All-pass filter

An all-pass filter , also known as an all-pass filter , is an electrical filter that ideally has a constant frequency response for all frequencies , while the phase shift depends on the frequency. All-passes are used, among other things, in communications systems for signal equalization or for generating transit times or dead times .

General

All-passes have the same amplitude for all frequencies; however, the phase shift is frequency-dependent. This property can be used for signal delay and phase equalization. It can be used to compensate or amplify frequency-dependent phase delays.

Furthermore, all-passes are used approximately for Hilbert transformation in order to generate an analytical signal from a real-valued signal . With the Hilbert transformation, the spectrum of a signal is rotated in phase by ^π / ₂ (90 °) in a certain frequency band . These all-passes are also known as "Hilbert transformers" and, since they can only be implemented with great effort in analog electrical circuits, are preferably used in the field of digital signal processing .

All-passes can be implemented as a passive filter , as an analog active filter , as a surface filter or as a time-discrete digital filter in the context of digital signal processing . Cables themselves are an all-pass, they delay signals by an amount that results from length, capacitance and inductance per unit length.

In general, all-passes have the following relationships:

• An all-pass filter of order n has n -phase rotations along the frequency axis.
• A non- minimum phase system can be divided into a minimum phase subsystem and an all-pass.

Pole and zeros

The transfer function of continuous all-pass filters, as implemented in the form of analog circuits, have zero and pole positions in pairs in the pole-zero diagram , which are symmetrical to the vertical jω-axis in the s-plane . All zeros are on the right half-plane (RHE), which causes the phase response that is not minimal.

In the case of time-discrete all-passes, used in the field of digital signal processing, all poles are inside and all zeros outside the unit circle in the complex Z-plane after the Z-transformation has been applied . The pole and zero points appear mirrored in pairs on the unit circle. [Explanation: A pole outside the circle results in a zero within. Exactly the other way around. In the Z-plane: pole at 2 → zero at 1/2. In the s-plane: pole at 2 + 2j → zero 2-2j].

In the special case of real-valued coefficients, poles and zeros are either purely real or complex conjugated in pairs. For discrete real all-passes this means: If there is a zero, then there is also a zero. If there is a pole, then there is a zero. ${\displaystyle z_{0\nu }}$ ${\displaystyle z_{0\nu }^{*}}$ ${\displaystyle z_{\infty \nu }}$ ${\displaystyle {\frac {1}{z_{\infty \nu }^{*}}}}$

Transfer function

The transfer function H ₁ ( s ) of a time-continuous all-pass 1st order or H ₂ ( s ) of a all-pass 2nd order is:

${\displaystyle H_{1}\left(s\right)={\frac {1-{\frac {s}{\omega _{0}}}}{1+{\frac {s}{\omega _{0}}}}},\qquad H_{2}\left(s\right)={\frac {1-{\frac {2\xi }{\omega _{0}}}\cdot s+{\frac {s^{2}}{{\omega _{0}}^{2}}}}{1+{\frac {2\xi }{\omega _{0}}}\cdot s+{\frac {s^{2}}{{\omega _{0}}^{2}}}}}}$

Higher orders are created by adding more pairs of pole zeros to the transfer function. An all-pass of the nth order is given by:

${\displaystyle H_{n}\left(s\right)={\frac {\prod _{i=1}^{n}\left(1-a_{i}s+b_{i}s^{2}\right)}{\prod _{i=1}^{n}\left(1+a_{i}s+b_{i}s^{2}\right)}}=e^{j\varphi }}$

with the general coefficients a _i and b _i . The phase shift φ occurring in the second expression is a function of the angular frequency ω with s  = j ω with the following relation:

${\displaystyle \varphi \left(\omega \right)=-2\sum _{i=1}^{n}\operatorname {arctan} {\frac {a_{i}\omega }{1-b_{i}\omega ^{2}}}}$

Since the frequency response is ideally constant, it makes no sense to define a 3 dB cutoff frequency in the absolute frequency response. All-passes are characterized via the group delay T _gr as a function of the angular frequency ω. The drop in the group delay to the limit value of the group delay at a frequency of 0 Hz is usually defined as the limit frequency. ${\displaystyle 1/{\sqrt {2}}}$

species

Active analog implementation

The figure below shows, among other things, an active, analog all-pass 1st order with an operational amplifier . Its transfer function (s = σ + j ω) is given as:

${\displaystyle H(s)={\frac {sRC-1}{sRC+1}}}$

with a pole at ⁻¹ / _RC and a zero at ¹ / _RC . The phase response as a function of the angular frequency ω with σ = 0 shows the following curve in this circuit:

${\displaystyle \angle H(j\omega )=\pi -2\arctan(\omega RC)}$

The circuit of a 2nd order all-pass filter is a possible implementation with an operational amplifier and the difference is a modified feedback network.

• Active analog all-passes
• 

  1st order filter

• 

  2nd order filter

One advantage of the implementation using operational amplifiers is that no coils are used in the circuit .

Passive analog implementation

Passive all-pass filters can be implemented in different ways. For example, the implementation can (engl. In a lattice form Lattice ) take place or as a T circuit. The passive T-Allpass was used, among other things, in landline connections to compensate for phase distortions between the exchange and the end user connection.

• Passive analog all-passes
• 

  Lattice circuit or Boucherot bridge
  filter 2nd order

• 

  T-circuit
  filter 2nd order

Discrete-time implementation

The discrete transfer function H ( z ) for a 1st order all-pass filter with a complex conjugate pole zero at a has the following form:

${\displaystyle H(z)={\frac {z^{-1}-a^{*}}{1-az^{-1}}}}$

literature

• T. Deliyannis, Yichuang Sun, JK Fidler: Continuous-Time Active Filter Design . 1st edition. CRC Press, 1999, ISBN 0-8493-2573-0 .
• Karl-Dirk Kammeyer, Kristian Kroschel: Digital signal processing . 6th edition. Teubner, 2006, ISBN 3-8351-0072-6 ( limited preview in Google book search).

Web links

• Example of All-Passes