path lengths between the transmit and receive antennas. The impact of a channel coefficient can be thought of as a rotation of the complex signal without altering its amplitude.
This means that the channel acts like a single tap filter and the process of convolution is reduced to simple multiplication (a reasonable assumption if the symbol length is much larger than the channel delay spread). The channel model does not accommodate for path loss and fading that are also inherent characteristics of the channel. But the techniques are general enough for these effects to be factored in later. Furthermore, it is assumed that the channel H is known at the receiver. This is a realistic assumption if the channel is slowly varying and can be estimated by sending pilot signals.
So going back to the linear model we see that we know x and H while s and w are unknown. Here w cannot be estimated since it’s random in nature (remember what the term AWGN stands for?) and we ignore it for the moment. The structure of s is known. For example if we are using BPSK modulation then the m symbols of the signal vector s can either be +1 or -1. So we can start the process of symbol detection by substituting all possible combinations of s1, s2…sm and determine the combination that minimizes
This is called the Maximum Likelihood (ML) solution as it determines the combination that was most likely to have been transmitted based upon the observation.
Although ML is conceptually very appealing and yields good results it becomes prohibitively complex as the constellation size or number of transmit antennas increases. For example for 2-Transmit case and BPSK modulation there are 2^(1 bit x 2 antennas)=2^2=4 combinations, which seems quite simplistic. But if 16-QAM modulation is used and there are 4-Transmit antennas the number of combinations increases to 2^(4 bits x 4 antennas)=2^16=65536. So we conclude that ML is not the solution we are looking for if computational complexity is an issue (which might become less of an issue as the processing power of devices increases).
Next we turn our attention to a technique popularly known as Zero Forcing or ZF (the origins of the name I still do not know). According to this technique the channel has a multiplicative effect on the signal. So to remove this effect we simply divide the signal by the channel or in the language of matrices we perform matrix inversion. Mathematically we have:
So we see that we get back the signal s but we also get a noise component enhanced by inverse of the channel matrix. This is the well-known problem of ZF called Noise Enhancement. Then there are other problems such as non-existence of the inverse when the channel H is not a square matrix (which only happens when the number of transmit and receive antennas is the same). The inverse of H also cannot be calculated if H is not full rank or determinant of H is zero. So we now introduce another technique called Least Squares (LS). According to this the signal vector can be estimated as
This is also sometimes referred to as the Minimum Variance Unbiased Estimator, as described by Steven M. Kay in his classical book on Estimation Theory [Fundamentals of Statistical Signal Processing Vol-1]. This can be easily implemented in MATLAB using Moore Penrose Pseudo Inverse or pinv(H). This is much more stable than going for the direct inversion methods.
We next plot the Bit Error Rate (BER) using the code below. The number of receive antennas is varied from two to ten while the number of transmit antennas is fixed at four. The transmit antennas are assumed to be positioned at 30, 40, 50 and 60 degrees from the axis of the receive array. The receive antennas are separated by λ/2 meters. The frequency of operation is 1GHz but it is quite irrelevant to the scenario considered as everything is measured in multiples of wavelengths. The Eb/No ratio (roughly the signal to noise ratio) is varied from 5dB to 20dB in steps of 5dB.
As expected the BER for the two methods, other than ML, is more or less the same and decreases rapidly once the number of receive antennas becomes greater than number of transmit antennas (or number of signals). The case where the number of receive antennas is less than number of signals (equal powered and with a small angular separation) is dealt with by Overloaded Array Processing (OLAP) techniques and have been discussed in detail by James Hicks [Doctoral Dissertation] a student of Dr. Reed at Virginia Tech.
Strangely enough it is seen that the overloaded case is not the worst part of the BER curve. The worst BER is observed when the number the number of transmit and receive antennas is the same (four in this case). In other words the BER gradually increases as the rank of the channel matrix increases and then decreases once it reaches its maximum value. This is quite interesting and obviously has to do with Noise Enhancement that we discussed earlier. This will be further investigated in future posts.
For further information on the above methods visit this interesting article.
So we struggled for a while to find out why the BER is worst at full rank and thought that there is something wrong in our model but ultimately we found that this has to do with how the pseudoinverse works and the way the tolerance limit (tol in MATLAB) for the singular values is set. We have found quite interesting results while experimenting with various inversion methods and the results are pending publication. Will keep you updated about the progress.
We experimented with the MATLAB function pinv by changing the tolerance parameter. Previously we had used the default tolerance that is built into the function pinv. The default tolerance (tol) is defined as:
tol = max (size (H)) * sigma_max (H) * eps
where sigma_max (H) is the maximal singular value of channel matrix H
and eps is the machine precision.
More precisely, eps is the relative spacing between any two adjacent numbers in the machine’s floating point system. This number is obviously system dependent. On machines that support IEEE floating point arithmetic, eps is approximately 2.2204e-16 for double precision and 1.1921e-07 for single precision.
So back to the subject we experimented with two values of tol; 1.0 and 0.1 while changing the signal to noise ratio. The number of transmit antennas (users) is fixed at 4 while number of receive antennas is varied from 2 to 8. For tol value of 1.0 it is seen that changing the value of EbNo does not change the results much up to 6 receive antennas but after that the BER results rapidly diverge. For tol value of 0.1 the results are quite unexpected. The BER drops with increasing number of antennas up to N=5 but then there is an unexpected increase in the BER for N=6. This needs to be further investigated.
MATLAB CODE USED TO GENERATE ABOVE PLOT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % MULTIUSER DETECTION USING % A UNIFORM LINEAR ARRRAY % AKA RECEIVE BEAMFORMING % COPYRIGHT RAYMAPS (C) 2018 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all % SETTING THE PARAMETERS FOR THE SIMULATION f=1e9; %Carrier frequency c=3e8; %Speed of light l=c/f; %Wavelength d=l/2; %Rx array spacing N=10; %Receive array length theta=([30 40 50 60])*pi/180; %Angular placement of Tx array (users) EbNo=10; %Energy per bit to noise PSD sigma=1/sqrt(2*EbNo); %Standard deviation of noise n=1:N; %Rx array vector n=transpose(n); %Converting row to column M=length(theta); %Tx array length % RECEIVE SIGNAL MODEL (LINEAR) s=2*(round(rand(M,1))-0.5); %BPSK signal of length M H=exp(-i*(n-1)*2*pi*d*cos(theta)/l); %Channel matrix of size NxM wn=sigma*(randn(N,1)+i*randn(N,1)); %AWGN noise of length N x=H*s+wn; %Receive vector of length N % PINV without tol % y=pinv(H)*x; % PINV with tol y=pinv(H, 0.1)*x; % DEMODULATION AND BER CALCULATION s_est=sign(real(y)); %Demodulation ber=sum(s!=s_est)/length(s); %BER calculation
We have previously discussed the theory of Planar Inverted F Antennas (PIFA), now let us look at a practical example. Shown below is the rear view of a Samsung Galaxy S phone with six antennas. The description of these antennas is given below.
1. 2.6 GHz WiMAX Tx/Rx Antenna
2. 2.6 GHz WiMAX Antenna Rx Only (as a diversity antenna)
3. WiFi/Bluetooth Tx/Rx Antenna
4. Cell/PCS CDMA/EVDO Tx/Rx Antenna
5. Cell/PCS CDMA/EVDO Rx Only (as a diversity antenna)
6. GPS Antenna Rx Only
The figure above shows the top conducting plane of the PIFAs. The bottom conducting plane (ground plane) is one large plane that extends throughout the length and breadth of the phone.
A Planar Inverted F Antenna or PIFA is a very common antenna type being used in cell phones. In fact a cell phone would have multiple PIFAs for LTE, WiMAX, WiFi, GPS etc. Furthermore, there would be multiple PIFAs for diversity reception and transmission. A PIFA is composed of 5 basic elements.
1. A large metallic ground plane
2. A resonating metallic plane
3. A substrate separating the two planes
4. A shorting pin (or plane)
5. A feeding mechanism
The resonant frequency of the PIFA can be calculated from the relationship between the wavelength of the antenna and the dimensions of the antenna. The relationship is given as:
It must be remembered that the wavelength here is the guided wavelength which is given as λg=λo/√εr. Here εr is the relative permittivity of the substrate and λo is the wavelength in free space. There exist two special cases of the above relationship. First is the case where the shorting plane has width W1. In this case the above relationship is reduced to:
In the second case the width of the shorting plane is reduced to zero i.e. the shorting plane is actually a shorting pin. In this case the relationship is reduced to:
In cell phones with multiple PIFAs the ground plane is actually one large ground plane for all the resonating surfaces and may include the body of the cell phone as well. Lastly, the input impedance of the PIFA is controlled by changing the distance of the feeding pin from the shorting plane. The impedance is zero at the shorting plane and is maximum at the other end (away from the shorting plane).
The Electric and Magnetic Field variations within a patch are sometimes a bit confusing and difficult to visualize. The figure below shows the E and H Field variations within a rectangular patch of length L and width W.
As can be seen the E-field varies along the length of the patch with minimum at the centre and maximum at the edges (maximum positive and maximum negative). The H-field also varies along the length is in a direction perpendicular to the E-field. The H-field is maximum at the center and minimum at the edges. Thus the impedance is zero at the center of the patch (using Z=V/I).
Another point to note is that the E-field does not completely terminate at the edges along the length of the antenna rather it extends at the outer periphery. These field extensions are known as fringing fields and cause the patch to radiate .
A microstrip antenna can be designed using either the transmission line model or the cavity model (more complex models also exist that suit a particular design). We here demonstrate the transmission line model since it is fairly simple to implement and results in antenna designs with reasonably good performance in terms of return loss and efficiency.
The design starts with selecting the operating frequency, selecting a substrate with the required permittivity, and defining the width of the substrate. Thick substrates with low permittivity result in antenna designs with high efficiency and large bandwidths. Thin substrates with high permittivity lead to a smaller antenna size but with a lower bandwidth and a high-radiation loss. The tradeoffs between substrate thickness and permittivity and antenna bandwidth and efficiency have been discussed in the literature.
According to the transmission line model, the length L and width W of the patch are calculated as
Although the design of the patch is quite simple, the design of the feeding mechanism is not that straightforward. There are four possible methods that can be used:
(1) Microstrip-line feed
(2) Probe feed
(3) Aperture-coupled feed
(4) Proximity-coupled feed
 Yasir Ahmed, Yang Hao, and Clive Parini, “A 31.5 GHz Patch Antenna Design for Medical Implants,” International Journal of Antennas and Propagation, vol. 2008, Article ID 167980, 6 pages, 2008.
A Microstrip Patch Antenna or simply a Patch Antenna is a very common antenna type used in cell phones and many other electronic devices. It basically consists of two metallic plates separated by a dielectric layer. The metallic plates are usually made of copper or some other highly conductive material. Another important feature of this antenna is the feeding mechanism, which is also made of a highly conductive material. A Microstrip Patch Antenna fed by a 50 ohm transmission line and a quarterwave transformer is shown below.
The E-field and H-field generated by the Patch Antenna can be calculated by using a simulation tool such as CST Microwave Studio. It is dependent on the length ‘L’ and width ‘W’ of the patch as well as on the wavelength ‘λ’ and the feeding mechanism. Other important parameters are the width of the substrate and the permittivity of the substrate. The E-field can be approximated as.
Part of the above equations is a sinc function which can be calculated by the MATLAB sinc function.
Antenna Pattern or Radiation Pattern is a three dimensional description of how the antenna radiates energy in to the space around it. All practical antennas are directional i.e. they radiate more energy in certain directions and lesser energy in other directions. Although the Radiation Pattern is a three dimensional quantity it can be described in two perpendicular planes known as the principal planes. Usually one of these planes is horizontal (azimuth plane) and the other is vertical (elevation plane).
As discussed previously antennas do not radiate uniformly in all directions. Antenna Gain is the ratio of the power transmitted in a certain direction to the power transmitted in that direction by an isotropic source. An isotropic source is an imaginary source which radiates power uniformly in all directions. The Antenna Gain is usually given in units of decibels and is given as dBi (dB above an isotropic source). The Antenna Gain may also be given with reference to a Dipole Antenna and this is labelled as dBd. When no direction is specified Antenna Gain refers to the maximum Gain.
The 3-dB Beamwidth is the angle between two points on the main lobe of the antenna that are 3dB down from the maximum value. A smaller Beamwidth means that an antenna is more directional and a higher Beamwidth means that the antenna is less directional. The 3-dB Beamwidth is usually defined in the two principal planes i.e. the azimuth plane and elevation plane.
The front to back ration (F/B) is the ratio of the Gain in the forward direction to the Gain in the backward direction within a principal plane (azimuth or elevation). It is an important metric in certain scenarios where radiation towards the back of the antenna is undesirable e.g. as in 120 degree sectored antennas used in cellular networks. The ratio is usually given on a logarithmic scale and labelled as dB.
The polarization of an antenna is a somewhat difficult concept to comprehend. The polarization of an antenna describes the orientation of the E-field that is generated by the antenna. Usually the orientation is horizontal or vertical but it could also be circular for the case where the E-field rotates (clockwise or counter clockwise) as it travels. If a transmit antenna is vertically polarized then the receive antenna should also be vertically polarized to achieve maximum power transfer through the medium.
The voltage standing wave ratio (VSWR) is defined as the ratio of the maximum voltage to the minimum voltage in a standing wave pattern. A ratio of 1:1 means that all power was transferred to the antenna and there are no reflections. Typically RF components have 50 or 75 ohms impedance and antennas must have the same impedance to achieve maximum power transfer.
The VSWR bandwidth is defined as the frequency range over which an antenna has a specified VSWR. Often, the 2:1 VSWR bandwidth is specified, but 1.5:1 is also common.
An omnidirectional antenna is an antenna that has a non-directional pattern (circular pattern) in a given plane with a directional pattern in any orthogonal plane. Examples of omnidirectional antennas are dipoles and collinear antennas.
 Cisco Aironet Antennas and Accessories, Antenna Patterns and Their Meaning.