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\section{Floor type 1 setup and decode} \label{vorbis:spec:floor1}

\subsection{Overview}

Vorbis floor type one uses a piecewise straight-line representation to

encode a spectral envelope curve. The representation plots this curve

mechanically on a linear frequency axis and a logarithmic (dB)

amplitude axis. The integer plotting algorithm used is similar to

Bresenham's algorithm.

\subsection{Floor 1 format}

\subsubsection{model}

Floor type one represents a spectral curve as a series of

line segments.  Synthesis constructs a floor curve using iterative

prediction in a process roughly equivalent to the following simplified

description:

\begin{itemize}

 \item  the first line segment (base case) is a logical line spanning

from x_0,y_0 to x_1,y_1 where in the base case x_0=0 and x_1=[n], the

full range of the spectral floor to be computed.

\item the induction step chooses a point x_new within an existing

logical line segment and produces a y_new value at that point computed

from the existing line's y value at x_new (as plotted by the line) and

a difference value decoded from the bitstream packet.

\item floor computation produces two new line segments, one running from

x_0,y_0 to x_new,y_new and from x_new,y_new to x_1,y_1. This step is

performed logically even if y_new represents no change to the

amplitude value at x_new so that later refinement is additionally

bounded at x_new.

\item the induction step repeats, using a list of x values specified in

the codec setup header at floor 1 initialization time.  Computation

is completed at the end of the x value list.

\end{itemize}

Consider the following example, with values chosen for ease of

understanding rather than representing typical configuration:

For the below example, we assume a floor setup with an [n] of 128.

The list of selected X values in increasing order is

0,16,32,48,64,80,96,112 and 128.  In list order, the values interleave

as 0, 128, 64, 32, 96, 16, 48, 80 and 112.  The corresponding

list-order Y values as decoded from an example packet are 110, 20, -5,

-45, 0, -25, -10, 30 and -10.  We compute the floor in the following

way, beginning with the first line:

\begin{center}

\includegraphics[width=8cm]{floor1-1}

\captionof{figure}{graph of example floor}

\end{center}

We now draw new logical lines to reflect the correction to new_Y, and

iterate for X positions 32 and 96:

\begin{center}

\includegraphics[width=8cm]{floor1-2}

\captionof{figure}{graph of example floor}

\end{center}

Although the new Y value at X position 96 is unchanged, it is still

used later as an endpoint for further refinement.  From here on, the

pattern should be clear; we complete the floor computation as follows:

\begin{center}

\includegraphics[width=8cm]{floor1-3}

\captionof{figure}{graph of example floor}

\end{center}

\begin{center}

\includegraphics[width=8cm]{floor1-4}

\captionof{figure}{graph of example floor}

\end{center}

A more efficient algorithm with carefully defined integer rounding

behavior is used for actual decode, as described later.  The actual

algorithm splits Y value computation and line plotting into two steps

with modifications to the above algorithm to eliminate noise

accumulation through integer roundoff/truncation.

\subsubsection{header decode}

A list of floor X values is stored in the packet header in interleaved

format (used in list order during packet decode and synthesis).  This

list is split into partitions, and each partition is assigned to a

partition class.  X positions 0 and [n] are implicit and do not belong

to an explicit partition or partition class.

A partition class consists of a representation vector width (the

number of Y values which the partition class encodes at once), a

'subclass' value representing the number of alternate entropy books

the partition class may use in representing Y values, the list of

[subclass] books and a master book used to encode which alternate

books were chosen for representation in a given packet.  The

master/subclass mechanism is meant to be used as a flexible

representation cascade while still using codebooks only in a scalar

context.

\begin{Verbatim}[commandchars=\\\{\}]

  1) [floor1\_partitions] = read 5 bits as unsigned integer

  2) [maximum\_class] = -1

  3) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{

        4) vector [floor1\_partition\_class\_list] element [i] = read 4 bits as unsigned integer

     \}

  5) [maximum\_class] = largest integer scalar value in vector [floor1\_partition\_class\_list]

  6) iterate [i] over the range 0 ... [maximum\_class] \{

        7) vector [floor1\_class\_dimensions] element [i] = read 3 bits as unsigned integer and add 1

	8) vector [floor1\_class\_subclasses] element [i] = read 2 bits as unsigned integer

        9) if ( vector [floor1\_class\_subclasses] element [i] is nonzero ) \{

             10) vector [floor1\_class\_masterbooks] element [i] = read 8 bits as unsigned integer

           \}

       11) iterate [j] over the range 0 ... (2 exponent [floor1\_class\_subclasses] element [i]) - 1 \{

             12) array [floor1\_subclass\_books] element [i],[j] =

                 read 8 bits as unsigned integer and subtract one

           \}

      \}

 13) [floor1\_multiplier] = read 2 bits as unsigned integer and add one

 14) [rangebits] = read 4 bits as unsigned integer

 15) vector [floor1\_X\_list] element [0] = 0

 16) vector [floor1\_X\_list] element [1] = 2 exponent [rangebits];

 17) [floor1\_values] = 2

 18) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{

       19) [current\_class\_number] = vector [floor1\_partition\_class\_list] element [i]

       20) iterate [j] over the range 0 ... ([floor1\_class\_dimensions] element [current\_class\_number])-1 \{

             21) vector [floor1\_X\_list] element ([floor1\_values]) =

                 read [rangebits] bits as unsigned integer

             22) increment [floor1\_values] by one

           \}

     \}

 23) done

\end{Verbatim}

An end-of-packet condition while reading any aspect of a floor 1

configuration during setup renders a stream undecodable.  In addition,

a \varname{[floor1\_class\_masterbooks]} or

\varname{[floor1\_subclass\_books]} scalar element greater than the

highest numbered codebook configured in this stream is an error

condition that renders the stream undecodable.  Vector

[floor1\_x\_list] is limited to a maximum length of 65 elements; a

setup indicating more than 65 total elements (including elements 0 and

1 set prior to the read loop) renders the stream undecodable.  All

vector [floor1\_x\_list] element values must be unique within the

vector; a non-unique value renders the stream undecodable.

\subsubsection{packet decode} \label{vorbis:spec:floor1-decode}

Packet decode begins by checking the \varname{[nonzero]} flag:

\begin{Verbatim}[commandchars=\\\{\}]

  1) [nonzero] = read 1 bit as boolean

\end{Verbatim}

If \varname{[nonzero]} is unset, that indicates this channel contained

no audio energy in this frame.  Decode immediately returns a status

indicating this floor curve (and thus this channel) is unused this

frame.  (A return status of 'unused' is different from decoding a

floor that has all points set to minimum representation amplitude,

which happens to be approximately -140dB).

Assuming \varname{[nonzero]} is set, decode proceeds as follows:

\begin{Verbatim}[commandchars=\\\{\}]

  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1\_multiplier]-1)

  2) vector [floor1\_Y] element [0] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer

  3) vector [floor1\_Y] element [1] = read \link{vorbis:spec:ilog}{ilog}([range]-1) bits as unsigned integer

  4) [offset] = 2;

  5) iterate [i] over the range 0 ... [floor1\_partitions]-1 \{

       6) [class] = vector [floor1\_partition\_class]  element [i]

       7) [cdim]  = vector [floor1\_class\_dimensions] element [class]

       8) [cbits] = vector [floor1\_class\_subclasses] element [class]

       9) [csub]  = (2 exponent [cbits])-1

      10) [cval]  = 0

      11) if ( [cbits] is greater than zero ) \{

             12) [cval] = read from packet using codebook number

                 (vector [floor1\_class\_masterbooks] element [class]) in scalar context

          \}

      13) iterate [j] over the range 0 ... [cdim]-1 \{

             14) [book] = array [floor1\_subclass\_books] element [class],([cval] bitwise AND [csub])

             15) [cval] = [cval] right shifted [cbits] bits

	     16) if ( [book] is not less than zero ) \{

	           17) vector [floor1\_Y] element ([j]+[offset]) = read from packet using codebook

                       [book] in scalar context

                 \} else [book] is less than zero \{

	           18) vector [floor1\_Y] element ([j]+[offset]) = 0

                 \}

          \}

      19) [offset] = [offset] + [cdim]

     \}

 20) done

\end{Verbatim}

An end-of-packet condition during curve decode should be considered a

nominal occurrence; if end-of-packet is reached during any read

operation above, floor decode is to return 'unused' status as if the

\varname{[nonzero]} flag had been unset at the beginning of decode.

Vector \varname{[floor1\_Y]} contains the values from packet decode

needed for floor 1 synthesis.

\subsubsection{curve computation} \label{vorbis:spec:floor1-synth}

Curve computation is split into two logical steps; the first step

derives final Y amplitude values from the encoded, wrapped difference

values taken from the bitstream.  The second step plots the curve

lines.  Also, although zero-difference values are used in the

iterative prediction to find final Y values, these points are

conditionally skipped during final line computation in step two.

Skipping zero-difference values allows a smoother line fit.

Although some aspects of the below algorithm look like inconsequential

optimizations, implementors are warned to follow the details closely.

Deviation from implementing a strictly equivalent algorithm can result

in serious decoding errors.

{\em Additional note:} Although \varname{[floor1\_final\_Y]} values in

the prediction loop and at the end of step 1 are inherently limited by

the prediction algorithm to [0, \varname{[range]}), it is possible to

  abuse the setup and codebook machinery to produce negative or

  over-range results.  We suggest that decoder implementations guard

  the values in vector \varname{[floor1\_final\_Y]} by clamping each

  element to [0, \varname{[range]}) after step 1.  Variants of this

    suggestion are acceptable as valid floor1 setups cannot produce

    out of range values.

\begin{description}

\item[step 1: amplitude value synthesis]

Unwrap the always-positive-or-zero values read from the packet into

+/- difference values, then apply to line prediction.

\begin{Verbatim}[commandchars=\\\{\}]

  1) [range] = vector \{ 256, 128, 86, 64 \} element ([floor1\_multiplier]-1)

  2) vector [floor1\_step2\_flag] element [0] = set

  3) vector [floor1\_step2\_flag] element [1] = set

  4) vector [floor1\_final\_Y] element [0] = vector [floor1\_Y] element [0]

  5) vector [floor1\_final\_Y] element [1] = vector [floor1\_Y] element [1]

  6) iterate [i] over the range 2 ... [floor1\_values]-1 \{

       7) [low\_neighbor\_offset] = \link{vorbis:spec:low:neighbor}{low\_neighbor}([floor1\_X\_list],[i])

       8) [high\_neighbor\_offset] = \link{vorbis:spec:high:neighbor}{high\_neighbor}([floor1\_X\_list],[i])

       9) [predicted] = \link{vorbis:spec:render:point}{render\_point}( vector [floor1\_X\_list] element [low\_neighbor\_offset],

				      vector [floor1\_final\_Y] element [low\_neighbor\_offset],

                                      vector [floor1\_X\_list] element [high\_neighbor\_offset],

				      vector [floor1\_final\_Y] element [high\_neighbor\_offset],

                                      vector [floor1\_X\_list] element [i] )

      10) [val] = vector [floor1\_Y] element [i]

      11) [highroom] = [range] - [predicted]

      12) [lowroom]  = [predicted]

      13) if ( [highroom] is less than [lowroom] ) \{

            14) [room] = [highroom] * 2

          \} else [highroom] is not less than [lowroom] \{

            15) [room] = [lowroom] * 2

          \}

      16) if ( [val] is nonzero ) \{

            17) vector [floor1\_step2\_flag] element [low\_neighbor\_offset] = set

            18) vector [floor1\_step2\_flag] element [high\_neighbor\_offset] = set

            19) vector [floor1\_step2\_flag] element [i] = set

            20) if ( [val] is greater than or equal to [room] ) \{

                  21) if ( [highroom] is greater than [lowroom] ) \{

                        22) vector [floor1\_final\_Y] element [i] = [val] - [lowroom] + [predicted]

		      \} else [highroom] is not greater than [lowroom] \{

                        23) vector [floor1\_final\_Y] element [i] = [predicted] - [val] + [highroom] - 1

                      \}

                \} else [val] is less than [room] \{

                    24) if ([val] is odd) \{

                        25) vector [floor1\_final\_Y] element [i] =

                            [predicted] - (([val] + 1) divided by  2 using integer division)

                      \} else [val] is even \{

                        26) vector [floor1\_final\_Y] element [i] =

                            [predicted] + ([val] / 2 using integer division)

                      \}

                \}

          \} else [val] is zero \{

            27) vector [floor1\_step2\_flag] element [i] = unset

            28) vector [floor1\_final\_Y] element [i] = [predicted]

          \}

     \}

 29) done

\end{Verbatim}

\item[step 2: curve synthesis]

Curve synthesis generates a return vector \varname{[floor]} of length

\varname{[n]} (where \varname{[n]} is provided by the decode process

calling to floor decode).  Floor 1 curve synthesis makes use of the

\varname{[floor1\_X\_list]}, \varname{[floor1\_final\_Y]} and

\varname{[floor1\_step2\_flag]} vectors, as well as [floor1\_multiplier]

and [floor1\_values] values.

Decode begins by sorting the scalars from vectors

\varname{[floor1\_X\_list]}, \varname{[floor1\_final\_Y]} and

\varname{[floor1\_step2\_flag]} together into new vectors

\varname{[floor1\_X\_list]'}, \varname{[floor1\_final\_Y]'} and

\varname{[floor1\_step2\_flag]'} according to ascending sort order of the

values in \varname{[floor1\_X\_list]}.  That is, sort the values of

\varname{[floor1\_X\_list]} and then apply the same permutation to

elements of the other two vectors so that the X, Y and step2\_flag

values still match.

Then compute the final curve in one pass:

\begin{Verbatim}[commandchars=\\\{\}]

  1) [hx] = 0

  2) [lx] = 0

  3) [ly] = vector [floor1\_final\_Y]' element [0] * [floor1\_multiplier]

  4) iterate [i] over the range 1 ... [floor1\_values]-1 \{

       5) if ( [floor1\_step2\_flag]' element [i] is set ) \{

             6) [hy] = [floor1\_final\_Y]' element [i] * [floor1\_multiplier]

 	     7) [hx] = [floor1\_X\_list]' element [i]

             8) \link{vorbis:spec:render:line}{render\_line}( [lx], [ly], [hx], [hy], [floor] )

             9) [lx] = [hx]

	    10) [ly] = [hy]

          \}

     \}

 11) if ( [hx] is less than [n] ) \{

        12) \link{vorbis:spec:render:line}{render\_line}( [hx], [hy], [n], [hy], [floor] )

     \}

 13) if ( [hx] is greater than [n] ) \{

            14) truncate vector [floor] to [n] elements

     \}

 15) for each scalar in vector [floor], perform a lookup substitution using

     the scalar value from [floor] as an offset into the vector \link{vorbis:spec:floor1:inverse:dB:table}{[floor1\_inverse\_dB\_static\_table]}

 16) done

\end{Verbatim}

\end{description}