POSITIVE REAL FUNCTIONS

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  

$$V(Z) \eq Y(Z) F(Z)$$ (14)

The filter Y(Z) is obviously causal. Since we believe we can do it the other way around, that is, prescribe the velocity and observe the force, there must exist a convergent causal I(Z) such that  

$$F(Z) \eq I(Z) V(Z)$$ (15)

Since Y and I are inverses of one another and since they are both presumed bounded and causal, then they both must be minimum phase.

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/(a² + b²) is also positive. Taking a + ib to represent a value of Y(Z) or I(Z) on the unit circle, we see the obvious fact that if either Y or I has the positive real property, then the other does, too.

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

$$\mbox{Power} \eq \cdots + f_0 v_0 + f_1 v_1 + f_2 v_2 + \cdots$$ (16)

This may be expressed in terms of Z transforms as

$$\begin{eqnarray} \mbox{Power} &= & {1\over 2}\; \mbox{coeff of}\; Z^0 \: \mbox{... ... Z\right) F(Z) \, + F\left(1\over Z\right)V(Z)\right] \, d\omega\end{eqnarray}$$ (17)

Using (14) to eliminate V(Z) we get

$$\mbox{Power} \eq {1 \over 2} \, {1 \over 2\pi} \int^{+\pi}_{... ...) \left[Y\left(1 \over Z\right) + Y(Z)\right] F(Z) \, d\omega$$ (18)

We note that Y(Z) + Y(1/Z) looks superficially like a spectrum because the coefficient of Z^k equals that of Z^-k, which shows the symmetry of an autocorrelation function. Defining  

$$R(Z) \eq Y(Z) + Y\left(1 \over Z\right)$$ (19)

(17) becomes  

$$\mbox{Power} \eq {1 \over 2} \, {1 \over 2\pi} \, \int^{+\pi}_{-\pi} R(Z)F\left(1 \over Z\right)F(Z) \, d\omega$$ (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 $\bar{F} F$), 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

$$\mbox{Re}[Y(Z)] \eq \mbox{Re} \left[ Y\left( 1 \over Z\right... ..., y_o + y_1 \;\cos \; \omega + y_2 \; \cos \; 2\omega + \cdots$$ (21)

Since the real part of the sum must be positive, then obviously the real part of each of the equal parts be positive.

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

 

$$f'_t \eq \left\{ \begin{array} {ll} f_t & t\leq 2 \\ 0 & t \gt 2 \end{array} \right.$$ (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  

$$W_2 \eq {1 \over 2}\, {1 \over {2\pi}} \, \int^{+\pi}_{-\pi} F'\left ( 1 \over Z\right)R(Z)F'(Z) \, d\omega$$ (23)

As we have seen, this integral merely selects the coefficient of Z⁰ of the integrand. Let us work this out. First, collect coefficients of powers of Z in R(Z)F'(Z). We have

$$\begin{eqnarray} Z^0 &: & \quad r_o f'_0 + r_{-1}f'_1 + r_{-2}f'_2 \nonumber \\ ... ...onumber \\ Z^2 &: & \quad r_2 f'_0 + r_1 f'_1 + r_0 f'_2 \nonumber\end{eqnarray}$$

To obtain the coefficient of Z⁰ in F'(1/Z)[R(Z)F'(Z)] we must multiply the top row above by f'₀, the second row by f'₁ and the third row by f'₂. The result can be arranged in a very orderly fashion by

$$\begin{eqnarray} W_2 &= & {1 \over 2} [f_0 \; f_1 \; f_2] \left[ \begin{array} ... ... \left[ \begin{array} {l} f_0 \\ f_1 \\ f_2 \end{array} \right]\end{eqnarray}$$ (24)

Not only must the 3 x 3 quadratic form (24) be positive (i.e., $W_2 \geq 0$ for arbitrary f_t) but all $t\times t$ similar quadratic forms W_t must be positive.

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

 

2-12
Figure 12 Important classes of time functions.

EXERCISES:

1. In mechanics we have force and velocity of a free unit mass related by dv/dt = f or $v = \int^t_{-\infty}fdt$. 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.]
2. Construct an example of a simple function which is minimum phase but not positive real.