Let us draw the diagram of the basic feedback filter. We now want to find the transfer function of the basic feedback filter, namely the first-order causal feedback filter. Its diagram is shown in Figure 12. We shall demonstrate how the stability of its impulse response is related to the minimum-delay property of this feedback filter. An alternative term for minimum delay is minimum phase (Ulrych and Laserre, 1966; Sacchi and Ulrych, 2000).
Let us describe the constant box of the basic feedback filter. The input to the feedback system is the wavelet , where n denotes time. According to our convention, the time index n is taken at discrete-time instants spaced one unit apart. We now analyze the feedback filter by tracing the path around the loop. We start at the beginning of the feedback loop (point A in Figure 12). We see that the filter output is fed back through the loop and enters the constant box as its input. The constant box carries out a constant multiplication. That is, the output of the constant box is equal to c times its input, where c is a fixed number. Thus, the impulse-response function of the constant box is the wavelet , so its Z-transform is
If the magnitude of c is less than 1 - that is, if - this box produces a constant attenuation. On the other hand, if the magnitude of c is greater than 1 - that is, if - this box produces a constant amplification. Because the input to the constant box is , the output from the constant box is (as shown at point B in Figure 12). The action of the constant box in terms of Z-transforms is
- Input to the constant box has the Z-transform Y(Z).
- The constant box has the Z-transform c.
- Hence, output from the constant box has the Z-transform cY(Z).
The output from the constant box enters the time-delay box (after point B in Figure 12). A unit-time-delay box produces a delay of one time unit from input to output. That is, the output of a unit-time-delay box is delayed one time unit with respect to its input. The impulse-response function of the unit-time-delay box is (), so its Z-transform is
If the time-delay box delays the input by T time units, we have a T-delay box, with the Z-transform .
Let us describe the T-delay box of the basic feedback filter. As we have just seen, the input to the T-delay box is ; thus, the output from this box is (as shown at point C in Figure 12). The action of this T-delay box in terms of Z-transforms is
- Input to the delay box has the Z-transform cY(Z).
- The delay box has the Z-transform .
- Hence, output from the delay box has the Z transform .
Next we describe the mixer of the basic feedback filter. The output from the unittime-delay box enters one input channel of the mixer (point C in Figure 12). At the same time, the closed-loop feedback system input enters the other input channel of the mixer (point D in Figure 12). The mixer produces an addition. That is, the mixer adds the input to the system input . Thus, the output of the mixer is (point E in Figure 12).
However, the output of the mixer is also the output of the closed-loop feedback system. Hence, we have the simple equation
for the feedback filter.
Suppose we wish to describe the transfer function of the basic feedback filter. In terms of Z-transforms, the equation just given becomes
Transposing terms, we obtain
The transfer function H(Z) of the feedback filter is defined to be the ratio of the Z-transform Y(Z) of the system output to the Z-transform W(Z) of the system input . Thus, the transfer function is
Next let us show that the feedback filter is stable only for the minimum-delay case. The Z-transform of the (T + 1)-length wavelet is . Therefore, the transfer function H(Z) of the feedback filter is the reciprocal of the Z-transform of the wavelet .
At this point, we will introduce the concepts of minimum delay and maximum delay. A thorough discussion of these concepts is given in Chapter 7. A wavelet is said to be a minimum-delay wavelet if it is front loaded, whereas it is said to be a maximum-delay wavelet if it is back loaded. The front coefficient in the wavelet is 1, and the back coefficient is c. Because all the other coefficients are zero, this wavelet is easy to clarify. If , then the front coefficient is greater in magnitude than the back coefficient, so the wavelet is front-loaded and hence minimum delay. If then the back coefficient is greater in magnitude than the front coefficient, so the wavelet is back-loaded and hence maximum delay.
We now state that the feedback filter is stable if the wavelet is a minimum-delay wavelet, and the feedback filter is unstable if the wavelet is a maximum-delay wavelet. That is, if , the constant box produces attenuation, and the effect of the feedback damps out with respect to time, thereby making the feedback filter stable.
According to still prevalent older terminology, negative feedback refers to a stable feedback system, whereas positive feedback refers to an unstable feedback system. However, it is simpler and more descriptive simply to say stable feedback and unstable feedback and not to use the terms negative or positive feedback (Treitel and Robinson, 1964).
- Ulrych, T. J., and M. Lasserre, 1966, Minimum-phase: Journal of the Canadian Society of Exploration Geophysicists, 2, 22–32.
- Sacchi, M. D., and T. J. Ulrych, 2000, Non-minimum-phase wavelet estimation using higher order statistics: The Leading Edge, 19, no. 1, 80–83.
- Treitel, S., and E. A. Robinson, 1964, The stability of digital filters: IEEE Transactions on Geoscience Electronics, GE-2, 6–18.
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