A lattice phase equaliser or lattice filter is an example of an all-pass filter. That is, the attenuation of the filter is constant at all frequencies but the relative phase between input and output varies with frequency. The lattice filter topology has the particular property of being a constant-resistance network and for this reason is often used in combination with other constant-resistance filters such as bridge-T equalisers. The topology of a lattice filter, also called an X-section, is identical to bridge topology. The lattice phase equaliser was invented by Otto Zobel using a filter topology proposed by George Campbell.
Characteristics The characteristic impedance of this structure is given by
Z o 2 = Z Z ′ {\displaystyle Z_{o}^{2}=ZZ'}
and the transfer function is given by
H ( ω ) = Z o − Z Z o + Z {\displaystyle H(\omega )={\frac {Z_{o}-Z}{Z_{o}+Z}}} .
Applications The lattice filter has an important application on lines used by broadcasters for stereo audio feeds. Phase distortion on a monophonic line does not have a serious effect on the quality of the sound unless it is very large. The same is true of the absolute phase distortion on each leg (left and right channels) of a stereo pair of lines. However, the differential phase between legs has a very dramatic effect on the stereo image. This is because the formation of the stereo image in the brain relies on the phase difference information from the two ears. A phase difference translates to a delay, which in turn can be interpreted as a direction the sound came from. Consequently, landlines used by broadcasters for stereo transmissions are equalised to very tight differential phase specifications. Another property of the lattice filter is that it is an intrinsically balanced topology. This is useful when used with landlines which invariably use a balanced format. Many other types of filter section are intrinsically unbalanced and have to be transformed into a balanced implementation in these applications, which increases the component count. This is not required in the case of lattice filters.
Design Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.
The essential requirement for a lattice filter is that for it to be constant resistance, the lattice element of the filter must be the dual of the series element with respect to the characteristic impedance. That is,
Z R 0 = R 0 Z ′ {\displaystyle {\frac {Z}{R_{0}}}={\frac {R_{0}}{Z'}}} . Such a network, when terminated in R0, will have an input resistance of R0 at all frequencies. If the impedance Z is purely reactive such that Z = iX then the phase shift, φ, inserted by the filter is given by
tan φ 2 = − X R 0 {\displaystyle \tan {\frac {\varphi }{2}}=-{\frac {X}{R_{0}}}} . The prototype lattice filter shown here passes low frequencies without modification but phase-shifts high frequencies. That is, it is phase correction for the high end of the band. At low frequencies the phase shift is 0° but as the frequency increases the phase shift approaches 180°. It can be seen qualitatively that this is so by replacing the inductors with open circuits and the capacitors with short circuits, which is what they become at high frequencies. At high frequencies the lattice filter is a cross-over network and will produce 180° phase shift. A 180° phase shift is the same as an inversion in the frequency domain, but is a delay in the time domain. At an angular frequency of ω = 1 rad/s the phase shift is exactly 90° and this is the midpoint of the filter's transfer function.
Low-in-phase section
The prototype section can be scaled and transformed to the desired frequency, impedance and bandform by applying the usual prototype filter transforms. A filter which is in-phase at low frequencies (that is, one that is correcting phase at high frequencies) can be obtained from the prototype with simple scaling factors. The phase response of a scaled filter is given by
tan φ 2 = − ω ω m {\displaystyle \tan {\frac {\varphi }{2}}=-{\frac {\omega }{\omega _{m}}}} , where ωm is the midpoint frequency and is given by
ω m = 1 L C {\displaystyle \omega _{m}={\frac {1}{\sqrt {LC}}}} .
High-in-phase section
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