Allpassfilter

An all-pass filter, also referred to as all-pass only, is an electrical filter for all frequencies having a constant magnitude frequency response of the ideal case, while the phase shift depends on the frequency. All-pass filters are used inter alia in communication engineering systems for signal equalization or for the generation of maturity or dead times.

  • 2.1 Active analog implementation
  • 2.2 Passive analog implementation
  • 2.3 Discrete-time implementation

General

All-pass filters have a frequency-dependent group transit time which can be used to frequency-dependent signal delay and phase equalization. It can thus be compensated transmission channels are frequency -dependent, usually unwanted signal delays. Since the group delay is the negative derivative of the phase response, this is not constant and depends on the frequency. With the aid of all-passes also an impedance transformation can be carried out.

Further all-pass filters are used for the Hilbert transform in order to generate an analytical signal from a real-valued signal. With the Hilbert transform, the spectrum of a signal in a given frequency band around π / 2 (90 °) is rotated in the phase. Those all-pass filters are also referred to as a " Hilbert Transformer" and since they are to be realized in analog electrical circuits only with great effort, preferably in the range of digital signal processing.

All-pass filters can be realized as a passive filter as an analog active filter, or in the context of digital signal processing as a time discrete digital filter.

General all-pass filters have the following relationships:

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

Poles and zeros

The transfer function of continuous all-pass filters, as they are realized in the form of analog circuits have occurring in pairs zeros and poles in the pole - zero plot, which are symmetrical to the vertical j.omega axis in the s-plane. In this case, all zeros lie on the right half-plane (RHE ), which causes the non- minimum- phase response.

In discrete-time all-pass networks, applied in the field of digital signal processing, occur after application of the Z transform all poles inside and all the zeros outside the unit circle in the complex z- plane. The poles and zeros occur while mirrored pairs on the unit circle.

In the special case of real-valued coefficients poles and zeros are either purely real or pairwise complex conjugate. For discrete real -passes the means: If a zero, then is also a root. Is a pole, then a zero.

Transfer function

The transfer function H1 ( s ) of a continuous-time all-pass first order or H2 ( s ) of a 2nd order all-pass filter is:

Higher orders arise by further pairs are added by pole-zero to the transfer function. An all-pass filter of order n is given by:

With the general coefficients ai and bi. The phase shift occurring in the second term φ is a function of angular frequency ω s = j · ω with the following relation:

Since the frequency response is ideally constant, the definition of a 3- dB cutoff frequency in the magnitude frequency response makes no sense. Ω -passes can be characterized on the group delay as a function of angular frequency Tgr. As a cut-off frequency in this case normally the sinking of the group delay is set to with respect to the limit of the group delay at the frequency 0 Hz.

Species

Active analog implementation

In the figure below is shown with an operational amplifier including an active, analog all-pass first order. Its transfer function ( s = σ j · ω ) is given as:

With a pole at -1/RC and a zero at 1/RC. The phase response as a function of angular frequency ω with σ = 0 has, in this circuit the following course:

The circuit of a 2nd order all-pass filter represents a possible embodiment of an operation amplifier and has a feedback network as modified difference on.

  • Active analog all-pass filters

2nd order

An advantage of the implementation by means of op-amps is that the circuit does not coils are used.

Passive analog implementation

Passive all-pass filter can be realized in various ways. For example, the implementation carried out in a lattice form (English Lattice ) or as a T circuit. The passive T- allpass was used for instance in fixed lines for compensation of phase distortions between the exchange and Endteilnehmeranschluss.

  • Passive analog all-pass filters

Lattice - circuit or Boucherot bridge

Discrete-time implementation

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

Telecommunications engineering Transmission delay Group delay and phase delay Analytic signal Z-transform Landline
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