Lattice delay network

Although there is duplication of impedances in this arrangement, it offers great flexibility to the circuit designer so that, in addition to its use as delay network (as featured here) it can be configured to be a phase corrector,[1] a dispersive network,[2] an amplitude equalizer,[3] or a low pass (or bandpass) filter,[4] according to the choice of components for the lattice elements .

Such a lattice is a constant resistance network and an all-pass filter, and it has a phase response determined by the properties of Za.

In order to achieve a desired delay, it is necessary to choose specific components for Za and Zb, and the design methods to do this are given in later sections.

Sometimes, the design procedures can result in Za and Zb being highly complicated networks, but it is always possible to derive a cascade of simpler lattices with identical electrical characteristics,[4] should that be preferred.

Although there are other ways of achieving signal delays, such as by a long length of coaxial cable, or by lumped element ladder networks, such solutions have either greater physical bulk, or they make inefficient use of a frequency band, or they have poor phase linearity.

Initially, the designs for lattice delays were based on image theory[4][6] in which the aim was to simulate a finite length of transmission line.

[7] This delay response is ripple free and is perfectly smooth over the passband, only deviating from the mean value as the band edge is reached.

In situations where a balanced network is not appropriate, a single ended circuit operating with a ground plane is required.

(In fact, the derivations of za by the continued fraction method result in family of lattices all of which have a maximally flat group delay characteristic).

The potential analogue method was proposed by Darlington[12] as a simple way to choose pole-zero positions for delay networks.

The method allows the designer to implement a delay characteristic by locating poles and zero on the complex frequency plane intuitively, without the need for complicated mathematics or the recourse to reference tables.

Other analogue methods, which were devised to aid the designer to choose the pole zero positions for his networks, include the "rubber sheet model"[13][14] and the "electrolytic tank".

A typical arrangement of poles and zeros to give, nominally, an electrical circuit with constant group delay follows the pattern shown in the figure below (see also Stewart[1]).

The poles and zeros lie in two lines, of finite length, parallel to the jω axis at a distance ‘a’ from it.

For frequencies beyond the end of the pole zero pattern, the group delay suffers a truncation error, but the band edge performance of a characteristic can be improved, by repositioning the outer poles and zeros slightly, to compensate for this sudden termination of the pattern.

The current example does not have a pole-zero pair located on the real axis, so a first order network is not required.

This is described as a maximally flat characteristic when as many as possible of the coefficients of ω in the power series equate to zero, by appropriate choice of values for a, b, c, d, etc.

In this particular expression, the maximally flat response is of order n. With the maximally flat characteristic, the delay remains constant, equal to the zero-frequency value, over a finite range of frequencies, but beyond this range the delay decreases smoothly with increasing frequency.

Such a procedure solves the problem of poor passband responses of the low pass filters, with the added bonus that the resulting networks have the constant resistance property.

Introducing zeros, in this way, gives double the delay of an all-pole low pass filter, but the phase characteristic still retains the desired maximally flat feature.

As an example of how a typical derivation proceeds, consider a 6th order low pass filter function.

is This expression for T(p) is identical to the one derived earlier, for a sixth order delay, by the continued fraction method.

In the case of analogue television waveforms, for example, the picture content also has a bearing on the acceptable levels of system distortion.

[26] In making a judgement of permitted distortion, limits can be set on waveform asymmetry, the level of overshoots and pre-shoots, and rise-time degradation and this is discussed in the section on ‘Transient Testing’ later.

A suitable ripple characteristic is obtained by taking power series approximations of sinh(x) and cosh(x),[1][10] rather than deriving the continued fraction expansion of tanh(x), as was done earlier.

These results are similar to those obtained by the ‘Forced Ripple Method’,[9][28] where a technique of curve fitting, at a finite number of frequencies of the phase response, is employed.

Comparing the bandwidths of networks with passband ripple to those with a maximally flat response, an increase of approximately 50% is achieved.

For example, in the case of television signals, sine-squared pulses may be used for the purpose[29][30] All networks given below are normalized for unit delay and one ohm terminations.

The circuit is where component values for a normalised 1 ohm network, with 1 second delay at low frequencies, are:

From the tables for power product approximation, given above, find the pole-zero positions: xA = 3.4659   yA = 2.1027      xB = 2.0857   yB = 6.9997