Broadband Delay FilterUniversity of Southern California213–740–7581[USC Fax]Original: June 20061.0.INTRODUCTION2.0.DELAY FILTER TRANSFER CHARACTERISTICS3.0.FILTER REALIZATION STRATEGY3.1.CONSTANT RESISTANCE BRIDGE3.2.CONSTANT RESISTANCE L-SECTIONS3.3.CONSTANT RESISTANCE TEE-SECTION4.0.DELAY FILTER REALIZATION4.1.FILTER SYNTHESIS4.2.FORMULATION OF FILTER DESIGN STRATEGY5.0.DESIGN EXAMPLE6.0.CONCLUSION6.0.REFERENCESCourse Notes #3 USC Viterbi School of Engineering Choma EE 541, Fall 2006: Course Notes #3 Passive, Constant Resistance, Broadband Delay Filter Dr. John Choma Professor of Electrical Engineering & Systems Architecture Engineering University of Southern California Department of Electrical Engineering-Electrophysics University Park: Mail Code: 0271 Los Angeles, California 90089–0271 213–740–4692 [USC Office] 213–740–7581 [USC Fax] [email protected] ABSTRACT: This paper addresses the synthesis of an alternative to the Bessel-Thomson delay filter. The new filter is forged of building blocks familiar to filter designers, while affording the RF designer the luxury of a designable delay that is not inversely dependent on filter bandwidth. Furthermore, the architecture has the desirable attribute of the relative simplicity and low device count that derives from only a second-order realization. In the case of a monolithic realization, excessive chip surface area is therefore not consumed, and the matching error inherent to large numbers of passive devices is minimized. Fi-nally, the new filter has a range of designable delay that is larger than its Bessel-Thomson counterpart due to the purposeful incorporation of right half plane zeros in the transfer function. The paper begins with a tutorial to ensure reader understanding of the building blocks for the proposed filter. The tutorial is followed by a design example. Original: June 2006 August 2006 - 102 - Delay FilterCourse Notes #3 USC Viterbi School of Engineering Choma 1.0. INTRODUCTION In the high performance linear amplifiers, filters, and digital signal processing cells pervasive of modern communication systems, distortionless transmission between the applied signal and resultant output response is an omnipresent engineering goal. “Distortionless” signal transmission is herewith taken to mean that the wave shape of the output response is identical to that of the applied input excitation to within a factor of a multiplicative constant. System output responses that are delayed in the time domain by a constant amount, but otherwise mirror the in-put excitation, are also viewed as distortionless. It follows that the idealized design goal of any linear distortionless network is the assurance that the output response, say y(t), to an applied in-put signal, x(t), is given in the steady state by the simple relationship, doy(t) Kx( t T ) ,=− (1) where K, the gain of the system, and Tdo, the time delay implicit to transmitting the input signal to the network output port, are constants. Specifically, K and Tdo are invariant with the fre-quency spectrum implicit to the input signal, x(t). While a system projecting the idealized input -to- output (I/O) relationship of (1) is physically unrealizable, specific system applications allow invoking acceptable approximations of the subject relationship. For example, constant I/O delay is relatively unimportant in electronic audio channels because the human ear can readily perceive only signal amplitude fluctuations, thereby rendering constant K far more important than con-stant Tdo. In video systems, the operational situation is the direct opposite of audio channels; that is, constant Tdo is a significantly more critical design objective than is constant K. On the other hand, non-constant time delay is a serious problem in almost all digital communication systems, in that delay variations with input signal frequency incur potentially significant pulse dispersion, which causes a time domain interference of pulses of interest with neighboring pulses[1]. If Y(s) is the Laplace transform of y(t) and if X(s) designates the Laplace transform of x(t), the frequency domain equivalent of (1) is dosTY(s) K X(s) ,e−= (2) which suggests a network transfer function, H(jω), in the sinusoidal steady state of dojωTY(jω)H(jω)KX(jω)e−=. (3) The suggested constant gain magnitude and linear phase response in the frequency domain sup-ports the contention that (1) infers a physically unrealizable network or system. The implication of this engineering reality is that the subject frequency domain transfer characteristic can be emulated only by the transfer function, jφ(ω)H(jω)H(jω)e= , (4) where |H(jω)| is understood to be constant K if |H(jω)| is independent of radial frequency ω, and φ(ω) is the I/O phase angle response of the considered system. A broadband system achieves |H(jω)| ≈ K to within a user-defined error over a designable, finitely wide frequency passband. Moreover, since the steady state delay response, D(ω), which is commonly referenced as group delay or envelope delay, is August 2006 - 103 - Delay FilterCourse Notes #3 USC Viterbi School of Engineering Choma dφ(ω)D(ω)dω=− , (5) constant I/O delay is seen to require a phase response exhibiting linear phase lag. Specifically, constant I/O time delay in the amount of Tdo is adequately approximated in the steady state if the system is designed to produce the lagging phase response, φ(ω) ≈ –ωTdo over an acceptably broad passband. The most common approximation of constant delay in electrical and electronic circuits is the Bessel-Thomson filter, which realizes a maximally flat delay (MFD) response over the passband of interest[2]. In a lowpass, nth order network realization of a transfer function deliver-ing MFD, the first (n – 1) frequency derivatives of delay response D(ω) are zero at ω = 0. Accordingly, the nominally constant, and indeed maximum, delay produced by such a realization is the zero frequency value, D(0) = Tdo, of the delay response. Despite its laudable delay response attributes and widespread utilization, the Bessel-Thomson filter suffers from a serious shortcoming. In particular, the observable 3-dB bandwidth of the Bessel-Thomson filter is, to within crude first order, inversely proportional to the desired value of the zero frequency delay. This shortfall stems from the fact that the Bessel-Thomson filter is a minimum phase