# Feedback filters

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 5 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | ISBN 9781560801481 |

Store | SEG Online Store |

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^{[1]}; Sacchi and Ulrych, 2000^{[2]}).

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^{[3]}).

## References

- ↑ 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.

## Continue reading

Previous section | Next section |
---|---|

Convolution | Inverse filters |

Previous chapter | Next chapter |

Sampling | Frequency |

## Also in this chapter