A frequency-domain approach to frequency-weighted balanced realization

The Ho-Kalman/Kung algorithm for balanced realization by singular-value decomposition of the Hankel operator has a natural frequency-domain equivalent. The controllability operator and observability operator are represented by intermediate transfer functions, called the gain to states and noise gain respectively. The product of these, the Hankel operator, can be expressed in terms of the input-output transfer function using an identity which the author refers to as the dynamic-range limitation. This frequency-domain theory extends naturally to the frequency-weighted case, and further to non-integrator-based linear fractional systems. It also provides additional design insight into the filter dynamic-range problem.