[go: up one dir, main page]

CN1667945A - Linear audio power amplifying system with predistortion - Google Patents

Linear audio power amplifying system with predistortion Download PDF

Info

Publication number
CN1667945A
CN1667945A CN 200510033974 CN200510033974A CN1667945A CN 1667945 A CN1667945 A CN 1667945A CN 200510033974 CN200510033974 CN 200510033974 CN 200510033974 A CN200510033974 A CN 200510033974A CN 1667945 A CN1667945 A CN 1667945A
Authority
CN
China
Prior art keywords
predistortion
distortion
processor
audio
signal
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN 200510033974
Other languages
Chinese (zh)
Inventor
李朝伟
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
SHENZHEN HUITIAN COMMUNICATION TECHNOLOGY Co Ltd
Original Assignee
SHENZHEN HUITIAN COMMUNICATION TECHNOLOGY Co Ltd
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by SHENZHEN HUITIAN COMMUNICATION TECHNOLOGY Co Ltd filed Critical SHENZHEN HUITIAN COMMUNICATION TECHNOLOGY Co Ltd
Priority to CN 200510033974 priority Critical patent/CN1667945A/en
Publication of CN1667945A publication Critical patent/CN1667945A/en
Pending legal-status Critical Current

Links

Images

Landscapes

  • Amplifiers (AREA)

Abstract

This invention relates to a pre-distortion audio linear power amplification system including a D/A processor, a filter and an audio power amplification system, among which, a pre-distortion processor is set between an audio digital signal source and a D/A processor for processing the pre-distortion to the audio digital signal source in a digital domain, the non-linearity and memory effect property expressed by this pre-distortion is opposite to those brought with the audio power amplification system itself, so the distortion brought with said system is cancelled out.

Description

Linear audio power amplifying system with predistortion
Technical field
The present invention relates to adopt the sound system of digital audio source, particularly the audio power amplification system part in the sound system.
Background technology
Along with the development of digital technology, adopt the more and more of digital audio source in the sound system, for example meet the digital audio and video signals CD player, MAT machine, MD machine, computer sound card of international S/PDIF interface standard etc.The signal flow of such sound system as shown in Figure 1, as described below: digital audio source 11, finish digital analogue conversion through D/A processor 12, the output analog signal is to low pass filter 13.Obtain exporting audio amplifier system to after the audio signal behind the out of band signal beyond the low pass filter 13 filtering audio frequency.Audio amplifier system comprises preamplifier 14, power amplifier 15, and audio signal is amplified through power amplifier 15, and output has the audio signal of certain power, and the audio-signal-driven loudspeaker vibration is sounded.
In order to obtain good linearty, enough big dynamic range and high efficient, existing audio power amplification system has adopted a lot of methods, but regrettably up to the present, each class methods is difficult to take into account simultaneously to this three, specifies as follows.
Class A (Class-A) power amplifier, owing to be operated in the Class A, its linearity is good, distortion is little, no intermodulation distortion and switch distortion, show on the index then be that harmonic distortion is little, good frequency response, intermodulation distortion be little, but power amplification efficiency is low, and the device heating amount is big, so volume is big, the cost height, the heat radiation difficulty.
The biasing of Class B (Class-B) power amplifier makes the transistor (or electron tube) of symmetrical operation when no drive signal, is in low current condition.When adding drive signal, an electric current rising in the half period in a pair of pipe, the another pipe then tends to end, and to another half period, situation is opposite.Since two pipes in turn by push-pull circuit work with synthetic complete signal waveform, therefore, the advantage of class B amplifier is that efficient is higher, shortcoming is that distortion is bigger.
AB power-like amplifier pluses and minuses get involved between category-A and the category-B, and efficient is than category-A height, and is lower than category-B, but though distortion is littler than category-B, but bigger than category-A distortion, be difficult to guarantee high power amplifier quality.
D class power amplifier is because of having used the pwm switch technology, so the efficient height.Because of the intrinsic dead time effect of its metal-oxide-semiconductor field effect transistor complementary output level (opening preceding shutoff), when more high-power output, produce the signal distortion that is difficult to accept inevitably.
Equally, the response speed of electron tube is fast, stability is high, harmonic distortion is little, but it is extremely low that shortcoming is the power utilization rate, and the efficient that the electron tube category-A amplifies is less than 10%, and the C class is 15%~17%, most of electric energy becomes heat and dissipates, so electron tube has, and power consumption is big, heating is high, shortcomings such as volume and weight is big, consumptive material is many, cost height.
Summary of the invention
The objective of the invention is to the problems referred to above of existing at existing all kinds of audio power amplification systems, provide a kind of can be at the linear audio power amplifying system with predistortion that obtains good linearty, enough well taken into account aspect big dynamic range and the high efficient three.
The object of the present invention is achieved like this: linear audio power amplifying system with predistortion of the present invention comprises the D/A processor, filter and audio power amplification system, between audio digital signals source and D/A processor, be provided with the predistortion processor that in numeric field, to carry out pre-distortion to the audio digital signals source, it is non-linear that this predistortion table reveals, the memory effect characteristic is opposite with nonlinear distortion and memory effect that described audio power amplification system itself is brought, thereby the distortion of distortion that the audio digital signals source was anticipated and audio power amplification system is cancelled out each other, thereby has improved the linear characteristic of audio power amplification system.The distortion of audio power amplification system is mainly brought by non-linearity of power amplifier and memory effect.
Predistortion processor of the present invention inside mainly comprises power amplification distortion Mathematical Modeling, check and show, address generator, digital filter, described power amplification distortion Mathematical Modeling can reflect the non-linear and memory effect characteristic of power amplifying system, produce the table of checking thus, described inspection table leaves in the memory of predistortion processor, this inspection table has been deposited the relevant value of non-linear and memory effect of reflection power amplifying system, the inspection table can be various dimensions, the amplitude that input signal can be arranged, phase place, the power amplifier direct current biasing, a plurality of dimensions such as temperature, the parameter of inspection table has determined the filter factor of digital filter, described address generator can be according to the amplitude of input audio signal, produce corresponding address, described predistortion processor can be according to the address, in conjunction with the power amplifier direct current biasing, the factor of other dimension such as temperature, inquiry is checked and is shown, obtain corresponding pre-distortion parameters, this parameter has determined the coefficient of digital filter.
Described power amplification distortion Mathematical Modeling is a kind of model of setting up at the non-linear of power amplifying system and memory effect characteristic.The nonlinear characteristic of power amplifying system can be approached with nonlinear mathematical model, for example adopts power series (Power Series) to represent that its nonlinear characteristic, the mathematic(al) representation of power series are f p(x)=x pThe memory effect of so-called power amplifying system is because power amplifying system non-linear not only relevant with the nonlinear characteristic of present moment, and front power amplifying system nonlinear characteristic constantly also can have influence on the power characteristic of present moment, this memory characteristic also can approach with multiple Mathematical Modeling, for example adopts dimension to receive model (Weiner Model).In fact, the model that the non-linear band memory characteristic of description of a lot of classics is arranged on the mathematics, we set up or adopt the principle of which kind of Mathematical Modeling is the non-linear and memory characteristic of power amplifying system that uses according to reality, so that the non-linear and memory characteristic of Mathematical Modeling and reality approaches as far as possible.
Predistortion processor of the present invention inside can also comprise comparator and distortion evaluator, also between predistortion processor and audio power amplification system, be provided with a feedback loop that constitutes by A/D processor and coupler, described coupler is used for the audio signal bypass to the output of audio power amplification system, output signal to the A/D processor, described A/D processor is used to finish the analog-to-digital conversion to this audio signal, the output digital signal feeds back to predistortion processor, described comparator is used for the signal and the input signal of the output of audio power amplification system are compared, obtain error signal, described distortion evaluator is used for this signal is carried out the distortion assessment, as foundation the parameter of power amplification distortion Mathematical Modeling is adjusted, thus the inspection table is adjusted, set up and correction predistortion model parameter according to this signal, to choose suitable pre-distortion parameters.
Predistortion processor of the present invention can be functional entity, also can be physical entity.Physical entity for example adopts digital signal processing chip (being dsp chip), fpga chip, predistortion processor special chip etc., and predistortion processor also can be passed through integrated this pre-distortion function of other types chip.
Audio power amplification system of the present invention is not limited to preamplifier and power amplifier, also is fit to comprise the audio power amplification system of loud speaker.
Technique effect of the present invention is: the present invention has adopted digital pre-distortion technology first in audio amplifier system, can improve the linearity of audio amplifier system, eliminate memory effect preferably, the index such as harmonic distortion, intermodulation distortion, transient intermodulation, frequency response of sound system is improved, and power amplification efficiency is improved.The present invention is applicable to that all audio-source are the sound system of digital signal, comprises category-A, category-B, AB class, D class constant power amplifier etc.
The invention will be further described below in conjunction with drawings and Examples.
Description of drawings
Fig. 1 is the structural principle block diagram of existing sound system.
Fig. 2 is the structural principle block diagram of the embodiment of the invention.
Fig. 3 is the cut-away view of predistortion processor among the embodiment.
Fig. 4 is the flow chart that predistortion processor is carried out pre-distortion among the embodiment to input signal.
Fig. 5 is the flow chart that predistortion processor is upgraded the table of checking among the embodiment.
Fig. 6 is that dimension is received the schematic diagram of model.
Embodiment
As shown in Figure 2, present embodiment comprises D/A processor 12, low pass filter 13, preamplifier 14, power amplifier 15 and loud speaker 16, present embodiment adopts common Class B audio-frequency power amplifier (ClassB), the present embodiment place different with prior art shown in Figure 1 mainly is to be provided with predistortion processor 101 between audio digital signals source 11 and D/A processor 12, the another one difference is also to be provided with a feedback loop that is made of A/D processor 103 and coupler 102 between predistortion processor 101 and power amplifier 15, present embodiment has adopted ceramic condenser as coupler 102, adopt 12 A/D processor, 3.58MHz crystal oscillator is sampled as clock, finishes the conversion to the analog to digital of feedback signal.Described predistortion processor 101 can be carried out pre-distortion to the audio digital signals source in numeric field, it is non-linear that this predistortion table reveals, the memory effect characteristic with by preamplifier 14, nonlinear distortion and memory effect that the audio power amplification system that power amplifier 15 is formed itself is brought are opposite, than original audio signal, audio signal had been the signal of distortion before being input to the audio power amplification system, its distorted characteristic is just in time opposite with the distortion that the audio power amplification system is brought, thereby offset the distortion that the audio power amplification system is brought, improved the linearity and the efficient of audio power amplification system.
The workflow of embodiment is described as follows referring to Fig. 2:
1, audio digital signals source 11 output digital signals are to predistortion processor 101;
2, predistortion processor 101 superposes the characteristic opposite with the power amplifying system distortion on input signal;
3, signal is analog signal through D/A processor 12 by digital translation;
4, filter 13 filtering high fdrequency components obtain audio signal;
5, through preamplifier 14, power amplifier 15, the audio power signal drives loud speaker 16 and sounds;
6, simultaneously, the audio signal of audio power amplification system output is through the audio signal of coupler 102 bypass segments, through A/D processor 103 to predistortion processor 101.
Predistortion processor 101 is cores of above workflow, and present embodiment adopts universal dsp chip to realize the function of predistortion processor.As shown in Figure 3, predistortion processor 101 inside of present embodiment comprise power amplification distortion Mathematical Modeling 206, inspection table 203, address generator 202, digital filter 201, comparator 204 and distortion evaluator 205.The power amplification distortion Mathematical Modeling of present embodiment adopts dimension to receive model (Weiner Model), and its model divides linear band memory section and the non-linear memory section of not being with, as Fig. 6.This model belongs to one of Mathematical Modeling of the classical non-linear band memory system of description, and wherein nonlinear characteristic adopts power series (Power Series) to approach.Digital filter 201 adopts the FIR filter on 32 rank.
The internal signal handling process of predistortion processor 101 comprises two parts: 1, to the flow process 2 of input signal pre-distortion, the flow process that renewal is checked and shown.Be respectively described below.
The flow process of 101 pairs of input signal pre-distortion of predistortion processor is as shown in Figure 4, and is as described below:
1, in predistortion processor 101, sets up and to reflect the non-linear power amplification distortion Mathematical Modeling 206 with the memory effect characteristic of power amplifying system.
2, power amplification distortion Mathematical Modeling 206 produces inspection table 203 thus, and inspection table 203 leaves in the memory of predistortion processor 101.This table has been deposited the relevant value of non-linear and memory effect of reflection amplification system.
3, the parameter of inspection table 203 has determined the filter factor of digital filter 201.
4, predistortion processor 101 received audio signals.
5, according to the amplitude of input signal, address generator 202 produces corresponding address.
6, predistortion processor 101 is according to the address, and in conjunction with the factor of other dimensions such as power amplifier direct current biasing, temperature, inquiry inspection table 203 obtains corresponding pre-distortion parameters.
7, this parameter has determined the coefficient of digital filter 201.
8, input signal is through digital filter 201 filtering, and its amplitude, phase place have obtained pre-distortion.
Inspection table 203 is cores of predistortion processor 101, and it has determined the coefficient of digital filter 201, thereby has determined the characteristic to the distorted signals processing.The nonlinear characteristic of audio power amplification system will be along with factors such as ambient temperature, operating time, input signal power dynamic ranges and is changed, therefore, in order to adapt to the characteristic variations of audio power amplification system, 101 pairs of inspection tables 203 of predistortion processor carry out regular update, are in the optimum linear operating state to guarantee whole sound system.Its basic principle is; Predistortion processor 101 compares the signal and the input signal of the output of audio power amplification system, obtain error signal, this signal is carried out the distortion assessment, and, thus inspection table 203 is adjusted as according to the parameter of power amplification distortion Mathematical Modeling 206 is adjusted.
The flow process that upgrading checks shows is as shown in Figure 5, and is as described below:
1, the output signal of audio power amplification system by feedback loop to predistortion processor 101.Feedback loop is made up of coupler 102, A/D processor 103, and the audio signal of 102 pairs of power amplifier 15 outputs frequently of coupler is carried out bypass, and coupler 102 outputs signal to A/D processor 103.A/D processor 103 is finished analog-to-digital conversion, so that predistortion processor 101 can be handled this signal at numeric field.
2, the comparator 204 of predistortion processor 101 inside compares feedback signal and input signal, obtains error signal.
3, predistortion processor 101 is carried out distortion assessment, relatively its distortion threshold by the distortion evaluator to error signal.
If 4 surpass distortion threshold, predistortion processor 101 is corrected the parameter of power amplification distortion Mathematical Modeling 206.
5, inspection table 203 parameter will correspondingly be adjusted along with the parameter adjustment of power amplification distortion Mathematical Modeling.
After adopting above-mentioned pre-distortion technology, common Class B audio-frequency power amplifier all has clear improvement in harmonic distortion, intermodulation distortion, linear dynamic range.

Claims (7)

1. linear audio power amplifying system with predistortion, comprise D/A processor, filter and audio power amplification system, it is characterized in that: be provided with the predistortion processor that can carry out pre-distortion in numeric field to the audio digital signals source between audio digital signals source and D/A processor, that this predistortion table reveals is non-linear, the memory effect characteristic is opposite with nonlinear distortion and the memory effect that described audio power amplification system itself is brought.
2. linear audio power amplifying system with predistortion according to claim 1, it is characterized in that: described predistortion processor inside mainly comprises power amplification distortion Mathematical Modeling, check and show, address generator, digital filter, described power amplification distortion Mathematical Modeling can reflect the non-linear and memory effect characteristic of power amplifying system, produce the table of checking thus, described inspection table leaves in the memory of predistortion processor, this inspection table has been deposited the relevant value of non-linear and memory effect of reflection power amplifying system, the parameter of inspection table has determined the filter factor of digital filter, described address generator can be according to the amplitude of input audio signal, produce corresponding address, described predistortion processor can be according to the address, in conjunction with the power amplifier direct current biasing, the factor of other dimension such as temperature, inquiry is checked and is shown, obtain corresponding pre-distortion parameters, this parameter has determined the coefficient of digital filter.
3. linear audio power amplifying system with predistortion according to claim 2, it is characterized in that: described predistortion processor inside also comprises comparator and distortion evaluator, between predistortion processor and audio power amplification system, be provided with a feedback loop that constitutes by A/D processor and coupler, described coupler is used for the audio signal bypass to the output of audio power amplification system, output signal to the A/D processor, described A/D processor is used to finish the analog-to-digital conversion to this audio signal, the output digital signal feeds back to predistortion processor, described comparator is used for the signal and the input signal of the output of audio power amplification system are compared, obtain error signal, described distortion evaluator is used for this signal is carried out the distortion assessment, as foundation the parameter of power amplification distortion Mathematical Modeling is adjusted, thus the inspection table is adjusted, set up and correction predistortion model parameter according to this signal, to choose suitable pre-distortion parameters.
4. according to claim 1 or 2 or 3 described linear audio power amplifying system with predistortion, it is characterized in that: described predistortion processor is a functional entity.
5. according to claim 1 or 2 or 3 described linear audio power amplifying system with predistortion, it is characterized in that: described predistortion processor is a physical entity.
6. according to claim 1 or 2 or 3 described linear audio power amplifying system with predistortion, it is characterized in that: described audio power amplification system comprises preamplifier and power amplifier.
7. linear audio power amplifying system with predistortion according to claim 6 is characterized in that: described audio power amplification system also comprises loud speaker.
CN 200510033974 2005-04-01 2005-04-01 Linear audio power amplifying system with predistortion Pending CN1667945A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN 200510033974 CN1667945A (en) 2005-04-01 2005-04-01 Linear audio power amplifying system with predistortion

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN 200510033974 CN1667945A (en) 2005-04-01 2005-04-01 Linear audio power amplifying system with predistortion

Publications (1)

Publication Number Publication Date
CN1667945A true CN1667945A (en) 2005-09-14

Family

ID=35038870

Family Applications (1)

Application Number Title Priority Date Filing Date
CN 200510033974 Pending CN1667945A (en) 2005-04-01 2005-04-01 Linear audio power amplifying system with predistortion

Country Status (1)

Country Link
CN (1) CN1667945A (en)

Cited By (10)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102064772A (en) * 2010-11-18 2011-05-18 瑞声声学科技(深圳)有限公司 Power amplifier device and sound field calibration method thereof
CN102723922A (en) * 2012-06-15 2012-10-10 钜泉光电科技(上海)股份有限公司 Automatic amplitude control loop and control method thereof
CN103873407A (en) * 2012-12-07 2014-06-18 马克西姆综合产品公司 Digital pre-distortion system for radio frequency transmitters with reduced sampling rate in observation loop
CN104510471A (en) * 2013-09-30 2015-04-15 上海联影医疗科技有限公司 Signal processing method, device and magnetic resonance system
CN105100489A (en) * 2015-08-07 2015-11-25 努比亚技术有限公司 Harmonic distortion reducing device and method
CN107852136A (en) * 2015-05-08 2018-03-27 思睿逻辑国际半导体有限公司 Audio-frequency amplifier with predistortion
CN110470935A (en) * 2019-09-18 2019-11-19 广州小鹏汽车科技有限公司 A kind of vehicle load measurement system, measurement method and vehicle
CN111796791A (en) * 2020-06-12 2020-10-20 瑞声科技(新加坡)有限公司 Bass enhancement method, system, electronic device and storage medium
CN112763769A (en) * 2021-04-08 2021-05-07 深圳市鼎阳科技股份有限公司 Signal generator with ultralow harmonic distortion and calibration method thereof
CN114218151A (en) * 2021-12-16 2022-03-22 深圳市领创星通科技有限公司 Digital predistortion correction method and device and computer equipment

Cited By (16)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102064772A (en) * 2010-11-18 2011-05-18 瑞声声学科技(深圳)有限公司 Power amplifier device and sound field calibration method thereof
CN102723922A (en) * 2012-06-15 2012-10-10 钜泉光电科技(上海)股份有限公司 Automatic amplitude control loop and control method thereof
CN102723922B (en) * 2012-06-15 2015-01-21 钜泉光电科技(上海)股份有限公司 Automatic amplitude control loop and control method thereof
CN103873407A (en) * 2012-12-07 2014-06-18 马克西姆综合产品公司 Digital pre-distortion system for radio frequency transmitters with reduced sampling rate in observation loop
CN103873407B (en) * 2012-12-07 2018-07-31 马克西姆综合产品公司 Digital pre-distortion system for having the radiofrequency launcher of the sample rate reduced in observation ring
CN104510471B (en) * 2013-09-30 2018-05-04 上海联影医疗科技有限公司 signal processing method, device and magnetic resonance system
CN104510471A (en) * 2013-09-30 2015-04-15 上海联影医疗科技有限公司 Signal processing method, device and magnetic resonance system
CN107852136A (en) * 2015-05-08 2018-03-27 思睿逻辑国际半导体有限公司 Audio-frequency amplifier with predistortion
CN107852136B (en) * 2015-05-08 2022-07-26 思睿逻辑国际半导体有限公司 Audio amplifier with predistortion
CN105100489B (en) * 2015-08-07 2017-02-15 努比亚技术有限公司 Harmonic distortion reducing device and method
CN105100489A (en) * 2015-08-07 2015-11-25 努比亚技术有限公司 Harmonic distortion reducing device and method
CN110470935A (en) * 2019-09-18 2019-11-19 广州小鹏汽车科技有限公司 A kind of vehicle load measurement system, measurement method and vehicle
CN110470935B (en) * 2019-09-18 2022-03-18 广州小鹏汽车科技有限公司 An in-vehicle measurement system, measurement method and vehicle
CN111796791A (en) * 2020-06-12 2020-10-20 瑞声科技(新加坡)有限公司 Bass enhancement method, system, electronic device and storage medium
CN112763769A (en) * 2021-04-08 2021-05-07 深圳市鼎阳科技股份有限公司 Signal generator with ultralow harmonic distortion and calibration method thereof
CN114218151A (en) * 2021-12-16 2022-03-22 深圳市领创星通科技有限公司 Digital predistortion correction method and device and computer equipment

Similar Documents

Publication Publication Date Title
CN1667945A (en) Linear audio power amplifying system with predistortion
CN1929300A (en) Apparatus and method to control audio volume in D class amplifier
CN1272903C (en) High efficiency linear power amplifier
CN102100001B (en) Switching power amplifier and method of controlling the same
CN1142625C (en) Method of attenuating zero crossing distortion and noise in a amplifier, amplifier and uses of the method and amplifier
CN101390287B (en) Digital PWM amplifier with simulation-based feedback
JP5143748B2 (en) ΔΣ modulation type digital-analog converter, digital signal processing method, and AV apparatus
CN101079602A (en) Automatic input-gain control circuit and method thereof
EP1562286A3 (en) Distortion compensating device and power amplifying device with distortion compensating function
CA2488472A1 (en) Digital memory-based predistortion technique
EP1791390A3 (en) Ultra directional speaker system and signal processing method thereof
EP1865598A3 (en) Input-gain control apparatus and method
ATE341855T1 (en) METHOD AND DEVICE FOR GENERATING A RADIO FREQUENCY SIGNAL
CN1835393A (en) Audio amplifier with high power and high efficiency
KR101060359B1 (en) Pulse Width Modulated Amplifier and Power Signal Elimination Device and Method for Automatic Gain Control
CN103534936B (en) Current buffer
CN1614879A (en) Power amplification circuit
CN1770623A (en) Method for high efficiency audio amplifier
CN1643781A (en) Arrangement for reducing non-linear distortions in an output signal of an amplifier stage
CN204316741U (en) A kind of automatic calibration noise reduction pickup
CN1504062A (en) Improvements in noise cancellation
CN1694354A (en) Method and apparatus for self-oscillating differential feedback class-D amplifier
ATE310335T1 (en) METHOD FOR LINEARIZING A HIGH FREQUENCY AMPLIFIER
CN2746651Y (en) Direct feedback linear power amplifier
Bellili et al. Low power class D audio amplifier with high performance and high efficiency

Legal Events

Date Code Title Description
C06 Publication
PB01 Publication
C02 Deemed withdrawal of patent application after publication (patent law 2001)
WD01 Invention patent application deemed withdrawn after publication