The size of the class of filters called impedances will be seen to be large, because impedances are derived by transformation from an easily specified family of filters called reflectances, say, ct and its Fourier transform .To be a reflectance, the time function must be strictly causal, and the frequency function must be strictly less than unity. By strictly causal it is meant that the time function vanishes both at zero time and before. For example, take and the reflectance ct to be an impulse of size after a time .The Fourier transform is
An impedance has been defined to be a causal filter with a causal inverse and a Fourier transform whose real part is positive. It will be shown that from any reflectance C the expression
The expression for R(C) is easily back-solved for C(R), but the converse theorem, that every R generates a reflectance, is harder to show. Nevertheless, it will be proved, along with a deeper theorem. A filter that is both causal and PR is said to be CPR. The deeper theorem is that every CPR has an inverse and hence is an impedance. This will be proved by showing that every CPR--say, , --can be used to construct a reflectance , which, since it is a reflectance, implies that the CPR is an impedance R. Backsolving gives
Proof requires that two things be shown--first, that the magnitude of is less than unity. To show this, form the magnitude of the denominator and subtract the magnitude of the numerator. The result is four times the real part of , which is positive. Second, must be proved causal. This is harder. can be expanded into a sum of positive powers of and hence of positive powers of the delay operator. But the convergence of the series is not assured, because nothing requires to be less than unity.
To prove that is causal, we will take advantage of rule 1, namely, that an impedance can be scaled by any real positive number that you like, and it will still be an impedance. Consider a function that is similar to .
So impedances arise more easily than you might think. It is not necessary to have a reflectance C to insert into the relation R = (1-C)/(1+C). We only need to have a CPR.