Two similar types of functions
called *admittance functions* *Y*(*Z*) and
*impedance functions* *I*(*Z*) occur in many physical problems.
In electronics, they are ratios of
current to voltage and of voltage to current;
in acoustics, impedance is the ratio of pressure to velocity.
When the appropriate electrical network or acoustical region
contains no sources of energy, then these ratios have
the positive real property. To see this in a mechanical example,
we may imagine applying a known force *F*(*Z*) and
observing the resulting velocity *V*(*Z*).
In filter theory, it is like considering that *F*(*Z*) is input
to a filter *Y*(*Z*) giving output *V*(*Z*). We have

(14) |

(15) |

First, before we consider any physics,
note that if the complex number *a* + *ib*
has a positive real part *a*,
then the real part of (*a* + *ib*)^{-1} namely *a*/(*a ^{2}* +

Power dissipated is the product of force times velocity, that is

(16) |

(17) |

(18) |

(19) |

(20) |

The integrand is the product of the arbitrary positive
input force spectrum and *R*(*Z*). If the power dissipation
is expected to be positive at all frequencies (for all ),
then obviously *R*(*Z*) must be positive at all frequencies;
thus *R* is indeed a spectrum. Since we have now discovered
that *Y*(*Z*) and *Y*(1/*Z*) must be positive for all frequencies,
we have discovered that *Y*(*Z*) is not an arbitrary
minimum-phase filter. The real part of both
*Y*(*Z*) and *Y*(1/*Z*) is

(21) |

Now if the material or mechanism being studied is passive
(contains no energy sources) then we must have positive
dissipation over a time gate from minus infinity up to
any time *t*. Let us find an expression for dissipation
in such a time gate. For simplicity take both the force
and velocity vanishing before *t* = 0. Let the end of
the time gate include the point *t* = 2 but not *t* = 3.

Define

(22) |

To find the work done over all time we may integrate (20)
over all frequencies. To find the work done in the selected
gate we may replace *F* by *F*' and integrate over all
frequencies, namely

(23) |

(24) |

In conclusion, the positive real property in the frequency
domain means that *Y*(*Z*) + *Y*(1/*Z*) is positive for any real
and the positive real property in the time domain
means that all matrices like that of (24)
are positive definite. Figure 12 summarizes
the function types which we have considered.

Figure 12

- In mechanics we have force and velocity of a
free unit mass related by
*dv*/*dt*=*f*or . Compute the power dissipated as a function of frequency if integration is approximated by convolution with (.5, 1., 1., 1., ...). [HINT: Expand (1 +*Z*)/ 2(1 -*Z*) in positive powers of*Z*.] - Construct an example of a simple function which is minimum phase but not positive real.

10/30/1997