Compact MIMO receive antennas

Compact MIMO Receive Antennas Ralf R. M¨ uller, Robert Bains, Jon A. Aas Department of Electronics and Telecommunications Norwegian University of Science and Technology 7491 Trondheim, Norway [email protected]

Abstract A MIMO receive antenna is proposed that is composed of only a single dipole antenna surrounded by parasitic elements in close vicinity (can be much closer than a wavelength). Steering the conductivity of these parasitic elements within fractions of a symbol duration and oversampling the received signal at the dipole at an appropriate rate, a similar effect as with a multi-element MIMO antenna can be achieved. An example of a MIMO antenna is given which is λ/2 long and less than λ/32 wide while creating three dimensions in signal space. This reduction in physical size is achieved at the expense of some drawbacks with respect to complexity of the required signal processing, power efficiency and resistance to adjacent channel interference.

1

Introduction

MIMO technology offers the possibility to boost data rate on a band-limited wireless channel by just adding antenna elements to both the transmitter and receiver. However, the MIMO effect occurs only if the antenna elements are placed sufficiently distant from each other. The minimum required distance is approximately half a wavelength. Considering the downlink channel as the bottleneck in wireless communications, multiple antenna elements at the transmitter do not pose real problems, as the base station is unlikely restricted in size. In contrast, carrier frequencies below 10 GHz and multiple antenna elements placed at distances of half a wavelength or more limit the number of antenna elements for hand-held receiving devices to just a few. The limit on the number of elements of the receiving array directly translates into a limit on the potential boost of the data rate. Several papers have addressed the question whether it is possible at all to build small MIMO receive arrays and found that there exist fundamental limits given by the laws of electric field theory [4, 2, 7, 10, 9]. However, these papers have considered only static receive arrays. In this paper, we show how the problem can be overcome by the use of a dynamic receive antenna. While being compact in size, irrespective of the number of non-vanishing eigenmodes of the MIMO channel, the proposed receive antenna configuration requires more intensive signal processing at both baseband and RF circuitry as well as shows lower resistance to co-channel interference. The general concept of the proposed MIMO receiver is laid out

in Section 2. Scaling issues related to capacity and physical size are addressed in Section 3. Problems related to co-channel interference and signal processing are discussed in Section 4. Conclusions are drawn in Section 5.

2

Proposed MIMO Antenna

MIMO technology only works with noticeable effect on data rate if the signal undergoes sufficient scattering on its propagation from the transmitter to the receiver. This is, as the only purpose of having multiple antenna elements at the receiver is to catch up several linearly independent mixtures of the signals transmitted at different transmitter elements. Clearly, if the antenna elements are placed too closely, differences in shifts of carrier phases at different elements of the receive array will be small and the signals received at different elements will hardly differ by more than thermal noise. Two directive antennas located at the same physical position, but aiming for different directions also receive differently weighted mixtures of the in-coming signal paths, since all paths are weighted by the angular characteristics of the two antennas. However, it is not possible to place an arbitrary number of directional antennas within a fixed amount of space. The problem that physical size needs to scale with data rate persists.

2.1

A Thought Experiment

Consider now the following thought experiment: Let the directional antenna rotate very fast—fast enough to catch significantly differently weighted mixtures of the in-coming paths of the same transmitted symbols. This is easily achieved for instance, if the antenna rotates 360 degrees within the duration of a symbol. Though for practically relevant data rates, such a system is technologically infeasible, it is very insightful to study its properties. Let sp (t) be the signal approaching the antenna rotating at constant angular speed ω from angular direction αp . Neglecting noise, the received signal is then given as the weighted superposition r(t) =

P X

sp (t)a (ωt + αp )

(1)

p=1

where a(α) is the angular characteristic of the direction antenna. Let the angular speed relate to the symbol rate 1/T as ω 1 ≤ ≪ fc (2) T 2π while being small compared to the carrier frequency fc . The time-modulation of the incoming narrow-band signal sp (t) results in a widened bandwidth of the signal r(t) where the actual bandwidth of r(t) depends on the spectral properties of the periodic function a(ωt), in particularly on the number 2L + 1 of non-vanishing coefficients in its Fourier series representation +L X a(ωt) = aℓ exp (jℓωt) . (3) ℓ=−L

In comparison to an array antenna which receives differently weighted overlays of the in-coming signal paths as parallel data streams, the rotating antenna, receives them all

multiplexed in a single data stream of larger bandwidth. Note that ω ≪ 2πfc is required for the antenna characteristic to be independent of the angular speed1 . Let Bx denote the space of real-valued signals limited to bandwidth x around the carrier frequency fc ≫ x. Then, the weighted superposition (1) is a projection from space BP1 onto the space B 1 + 2π . If the number of non-vanishing coefficients in the T ωL T Fourier series of the antenna pattern is at least as large as the number of propagation paths P , and the angular speed is large enough to avoid spectral aliasing2 , the mapping (1) is actually a bijection and, thus, r(t) is a sufficient statistic for the in-coming waves sp (t), p = 1..P . Note that not all modes of the in-coming wave field may be excited by the transmitter: if the number of elements at the transmit array is limited to nT < P , only nT spectral components of the antenna pattern are required to provide sufficient statistics for the transmitted data.

2.2

A Sampled Rotating Antenna

Though, we cannot convert this thought experiment into hardware for interesting data rates 1/T , we can try to approximate the effect of such a rotating antenna receiver by fixed (non-moving) electronic circuitry. A piece of electrically conducting material in close vicinity of an antenna dipole influences the antenna pattern of the dipole. This influence strongly depends on the geometry of the conducting material. If the conducting material is similar in size and shape to the dipole, it will have strong influence on the antenna pattern of the dipole, since the conducting material will, as the dipole itself, be in resonance with the in-coming waves. An example of an antenna pattern of a dipole antenna which is accompanied by a second disconnected dipole in close vicinity is given in Fig. 1. If the conducting material, for instance, is a dipole, but only half as long as the antenna dipole, it will have minor effects on the antenna pattern, since it will not be matched to the carrier frequency. Thus, cutting a disconnected dipole in close vicinity (also called a parasitic element) into two or more notably smaller pieces, will have a similar effect onto narrowband signals as letting it disappear. Cutting and re-connecting a parasitic element can be implemented easily by changing the conductivities of bridging wires via transistor circuits. In that way, we can have parasitic elements appear and disappear at any position in space and time we would like to. We can emulate the rotating directional antenna of Section 2.1 by placing parasitic elements at angular positions αi =

2πi , 2L + 1

i = −L . . . L.

(4)

and make their bridging circuits conducting in a time-division multiple-access (TDMA)like fashion. Note that 2L + 1 samples per rotation are sufficient to fulfill the sampling theorem, as the existence of a finite number of harmonics implies that the antenna pattern a(ωt) is a bandlimited function of time. Adequate post-processing of the signal at the antenna connector, e.g. sample-and-hold equalization and bandlimited interpolation, will generate the same radio-frequency signal as with a physically rotating antenna provided the antenna patterns are identical in both cases. 1 2

If the angular speed were close to the carrier frequency, relativistic effects would become relevant. Inequality (2) is a sufficient condition to prevent spectral aliasing.

90 120

60 +2 dBi +1 dBi

150

30

−1 dBi −2 dBi 180

0

330

210

Parasitic element closed Parasitic element open 300

240 270

Figure 1: Directional antenna pattern for a dipole of half wavelength accompanied by a parasitic dipole of identical length placed 1/64-th wavelength right of the active dipole. For comparison, the antenna pattern is also shown when the parasitic dipole is cut in the middle. The antenna pattern was calculated using the software package WIPL-D [6].

2.3

A Fluctuating Beam Antenna

The sampled rotating antenna is one possibility to implement a compact MIMO receive antenna in practice. However, there is no need to precisely emulate the effect of a physically rotating antenna, since we only aim for getting linearly independent weighted superpositions of the in-coming waves. These superspositions need not correspond to a rotation with constant angular speed, but can be merely arbitrary as long as they span sufficient dimensions in signal space. If we make several parasitic elements conduct at the same time, the resulting antenna pattern will not be a linear superposition of the antenna patterns corresponding to the individual parasitic elements due to coupling of induced currents in the elements. We are able to remove some parasitic elements and substitute their effects by the simultaneous activations of several others. Thus, we are able to create sufficient statistics for the transmitted signals with even (much) less than 2L + 1 parasitic elements. If the parasitic elements are either short-circuited or cut, even as few as ⌈log2 (2L+1)⌉ elements could, in principle, do. Fewer elements lead, however, to larger eigenvalue spread of the resulting channel matrix and, in presence of noise, will result in some degradation of performance. There is not any fundamental reason, why the conductivities of the parasitic elements should be either 0 or infinity. Any values in between would do as well and, due to the non-linear effect of the conductivity onto the wave field, could generate many linearly independent samples of the wave field with few parasitic elements changing conductivity at least 2L+1 times within each symbol duration. However, when having a finite positive conductivity, parasitic elements will absorb energy from the wave field and reduce the total received power.

2.4

A Virtually Rotating Antenna

The sampled rotating antenna discussed in Section 2.2 shows a severe drawback: Though it has, after equalization and filtering, the same effect onto the signal of interest as the physically rotating antenna, its treatment of unwanted signals in other frequency bands is somewhat different. Signals in other frequency bands are not irrelevant to us when using rotating beam antennas, as they can appear within the frequency range of interest due to frequency-shifts induced by the rotating beam pattern. For more details on this phenomenon the reader is referred to Section 4. Frequency-shifts occur for both the physically rotating antenna and the sampled rotating antenna. The conceptual difference between the two is induced by the sampling applied in the latter. While the rotating beam antenna creates a finite number of frequency-shifts for an antenna pattern with a finite number of harmonics, the sampled rotating antenna causes, irrespective of the antenna pattern, an infinite number of frequency-shifts due to sampling. While unwanted copies of the signal of interest at far frequency bands can easily be filtered away, copies of unwanted signals from frequency bands far away that are folded into the band of interest cannot. While for a finite number of those unwanted signals, appropriate interference mitigation techniques, e.g. interference cancellation can help, an infinite number of unwanted signals cannot be removed. To overcome this obvious disadvantage of the sampled beam antenna—the fluctuating beam antenna, in its general formulation, suffers from the same effect–we propose a virtually rotating antenna. The virtually rotating antenna shall avoid discontinuous change of the antenna pattern. A smooth change of the antenna pattern is possible, if the conductivity of the parasitic elements changes smoothly. Thus, not only a single parasitic dipole will be conducting at a time, but most or all of them will be conducting to various extents making the beam pattern smoothly sweep the desired directions. If the shape of the beam pattern is managed to be held constant while its direction changes, the effect of a rotating beam antenna as discussed in Section 2.1 is achieved. Clearly, the continuous evolution of conductivities over time needed to induce such a beam rotation is not trivial to calculate due to the nonlinearities mentioned in Section 2.3 and requires to solve systems of partial differential equations characterizing the wave fields. The more parasitic elements we have available, the more accurate the emulation of the rotating wave field will be and the less interference from far frequency bands will disturb the signal of interest. Forming beam patterns by simultaneous use of multiple parasitic elements also gives the opportunity to design the shape of the antenna pattern appropriately. The discussions in Sections 3 and 4, will give some answers to the question which properties antenna patterns should have to perform well.

2.5

Some Trade-Offs

Sections 2.2 to 2.4 have suggested different practical implementations of a compact MIMO antenna. The sampled beam antenna has the unique advantage among the three that its parasitic elements are either shortcut or open and, thus, do not extract and consume energy from the wave-field. On the other hand, it suffers from aliased copies of many adjacent frequency bands. It can reduce those interference by increasing the number of parasitic elements correspoding to a higher rate of oversampling. The fluctuating beam antenna saves on the number of parasitic elements at the expense of being even more vulnerable to interference and loosing signal power in parasitic

elements. The virtually rotating antenna gives a good trade-off between the ideas leading to the developments of the sampled beam antenna and the fluctuating beam antenna, but it requires sophisticated mathematics for its design. A sampled beam antenna which creates 24 beam patterns with only 6 parasitic elements is shown in [12]. It avoids loosing signal power by using purely capacitive loads for beam steering.

3

On Scaling

Though the rotating beam antenna cannot be built physically, Section 2.4 showed how it can be emulated by electronic circuitry. In this section, we will study its properties with respect to information-theoretic performance measures. Plugging (3) into (1) allows for the following decomposition of the signal at the antenna connector: r(t) =

+L X

exp (jℓωt) aℓ

P X

exp (jℓαp ) sp (t)

(5)

p=1

ℓ=−L

|

{z

rℓ (t)

}

Given that Inequality (2) is fulfilled with some margin coping for a non-vanishing roll-off factor of the transmit waveforms, the 2L + 1 components of the outer sum are orthogonal signals due to their separation in the spectral domain. Thus, the signal r(t) is a sufficient statistic for the signals rℓ (t), ℓ = −L . . . + L. In matrix notation       a−L 0 0 e−jLα1 · · · e−jLαP s1 (t) r−L (t)       ..  .. .. .. .. (6)   =  0 ... 0     . , . . . . +jLα1 +jLαP 0 0 a+L sP (t) r+L (t) e ··· e {z } | {z }| {z } | {z } | r(t)

A

V

s(t)

we find that the effect of the rotating antenna can be written as a multiplication of the vector of in-coming waves with a Vandermonde matrix V of angular phases and subsequent multiplication with a diagonal matrix A of Fourier coefficients of the antenna pattern. A Vandermonde matrix also occurs for propagation from scattering objects to a standard linear antenna array. Its occurence in this context does not surprise. For scatterers placed at random angular positions, it can be well approximated by a random matrix with independent identically distributed entries due to the argument given in [8, App. B]. For equidistant angular directions of the in-coming waves—a propagation model which was proposed and justified in [11, 3]—the matrix represents a discrete Fourier transform. In either case, the Vandermonde matrix is not the factor dominating mutual information: With orthogonal rows, it has no influence on the strengths of the eigenmodes of the channel, as a random matrix it will cause some minor eigenvalue spread, but will not lead to rank deficiencies, and thus allow for the well-known linear scaling of mutual information with the number of antenna elements. For standard multi-element receive array, mutual information is limited by the number of antenna elements (provided there are enough eigenmodes of the channel excited which requires a sufficient number of antenna elements at the transmitter and sufficiently rich

scattering). For a rotating beam antenna, mutual information is limited by the properties of the diagonal matrix A which contains the relative strengths of the spectral harmonics of the antenna pattern. Clearly, the more directional the antenna pattern the peakier the function a(ωt) and the more harmonics have non-negligible strengths. However, an antenna pattern need not be peaky to have many harmonics of significant strengths. A random-like antenna pattern shows similar spectral properties. In either case, the matrix A† A becomes a multiple of the identity matrix of infinite size and the rotating beam antenna is equivalent to an antenna array with an infinite number of elements spaced sufficiently far from each other3 . It follows that the scaling of mutual information with the physical size of the rotating beam antenna critically depends on the limits on the spectral shape of the antenna pattern given a fixed size of the directional antenna. Given a fixed size of an antenna, theoretical limits on its directivity do not exist [5], though its implementation may be hard. Superdirective implementations with parasitic elements are proposed in [1, 13].

3.1

Equally Strong Harmonics

Assume there are a finite number of 2L + 1 harmonics. Unless the interference and noise spectrum is colored, it is best if all these harmonics are of equal strengths, as they define the spectrum of the signal of interest and coloring the spectrum would reduce mutual information. For (approximately) equally strong harmonics, the same effect is obtained as having as many distantly spaced antenna elements at a static conventional MIMO receive array. Capacity would scale in the well-known way: linear with the minimum of the number of transmit antenna elements and significant harmonics of the pattern of the receive antenna.

3.2

Sampled Rotating Antenna

Consider a sampled rotating antenna consisting of an active dipole and parasitic elements close to it. The change of antenna pattern resulting from the change of conductivity of one of the parasitic elements is shown in Fig. 1. Here we study the scaling of mutual information when adding additional parasitic elements to change the direction of the beam as proposed in Section 2.2. We want to study solely the effect of the receiving array. In order to not be limited by saturation effects caused by the transmitting array, we chose the number of transmit antennas to be 20. Since limited scattering creates statistical dependencies in the channel matrix [8], the number of scatterers is chosen to be 120 which is large compared to the number of transmit antennas. Since the linear scaling of mutual information with degrees of freedom is best observed at high signal-to-noise ratio, we let the power of additive white Gaussian noise be 20 dB below the received signal power. We assume that the receiver has perfect channel state information and the transmitter is fully unaware of the channel and simulate the mutual information using ray tracing with ergodic random positions for the scatterers. The scaling of ergodic mutual information with the number of channel samples per rotation is shown for different distances between active dipole and parasitic elements 3

Limiting effects of a different nature will start to matter if the condition ωL ≪ 2πfc is no longer fulfilled due to a large number L of relevant harmonics of the antenna pattern.

16

Mutual Information, bit/s/Hz

14 12 10 8 6 4 2 0 1/64

Single dipole Discrete rotation 2 steps Discrete rotation 3 steps Discrete rotation 8 steps 1/32 1/16 1/8 1/4 1/2 1 2 Distance of the parasitic elements to the dipole in wavelengths

Figure 2: Ergodic mutual information vs. distance between the active dipole and the parasitic elements for 0, 2, 3, and 8 parasitic elements at a given noise level. At transmitter side a linear array with 20 elements was used. The channel propagation was modelled by ray tracing through 120 scatterers uniformly distributed in angular direction. Additive white Gaussian noise was 20 dB below signal level. in Fig. 2. At small element distance, the virtual antenna rotation is obviously capable of creating exactly 3 degrees of freedom which allows for more than doubling mutual information. This is why an increase of the angular sampling from 3 to 8 samples per rotation does not lead to improvements. This fact is understood from the Fourier series of the antenna pattern in Fig. 1 which happens to be almost exactly a squared sinusoid, thus having 3 significant harmonics. For larger distances between elements, the mutual information decreases, except for the case of 8 samples per rotation. The larger the distance, the smaller the influence of the parasitic element onto the wave field and the less directional is the antenna pattern. A remarkable effect occurs for 8 samples per rotation. For distances around a quarter wavelength, the antenna pattern has more than 3 significant harmonics which leads to an increased data rate in case of 8 samples per rotation. For only 3 samples per rotation or less, an improvement does not take place, since the additional harmonics cannot be exploited due to aliasing effects.

4

Adjacent Channel Interference

The fast time-variant nature of the rotating beam antenna and its discrete-space substitutes results in various frequency-shifts of the in-coming modulated waves. This frequency-shift does not only apply to the signals of interest, but also to signals in frequency bands adjacent to the signal of interest. Due to such frequency shifts, signals which were thought to be separated by frequency-division multiple-access loose orthogo-

nality and overlap in frequency-domain at the receiver. Adjacent channels are converted into co-channels leading to severe distortion by interference. Depending on the nature of the signals in adjacent channels and the countermeasures taken by receiver signal processing the interference can be negligible or cause serious degradation.

4.1

Operation in Unlicensed Spectrum

When operating in unlicensed spectrum, we can, in general, not hope for adjacent channel signals being much weaker in power than co-channel signals. The conversion of adjacent channel signals into co-channel signals will, therefore, not lead to severe increase of the interference level.

4.2

Operation in Licensed Spectrum

When operated in licensed spectrum, the co-channel interference level is, in general, lower than the power of the signal of interest and also lower than the signal power in adjacent channels. The impairing effects of moving adjacent channel signals into the received ω signal cannot be neglected. Denoting the adjacent channel signals at frequency fc + i 2π impinging from angle βp,i by qp,i(t), we have rℓ (t) = aℓ

P X

exp (jℓαp ) sp (t) +

p=1

X i6=0

aℓ−i

P X

exp (jℓβp,i) qp,i(t).

(7)

p=1

Assume that the signals in frequency band i originate from a single antenna element. This means qp,i (t) = qi (t)γp (8) and gives rℓ (t) = aℓ

P X p=1

exp (jℓαp ) sp (t) +

X i6=0

aℓ−i qi (t)

P X

exp (jℓβp,i) γp .

(9)

p=1

Considering K ≥ L frequency bands left and right of the carrier frequency, (9) is a linear system of 2K + 1 equations and 2K + P unknowns. The P signals sp (t) cannot be reconstructed exactly from the signals rℓ (t) unless P = 1, but can be approximated very closely if K ≥ P , e.g. by multiplying the vector r ℓ (t) by the pseudo-inverse of the system or using linear MMSE estimation.

5

Conclusions

Proposing the use of parasitic elements with changing conductivity, we have demonstrated how and why a MIMO receive antenna can be built that does not require its components to be spaced significant fractions of a wavelengths apart. In the given example 1/64 of a wavelength was sufficient to almost triple mutual information in comparison to a single receive antenna. Larger gains are expected using more sophisticated shapes for the parasitic elements than just dipole geometries. With superdirective antenna geometries, there is no theoretical limit on the scaling of mutual information for fixed physical size as long as Inequality 2 holds.

The savings in physical size come at the expense of increased sensitivity to adjacent channel interference. Since for operation in licensed frequency bands, significant effort is required for interference mitigation, those types of MIMO receive antennas are more suitable for operation in unlicensed spectrum.

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