Normal Equations

There has been recent interest in using orthonormalised forms of fixed denominator model structures for system identification. A key motivating factor in the employment of these forms is that of improved numerical properties. Namely, for white input perfect conditioning of the least-squares normal equations is achieved by design. However, for the more usual case of coloured input spectrum, it is not clear what the numerical conditioning properties should be in relation to simpler and perhaps more natural model structures. This paper provides theoretical and empirical evidence to argue that in fact, even though the orthonormal structures are only designed to provide perfect numerical conditioning for white input, they still provide improved conditioning for a wide variety of coloured inputs.