# Scaled Transfer Function Estimator

June 2, 2021

Back to: Fundamentals of Signal Processing

The scaled transfer function estimator Ĥ_{s}(f) with scaling factor s (where 0 ≤ s ≤ ∞) is an estimate of the true transfer function H(f) of a system.

The scaling factor is the ratio of output noise power to input noise power such that s^{2} = Ŝ_{yy}/Ŝ_{xx}. This is with the assumption that Ŝ_{xx}(f) and Ŝ_{yy}(f) are constant with respect to frequency *f*.

Ĥ_{s} is a generalization of the transfer function estimators Ĥ_{1}, Ĥ_{2}, and Ĥ_{v}. Ĥ_{s} is defined as:

(1)

### Alternate Forms

Using the technique of rationalizing the denominator—or, in this case, rationalizing the numerator—we can arrive at alternate and useful forms for Ĥ_{s}. Particularly, if we multiply numerator and denominator by the rationalizing factor, we can arrive at the following equation:

(2)

Moreover, if we divide the numerator and denominator by s^{2}, we get:

(3)

### Special Cases of Ĥ_{s}

Ĥ_{1}, Ĥ_{2}, and Ĥ_{v} are special cases of Ĥ_{s}. Using the definition of Ĥ_{s}, we see Ĥ_{s}=0=Ĥ_{2} and Ĥ_{s}=1=Ĥ_{v}. Using the second alternate form, we see Ĥ_{s}➜∞=Ĥ_{1}.