We now shall discuss minimum delay and its relationship to feedback. At this point, let us expand our discussion about minimum delay and its relationship to feedback systems. For example, suppose that the desired direction of a ship is set on the gyrocompass. A feedback mechanism indicates the error between the desired direction and the actual direction of the ship. The error activates the guidance system, which consists of power amplifiers forcing the rudder in the direction that decreases the error. Because it takes time to supply the power to change the ship’s course, there is a time delay in the guidance system. Suppose that the ship is off course to starboard. The feedback mechanism indicates an error to starboard, and the power amplifiers force the rudders to port. Because of the time delay, the ship overshoots the gyro direction to port. Now the feedback mechanism indicates an error to port, and the power amplifiers force the rudders to starboard. Because of the time delay, the ship again overshoots the gyro direction, this time to starboard. The feedback mechanism now indicates an error to starboard, and because of the time delay, a third overshoot is produced, this time to port. These oscillations about the gyro direction can either increase in magnitude on each successive swing or can decrease. If they increase, the guidance system is unstable. If they decrease, it is stable. Clearly, the minimum-delay guidance system is the one that is stable. For example, in the layer-cake model of seismic stratigraphy, downgoing waves from an impulsive source have the minimumdelay property (Robinson and Treitel, 1976; Loewenthal and Robinson, 2000).
We can describe a causal system by its gain and its delay. Any causal linear system can be described by its gain and its delay. Its gain is a measure of the increase or decrease of the magnitude of the output compared with the magnitude of the input. Delay is a measure of the time from the instant the input is activated to the instant that this input is significantly felt at the output. As we expect, both gain and delay depend on the frequency of the signal.
What system has the smallest delay for its gain? It is possible to have many systems, each with the same gain but with a different delay. In fact, it is always possible to have systems with very great delays because the largeness of the delay that can be incorporated into a system has no theoretical limit. On the other hand, the smallness of the delay that a system can possess is limited because a system always takes some time to respond significantly to an input. The system with the smallest possible delay for its gain is called the minimum-delay system.
What are the gain and delay of systems connected in series? Suppose that we have two causal systems, A and B, connected in series. The gain of the overall system is equal to the product of the gains of the component systems, whereas the delay of the overall system is equal to the sum of the delays of the component systems. Instead of considering the gain, we can consider the logarithm of the gain, called the log gain. The logarithm of a product is equal to the sum of the logarithms of the individual factors. Hence, the log gain of the overall system is equal to the sum of the log gains of the component systems. In summary, we have (1) the log gain of the overall system = the log gain of A + the log gain of B and (2) the delay of the overall system = the delay of A + the delay of B.
What is a fair price to pay for gain and delay? As we have just seen, a causal system can be described by its log gain and its delay. In a figurative sense, let us think of the log gain of a system as an asset and the delay of the system as the cost of that asset. That is, log gain always must be paid for in delay.
It turns out that the price that is paid is always larger than, or at best equal to, the amount of log gain received. Causal systems for which the price paid equals the amount of log gain received are minimum-delay systems. Hence, minimum delay is the fair price to pay for log gain. All other causal systems - that is, the nonminimum-delay systems - have delays greater than the minimum. For those systems, the price paid for the log gain is greater than the fair price, and the extra amount paid is the difference between the system’s actual delay and the minimum possible delay.
What are all-pass systems? Causal systems occur that have no log gain, and such systems are called all-pass systems. There are two kinds of all-pass systems. The first kind is the trivial all-pass system, which has no delay. Because a trivial all-pass system has zero log gain and zero delay, we pay nothing for nothing, which is fair. The other kind of all-pass system is the nontrivial all-pass system, and it does produce delay. Because a nontrivial all-pass system has no log gain but does have delay, we pay something for nothing, which is unfair (Robinson, 1962; Robinson and Treitel, 1965).
We can summarize the various types of systems as follows:
- minimum-delay system:
asset = log gain cost = minimum delay (fair price)
- nonminimum-delay system:
asset = log gain cost = minimum delay plus extra delay (unfair price)
- trivial all-pass system:
asset = none cost = none (fair price)
- nontrivial all-pass system:
asset = none cost = delay (unfair price)
A trivial all-pass system is one whose impulse response is the Dirac delta function, or unit spike; such a system passes input to output with no change at all. That is, a trivial all-pass system is the identity operator.
What is the canonical representation? From the above asset-cost tables, we are led to the following considerations: The overall system resulting from connecting a nontrivial all-pass system to a minimum-delay system is a nonminimum-delay system. Conversely, any nonminimum-delay system always can be decomposed into two systems connected in series: a nontrivial all-pass system and a minimum-delay system. These results comprise the so-called canonical representation.
Why is minimum delay the fair price for the gain attained? In other words, what is the fair price for an asset? The fair price for an asset is the price for which we can sell the asset and by so doing return to our original position. A minimum-delay system is a system for which the fair price (i.e., minimum delay) has been paid for the log gain. Consequently, for each minimum-delay system, a realizable inverse system exists. If a signal is the input to a minimum-delay system, then we can recover that signal in its original form by passing the output of the minimum-delay system through its stable inverse system, which is a minimum-delay system. The signal’s recovery is accomplished with no overall time delay. Hence, in transmitting information from input to output, a minimum-delay system neither destroys nor delays the information.
When is an unfair price paid for the gain attained? A nonminimum-delay system is a system for which an unfair price has been paid for the log gain. Thus, there is no causal inverse system for a nonminimum-delay system. Some nonminimum-delay systems do not destroy information about the signal but only delay the information. For those systems, the original signal can be recovered, at least approximately, but at the cost of an overall time delay. Other nonminimum-delay systems destroy information about the signal so that the original signal cannot be recovered, even with an indefinitely long time delay.
- Robinson, E. A., and S. Treitel, 1976, Net downgoing energy and the minimum-delay property of downgoing waves: Geophysics, 41, 1394–1396.
- Loewenthal, D., and E. A. Robinson, 2000, Relativistic combination of any number of collinear velocities and generalization of Einstein’s formula: Journal of Mathematical Analysis and Applications, 246, no. 1, 320–324.
- Robinson, E. A., 1962, Random wavelets and cybernetic systems: Charles Griffin and Co. and Macmillan.
- Robinson, E. A., and S. Treitel, 1965, Dispersive digital filters: Reviews of Geophysics, 3, 433–461.
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