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Abstract—An UWB-MIMO channel model with a novel antenna array model which includes mutual coupling is proposed. The array model uses equivalent circuits to consider antenna elements, and the ideas of mutual inductance and mutual capacitance to consider mutual coupling between antennas. The results show that mutual coupling changes the amount of power needed to send the same signals, leads to power transfer between antennas, and alters array’s radiation efficiency. The effect of mutual coupling is also impacted by the amplitude and phase relations of input signals even if the total input signal power is fixed. We then deduce the channel capacity of the proposed channel model. It is shown both in theory and numerical examples that mutual coupling can increase, decrease, or maintain capacity depending on the conditions. The condition of best mutual coupling of an n by n MIMO system is derived, which can be used to decide optimum power allocation for a given transmit array, and determine the frequency response of the transmit array that is best mutually coupled for a given transmit power allocation.
Index Terms—UWB-MIMO, mutual coupling, antenna array model, channel capacity
I. INTRODUCTION
OR a long time, MIMO channel models have been in the realm of information theory, signal processing and coding.
These models, such as the Rayleigh fading model in [1], scattering model in [2], and space-time model in [3], can provide insights on how to realize optimal channel capacity through power allocation [4], code design [5], and transceiver architecture [6-8], and guide our measurements and simulations to specify channel characteristics in different environments [9-13], such as indoor and outdoor scenarios. For mathematical conveniences, many of these models have simplified the physical essence of the transmit and receive arrays. But in the analysis of mutual coupling, some authors claim it can increase channel capacity due to decorrelation effect [14,15], while others believe it leads to capacity loss by causing additional correlation or reduced radiation efficiency [16,17]. It is recognized that mutual coupling arises
F
Manuscript received by This work is supported by Kun Chen is with School of Physical Electronics, University of Electronic Science and Technology of China (UESTC), Chengdu, 610054, China (email: [email protected])Shaoqiu Xiao is with the Institute of Applied Physics, UESTC (email: [email protected])Mengna Yang is with with School of Physical Electronics, UESTC (email:)Shanshan Gao is with the Institute of Applied Physics, UESTCBing-Zhong Wang is with
from the interactions between different antenna elements, that is, the field generated by one antenna changes the current distribution of the other antennas. As a result, the radiation pattern and input impedance of each element are disturbed [15]. This is from the perspective of physics instead of information theory. To figure out the effect of mutual coupling on channel capacity, we have to introduce the physical aspects of transmit and receive arrays into MIMO channel models in an appropriate way, the absence of which contributes to the above controversy.
Recently, researchers have made a great leap ahead in combining the physical and mathematical aspects of communication [18,19]. Regarding an antenna array with n elements as a 2n-port network, the authors construct the input and output relationship of the array via scattering or impedance matrix. The mathematical and physical layers of communication are then connected by inserting the transmit and receive arrays into the channel. However, the two outstanding works mostly focus on the deterministic relations of the channel, and don’t allow much independence of the spatial channel. In fact, the gains of signals traveling in the spatial channel are highly dependent on the realistic environments, and would be better considered from the angle of statistics, which is a usual convention. Besides, as it is difficult to get the analytic form of scattering or impedance matrix of an array, mutual coupling and its effects are not explicitly investigated in the works. In [20], the conclusion that mutual coupling uplifts capacity should be thought with care in that the given S-parameters are not measured ones and have an entry with magnitude greater than one, which lack generality.
In this paper, some modifications are made for the models aforementioned to account for the effect of mutual coupling explicitly and extensively. Firstly, a new antenna array model is adopted, in which the mutual coupling between antenna elements is treated through more basic ideas—mutual inductance and mutual capacitance. Secondly, the channel is divided into three successive sub-channels: the transmit array, the spatial channel and the receive array. The first and third sub-channels are investigated in a circuit model, and the spatial channel is considered from statistical views. Each part has a channel matrix, which is its frequency response matrix. Finally, the UWB channel capacity of the revised model is deduced, and the impact of mutual coupling on channel capacity is revealed both from theory and through numerical simulations.
The Effect of Antenna Mutual Coupling on UWB-MIMO Channel Capacity
Kun Chen, Shaoqiu Xiao, IEEE Member, Mengna Yang, Shanshan Gao and Bing-Zhong Wang, IEEE Member
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KUN CHEN et al: MUTUAL COUPLING ON UWB-MIMO CHANNEL CAPACITY
II. ARRAY MODEL WITH MUTUAL COUPLING
Figure 1 shows a channel model with a new antenna array model, which is based on the following assumptions about the transmit and receive arrays.
Assumption 1: All the elements in the array can be modeled by lumped parameters from the view of circuit.
Assumption 2: The input of each antenna element comes from a constant voltage source. In this figure, the left and right parts represent the proposed transmit and receive antenna array models, respectively, while in the middle is the spatial channel. There are n transmit antennas and m receive antennas. Each antenna element is modeled by a series of an inductor, a capacitor, and a resistor. The resistor in the transmit array element is chosen so that the power consumed by it is equal to the radiated power of the element. As for each receive array element, the resistor stands for the loaded resistance, and the power dissipated by it is the power taken away by the receiver for further processing. The source of the transmitter is prearranged, while that of the receiver is the received signals. Of course, based on the linearity of antennas, we can insert one more resistor in each element model to include the loss property, but the influence of this operation on the output voltage of each antenna is just multiplying its output without the loss resistor by a scalar. When all the antenna elements are the same, the output of the array is just multiplying its output without the loss resistors by a scalar. To simplify the analysis, we first consider the case when all antenna elements are the same. Therefore the loss resistors are neglected and the array is regarded lossless. We will later discuss the case with loss.
Fig.1 The MIMO channel model with new antenna array model.
It is custom to regard an array as a multi-port network and use impedance matrix to depict its properties. The non-diagonal elements of the impedance matrix originate from the interactions between antenna elements, i.e., mutual coupling. But due to the difficulty in getting the analytic form of impedance matrix of an array, we resort to more basic ideas—mutual inductance and mutual capacitance—to help us treat mutual coupling. Mutual inductance is an old idea, and according to [21], the mutual capacitance concept evolves from the duality principle as the correspondent part of mutual inductance.
We focus on the transmit array, and the case of the receive array is similar. The voltage relationship of the transmit array is
(1)
where , , , are
vectors, representing source voltages, voltages between two ends of inductors, capacitors, and resistors in the frequency domain, respectively. We also have
(2)
(3)
(4)
where is the current vector,
represents the radiation resistors, and
, (5)
and represent the self-inductance and self-
capacitance when ; the mutual inductance and
mutual capacitance when . It’s easy to know that both
and are symmetric, namely and .
Combining (1) to (4), we arrive at the input-output relationship of the array as
(6)
where is the frequency response and has the form
(7)
If we think of the array as a channel, can then be
viewed as a channel matrix. Similarly, we can also find a
channel matrix for the receive array, which we omit
for brevity.
Let be the input voltage signals in
time domain, the output. It can be seen
that the power instantly supplied to and radiated by the i-th
transmit antenna are and ,
respectively. The power received and taken away by the i-th
receive antenna are and ,
correspondingly.
III. EFFECT OF MUTUAL COUPLING ON ARRAY PERFORMANCE
To illustrate the effect of mutual coupling on the input and output of the array, we consider a 2-element transmit array. Suppose the array works at the angular frequency
, and has the specifications
1tu
tnu
1tc
1tr 1otu
1tl
tnl tncotnutnr
1ru1rc
1rr1oru
1rl
rmrormu rmu
rmlrmc
2
2
, , and
. The non-diagonal elements of and are
given as , , where and
can be viewed as relative mutual inductance and
capacitance. Let , be
the input voltages, where is the phase difference of the two signals, and are constrained by the condition
so that the total average input signal power is
. It should be mentioned that both signal power and physical power appear in this paper, but in contrast to what has been done in [19], they are distinguished here. Physical power is called power for simplicity.
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rage
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Ant. 2 Pin
Array PinAnt. 1 Pout
Ant. 2 Pout
Array Pout
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& P
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Ant. 2 Pin
Array PinAnt. 1 Pout
Ant. 2 Pout
Array Pout
(b)
Fig. 2 Input and output power of the two elements and the array as a function
of (a) or (b) when
Fig. 2 shows the average input and output power of the two elements and the whole array. In both (a) and (b), it’s easy to see that the total average input and output powers of the array are equal. This satisfies the assumption that the antenna array is lossless. Then we find the input/output power of the array
changes with and . Since the input voltages do not
change with or , this fact can be interpreted as: with
different mutual coupling, different amount of powers are dissipated to send the same voltage signals, which are regarded as information carrier. Compared with the case where no mutual coupling exists, situations with mutual coupling may require more, or less, or the same amount of power. As the radiation resistors of the two elements are equal and the array is lossless, the output signal power of each element is proportional to its input power. Therefore, the total output signal power of the array can be increased, decreased, or maintained under the influence of mutual coupling.
The figure also says that in cases when mutual coupling does not change the amount of physical input or output power of the array, the signal power allocation between the two elements when there is mutual coupling can be different from when no mutual coupling is present. We also see that at some
regions of and , the input power of one element can be
below zero, yet its output above zero. The two phenomena indicate that there is power transfer between the two elements because of mutual coupling. Possibly the power transfer is so large that one element may not be able to get input power from its own source, but only from the source of the other element.
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er t
rans
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(b) Fig. 3 Total average input/output power (a) and power transferred from
Antenna 1 to Antenna 2 (b) as a function of both or when
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KUN CHEN et al: MUTUAL COUPLING ON UWB-MIMO CHANNEL CAPACITY
.
The combined effect of mutual inductance and capacitance on the array input/output power and power transfer when
is displayed in Fig. 3. We can see that at some
pairs of mutual inductance and mutual capacitance, the array output power and power transferred can be extremely large. That is to say, mutual coupling has a particularly great impact on the performance of the array in these situations.
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(b)Fig. 4 Output power of the array versus relative mutual inductance for different values of . (a) ; (b)
Fig. 4 and Fig. 5 show the output power of the array and power transfer between the two elements when different input voltage pairs are supplied as a function of relative mutual inductance (mutual capacitance is assumed to be zero). Like previous cases, the total average input signal power is fixed at 100. It is observed that, with the same mutual inductance, array output power and power transfer will be different if the amplitude combinations and phase differences of the input signals are in different cases, respectively. Taking both mutual inductance and capacitance into account, we can further claim that the effect of mutual coupling on the performance of the
array is closely related to the input signals.
In above analysis, . To consider the case
with loss, we can insert one more resistor in each element
model. It is of equal effect if we divide both and into
two parts: one for loss; the other for radiation. In this way, we can analyze array radiation efficiency. Suppose Antenna 1 has a higher radiation efficiency than Antenna 2. As we can see from above results, under the impact of mutual coupling, power can be transferred both from Antenna 1 to 2 and in the reverse direction. If it transfers from Antenna 1 to 2, the radiation efficiency of the array is lowered; from Antenna 2 to 1, the radiation efficiency of the array is improved.
We can generalize the conclusions drawn from this example to the case of arrays with more than two antennas and fed with voltage signals that have fixed total signal power. Briefly, to input the same voltage signals into an antenna array, mutual coupling influences the power needed by each antenna. If mutual coupling exists, more power, or less power, or the same amount of power may be required in comparison to the situation with mutual coupling absent. Mutual coupling will lead to power transfer between antenna elements, and alter power distribution among elements even if it doesn’t change the total power. As a result of power transfer between antenna elements with different radiation efficiencies, the radiation efficiency of the array will also be impacted. The effect of mutual coupling on the input and output power of the array and power transfer among elements are affected by the input signals’ amplitude and phase relations.
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(b)Fig. 5 Power transferred from Antenna 1 to 2 versus relative mutual inductance for different values of . (a) ; (b)
IV. EFFECT OF MUTUAL COUPLING ON CHANNEL CAPACITY
The proposed channel model has been shown in Fig. 1. We have dealt with the input-output relationship of the transmit array, and it is similar with the receive array. For the spatial channel, we assume it to be the UWB indoor channel, which can be depicted by the following equation [22]:
(8)
where is the sampling period, is the additive white
Gaussian noise. ( ) stand for the amplitude
fading matrices of paths to , which are assumed to be independent of each other and have independently identically
distributed entries , where
and . takes values of and with equal
probability, and accords to Nakagami distribution
(9)
where ， .In this
paper, is assumed to be constant. As the power of the
amplitude fading is exponentially decreasing with the excess
delay [22], we further assume
(10)
where is also a constant.In frequency domain, we have the input-output relationship
of the spatial channel as
(11)
where . Now we can reach the
overall input-output relationship of the channel in frequency domain, which is
(12)
where
(13)
is the overall channel matrix.
Let denote a realization of
amplitude fading, the power spectral density of
, and the noise is assumed to be additive white
Gaussian with power spectral density . A power
constraint is exerted on the input, which comes as
(14)
The mutual information of the MIMO system is
(15)
In what follows, we limit our discussion to the case when (1) there are equal number of antennas at the transmitter and receiver, namely ; (2) the receiver has identical antenna elements and suffers from no mutual coupling so that
is a scalar of unit matrix. This limit offers
conveniences yet in essence does not affect the generality of
the conclusions. Given that ,
the above mutual information can be written as
(16)
where is the power spectral
density of .
When the transmitter has no information of the spatial channel matrix, the best transmitting policy is to make the output signal power of the transmit array uniformly distributed across the frequency band and transmit antennas, namely
(17)
or
(18)
where is the total output signal power of the transmit array.
The capacity is then
(19)The condition （ 18） can be utilized in two approaches:
first, for a given transmit system, we can decide the power allocation strategy that maximizes the transmission rate; second, for a specific power allocation, we can determine the frequency response of the transmit array with best mutual coupling. It should be mentioned that to achieve the largest capacity, we have to make sure that the total output signal power of the transmit array is the largest.
If the spatial channel matrix is available at the transmitter,
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KUN CHEN et al: MUTUAL COUPLING ON UWB-MIMO CHANNEL CAPACITY
we can use the water filling algorithm to find the transmit policy that achieves capacity by maximizing (16). According to [22], when the signal to noise ratio at the transmitter is high enough, say 10dB, there is little difference between the performance of uniform power allocation and that of water filling algorithm. We have used water filling to simulate the ergodic channel capacity of a MIMO system. The transmit array is just that in the previous section, and receive array is assumed to be identical with the transmit array except that the receive array has no mutual coupling. We also set
, , , .The signal to noise ratio is
10dB at the transmitter, and the sampling period . The spatial channel matrix is produced randomly according to corresponding distributions. Each value of capacity is the average of 500 samples. The results are shown in Fig. 6. It is clearly demonstrated that mutual coupling can make channel capacity increase, decrease, or remain unchanged. Also from this figure, the potential of mutual coupling to improve the capacity is very significant, which gives rise to the problem that best mutual coupling at the transmitter and receiver has to be realized for a system to achieve maximum capacity.
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(b)Fig. 6 Mean capacity versus relative mutual inductance and mutual
capacitance. (a) 3-D view; (b) or .
Ⅴ. CONCLUSION
The effect of mutual coupling on the performance of
antenna arrays is investigated through a new array model. When the total input signal power is fixed, mutual coupling can affect the amount of physical power needed to send the signals. Compared with the case when no mutual coupling is present, cases with mutual coupling may require more, less, or equal amount of physical power. Mutual coupling also leads to power transfer among different antenna elements and influences array’s radiation efficiency. As a result, array output signal power is also impacted by mutual coupling. In addition, the effect of mutual coupling on arrays is closely related to the amplitude and phase of the input signals.
By combining the array model and UWB-MIMO channel model, we construct a new channel model that helps us study the impact of mutual coupling on channel capacity extensively and explicitly. We find that mutual coupling may increase, decrease, or maintain capacity according to specific conditions. To maximize channel capacity, we have to make sure that the transmit array and receive array are best mutually coupled. The condition of best mutual coupling for the transmit array is drawn, which can be used to decide power allocation strategy that maximizes capacity when the frequency response of transmit array is given. For a given power transmit scheme, the condition can be also used to determine theoretically the frequency response of the transmit array with best mutual coupling.
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