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Condition Numbers and MIMO Insights

Blog Post created by benz on Sep 13, 2016

Originally posted Nov 10, 2014

 

What condition is your condition in?*

As we know all too well, the RF spectrum is a limited resource in an environment of ever-increasing demand. Engineers have been working hard to increase channel capacity, and one of the most effective techniques is spatial multiplexing via multiple-input, multiple-output (MIMO) transmission.

MIMO allows multiple data streams to be sent over a single frequency band, dramatically increasing channel capacity. For an intuitive approach to understanding the technique, see previous posts here: MIMO: Distinguishing an Advanced Technology from Magic and Hand-Waving Illustrates MIMO Signal Transmission. And if MIMO sounds a little like CDMA, the difference is explained intuitively here as well.

Intuitive explanations are fine things, but engineering also requires quantitative analysis. Designs must be validated and optimized. Problems and impairments must be isolated, understood and solved. Tradeoffs must be made, costs reduced and yields increased.

Quantitative analysis is a special challenge for MIMO applications. For example, a 4×4 MIMO system using OFDM has 16 transmit paths to measure, with vector results for each subcarrier and each path.

The challenge is multiplied by the fact that successful MIMO operation requires more than an adequate signal/noise ratio (SNR). Specifically, the capacity gain depends on how well receivers can separate the simultaneous transmissions from each other at each antenna. This separation requires that the paths be different from each other, and that SNR be sufficient to allow the receiver to detect the differences. Consider the artificially-generated example channel frequency responses shown below.

A 2 MHz bandpass filter has been inserted into one channel frequency response of a MIMO WiMAX signal with 840 subcarriers. The stopband attenuation of the filter will reduce SNR at the receiver and impair its ability to separate the MIMO signals from each other.

A 2 MHz bandpass filter has been inserted into one channel frequency response of a MIMO WiMAX signal with 840 subcarriers. The stopband attenuation of the filter will reduce SNR at the receiver and impair its ability to separate the MIMO signals from each other.

The bandpass filter applied to one channel will impair MIMO operation for the affected OFDM subcarriers in some proportion to the amount of attenuation. The measurement challenge is then to quantify the effect on MIMO operation.

The answer is a measurement called the MIMO condition number. It’s a ratio of the maximum to minimum singular values of the matrix derived from the channel frequency responses and used to separate the transmitted signals. You can find a more thorough explanation in the application noteMIMO Performance and Condition Number in LTE Test, but from an RF engineering point of view it’s simply a quantitative measure of how good the MIMO operation is.

Condition number quantifies the two most important problems in MIMO transmission: undesirable signal correlation and noise. I’ll discuss signal correlation in a future post; here I’ll focus on the effects of SNR by showing the condition number resulting from the filtering in the example above.

Condition number is a ratio of singular values and always a positive real number greater than one. It is often expressed in dB form, plotted for each OFDM subcarrier. The ideal value for MIMO is 1:1 or 0 dB, and values below 10 dB are desirable. In this measurement example the only signal impairment is SNR, degraded by a bandpass filter.

Condition number is a ratio of singular values and always a positive real number greater than one. It is often expressed in dB form, plotted for each OFDM subcarrier. The ideal value for MIMO is 1:1 or 0 dB, and values below 10 dB are desirable. In this measurement example the only signal impairment is SNR, degraded by a bandpass filter.

Condition-number measurements are an excellent engineering tool for several reasons:

  • They effectively measure MIMO operation by combining the effects of noise and undesirable channel correlation.
  • They are measured directly from channel frequency response, without the need for demodulation or a matrix decoder.
  • They are a frequency- or subcarrier-specific measurement, useful for uncovering frequency response effects.
  • They relate a somewhat abstract matrix characteristic to practical RF signal characteristics such as SNR.

The last point above is especially significant for understanding MIMO operation: With condition number expressed in dB, if the condition number is larger than the SNR of the signal, it’s likely that MIMO separation of the multiple data streams will not work correctly.

*I don’t know about you, but I can’t hear the phrase “condition number” without thinking of the Mickey Newbury song Just Dropped In (To See What Condition My Condition Was In), made famous by Kenny Rogers & the First Edition way back in 1968. Did Newbury anticipate MIMO?

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