EE 541 , Fall 2009 : Course Notes # 3 Passive , Constant Resistance , Broadband Delay Filter

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. Finally, the new filter has a range of designable delay that is larger than its BesselThomson 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: August 2006 Course Notes #3 USC Viterbi School of Engineering Choma August 2006 103 Delay Filter 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 input 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 input signal, x(t), is given in the steady state by the simple relationship, do y(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 frequency spectrum implicit to the input signal, x(t). While a system projecting the idealized input tooutput (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 constant 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. 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 do sT Y(s) K X(s) , e− = (2) which suggests a network transfer function, H(jω), in the sinusoidal steady state of do jωT Y(jω) H(jω) K X(jω) e− = . (3) The suggested constant gain magnitude and linear phase response in the frequency domain supports 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 Course Notes #3 USC Viterbi School of Engineering Choma August 2006 104 Delay Filter 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. In a lowpass, n order network realization of a transfer function delivering 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 BesselThomson 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 network having no finite frequency zeros. Because of the lack of finite frequency zeros, the phase response, and thus the delay characteristic, of the Bessel-Thomson structure derives solely from its pole locations in the complex frequency plane. Indeed, the zero frequency delay in a lowpass Bessel-Thomson filter is precisely equal to the coefficient of the first order term in a monic representation of its characteristic polynomial. It is easily shown that this coefficient is, in turn, identically equal to the sum of inverse pole frequencies, which clearly gives rise to the nominal inverse dependence between zero frequency delay and 3-dB bandwidth. This paper exploits constant resistance, lowpass lattice networks to forge a synthesis procedure for a viable alternative to the classic Bessel-Thomson delay filter. Although the alternative proposed herein does not necessarily deliver a maximally flat delay response, it does afford its user the flexibility of specifying an allowable positive or negative delay error over frequency with respect to the desired zero frequency delay. The absence of a strictly maximally flat delay characteristic is mitigated by two engineering attributes that derive from a filter realization premised on a non-minimal phase architecture. First, the designable amount of zero frequency delay can be larger than that afforded by the Bessel-Thomson filter since both the poles and incorporated right half plane zeros of the circuit realization contribute to the phase and delay responses. Second, and because of the presence of right half plane zeros, acceptable delays over relatively broad frequency ranges can be achieved with only a second order structure. In contrast, broadband performance in the Bessel-Thomson filter mandates progressively higher filter order. The third, and perhaps most significant, attribute of the new delay filter is that through a judicious selection of the natural frequency associated with the right half plane zeros, the desired zero frequency delay is rendered relatively insensitive to the 3-dB bandwidth of the filter magnitude response. 2.0. DELAY FILTER TRANSFER CHARACTERISTICS The transfer function, H(s), of a non-minimal phase, allpass, second order circuit is expressible as Course Notes #3 USC Viterbi School of Engineering Choma August 2006 105 Delay Filter 2