Newly-Defined Radiation Efficiency

Equation (4.4) shows the physical meaning that how much power turns into radiation can be computed by the total available power minus the power reflected back to the excitation ports for a multiport antenna system.

For a desired port excitation, the total active reflection coefficient (TARC) is defined as the square root of the available power generated from all excitations minus radiated power, divided by the available power as

For example, if an N-port antenna is excited at the i-th port and the other ports are connected to termination, the TARC can be calculated as

( )

For multiport excitation, the TARC is therefore in the form of

∑

The TARC is a real number between 0 and 1. When the value of the TARC is equal to 0, all the delivered power is radiated and when it is 1, all the power either reflects back or goes to the other ports.

4.1.2 Newly-Defined Radiation Efficiency

A general definition of radiation efficiency in multiple antenna systems is also introduced in Section 2.3. Different from the previous definition, we derived the newly-defined radiation efficiency erad as

Figure 4.2 MIMO-OFDM system block diagram.

) Γ (1 a b

a

b

1 i

2 e,i N

1 i

2 e,i N

1 i

2 ei N

1 i

2

r,i

= ∑ − ∑ = ∑ −

∑

(4.8)

) Γ (1 a

b

e

1 i

2 i e, N

1 i

2 i r,

rad

= = −

∑

∑

=

= (4.9)

It is wondered what is the difference between Equation (4.9) and Equation (2.11).

Equation (2.11) is considered the composite power efficiency representation for it includes not only the reflection caused by input mismatch of the excitation port but also the power absorption resulting from the termination at the other unexcited antenna branch. Based on this definition, we may find Equation (2.11) is actually a special case of Equation (4.9). Take a dual-antenna system for example. Equation (2.11) will let one branch of the dual antenna system excite signals and the other terminated with impedance load. While in Equation (4.9), two ports of the antenna system simultaneously excite signal with their own port impedances. That exactly means if we determine the radiation efficiency using Equation (4.9) but with one branch feeding signals of zero amplitude, the analysis result will be the same as that using Equation (2.11). Moreover, if the phase difference between two signals is ± 90°,

Equation (4.9) will have the equivalent result as Equation (2.11) as well.

The most important advantage of the TARC-based definition of radiation efficiency is it takes into account the effect when ports of the multiple antenna system are fed with signals of different phases. TARC is originally developed for signals with various phase delays for multi-polarization operations, and this concept can be further extended to the multiple antenna system [23]. It is well known that mutual coupling causes some portion of signal power within each element to be radiated and absorbed by the other elements. The combination of each antenna port’s primary reflected signal with the coupled signals can be constructive or destructive depending on the phase of the component signals. MIMO array efficiency is therefore a function of constructive or destructive signal combination. A more detailed analysis is given as follows. Figure 4.2 represents the MIMO-OFDM system block diagram.

Mathematically, the OFDM signal is expressed as a sum of the prototype pulses shifted in the time and frequency directions and multiplied by the data symbols. In continuous-time notation, the k-th OFDM symbol at the n-th transmission branch is given by

where T is symbol duration, TFFT is FFT time, N is number of FFT points, fc is RF central frequency, w(t-kT) is transmitter pulse shaping function, and xi,k,n is digitally modulated signals. We may observe from Equation (4.10) that the RF cosine-modulated signals contain different phase information of the data symbols, which results from the OFDM operation of digitally modulated signals at different transmission branches. This matches the condition that feeding signals of different phases at different antenna elements will result in constructive or destructive performance of radiation efficiency. Therefore, the TARC-based definition of radiation efficiency is considered able to take into account the effect when ports of the multiple antenna system are fed with signals of different phases.

Figure 4.3 and Figure 4.4 represent the radiation efficiency analysis using Equation (2.11) and Equation (4.9) respectively. The whole HFSS simulation setup is the same as in Chapter 3, and simulation programs are written in MATLAB^® and run on the PC with an Intel^® Pentium IV 3-GHz CPU. Because the phase information of the data symbols depends on many factors, we assume port 1 is excited with unity-amplitude and signal port 2 with exp(jxπ/180°) where x={10°, 20°, …,360°}.

This set up contains a range of excitations with unity-amplitude but different phase offset distribution. Compare these two figures, and we may find the phase difference between antenna elements indeed deeply affects the radiation performance. Figure 4.3 only shows -2.7 dB at worst when antenna element spacing is 0.1 λ, but the worst radiation efficiency will be -5 dB which means only about 30 % of the power radiates using the TARC-based radiation analysis.

Another observation from Figure 4.4 is if the antenna element of MIMO is very close to the neighboring element, the radiation efficiency performance will have larger swing margin which means the radiation efficiency may be either very good or very bad at a given close antenna element spacing. The swing margin will become

Figure 4.3 Radiation efficiency analysis using Equation (2.11).

Figure 4.4 Radiation efficiency analysis using Equation (4.9).

smaller as the antenna element spacing increases, where it means the radiation performance would be better and more stable as mutual coupling between antenna elements is less strong.

4.2 Impact of Termination Networks on Radiation