Analysis and Design of Systems with a Non-negative Impulse Response

Doctoral Dissertation


Non-negativity of the impulse response of a system (a.k.a. a monotonic step response system) is a widely required feature in many industrial applications. Although the importance of this feature has long been acknowledged, an in-depth understanding of the influence of this feature on the time and frequency domains is still not available. The problem of designing a high-performance filter with a non-negative impulse response (NNIR) therefore remains unsolved.

This dissertation provides some studies on the influence of the NNIR feature on both the time and frequency domains. The problem is approached from two angles: 1.) Restrictions on the frequency domain response imposed by a nonnegative impulse response; 2.) The pole-zero pattern of a transfer function that ensures a non-negative impulse response.

In characterizing the frequency domain response of a linear NNIR system, it is found that due to the non-negativity constraint in the time-domain, the frequency domain has some unique fundamental properties that a conventional linear system does not possess. The significance of these fundamental properties is discussed. As a result, the gains at frequencies along the frequency axis are found to be related to each other inherently.

Upper/lower bounds on the magnitude response within critical regions (frequency regions within which hard specifications are given) are derived. Based on these bounds, limitations in the frequency selectivity of various types of filters are analyzed and illustrated. The results explain the difficulties associated with non-negative impulse response filter design for systems other than lowpass filters. An approximation-based approach is presented for designing high-performance non-negative finite impulse response (NNFIR) lowpass filters, while NNFIR nonlowpass filters of various types can be obtained from a lowpass design via proposed transformations that preserve the NNIR feature.

In exploring pole-zero patterns, this dissertation presents a set of sufficient conditions that ensure the non-negativity of the impulse response for several classes of arbitrary-order transfer functions. Such conditions are given for both continuoustime and discrete-time systems. Most of the existing work can be classified as a special case of these sufficient conditions. Also, for an arbitrary-order discretetime system with complex poles and zeros, this dissertation presents a new set of sufficient conditions that ensures the nonnegativity of its impulse response. With these sufficient conditions, the class of zero-pole patterns known to exhibit a non-negative impulse response is significantly expanded.


Attribute NameValues
  • etd-04152011-020008

Author Yuzhe Liu
Advisor Peter H. Bauer
Contributor Ken Sauer, Committee Member
Contributor Panos Antsaklis, Committee Member
Contributor Peter H. Bauer, Committee Chair
Contributor Yih-Fang Huang, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Electrical Engineering
Degree Name PhD
Defense Date
  • 2011-03-31

Submission Date 2011-04-15
  • United States of America

  • monotonic step response

  • non-negative impulse response

  • filter design

  • University of Notre Dame

  • English

Record Visibility and Access Public
Content License
  • All rights reserved

Departments and Units


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