##### In designing, troubleshooting and optimizing WiMAX Wave 2 systems, a variety of dual-channel measurements can provide essential insight into their operation and performance.

click to enlarge Table 1. VSA measurement configurations and results for Matrix A and Matrix B WiMAX waveforms. |

##### Matrix A and Matrix B Configurations

Multiple antenna operation in a DL transmission of a WiMAX Wave 2 system may include Space-Time Coding (STC), defined as Matrix A, or MIMO, defined as Matrix B. Typical downlink configurations for the 2x1 STC and 2x2 MIMO are shown in Figure 1.click to enlarge Figure 1. Matrix A (STC) and Matrix B (MIMO) Implementations for WiMAX Wave 2. |

Alternately, Matrix B (MIMO) systems may achieve higher data rates and improved spectral efficiency by simultaneously transmitting different data streams from each antenna over the same frequency channel. For the Matrix B configuration shown in figure 1, the measured received signals in a noiseless system are:

R

_{ x0}= h

_{00}T

_{ x 0}+ h

_{10}T

_{ x 1}(1)

R

_{ x1}= H

_{01}T

_{ x0}+h

_{11}T

_{ x1}(2)

The Matrix B receiver, having knowledge of the four channel coefficients, can differentiate and recover the transmitted waveforms using the following simplified technique.

(3)T

_{x0}= B(H

_{11}R

_{x0}- h

_{10}R

_{x1}

(4)T

_{x1}= B(-h

_{01}R

_{x0}+H

_{00}R

_{x1})

(5)where B= 1 H

_{00}h

_{11}- h

_{10}h

_{01}

These equations may also be represented in matrix form as

(6)[T

_{x}]=[H]

^{-1}[R

_{x}]

It is the function of a matrix decoder to perform the channel matrix [H] inversion and associated mathematical operations to recover the original transmitted data streams and pass this information to the demodulator. Note that matrix decoding operation is separate from, and performed prior to, the demodulation operation.

click to enlarge Figure 2. Signal flow and metrics in a vector signal analyzer for STC/MIMO measurement system. |

In the uplink, MIMO can be implemented with coordinated simultaneous transmissions from two separate MS (handsets) operating on the same frequency channel. This technique, referred to as Uplink Collaborative Spatial Multiplexing (UL-CSM), uses two or more receive antennas at the BS and a single antenna at each MS for 2x2 MIMO operation[2]. Note that in this configuration the MIMO operation is on the uplink only. DL-MIMO requires two antennas and receiver channels for each MS.

##### Channel Estimation, Matrix Decoding and Demodulation

Figure 2 shows the measurement flow through a typical dual-channel VSA with WiMAX MIMO measurement capability, such as the Agilent 89600-series analyzer with option B7Y. For Matrix B configurations, MIMO signal analysis begins with estimating the complex channel coefficients using measurements made on a large number of known pilot subcarriers received from the two input signals, shown as Rx0 and Rx1 in the figure. These four channel coefficients, displayed as a function of subcarrier frequency, can be very useful as an analysis tool when optimizing and troubleshooting MIMO systems. The estimated channel coefficients are primarily used by the matrix decoder to recover the two independent data streams from the 2x2 MIMO signal. The matrix decoder is designed to reverse the channel effects and does not perform data demodulation. As shown in the figure, the recovered Matrix B data streams are then sent to the OFDM demodulator for further signal analysis.

Matrix A signal analysis follows the same signal path through the VSA as a Matrix B configuration but only requires a single-channel analyzer. Table 1 shows a brief list of typical measurement configurations when testing Matrix A and Matrix B waveforms using a single and dual-input solution such as the Agilent 89600-series VSA. The table lists the effects of the matrix decoder on the OFDM demodulation results. The table also shows the displayed channel coefficients when selecting the Equalizer and MIMO Channel Frequency Response functions on the VSA.

##### Uncovering Signal Impairments

Example measurement results are shown in Figure 3 for a simulated pair of Matrix B waveforms using a dual-channel VSA. In this example, the influence of significant crosstalk between the transmitter channels is shown with and without the use of the matrix decoder. The left plots show a portion of the demodulated IQ constellation with one pilot and one data symbol point enlarged to show detail. With the matrix decoder off, shown in the upper left, there is a spreading in the data constellation as the other transmit channel couples into this measurement at a relative level of -29 dB. This high level of crosstalk results in a measured Relative Constellation Error (RCE) of 2.9%. The error from this crosstalk alone would be enough to fail the RCT requirement for a WiMAX Wave 2 waveform. Also shown in the upper right of this figure is the associated Error Vector Spectrum - the OFDM error plotted vs. subcarrier frequency. This measurement display is an excellent tool for troubleshooting timing errors in the system as will be shown in the next example.While the matrix decoder is not used for RCT testing it is an excellent troubleshooting tool for measuring and removing the effects of crosstalk that could obscure additional signal impairments. For example, Figure 4 shows how the matrix decoder removes crosstalk to uncover a symbol timing error present in the system. As before, the upper plots show the constellation and error vector spectrum of a signal with a crosstalk level of 29 dB. Without the matrix decoder the error spectrum is dominated by the crosstalk, making it difficult to see the timing error in the waveform. With the matrix decoder enabled, the crosstalk is removed from the measurement and a timing error can easily be observed. In the measurement on the lower right the error spectrum now shows the familiar “V” pattern that is characteristic of a symbol timing error [4].

##### Channel Frequency Response Measurements

The equalizer and MIMO channel responses are other useful diagnostic tools for characterizing Matrix A and Matrix B waveforms. The magnitude and shape of these responses can provide an understanding into the quality of received waveforms prior to demodulation. For example, it is known that MIMO systems operating in rich multipath environments will generally experience low correlations between the channel coefficients, making better data recovery possible at the receiver. When the coefficients are highly correlated, the system performance rapidly degrades. Figure 5 shows the magnitude of the measured channel coefficients for two different MIMO channels, one with relatively high correlation (left) and the other with low correlation (right). Both measurements are made with the matrix decoder enabled. For the high correlation case the pairs of coefficients have a similar complex frequency response and it would be expected that the system performance would be reduced. As shown in the inset on the lower plot, the measured 64-QAM constellation shows a high degree of signal distortion. As a comparison, the measurement in the upper right shows the measured channel coefficients having low correlation. In this case, the coefficients have dissimilar frequency responses resulting in an improvement in the data recovery as shown by the measured constellation in the lower right of the figure.##### Condition Number

Another useful troubleshooting tool is the “MIMO condition number” which is calculated from an eigen-decomposition of the channel matrix [H] and taking a ratio of the maximum singular value to the minimum singular value at each subcarrier. It is a measure of how ill-conditioned the matrix in the receiver is. The ratio is usually displayed on a log scale, and the ideal ratio of singular values for a well-conditioned matrix is 1, or 0 dB. As a general guide, when the condition number of the signal is larger than its signal/noise ratio, the matrix decoder will not be able to effectively separate the signals and demodulation performance will be poor. This is evident by the condition number response shown in the lower left plot of Figure 5. In this case, the condition number is close or above a 20 dB value and the demodulated constellation is very poor. As a comparison, the plot on the right shows a condition number that is generally below 10dB and the associated constellation plot is greatly improved.##### Conclusion

WiMAX Wave 2 systems using either Matrix A or Matrix B configurations can greatly improve system performance by taking advantage of the rich multipath characteristics of the wireless environment. In designing, troubleshooting, and optimizing these systems a variety of dual-channel measurements can provide essential insight into their operation and performance.Ben Zarlingo has a BSEE from Colorado State University and has been with HP/Agilent for 28 years. His primary focus for the past dozen years has been emerging communications technologies and measurements using spectrum and vector signal analyzers.

##### References

[1] WiMAX System Evaluation Methodology, Version 2.1, July 7, 2008. Available at the WiMAX Forum website, www.wimaxforum.org[2] Agilent Technologies, Webcast, “WiMAX Wave 2 Testing - MIMO & STC”, January 17, 2008, www.techonline.com

[3] “Matrix A and B re-measured; Single channel measurements for WiMAX™ Wave 2 reduce the need for multi-channel analysis” WiMAX Daily, June 18, 2008

[4] Testing and Troubleshooting Digital RF Communications Transmitter Designs, Agilent Application Note 1313, literature number 5968-3578E http://cp.literature.agilent.com/litweb/pdf/5968-3578E.pdf [7]