Properties of Passive Impedances
It is well known that a real impedance (in Ohms, for example) is passive so long as . A passive impedance cannot create energy. On the other hand, if , the impedance is active and has some energy source. The concept of passivity can be extended to complex frequency-dependent impedances as well: A complex impedance is passive if is positive real, where is the Laplace-transform variable. The positive-real property is discussed in §C.11.2 below.
This section explores some implications of the positive real condition for passive impedances. Specifically, §C.11.1 considers the nature of waves reflecting from a passive impedance in general, looking at the reflection transfer function, or reflectance, of a passive impedance. To provide further details, Section C.11.2 derives some mathematical properties of positive real functions, particularly for the discrete-time case. Application examples appear in §9.2.1 and §9.2.1.
where is the wave impedance connected to the impedance , and the corresponding velocity reflectance is . As mentioned above, all passive impedances are positive real. As shown in §C.11.2, is positive real if and only if is stable and has magnitude less than or equal to on the axis (and hence over the entire left-half plane, by the maximum modulus theorem), i.e.,
In particular, for all radian frequencies . Any stable satisfying Eq.(C.77) may be called a passive reflectance.
If the impedance goes to infinity (becomes rigid), then approaches , a result which agrees with an analysis of rigid string terminations (p. ). Similarly, when the impedance goes to zero, becomes , which agrees with the physics of a string with a free end. In acoustic stringed instruments, bridges are typically quite rigid, so that for all . If a body resonance is strongly coupled through the bridge, can be significantly smaller than 1 at the resonant frequency .
Solving for in Eq.(C.77), we can characterize every impedance in terms of its reflectance:
where denotes admittance, with
Mathematically, any stable transfer function having these properties may be called a Schur function. Thus, the discrete-time reflectance of an impedance is a Schur function if and only if the impedance is passive (positive real).
In the limit as damping goes to zero (all poles of converge to the unit circle), the reflectance becomes a digital allpass filter. Similarly, becomes a continuous-time allpass filter as the poles of approach the axis.
Recalling that a lossless impedance is called a reactance (§7.1), we can say that every reactance gives rise to an allpass reflectance. Thus, for example, waves reflecting off a mass at the end of a vibrating string will be allpass filtered, because the driving-point impedance of a mass () is a pure reactance. In particular, the force-wave reflectance of a mass terminating an ideal string having wave impedance is , which is a continuous-time allpass filter having a pole at and a zero at .
It is intuitively reasonable that a passive reflection gain cannot exceed at any frequency (i.e., the reflectance is a Schur filter, as defined in Eq.(C.79)). It is also reasonable that lossless reflection would have a gain of 1 (i.e., it is allpass).
Note that reflection filters always have an equal number of poles and zeros, as can be seen from Eq.(C.76) above. This property is preserved by the bilinear transform, so it holds in both the continuous- and discrete-time cases.
Consider the special case of a reflection and transmission at a yielding termination, or ``bridge'', of an ideal vibrating string on its right end, as shown in Fig.C.28. Denote the incident and reflected velocity waves by and , respectively, and similarly denote the force-wave components by and . Finally, denote the velocity of the termination itself by , and its force-wave reflectance by
The bridge velocity is given by
Power-Complementary Reflection and Transmission
The average power incident at the bridge at frequency can be expressed in the frequency domain as . The reflected power is then . Removing the minus sign, which can be associated with reversed direction of travel, we obtain that the power reflection frequency response is , which generalizes by analytic continuation to . The power transmittance is given by
Any passive driving-point impedance, such as the impedance of a violin bridge, is positive real. Positive real functions have been studied extensively in the continuous-time case in the context of network synthesis [68,524]. Very little, however, seems to be available in the discrete time case. This section (reprinted from ) summarizes the main properties of positive real function in the plane (i.e., the discrete-time case).
Definition. A complex valued function of a complex variable is said to be positive real (PR) if
We now specialize to the subset of functions representable as a ratio of finite-order polynomials in . This class of ``rational'' functions is the set of all transfer functions of finite-order time-invariant linear systems, and we write to denote a member of this class. We use the convention that stable, minimum phase systems are analytic and nonzero in the strict outer disk.C.8 Condition (1) implies that for to be PR, the polynomial coefficients must be real, and therefore complex poles and zeros must exist in conjugate pairs. We assume from this point on that satisfies (1). From (2) we derive the facts below.
Property 1. A real rational function is PR iff .
Proof. Expressing in polar form gives
since the zeros of are isolated.
Property 2. is PR iff is PR.
Proof. Assuming is PR, we have by Property 1,
Property 3. A PR function is analytic and nonzero in the strict outer disk.
Proof. (By contradiction)
Without loss of generality, we treat only order polynomials
The general (normalized) causal, finite-order, linear,
time-invariant transfer function may be written
where is the number of distinct poles, each of multiplicity ,and
Suppose there is a pole of multiplicity outside the unit circle. Without loss of generality, we may set , and with . Then for near , we have
Consider the circular neighborhood of radius described by . Since we may choose so that all points in this neighborhood lie outside the unit circle. If we write the residue of the factor in polar form as , then we have, for sufficiently small ,
Therefore, approaching the pole at an angle gives
Corollary. In equation Eq.(C.80), .
Proof. If , then there are poles at infinity. As , , we must have .
Corollary. The log-magnitude of a PR function has zero mean on the unit circle.
Corollary. A rational PR function has an equal number of poles and zeros all of which are in the unit disk.
This really a convention for numbering poles and zeros. In Eq.(C.80), we have , and all poles and zeros inside the unit disk. Now, if then we have extra poles at induced by the numerator. If , then zeros at the origin appear from the denominator.
Corollary. Every pole on the unit circle of a positive real function must be simple with a real and positive residue.
Proof. We repeat the previous argument using a semicircular neighborhood of radius about the point to obtain
In order to have near this pole, it is necessary that and .
Corollary. If is PR with a zero at , then
Proof. We may repeat the above for .
Property. Every PR function has a causal inverse z transform .
Proof. This follows immediately from analyticity in the outer disk [342, pp. 30-36] However, we may give a more concrete proof as follows. Suppose is non-causal. Then there exists such that . We have,
Hence, has at least one pole at infinity and cannot be PR by Property 3. Note that this pole at infinity cannot be cancelled since otherwise
which contradicts the hypothesis that is non-causal.
Property. is PR iff it is analytic for , poles on the unit circle are simple with real and positive residues, and re for .
Proof. If is positive real, the conditions stated hold by virtue of Property 3 and the definition of positive real.
To prove the converse, we first show nonnegativity on the upper semicircle implies nonnegativity over the entire circle.
Alternatively, we might simply state that real implies re is even in .
Next, since the function is analytic everywhere except at
, it follows that
is analytic wherever
is finite. There are no poles of outside the unit
circle due to the analyticity assumption, and poles on the unit circle
have real and positive residues. Referring again to the limiting form
Eq.(C.81) of near a pole on the unit circle at ,
we see that, as
, we have
since the residue is positive, and the net angle does not exceed . From Eq.(C.83) we can state that for points with modulus , we have For all , there exists such that . Thus is analytic in the strict outer disk, and continuous up to the unit circle which forms its boundary. By the maximum modulus theorem ,
For example, if a transfer function is known to be asymptotically stable, then a frequency response with nonnegative real part implies that the transfer function is positive real.
Note that consideration of leads to analogous necessary and sufficient conditions for to be positive real in terms of its zeros instead of poles.
By the representation theorem [19, pp. 98-103] there exists an asymptotically stable filter which will produce a realization of when driven by white noise, and we have . We define the analytic continuation of by . Decomposing into a sum of causal and anti-causal components gives
where is found by equating coefficients of like powers of in
Since the poles of and are the same, it only remains to be shown that re.
Since spectral power is nonnegative, for all , and so
Property. The function
is a Schur function if and only if is positive real.
Suppose is positive real. Then for , rere is PR. Consequently, is minimum phase which implies all roots of lie in the unit circle. Thus is analytic in . Also,
Conversely, suppose is Schur. Solving Eq.(C.84) for and taking the real part on the unit circle yields
If is constant, then is PR. If is not constant, then by the maximum principle, for . By Rouche's theorem applied on a circle of radius , , on which , the function has the same number of zeros as the function in . Hence, is minimum phase which implies is analytic for . Thus is PR.
Property. re for whenever
Proof. We shall show that the change of variable , provides a conformal map from the z-plane to the s-plane that takes the region to the region re. The general formula for a bilinear conformal mapping of functions of a complex variable is given by
In general, a bilinear transformation maps circles and lines into circles and lines . We see that the choice of three specific points and their images determines the mapping for all and . We must have that the imaginary axis in the s-plane maps to the unit circle in the z-plane. That is, we may determine the mapping by three points of the form and . If we predispose one such mapping by choosing the pairs and , then we are left with transformations of the form
Letting be some point on the imaginary axis, and be some point on the unit circle, we find that
There is a bonus associated with the restriction that be real which is that
We have therefore proven
The class of mappings of the form Eq.(C.85) which take the exterior of the unit circle to the right-half plane is larger than the class Eq.(C.86). For example, we may precede the transformation Eq.(C.86) by any conformal map which takes the unit disk to the unit disk, and these mappings have the algebraic form of a first order complex allpass whose zero lies inside the unit circle.
where is the zero of the allpass and the image (also pre-image) of the origin, and is an angle of pure rotation. Note that Eq.(C.88) is equivalent to a pure rotation, followed by a real allpass substitution ( real), followed by a pure rotation. The general preservation of condition (2) in Def. 2 forces the real axis to map to the real axis. Thus rotations by other than are useless, except perhaps in some special cases. However, we may precede Eq.(C.86) by the first order real allpass substitution
Riemann's theorem may be used to show that Eq.(C.89) is also the largest such class of conformal mappings. It is not essential, however, to restrict attention solely to conformal maps. The pre-transform , for example, is not conformal and yet PR is preserved.
Property. is PR if is positive real in the analog sense, where is interpreted as the sampling period.
Proof. The mapping takes the right-half -plane to the outer disk in the -plane. Also is real if is real. Hence PR implies PR. (Note, however, that rational functions do not in general map to rational functions.)
These transformations allow application of the large battery of tests which exist for functions positive real in the right-half plane .
- The sum of positive real functions is positive real.
- The difference of positive real functions is conditionally positive real.
- The product or division of positive real functions is conditionally PR.
- PR not PR for .
All properties of MP polynomials apply without modification to marginally stable allpole transfer functions (cf. Property 2):
- Every first-order MP polynomial is positive real.
- Every first-order MP polynomial
is such that
is positive real.
- A PR second-order MP polynomial with complex-conjugate zeros,
- All polynomials of the form
- If all poles and zeros of a PR function are on the unit circle,
then they alternate along the circle. Since this property is
preserved by the bilinear transform, it is true in both the
and planes. It can be viewed as a consequence of the
phase bounds for positive-real functions.
- If is PR, then so is , where the prime denotes differentiation in .
Loaded Waveguide Junctions
``Traveling Waves'' in Lumped Systems