Prepared By: Ian McLeod ~ 'If/Ref: Issue/Revision: Date: PX-TN-51-0756 1/0 JAN.07,2000 Technical Note: ENVISAT ASARData Decoding Project Manager: Tom Henry Prepared By: Ian McLeod

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PX-TN-51-07561/0

JAN. 07, 2000

Technical Note: ENVISAT ASAR Data Decoding

Project Manager: Tom Henry

Prepared By: Ian McLeod

Quality Assurance: Rob McMillin

~ _ 'If/ loo(J~-

Checked By: Peter Meisl

I '-I ,TOMZcoo

Summary: This technical note describes the algorithms for decoding ENVISA TLevel 0 ASAR data. (Contract 13815/99/NL/GD)

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TABLE OF CONTENTS

1 INTRODUCTION .1.1 Purpose of the Document .

1-11-1

1.2 Scope of the Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11.3 Definitions, Acronyms and Abbreviations ..... ,. . . . . . . . . . . . . . . . . . . . . . . . . 1-11.4 Applicable and Reference Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 ·

1.4.1 Applicable Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21.4.2 Reference Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

1.5 Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

2 ASAR DATA ENCODING . 2-12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12.2 Data Encoding Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2.1 Flexible Block Adaptive Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32.2.2 Sign+ Magnitude Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-32.2.3 Full-8 Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

3 ASAR DATA FORMATS . 3-13.1 Instrument Characterisation File. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13.2 ASAR level 0 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

3.2.1 Preparing for Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

4 FBAQ DECODING FOR ASAR . 4-14.1 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14.2 Characteristics of ASAR Signal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24.3 Derivation of FBAQ Thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24.4 Determination of Block Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-44.5 Steps in Basic Block Adaptive Quantisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-54.6 ASAR FBAQ Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-54.7 Steps in Flexible Block Adaptive Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

4.7.1 FBAQ Decoding Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-94.7.2 Unpacking the Encoded Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-94.7.3 Accessing the Decoder LUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

4.8 RMS Equalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-104.8.1 RMS Equalisation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-114.8.2 Effect On Reconstructed SAR Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-124.8.3 FBAQ Decoding Steps When Using RMS Equalisation.............. 4-12

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5 SIGN AND MAGNITUDE COMPRESSION . 5-1


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6 ANALOG TO DIGITAL CONVERTER CORRECTION (FULL-8 DECODING) 6-1

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LIST OF FIGURES

Figure 4-1Figure 6-1

Reconstruction Look-Up Table Layout For FBAQ 8/2 Compression .ADC Output Characteristics .

4-106-1

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Table 2-1Table 3-1Table 3-2Table 3-3Table 4-1Table 4-2Table 4-3Table 4-4Table 5-1Table 6-1

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LIST OF TABLES

Default Compression Type For Each Data Type And Mode .Level 0 Product Structure .Level 0 MDSR Structure .Contents of the FEP Annotations .Constants for Threshold and Reconstruction Levels for 4-bit BAQ .Constants for Threshold and Reconstruction Levels for 3-bit BAQ .Constants for Threshold and Reconstruction Levels for 2-bit BAQ .FBAQ Encoded Data Format .Sign and Magnitude Reconstruction Table .ADC Characterisation Look-Up Table .


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1 INTRODUCTION

1.1 Purpose of the Document

The purpose of this document is to provide a description of how to decode ENVISA TLevel 0 ASAR data. In particular, the decoding steps for the Flexible Block AdaptiveQuantisation (FBAQ) data reduction algorithm are described.

1.2 Scope of the Document

This document is intended for those who wish to understand the processing needed todecode the ASAR Level 0 data products.

1.3 Definitions, Acronyms and Abbreviations

ADC

APM

ASIC

ASAR

ASQNR

BAQ

CRC

Analog to Digital Convertor

Alternating Polarization Mode

Application Specific Integrated Circuit

Advanced Synthetic Aperture Radar

Average Signal to Quantisation Noise Ratio

Block Adaptive Quantisation

Cyclic Redundancy Code

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1-1


ESA

FBAQ

GMM

IM

ISP

LSB

LUT

MJD

MSB

MSE

PF-ASAR

QSC

RMS

SAR

VCDU

WM

WSM

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European Space Agency

Flexible Block Adaptive Quantisation

Global Monitoring Mode

Image Mode

Instrument Source Packet

Least Significant Bit

Look-Up Table

Modified Julian Day

Most Significant Bit

Mean Squared Error

Processing Facility Advanced Synthetic Aperture Radar

Quantizer Selection Code

Root Mean Squared

Synthetic Aperture Radar

Virtual Channel Data Unit

Wave Mode

Wide Swath Mode

1.4 Applicable and Reference Documents

1.4.1 Applicable Documents

Document Title Identifier Intern.Ref.

ENVISAT-1 ASAR Interpretation of PO-TN-MMS-SR-0248 A-1Source Packet Data, Issue B

ENVISAT-1 Products Specifications Issue PO-RS-MDA-GS-2009 A-23, Revision K

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1.4.2 Reference Documents

Document Title Identifier Intern.Ref.

FBAQ Extended Study Technical Report, DC-TN-50-6904 R-1Volume 1, Issue 1, Revision 1

J. Max. Quantizing for Minimum n/a R-2Distortion, IRE Trans. Inform. Theory IT-6,1960, pp. 7-12.

A.K. Jain. Fundamentals of Digital Image n/a R-3Processing, Prentice Hall, 1989.

P. Dubois, "Block Floating Point Quantizer: n/a R-4Performance Study for SIR-C'', JPLInternal Report, Feb. 16, 1990.

R. Kwok, W. Johnson. Block Adaptive n/a R-5Quantization of Magellan SAR Data. IEEETransactions on Geoscience and RemoteSensing, Vol. 27, No. 4, July 1989. pp. 375- 383.

1.5 Document Overview

Section 1contains the introductory material and lists applicable and reference documents.

Section 2 gives an overview of ASAR data encoding.

Section 3 gives a description of the ASAR data formats, which includes the ASAR Level0 data and the look-up tables used for decoding.

Section 4 describes the algorithm for decoding FBAQ-encoded ASAR echo data.

Section 5 describes the Sign + Magnitude decompression algorithm.

Section 6 describes 8-bit Analog to Digital Converter (ADC) correction, which is also thedecoding algorithm applied to "Full-S" bit encoded data.




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2 ASAR DATA ENCODING

2.1 Introduction

The ENVISA T ASAR instrument produces several different types of data and supportsseveral different types of data encoding. As a result, the ASAR Level 0 products, whichcontain the raw instrument source packets received from the satellite, require a certainamount of decoding before the data is in a form suitable for SAR processing. This sectionintroduces the data formats and encoding techniques used on the ASAR instrument, andprovides an overview of the corresponding decoding methods. A detailed description ofthe decoding techniques is provided in later sections.

2.2 Data Encoding Methods

ENVISA T ASAR produces three primary types of data:

1. Echo data, which is the raw SAR echoes received by the instrument

2. Calibration data

3. Noise data

Each of these data are initially quantized using an 8-bit Analog to Digital Converter(ADC). At this point, the data is in "offset binary" format, or "Full-8" bit quantization.Subsequently, the data may then be compressed using one of 2 possible methods: "Sign+ Magnitude" (S+M) encoding, which compresses the data to 4 bits/sample, or "FlexibleBlock Adaptive Quantization" (FBAQ) encoding, which can compress the data to 4, 3, or2 bits/sample.

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The choice of compression method depends on the type of data. FBAQ is used tocompress echo data, while S+M encoding is used for noise data. Calibration data is notcompressed, and remains in "Full-S" format. In addition, the level of FBAQ compressionused on the echo data depends upon which ASARmode was used to gather the echo data.Table Table 2-1 lists the types of ASAR data and their current compression method(subject to change by the European Space Agency).

Table 2-1 Default Compression Type For Each Data Type And Mode

Acquisition Mode Data Type Default Compression Type

Echo 4-bitFBAQImage Mode Noise S+M

Calibration Full-8

Echo 4-bit FBAQAlternating Noise S+M

Polarization ModeCalibration Full-8

Echo 4-bit FBAQWide Swath Mode Noise S+M

Calibration Full-8

Echo 4-bit FBAQGlobal Monitoring Noise S+M

ModeCalibration Full-8

Echo 2-bit FBAQWave Mode Noise S+M

Calibration Full-8

A brief description of each of the different data encoding techniques is provided below.A more thorough description of each is presented later in this document.


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2.2.1

2.2.2

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Flexible Block Adaptive Quantization

FBAQ compresses the raw SAR echo data from 8 bits/sample to 4, 3, or 2 bits/sample,generally abbreviated as 8/4, 8/3, and 8/2 FBAQ. FBAQ is a "lossy" compressionalgorithm, meaning the algorithm introduces distortion into the data in the form of"quantization noise". The level of the distortion is proportional to the compression ratio,meaning a 8/2 FBAQ compression introduces more distortion than 8/4 FBAQcompression. For this reason, the level of FBAQ encoding used by the ASAR instrumentdepends on the imaging mode, and has been chosen based on a trade-off betweenavailable bandwidth and allowable distortion ..The current FBAQ compression settings vs.ASAR mode are shown in Table 2-1. Once encoded, the compressed samples are"packed," and down-linked to the ground.

On the ground, the FBAQ data is unpacked and decompressed. Decompression of FBAQencoded data is also accomplished via the use of LUTs. In this case the LUTs map theFBAQ codewords to corresponding normalized 4-byte floating point values. ADCcorrection is included in the LUTs, so the output of the FBAQ decoding is ADC corrected.As for the other decoding methods, there is one LUT for the I-channel and one for the Q-channel, since different ADCs are used for each channel.

Note that in some circumstances an "RMS-Equalization" option may be chosen fordecompressing the FBAQ data. Selection of this option introduces several changes in thedecoding procedure which are described in detail in later sections of this document.FBAQ decoding is described in Section 4.

Sign + Magnitude Encoding

Sign + Magnitude encoding is a reduction of the 8-bit data to 4 bits/sample, whilemaintaining the sign of the data. Essentially this results in a 4-bit sample consisting of thesign bit, plus the 3 least significant bits of the 8-bit sample. These 4-bit samples are"packed" together (2 samples per byte) prior to downlink.

To decode S+M data the data is first unpacked, then LUTs are used that map the S+Mcodewords to normalized 4-byte floating point values. ADC correction is included in theLUTs, so the output of the S+M decoding is ADC corrected. There is one LUT for the l-channel and one for the Q-channel, since different ADCs are used for each channel.

The Sign+ Magnitude encoding/decoding is described in greater detail in Section 5 of thisdocument.




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2.2.3 Full-8 Data

Full-8 data is 8 bits/sample data which is in the raw format received from the ASAR ADC.Since it is not compressed, no decoding algorithm is needed to decompress the data on theground. However, the Full-8 data is still subject to some processing on the ground sinceit must be corrected for known non-linearities in the ADC converter. This ADC correctionis done on the ground via Look-Up Tables (LUTs) which map each raw 8-bit binary offsetvalue to an ADC corrected normalized 4-byte floating point value. There is one LUT forthe I-channel and one for the Q-channel, since different ADCs are used for each channel.

The Full-8 encoding/decoding is described in greater detail in Section 6 of this document.



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3 ASAR DATA FORMATS

Two inputs are required to decode ASAR Level 0 data: the Instrument CharacterisationFile, and the Level 0 product itself. The following sections describe the format of bothinputs, specific to ENVISA T ASAR.

Note that this document is not the applicable document for ASAR data formats and thedescriptions provided below are for information purposes only. The applicable documentfor ASAR level 0 format and Instrument Characterization File format is the ENVISA TProduct Specifications, Volume 6 and Volume 8 respectively [Document A-2]. Theapplicable document for ASAR source packet format is ENVISAT-1 ASARInterpretation of Source Packet Data [Document A-1 ].

3.1 Instrument Characterisation File

The Instrument Characterization File is a binary file which contains data which describesthe instrument, plus certain Look-Up Tables needed to decode the ASAR data. The exactformat of the Instrument Characterization File is described in document A-2 and is notrepeated here. Rather, this section highlights the fields contained in the InstrumentCharacterization File needed for decoding ASAR Level 0 data.

These fields are:

l. The ADC characterisation table

2. The reconstruction table for "Full-S" quantization

3. The reconstruction table for FBAQ decoding (includes ADC correction)

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4. The reconstruction table for "FBAQ (no ADC)" decoding (used during the RMSEqualization option for FBAQ decoding).

5. The reconstruction table for "Sign + Magnitude" quantization

The following list describes the format of the LUTs in the Instrument CharacterizationFile (taken from Document A-2, Volume 8):

• Look Up Tables for ADC Characterization (one for I channel, one for Qchannel): Contains 255 normalized amplitude levels corresponding to voltagethresholds. First value in LUT is for -127, last value is for+ 127.Format as is givenin Document A-1.

• Reconstruction Look Up Tables for Full 8-bit Quantization (one for I channel,one for Q channel): Contains normalized amplitude levels corresponding tosample codewords. First value in LUT is for codeword 0, last value is for +255(binary offset format). Table values account for ADC correction. Format as is givenin Document A-1.

• Reconstruction Look Up Tables for FBAQ 4-bit Quantization (one for Ichannel, one for Q channel): Gives 4096 normalized amplitude reconstructionlevels which include ADC correction. Format as is given in Document A-1. Tablevalues account for ADC correction.



• Reconstruction Look Up Table for FBAQ 4-bit Quantization (no ADC): This isthe FBAQ reconstruction LUT which does not have ADC correction incorporatedinto it. It gives 4096 reconstruction levels which decodes FBAQ codewords tofloating point values on the 8-bit range (-127 to +127).This table is used for RMSEqualization. The format of this table is identical to that of the FBAQ 4-bitReconstruction LUT for the I channel, as given in Document A-1.

• Reconstruction Look Up Table for FBAQ 3-bit Quantization (no ADC): This isthe FBAQ reconstruction LUT which does not have ADC correction incorporatedinto it. It gives 2048 reconstruction levels which decodes FBAQ codewords tofloating point values on the 8-bit range (-127 to +127).This table is used for RMS



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Equalization. The format of this table is identical to that of the FBAQ 3-bitReconstruction LUT for the I channel, as given in Document A-1.

• Reconstruction Look Up Table for FBAQ 2-bit Quantization (No ADC): Thisis the FBAQ reconstruction LUTwhich does not have ADC correction incorporatedinto it. It gives 1024 reconstruction levels which decodes FBAQ codewords tofloating point values on the 8-bit range (-127 to +127).This table is used for RMSEqualization. The format of this table is identical to that of the FBAQ 2-bitReconstruction LUT for the I channel, as given in Document A-1.

• Reconstruction Look Up Table for Sign+ Magnitude Quantization (one or Ichannel, one for Q channel): Contains normalized amplitude reconstruction levelscorresponding to sample codewords. First value in LUT is for threshold -8, lastvalue is for threshold+ 7. Format as is given in Document A-1. Table values accountfor ADC correction.

More detailed information regarding the table formats is provided in later sections.

3.2 ASAR level Odata

ASAR level 0 data is described in the ENVISAT Product Specifications, Volume 6[Document A-2]; the details of the source packet data are described in the ENVISAT-1ASAR Interpretation of Source Packet Data [Document A-1].

The information Table 3-1 shows a general summary of a level 0 product as taken fromA-2:

Table 3-1 Level 0 Product Structure

Main Product Header (MPH) - 1247 bytes

Specific Product Header (SPH) - 1956 bytes

Measurement Data Set (MDS) (variable size)

MDSR#l

....

MDSR #N (last MDSR)




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JAN. 07, 2000 IllMACDONALDDEITWILERThe product name in the Main Product Header (MPH) contains information to determinethe acquisition mode (e.g., image, wide swath, wave).

The Specific Product Header (SPH) contains the total size of the Measurement Data Set(MDS) in bytes.

Inside the MDS, the actual data is stored in the Measurement Data Set Records (MDSRs),which consists of two parts: the annotations and the Instrument Source Packet (ISP), asshown in Table 3-2.

The annotations, in tum, consist of Front End Processor (FEP) annotations, and Level 0Processor Annotations.

The ISP length field of the FEP annotations (see Table 3-3) is a particularly useful field,and it should be used to determine where the next source packet is, as opposed to thesource packet length in the source packet itself. This is because the FEP annotation ismore likely to be correct, since it is determined on ground after the source packets arereconstructed from the downlink, whereas the source packet itself may contain bit errorsthat occurred during downlink.

Table 3-2 Level 0 MDSR Structure

Annotations Instrument Source Packet

LevelOProcessor FEP Annotations Packet Header Packet Data Field

Annotations

FEP QualityPacketISP Sensing Data and Packet Packet Data Field Source

Time (MJD) Reception Time Identification Sequence Length Header DataStamp Control

12bytes 20 bytes 2 bytes 2 bytes 2 bytes 15bytes Varies

Table 3-3 Contents of the FEP Annotations

FEP Annotations

MJD 2000 Length of ISP No. ofVCDUs in No. ofVDCUs in the ISP forTime Stamp (length of ISPminus ISP which contain which a Reed-Solomon error Spare

6 byte header - I) a CRC error correction was performed

12bytes 2 bytes 2 bytes 2 bytes 2 bytes



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The ISP itself also has a header which contains information on the source packet. Theexact format is provided in Document A-1. The following information is used duringpacket decoding:

1. packet length (note: the packet length value in the FEP header should be usedinstead, as it is more· reliable)

2. antenna beam set number (depends on the mode and the swath)

3. compression ratio (8/2, 8/3, or 8/4 -- only valid if the source packet contains echodata)

4. echo flag (bit field which indicates the presence of echo data in this source packet)

5. window length (in samples)

6. resampling factor (used to determine the window length in bytes)

For detailed instructions on how these fields are interpreted, refer to A-1.

Preparing for Decoding

Each source packet may contain more than one sampling window (equivalent to a rangeline of data), depending on the mode in which the data was acquired. Each samplingwindow should be decoded separately. The size of each sampling window in bytes can bedetermined from the acquisition mode, compression ratio, sampling window length (insamples), and the resampling factor. The formula for determining this is given inDocument A-1, in Annex D.

The following sections, which describe the decoding algorithms, assume that the inputdata received by the algorithm is that of one sampling window.




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4 FBAQ DECODING FOR ASAR

The most complex type of decoding performed for ASAR data is FBAQ decoding. Thissection provides some background information describing the algorithm itself, plusdetailed instructions for decompressing the FBAQ data.

4.1 Algorithm Overview

The Block Adaptive Quantization (BAQ) algorithm is based on the principles ofminimum mean-squared error quantization first put forward by J. Max in 1960 (seeDocument R-2). The theory can be summarised by two important points:

1. The "optimal" quantiser for a given input signal is chosen as the quantiser whichminimizes the mean squared error between the original signal and the quantisedsignal.

2. In order to achieve this minimum mean squared error condition, the thresholds andreconstruction levels of the quantiser must be optimised to match the probabilitydistribution function of the data.

While the choice of minimizing the mean squared error is a somewhat arbitrary criterionfor "optimal" quantisation, the importance of adapting the threshold levels of thequantiser to the characteristics of the input data is noteworthy. The implication of thisstatement is that the traditional uniform quantiser (a quantiser with evenly spacedthresholds across the range of the input signal) is only the "optimal" quantiser when theinput signal is characterised by a uniform distribution. If the input signal is not uniformlydistributed, the thresholds used to quantise the signal could be positioned differently inorder to reduce the mean squared error. Therefore, BAQ works by re-quantising the SAR



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data to fewer bits per sample, using thresholds and reconstruction levels which have beenchosen such that they minimize the mean square error of the resulting quantisation noise.

For ASAR, Flexible BAQ (FBAQ) is used, which allows re-quantization of the 8-bits persample input data to 4, 3 or 2 bits per sample.

Note that this form of data quantisation is lossy data compression. Distortion is introducedinto the data in the form of quantisation noise. The goal of the BAQ algorithm is tominimize the mean squared error of the quantisation noise.

4.2 Characteristics of ASAR Signal Data

In order to adapt the BAQ thresholds and reconstruction levels to the ASAR data, thestatistical characteristic of the data must be understood. ASAR signal data (before BAQencoding) is characterized by the following properties:

1. Complex data at baseband (8-bit I, 8-bit Q)

2. Zero-mean circular Gaussian distribution

3. Slowly changing variance in both range and azimuth

4. Zero I, Q sample correlation

5. Low inter-sample correlation

6. Data spectrum is relatively constant over both range and azimuth

The second property (data follows a Gaussian distribution) implies that non-uniformthresholds are required. The third property (slowly changing variance) allows the usageof blocks to track the data statistics. The fourth and fifth properties imply that I and Qsamples are treated equally when computing block statistics (i.e. separate statistics are notrequired for each of the I and Q components of the data).

4.3 Derivation of FBAQ Thresholds

Given that the SAR input signal is zero-mean and Gaussian distributed, the position of theoptimal thresholds for any level of quantization can be determined (for a mathematicaltreatment refer to Documents R-2 or R-3). Since a Gaussian distribution can becompletely characterized by two parameters, the mean and variance, it is not surprisingthat the threshold positions are also dependent on these two values. In fact, each threshold



,,..·

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is calculated as Tn = (Cn · o) +µ,where en is a constant associated with the nmthreshold, and o is the standard deviation of the distribution (which is equal to the squareroot of the variance) and µ is the mean of the distribution. The reconstruction levels usedduring the data decompression are determined in a similar manner, withRn = (Dn -o) +µ,where D; is a constant associated with the n'"reconstruction level. Thevalues of en and D, for BAQ are tabulated in the tables below, as taken from R-3. Thethreshold and reconstruction levels are symmetrical about the data mean (assumed to bezero) so there is a positive set of thresholds/reconstruction levels, and a negative set ofthresholds/reconstruction levels (found using -D, and -Cn).

Table 4-1 Constants for Threshold and Reconstruction Levels for 4-bit BAQ

n en Dn

0 0 0.1284

1 0.2583 0.3882

2 0.5226 0.6569

3 0.7998 0.9426

4 1.0995 1.2565

5 1.4374 1.6183

6 1.8338 2.0693

7 2.4011 2.7328


n en Dn

0 0 0.2451

1 0.5006 0.7561

2 1.0500 1.3440

3 1.7480 2.1520




',IIMACDONALDDETIWILER


n en Dn0 0 0.4528

1 0.9816 1.5104

Thus, the thresholds and reconstruction levels for optimal quantisation are determineduniquely from the standard deviation and mean of the input SAR data.

4.4 Determination of Block Size

Point 3 in Section 4.2 indicates that the standard deviation of the ASAR data is notstationary. Therefore, in order to ensure good performance of the algorithm, the data setis sub-divided into smaller blocks. Within each block, it is assumed that the statistics ofthe data are stationary, and thus the optimal thresholds can be determined by estimatingthe standard deviation for the block. The thresholds are then re-calculated for each blockof data. Thus, the algorithm can adapt to changes in the standard deviation over the fullSAR data set.

The choice of block size is a trade-off: A smaller block allows the algorithm to adapt morequickly to changes in signal standard deviation. However, there is an encoding overheadto each block (to be explained later) so the use of smaller blocks reduces the achievablecompression ratio of the algorithm. The use of larger blocks results in a highercompression ratio, and a more statistically significant estimate of standard deviation, butreduces the ability of the algorithm to adapt to changes in the standard deviation of thedata. Therefore the goal is to choose a block size that is as large as possible, while stillresponsive to changes in the SAR data. Hardware and throughput issues must also beconsidered when selecting a block size.

For ENVISAT ASAR, an experimental study involving simulated ENVISAT data wasperformed to determine the block sizes. They block sizes which were eventually selectedare:

• 2-bit FBAQ: 126complex samples from one range line

• 3-bit FBAQ: 84 complex samples from one range line

• 4-bit FBAQ: 63 complex samples from one range line



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4.5 Steps in Basic Block Adaptive Quantisation

Based on the above discussion, the basic BAQ algorithm is summarised below.

Encoding is performed via the following steps:

1. Divide the data set into blocks

2. For each block:

a) Estimate the standard deviation of the samples in the block

b) From the estimate of the standard deviation, determine the "optimal"thresholds (assuming zero mean)

c) Quantise the data using the threshold values and pack them

d) Write the value of the standard deviation to the front of the block. This iscalled the "block ID"

Decoding of the data is performed as follows:

1. For each block:

a) Read the standard deviation of the samples from the "block ID"

b) From the estimate of the standard deviation, determine the "optimal"reconstruction levels

c) Unpack the BAQ codewords

d) Reconstruct the quantised data using the reconstruction levels

4.6 ASAR FBAQ Implementation

The description of the BAQ algorithm above is "theoretical" description. For a practical,real-time implementation of the BAQ algorithm several modifications must beintroduced. For the ENVISA T FBAQ implementation of the algorithm, several steps havebeen taken to increase the throughput and compression ratio beyond what is achievableusing the "basic" BAQ algorithm described previously. These steps come at the cost of adecrease in the accuracy of the algorithm (i.e. an increase in mean squared error). Thesesteps include:

1. Use mean absolute value. Instead of calculating the standard deviation, the meanabsolute value is calculated (in actual fact it is the truncated absolute which is

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Ref: PX-TN-51-0756Issue/Revision: 1/0Date: JAN.07, 2000 IIMACDONALDDETIWILER

calculated for ENVISAT as explained below, but this is simply an extension of themean absolute theory presented here). The advantage of this is a throughputincrease since the calculation of standard deviation requires the use of the squareroot operation, which can be slow to compute. It can be shown (see R-5) that for an8-bit input signal the mean absolute value of a block can be mapped to the blockstandard deviation using the equation:

(4-1)

where erf() is the error function and a is the signal standard deviation. The resultcan be extended to other levels of input quantization easily.

Therefore, by inverting equation (4-1) the value of a can be determined for acalculated mean absolute value. Due to the overhead of computing the errorfunction however, a throughput savings is only realized if this procedure iscombined with the technique described in point (2) below (use of a LUT). In sucha case, the LUT of pre-computed threshold values can be indexed directly bydiscrete values of mean absolute value rather than discrete values of standarddeviation.

2. Use Look-up Tables. Instead of using floating point values to determinethresholds, the allowable values of mean absolute were quantized. Due to on-boardmemory constraints, 256 possible values were chosen for 4-bit encoding, and 64values for 3-bit and 2-bit encoding. Therefore the computed mean absolute value isrounded to the nearest allowable value. This procedure has 2 major advantages:

a) The threshold values at each allowable value can be pre-computed and storedin a look-up table (LUT). This results in a dramatic increase in throughput.The reconstruction values for each set of thresholds can also be pre-computedand stored in a LUT which increases the throughput of the decoder.

b) The size of the block ID is reduced to 8 bits (for the 256 entry LUT). If themean absolute value was stored as a floating point value, 32 bits would beneeded. In addition to improving throughput, this reduces the overhead ofeach block, and hence increases the compression ratio.

3. Use integer arithmetic.Another modification used on the ENVISAT encoder is touse the truncated absolute sum rather than the actual mean absolute in order to indexthe LUTs. This can be accomplished without loss of accuracy since the block sizesare fixed. The main advantage of this approach is that it allows the complete FBAQencoding to be done using integer arithmetic, which is a significant advantage sincethe ENVISAT FBAQ encoding algorithm is implemented on a single ASIC chip.



MACDONALDDETIWILER


Therefore for each compression ratio, the absolute values of the I and Q samples fora block are summed. For 4-bit compression, the 8 Most Significant Bits (MSBs) ofthe sum are used to form the "Quantiser Selection Code (QSC) which is used toindex to the 4-bit encoding LUT. For 3-bit and 2-bit FBAQ the 6 MSBs of the sumare used to form the QSC for the 3-bit or 2-bit encoding LUT. It is important to notehowever, that in all cases, the 8 MSBs of the sum are stored to the block ID.Therefore, for 4-bit encoding the block-ID is equal to the QSC, but for 3-bit and 2-bit encoding the block-ID is actually has 2-bits greater accuracy than the QSCwhich was used to do the encoding. This fact is exploited during decompression of3-bit and 2-bit FBAQ, since the reconstruction LUTs can be made larger than theencoding LUTs.

4. Simplify Partial Block Handling. Since the block size is fixed, but the range linesize may not be, the possibility exists that the last block in a range line will be apartial block, and will thus have less samples in it than the other blocks. In such acase the FBAQ algorithm does not compute the absolute sum, but instead simplyuses the QSC from the previous block to index the encoder LUT, and thus uses theblock ID from the previous block as the block ID for the partial block.

4.7 Steps in Flexible Block Adaptive Quantisation

Based on the above discussion, the FBAQ algorithm steps are summarised below

Encoding is performed via the following steps:

1. For each range line:

a) For each block:

1. If full block:

a) Sum the absolute values of the samples

b) Use either the 8 MSBs or 6 MSBs of the sum (depending on thecompression level) to index the encoder LUT containing the pre-computed threshold values

c) Write the 8 MSBs of the sum to the front of the output block. Thisis called the "block ID"

d) Quantise the data using the threshold values and pack them afterthe block ID

2. Else, if a partial block:

a) Use the QSC from the previous block to access the encoder LUT




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JAN. 07, 2000 IIMACDONALDDEITWILERb) Write the block ID from the previous block as the block ID of this

block

c) Quantise the data using the threshold values and pack them afterthe block ID

3. Once all blocks are encoded and packed, zero-pad the encoded rangeline out to the nearest 16-bitword boundary (i.e. make sure total numberof bytes in the encoded range line is even).

Therefore, once encoding is complete, the encoded SAR echo data for one range line willbe in the format shown in the table below.

Table 4-4 FBAQ Encoded Data Format

8-bit block ID of first block

63 bytes of packed samples, I,Q,I,Q etc."

8-bit block ID of second block

63 bytes of packed samples, I,Q,I,Q etc.

........

8-bit block ID of last block (may be a partial block)

X bytes of packed samples, l,Q,1,Q etc. where X

4.7.1

4.7.2

\I

4.7.3

NIACDONAlDDETIWILER


3. Unpack the FBAQ codewords

4. Reconstruct the quantised data using the reconstruction levels

FBAQ Decoding Details

Based on the FBAQ algorithm description above, the steps needed to decode the data areexplained in greater detail in the following sections.

Unpacking the Encoded Data

Each byte following the block ID contains several n-bit codewords (I value followed byQ value); the value of n is 4, 3, or 2 depending on the code type. These codewords mustbe unpacked into 8-bit format. This can be accomplished in various ways; for example, abit mask (i.e., 1111000 and 00001111 for 4-bit encoding)) and the binary AND operationcould be used.

Accessing the Decoder LUT

The decoder LUT is indexed by the sample code word and the block ID; each sample codeword corresponds to a different reconstruction level. There are 16reconstruction levelsfor 4-bit encoding, 8 reconstruction levels for 3-bit encoding, and 4 reconstruction levelsfor 2-bit encoding. At each reconstruction level, there are 256 reconstruction values storedas 32-bit floating point values, with each value corresponding to a different block ID.Note that there are 256 values for 3-bit and 2-bit decoding even though the QSC for thesemodes was only 6-bits. This is because we can use the full 8'-bitaccuracy of the block IDduring decoding. As an example, the table format for 8/2 compression is illustrated inFigure 4-1 below.

There is a separate reconstruction look-up table for each FBAQ compression ratio. Thetables which also account for ADC correction have separate tables for each of the I and Qchannels since the two channels use independent ADCs. Each table contains normalized

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4-byte floating point reconstruction values for then levels of 256 quantizers (n=4 for 8/2,n=8 for 8/3, and n=16 for 8/4).

quantiser level (block ID)

0 1 254 255

11

sample ) 10 1---1--~codeword l or

10

Figure 4-1 Reconstruction Look-Up Table Layout For FBAQ 8/2 Compression

The layout for the 8/4 and 8/3 reconstruction look-up tables have a similar format. Thedifference is that the sample codeword ranges from 111 to 110 for 8/3, and from 1111 to1110 for 8/4 encoding.

For the applicable description of the FBAQ table layout refer to Document A-2.

4.8 RMS Equalisation

The algorithm description provided in Section 4.7 is for the basic FBAQ algorithm, whichis used in the majority of the cases. However, a second option, called Root Mean Squared(RMS) Equalization is possible when decoding FBAQ data. Note this option has noimpact on the encoding of the data, it is strictly a decoding method.

RMS Equalization is a method used to compensate for one of the fundamental propertiesof a minimum mean-squared error quantiser. This property is that some energy is lost dueto the encoding/decoding process. That is, the RMS value of a block of data after FBAQencoding and decoding is less than the RMS value of the original block of data. Theenergy loss is small for 4-bit encoding, but significant for 2-bit encoding, which is usedfor Wave Mode (see Document R-1). Therefore, it may be desirable in some cases tosacrifice somemean squared error performance in order to improve the energy conservingperformance of FBAQ by scaling the output pixel values so that the input and outputquantized block RMS values are equal (or close to equal). This procedure is referred to asRMS Equalisation.



4.8.1

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Ref: PX-TN-51-0756-Issue/Revision: 1/0Date: JAN. 07, 2000

Thus, the user has the option of doing FBAQ decoding under the constraint that the meansquare error is minimized (normal FBAQ), or under the constraint that the input andoutput block RMS (and hence, energies) values are equalised (RMS-Equalisation FBAQ).Each of these cases may be appropriate for different applications; it is up to the user todecide what type of FBAQ algorithm is best for his/her particular application.

The following sections describe the RMS Equalization procedure, and its effects on thereconstructed data.

RMS Equalisation Procedure

The difficulty in doing the RMS Equalization procedure is that the input block mean-absolute value (in the form of the truncated absolute sum) is transferred in the encodeddata to the decoder instead of the input quantized block RMS; consequently, the inputquantized block RMS value must be derived so that the scaling can be done in the decoder.The RMS equalization procedure is as follows:

1. Calculate the input continuous block RMS from the input block ID value. The moststraightforward approach is to simply back-out the standard deviation value used tocreate the reconstruction LUT for the given block ID using the relationR = (D · o) + µ described in Section 4.3. Using the block ID, one can extract an Tlvalue of R11 from the reconstruction LUT for an arbitrary value of n. Since Dn is aknown constant, and µ= 0, then

R,,()c1111f = D n (4-2)

2. Calculate the input quantised block RMS from the input continuous block RMS:

3. Calculate the output block RMS in the decoder, after the data is reconstructed:

o _ I ; ( 2 2)q ua n t jout - _/2N X L.J f; +Qi (4-4)

i = 0



Ref: PX-TN-51-0756Issue/Revision: 1/0Date: JAN. 07, 2000 IIMACDONALDDETIWILER

4. Create a set of block scaling factors equal to the derived input quantised block RMSdivided by the output block RMS (i.e. CJquant_in I C5quant_ou1).

5. Apply these scaling factors in the decoder to the reconstructed complex samples togive a new output data set that has much closer block RMS values to the original 8-bit data before FBAQ.

The accuracy of this RMS equalisation procedure is dependent on the accuracy of therelationships relating the various quantised and continuous SAR data statistics, sincethese relationships are used to calculate the scaling factors used in the procedure. Thesuccess of these equations is in turn dependent on how close the assumptions used toderive these equations correspond to reality (i.e. how close the SAR data within a blockis to a zero mean Gaussian distribution).

4.8.2 Effect On Reconstructed SAR Data

It was found (see Document R-1) that FBAQ decoder RMS Equalization has thefollowing effects on reconstructed SAR data:

• raise the mean square error slightly

• lower the Average Signal to Quantisation Noise Ratio (ASQNR) slightly

• raise the absolute peak error significantly 1

Note that the first effect is expected because FBAQ without the RMS equalisation is aminimum mean-square error algorithm; if the values are scaled, the modified FBAQalgorithm will no longer be optimal in this sense.

4.8.3 FBAQ Decoding Steps When Using RMS Equalisation

The procedure for decoding data with RMS equalization is very similar to the procedurediscussed in Section 4.7, except that a different set of LUTs is used, an additional step forRMS equalization is introduced, and ADC correction must be performed explicitly at theend of the decoding, rather than as a combined step included in the FBAQ LUTs. Theresult is usually a large throughput decrease for RMS Equalization as compared to regularFBAQ decoding. The algorithm steps are:

1. For each range line of data

I. The absolute peak error is not viewed as being a very reliable statistic for judging the performance of FBAQsince it only represents the reproduction of one pixel magnitude.



I....,

MACDONALDDEITWILER


a) For each encoded block:

1. Read the block ID from the first byte of the block

2. Use the block ID to index the FBAQ table which do not have ADCcompensation included. This table provides optimal reconstructionlevels in the 8-bit range (i.e. from -127 to +128)

3. Unpack the samples

4. Decode the samples using the reconstruction levels

5. Calculate the output block RMS in the decoder, after the data isreconstructed (see equation (4-4)) to yield


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MACDONALDDETIWILER

Use. duplication. or disclosure of this document or any of the informationcontained herein is subject to the restrictions on the title page of this document.

MACDONALDDETIWILER


5 SIGN AND MAGNITUDE COMPRESSION

Typically, only noise data will be compressed in sign and magnitude (fixed exponent)format. Each 8-bit sample from the ADC comes in offset binary format (see Section 6).Given an 8-bit sample R7R6R5R4R3R2R1Ro, the 4-bit sample C3C2C1C0 will be compressedin the following manner (where inverse() is a bit-wise inversion operator; e.g. inverse(Ol)= 10):

1. Convert to 8-bit sign and magnitude (S7S6S5S4S3S2S1S0):

S7 = inverserk-)if (R7 == 1) S6S5S4S3S2S1S0 = R6R5R4R3R2R1Roelse S6S5S4S3S2S1S0 = inverse(~R5R4R3R2R1Ro)

2. Truncate to 4-bit sign and magnitude

C3 = S7if (S6S5S4S3 == 0000) C2C1C0 = S2S1S0else C2C1C0 = 111

To decode sign and magnitude compressed data, the Reconstruction LUT for Sign+Magnitude Quantization from the Instrument Characterization File should be used. As forFBAQ decoding, there is one table for the I channel and one table for the Q channel. Notethat ADC correction is built into the table, so once the decoding is complete no furthercorrection need be applied. The table is simply indexed by the S+M codeword to yield acorresponding 4-byte floating point value as illustrated in Table 5-1.




5-2

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Table 5-1 Sign and Magnitude Reconstruction Tablea

Codeword Reconstructed Valueb

1111 -0.059

1110 -0.051

...

1000 -0.004

0000 0.004

...

0110 0.051

0111 0.059

a. Two tables are present in the Instrument Characteriza-tion File, one for the I channel and one for the Q channel

b. Reconstruction values are for illustration purposesonly and do not represent true reconstruction values


MACDONALDDETIWILER


6 ANALOG TO DIGITAL CONVERTER CORRECTION(FULL-8 DECODING)

Calibration data is typically not compressed, and the received data is exactly what comesfrom the satellite. This format is referred to as Full-8 encoding, or offset binary format.The 8-bit values range from 0 to 255. ADC quantization is illustrated in Figure 6-1.

Codeword111111111111111011111101

000000100000000100000000

1000000001111111input voltage

Figure 6-1 ADC Output Characteristics

While not compressed, the Full-8 data must be corrected for known non-linearities in theADC. This correction is done using the LUT for ADC Characterisation from theInstrument Characterisation File. Since the I and Q channels use different ADCs, there isa different LUT for each channel. The LUT is simply indexed using the 8-bit binary offset




MACDONALDDETIWILER

values to yield a corresponding 4-byte floating point value. The following table illustratesthe format and contents of the LUT.

a. There is one table for the I channel and one for the Qchannel. Table format is the same.

Table 6-1 ADC Characterisation Look-Up Tablea

Codeword Reconstructed Valueb

00000000 -1.004

00000001 -0.996

...

01111111 -0.004

10000000 0.004

...

11111110 0.996

11111111 1.004

b. Reconstruction values listed are for illustration only.Actual values may differ.



Page 1TitlesTechnical Note: ENVISAT ASAR Data Decoding , ~ _ 'If/ loo(J ~- I '-I ,TOM Zcoo MACDONALD

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Page 3TitlesMACDONALD CHANGE RECORD Ref: PX-TN-51-0756 Issue/Revision: 1/0 Date: JAN. 07, 2000 ISSUE DATE 1/0 Dec. 07, 1999 PAGE(S) DESCRIPTION All SCR #46, CR #46 (iii)


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Page 5TitlesMACDONALD TABLE OF CONTENTS Ref: PX-TN-51-0756 Issue/Revision: 1/0 Date: JAN. 07, 2000 1 INTRODUCTION . 1.1 Purpose of the Document . 1-1 1.2 Scope of the Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1 1.3 Definitions, Acronyms and Abbreviations ..... ,. . . . . . . . . . . . . . . . . . . . . . . . . 1-1 1.4 Applicable and Reference Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 · 1.4.1 Applicable Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 1.4.2 Reference Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 1.5 Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 2 ASAR DATA ENCODING . 2-1 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2 Data Encoding Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2.1 Flexible Block Adaptive Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 2.2.2 Sign+ Magnitude Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 2.2.3 Full-8 Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 3 ASAR DATA FORMATS . 3-1 3.1 Instrument Characterisation File. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.2 ASAR level 0 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3 3.2.1 Preparing for Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 4 FBAQ DECODING FOR ASAR . 4-1 4.1 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 4.2 Characteristics of ASAR Signal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 4.3 Derivation of FBAQ Thresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 4.4 Determination of Block Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 4.5 Steps in Basic Block Adaptive Quantisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 4.6 ASAR FBAQ Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 4.7 Steps in Flexible Block Adaptive Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 4. 7 .1 FBAQ Decoding Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 4.7.2 Unpacking the Encoded Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 4.7.3 Accessing the Decoder LUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 4.8 RMS Equalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 4.8.1 RMS Equalisation Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 4.8.2 Effect On Reconstructed SAR Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 4.8.3 FBAQ Decoding Steps When Using RMS Equalisation.............. 4-12

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Page 7TitlesMACDONALD LIST OF FIGURES Ref: PX-TN-51-0756 Issue/Revision: 1/0 Date: JAN. 07, 2000 Figure 4-1 Reconstruction Look-Up Table Layout For FBAQ 8/2 Compression . ADC Output Characteristics . 4-10 (vii)

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Page 9TitlesMACDONALD LIST OF TABLES

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Page 11TitlesMACDONALD 1 INTRODUCTION 1.1 Purpose of the Document 1.2 Scope of the Document 1.3 Definitions, Acronyms and Abbreviations

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Page 12TitlesMACDONALD 1.4 1.4.1 Applicable and Reference Documents Applicable Documents


TablesTable 1

Page 13Titles1.4.2 MACDONALD Reference Documents 1.5 Document Overview

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Page 15TitlesMACDONALD 2 ASAR DATA ENCODING 2.1 Introduction 2.2 Data Encoding Methods

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Page 17Titles2.2.1 MACDONALD Flexible Block Adaptive Quantization 2.2.2 Sign + Magnitude Encoding

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Page 19TitlesMACDONALD 3 ASAR DATA FORMATS 3.1 Instrument Characterisation File

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Page 21TitlesMACDONALD 3.2 ASAR level O data

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TablesTable 1Table 2

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Page 25TitlesMACDONALD 4 FBAQ DECODING FOR ASAR 4.1 Algorithm Overview

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Page 26TitlesII MACDONALD 4.2 Characteristics of ASAR Signal Data 4.3 Derivation of FBAQ Thresholds ,,..·



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Page 29TitlesMACDONALD 4.5 Steps in Basic Block Adaptive Quantisation 4.6 ASAR FBAQ Implementation

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Page 31TitlesMACDONALD 4. 7 Steps in Flexible Block Adaptive Quantisation

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Page 33TitlesNIACDONAlD 4.7.1 4.7.2 4.7.3 FBAQ Decoding Details Unpacking the Encoded Data Accessing the Decoder LUT ...

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Page 35TitlesMACDONALD 4.8.1 RMS Equalisation Procedure


Page 36TitlesII MACDONALD 4.8.2 4.8.3 Effect On Reconstructed SAR Data FBAQ Decoding Steps When Using RMS Equalisation

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Page 39TitlesMACDONALD 5 SIGN AND MAGNITUDE COMPRESSION

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Page 41TitlesMACDONALD 6 ANALOG TO DIGIT AL CONVERTER CORRECTION

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TablesTable 1