### From The MPEG-4 Structured Audio Book by John Lazzaro and John Wawrzynek.

# Part IV/3: Signal Processing Core Opcodes

## Sections

- Introduction.
- Level Matching. Absolute and relative.
- Specialops. Blending a-rate and k-rate semantics.
- Window Wavetables. Generating window function tables.
- Sample Rate Conversion. Map between a-rate and k-rate.
- Gain Control. Compressor/limiter/expander/noise-gate opcode.
- Fourier Analysis. Core opcodes fft and ifft.
- Portamento. Smooth pitch changes.

## Core Opcodes:

balance compressor decimate downsamp fft gain ifft port rms samphold sblock upsamp

## Wavetable Generator:

window

## Introduction

In this chapter, we complete our description of the core opcode library.

We describe opcodes that perform signal processing operations on a buffer of a-rate signal values. These opcodes perform operations such as gain control, sample-rate conversion, and Fourier analysis on the buffer.

We describe the specialop semantics that govern several of the opcodes described in the chapter. A specialop opcode computes at the a-rate, but returns values at the k-rate.

We describe the core wavetable generator window, that computes popular windowing functions used in block-based signal processing. Several of the opcodes in this chapter have wavetable parameters that window buffer data.

## Level Matching

The a-rate core opcodes gain and balance act as simple automatic gain control systems. See the right panel for header syntax.

On each call, these opcodes return the scaled copy `a*x` of the input signal parameter `x`, where `a` is an internal variable that sets the attenuation of the system.

The attenuation variable is initialized to 1 during the first opcode call, and is updated as specified by the gain control algorithm for each opcode.

#### `gain`

The gain opcode returns a signal whose RMS power approximates the power level specified by the parameter `g`.

To perform this task, gain periodically recalculates the attenuation variable, using a formula (shown on the right panel) that measures the power level of recent values of the signal parameter `x`.

By default, the attenuation is updated once every control period (the inverse of the k-rate). The optional i-rate parameter `length` (units of seconds) overrides the default value for the attenuation period.

During the first call to gain, a buffer is created of sufficient size to hold the `x` values for an entire attenuation period, and the current `x` value is placed at the start of the buffer. Subsequent calls to gain fill successive positions in the buffer.

On the gain call that fills the buffer, a new attenuation value is computed, using the equation shown on the right panel. The buffer is cleared, and future calls to gain refill the buffer in preparation for the next attenuation update.

#### `balance`

The balance opcode returns a scaled copy of the parameter `x`. The returned signal has an RMS power level that approximates the power of the signal parameter `ref`.

To achieve this behavior, the opcode creates two buffers, to hold recent values of `ref` and `x`. The opcode periodically updates the attenuation parameter, to reflect the energy of the signals in the two buffers.

The control period (`1/k_rate`) sets the default length of balance's buffers, which may be overridden by the optional i-rate parameter `length`.

During the first call to balance, buffers are created for `ref` and `x` parameters, and the current values for `ref` and `x` are placed at the start of the buffers. Subsequent calls fill successive positions in the `ref` and `x` buffers.

On the opcode call that fills the buffers, a new attenuation value is computed, using the equation shown on the right panel. The buffers are cleared, and future calls to balance refill the buffers in preparation for the next attenuation update.

## `gain`

`

    aopcode gain(asig x, ksig g
                 [, ivar length])

    on every call, return a*x.

    on first call:

        set internal variable a
        to 1, and create buffer
        xh. the optional parameter
        length (units of seconds)
        sets the buffer size. if
        this parameter is not given,
        set length to the 1/k_rate.

        xh contains

        L = floor(length*s_rate)

        samples. insert x into xh[0].

    on subsequent calls:

        put x into next position
        in xh. if xh is filled,
        compute new value of a:

                 g*sqrt(L)
    a = -------------------------------
        sqrt(xh[0]^2 + ... + xh[L-1]^2)

        this completes one cycle of
        the algorithm. on the next
        call, insert x into xh[0],
        starting the next cycle.

`

## `balance`

`

    aopcode balance(asig x,
                    asig ref
                 [, ivar length])

    on every call, return a*x.

    on first call:

        set internal variable a
        to 1. create buffers xh
        and rh. optional parameter
        length (units of seconds)
        sets the buffer sizes. if
        this parameter is not given,
        set length to the 1/k_rate.

        xh and rh contains

        L = floor(length*s_rate)

        samples. insert x into xh[0]
        and ref into rh[0].

    on subsequent calls:

        put x into next position
        in xh, and ref into next
        position of rh. if buffers
        filled, compute the new
        value of a:

        sqrt(rh[0]^2 + ... + rh[L-1]^2)
    a = -------------------------------
        sqrt(xh[0]^2 + ... + xh[L-1]^2)

        this completes one cycle of
        the algorithm. on the next
        call, insert x into xh[0],
        and ref into rh[0], starting
        the next cycle.

`

## Specialops

The gain opcode, if called without the optional `length` parameter, fills its buffer by accepting new `x` values with each a-rate call, and computes the signal energy of its buffer once per k-rate.

The rms opcode also performs this function, but returns the signal energy as its k-rate return value. See the right panel for the header syntax and exact semantics for the rms opcode.

The rms opcode is an example of a SAOL specialop opcode, which has aspects of both aopcode and kopcode semantics.

Like an a-rate opcode, the rms opcode runs at the a-rate in order to fill the buffer. But like a k-rate opcode, it also runs at k-rate, and returns a k-rate value.

Specialop calls may only appear in instrument code, and in aopcode user-defined opcodes (described in Part IV). The rules below set the semantics of specialop opcodes:

1. A specialop returns values at the k-rate. For the purpose of evaluating the rate of expressions, a specialop is considered to be a kopcode.
2. A specialop is evaluated at both the a-rate and the k-rate. However, the expression returns a value, and the statement containing it executes, at the k-rate.
3. A specialop may appear in an a-rate statement. If so, its k-rate return semantics work in the same way as a normal k-rate opcode call.

Specialop calls may only appear in instrument code, and in aopcode user-defined opcodes (described in Part IV/4). Specialop calls are also restricted in these ways:

1. An expression containing a specialop opcode is considered a specialop expression.
2. A specialop expression may not appear in the code block or guard expression of a while statement.
3. A specialop expression may only appear in the code block of an if or if-else statement if the guard expression of the statement is also specialop.

The right panel shows several examples of specialop semantics, using the rms opcode.

## `rms`

`

    specialop rms(asig x,
                [, ivar length])

    as a specialop, it runs at the
    a-rate and k-rate, but only
    returns values at the k-rate.

    k-rate, first call:

       create the buffer xh, and
       initialize values to zero.
       the optional parameter length
       (units of seconds, must be > 0)
       sets the buffer size. if this
       parameter is not given, set
       length to the 1/k_rate.

       xh contains:

       L = floor(length*s_rate)

       samples. create buffer index,
       set it to zero (first element).

    k-rate, all calls:

       return the value

    sqrt(xh[0]^2 + ... + xh[L-1]^2)
    -------------------------------
               sqrt(L)

    a-rate, all calls:

       place the x value into the
       buffer xh, at the position
       of the buffer index. then
       increment buffer index. if
       index has value L, reset
       the index to 0.

`

## Examples

`

    asig x;
    ksig k;

    // legal, rms runs at a-rate
    // and k-rate, but returns
    // a value at k-rate that is
    // assigned to y.

    y = rms(x);

    // legal, both rms run at a-rate
    // and k-rate, if condition is
    // true at k-rate, assignment
    // is made. rms in assignment
    // returns same value as rms in
    // conditional

    if (rms(x) > y)
     {
       y = rms(x);
     }

`

## Sample Rate Conversion

The rms opcode converts the information carried by an a-rate parameter to a k-rate return value. In this sense, it performs a type of sample-rate conversion.

In this section, we describe other core opcodes that perform sample-rate conversion.

#### Downsampling Opcodes

Three other simple opcodes make a-rate signal information available at the k-rate. These opcodes are all specialop opcodes.

The decimate opcode returns (during its k-pass) one of the `in` parameter values that it received in the preceding set of a-pass calls. The opcode definition does not specify which `in` value is chosen.

The downsamp opcode buffers the `in` values of the last `s_rate/k_rate` opcode calls at the a-rate. At the k-rate call following the a-rate calls, it returns the mean of the buffer.

If the downsamp call includes the optional table parameter `win`, the wavetable values are multiplied with the buffer values point by point, and opcode returns the sum of of all multiplication results. If the `win` table is shorter than the buffer, zeros are used for the extra window values.

The sblock opcode buffers `in` values of the last `s_rate/k_rate` opcode calls at the a-rate. During the k-rate call, it places these buffer values in the table provided by parameter `t`, which must have at least `s_rate/k_rate` table values. The opcode always returns zero.

#### Upsampling Opcodes

The simplest way to upsample control information to the audio rate is to assign a k-rate value to an a-rate variable. The upsamp and samphold core opcodes offer more sophisticated methods of upsampling.

The upsamp opcode upsamples the k-rate parameter `in` to a-rate via a shift-and-add technique. An optional table parameter `win` controls the spectral properties of the upsampling. The upsamp opcode reduces the aliasing artifacts produced by assigning k-rate values to a-rate variables directly. See the right panel for a complete explanation of this opcode.

The polymorphic samphold opcode performs a sample-and-hold operation on the polymorphic input parameter `in`, under the control of the k-rate parameter `gate`. It acts as an upsampling system if the `in` parameter is a-rate.

The samphold opcode returns the value of an internal state variable, that is initialized to zero at the start of the first call to the opcode. If the `gate` parameter is non-zero, the internal state variable is updated to the value of the `in` parameter.

## Downsampling Opcodes

`

    specialop decimate(asig in)

    specialop downsamp(asig in
                       [,table win])

    specialop sblock(asig in,
    		 table t)

    see left panel for algorithms.

`

## Upsampling Opcodes

`

    opcode samphold(xsig in,
                    ksig gate)

    see left panel for algorithm.

    asig upsamp(ksig in
                [,table win])

    This opcode upsamples the
    k-rate in parameter to
    a-rate, using a smoothing
    buffer. In the interesting
    case, the buffer size is
    the size of the table win,
    and is several times greater
    than a_rate/k_rate in length.
    On the first call to upsamp,
    the buffer buf[] is created,
    and initialized to zeros.

    On the first a-pass call to
    upsamp in a given execution
    cycle, the contents of buf[]
    are shifted forward by
    a_rate/k_rate samples. The last
    a_rate/k_rate buff[] values are
    set to zero. Then, all buf[]
    values is updated using this
    formula:

    buf[i] = buf[i] +
             input*win[i]

    This first a-pass call returns
    buf[0]; future a-pass calls in
    the execution cycle return buf[1],
    buf[2], ...

    If the win table has fewer than
    a_rate/k_rate elements, the buf[]
    has a size a_rate/k_rate, and zeros
    are used for the extra win values in
    the formula.

    If no win table is provided, a win of
    size a_rate/k_rate is used, with all
    samples of value 1. The buf[] is also
    a_rate/k_rate.

`

## Window Wavetables

Several of the opcodes in the previous section let the programmer specify a windowing function as a wavetable of window values.

The core wavetable generator window simplifies the creation of windowing wavetables. The right panel shows the declaration syntax and algorithm for this wavetable generator.

The `size` parameter sets the number of samples in the window table, and must be greater than zero. The `type` parameter is an integer that sets the window type.

The window generator produces six window types.

1. Hamming window.
2. Hanning window.
3. Bartlett window.
4. Gaussian window.
5. Kaiser window.
6. Boxcar window

The numbering of the list indicates the value of the `type` parameter that produces the associated window shape.

The Kaiser window algorithm creates a family of windows, controlled by the optional parameter `p`.

## `window`

`

    table t(window, size, type[,p]);

    Type parameter is an integer that
    codes the window shape. Listing
    below shows algorithm, for samples
    that lie in range 0 <= x <= size-1.

    [1] Hamming window.

    0.54 - 0.46*cos(2*pi*x/(size-1))

    [2] Hanning window.

    0.54*(1 - cos(2*pi*x/(size-1)))

    [3] Bartlett window (triangle).

         2*fabs(x - ((size-1)/2))
    1 -  ------------------------
                (size-1)

    [4] Gaussian window:

            exp(-((m-x)^2)/a)

      where

      m = size/2   a = (size*size)/18

    [5] Kaiser window

       a = (size-1)/2

       Io[p*sqrt(a^2 - (x-a)^2)]
       -------------------------
               Io[p*a]

    [6] Boxcar window -- all table
        values are 1.

    Slib defines the constants
    WINDOW_HAMMING, WINDOW_HANNING,
    WINDOW_BARTLETT, WINDOW_GAUSSIAN,
    WINDOW_KAISER, and WINDOW_BOXCAR
    to use as the type parameter in
    the window wavetable generator.

`

## Gain Control

The compressor opcode implements a complete gain control system. The opcode may be configured to perform gain control functions such as compression, expansion, noise-gating, and limiting. The right panel shows the header syntax and algorithm for this opcode.

The opcode returns a scaled version of the a-rate input signal parameter `x`, with a latency of set by the parameter `look`. The scaling depends on the loudness of the a-rate signal parameter `comp`. For most uses, `comp` and `x` are set to the same value.

The opcode measures the loudness of `comp`, expressed in terms of decibels (dB), and changes the scaling of `x` in response to this loudness. The loudness is not computed as an instantaneous value, but by evaluating the signal over a short analysis window (set by the parameter `look`).

In this scale, 90 dB corresponds to a signal with a peak waveform value of 1, 70 dB corresponds to a signal with a peak waveform value of 0.1, etc. The noise floor of the system is set by the k-rate parameter `nfloor`, in units of dB.

The parameters `att` and `rel` set the attack and release times (in seconds) for the loudness measurement of `comp`. Short attack and release times let the loudness track quick signal transients, while longer attack and release times result in a smoother loudness estimate.

Given the loudness measurement of `comp`, the opcode calculates the scaling factor for the delayed version of `x` using the table shown on the right panel. The k-rate parameters `nfloor`, `thresh`, `loknee`, `hiknee`, and `ratio` control this scaling. All of these parameters have units of dB.

#### Noise gating

The `nfloor` and `thresh` parameters control noise gating. If the loudness of `comp` is above `thresh`, the noise gate is open, and the opcode returns a delayed replica of the `x` signal. If the loudness of `comp` is below `nfloor`, the noise gate is closed, and the opcode returns zero. Non-normative interpolation occurs in the transition regime between `nfloor` and `thresh`.

To turn off noise gating, both `nfloor` and `thresh` should be set to noise floor of this system (for most applications, a value of -40 dB yields good results).

#### Compression/expansion

If the loudness of `comp` is above `hiknee`, the opcode acts as a compressor or expander. The value of `ratio` determines the exact behavior in this regime. If the loudness of `comp` increases by `ratio` dB, the opcode returns a delayed version of `x` whose loudness has increased by 1 dB. Thus, `ratio` values greater than 1 dB result in compression, and ratio values between 0 and 1 dB result in expansion. Negative `ratio` values are prohibited.

If the loudness of `comp` is below `loknee`, the opcode performs as a "wire with latency", returning a replica of parameter `x` delayed by the analysis window time `look`. Non-normative interpolation occurs in the transition regime between `loknee` and `hiknee`.

#### Cross-signal effects

By choosing the `comp` signal to be different than the `x` signal, the opcode produces a version of the `x` signal whose dynamics are shaped by the `comp` signal.

## `compressor`

`

    aopcode compressor(asig x,
            asig comp, ksig nfloor,
            ksig thresh, ksig loknee,
            ksig hiknee, ksig ratio,
            ksig att, ksig rel,
            ivar look)

    The compressor opcode delays
    the signal parameter x for
    look seconds, and returns
    the delayed value after
    weighting it by R.

    R is determined by measuring
    the dB level of the signal
    parameter comp, as shown
    by the table. The parameters
    nfloor, thresh, loknee, and
    hiknee, and ratio are all in
    units of dB (90 dB corresponds
    to a signal amplitude of 1.0).

     comp (dB) |     R
    -------------------------
    less than  |     0
    nfloor     | (noise gate:
               |    closed)
    -------------------------
    between    |  0 < R < 1
    nfloor and | (noise gate:
    thresh     |  transition)
    -------------------------
    between    |     1
    thresh and | (noise gate:
    loknee     |     open)
    -------------------------
    between    |  transition
    loknee and |    regime
    hiknee     |
    -------------------------
    greater    | R is set so
    than       | that a ratio
    hiknee     | dB increase
               | in comp
               | yields a 1
               | dB increase
               | in x.
    -------------------------

    given that:

    nfloor <= thresh
    thresh <= loknee
    loknee <= hiknee
    ratio  >  0

    If ratio is < 1 dB, the
    opcode acts as an expander.
    If ratio > 1 dB, the opcode
    it acts as a compressor.

    To compute comp dB value,
    the opcode keeps a buffer of
    instantaneous dB values of
    the comp signal, using the
    equation:

    90 + 20*log_10(abs(comp))

    This buffer length is set by
    the parameter look. The comp dB
    signal is computed by extrapoling
    signal trends in this buffer,
    under the guidance of the
    attack and release times of
    the opcode, set by parameters
    att and rel (which have units of
    seconds). Short att and rel values
    produce in quick changes in R,
    longer att and rel produce slower
    changes in R.

`

## Fourier Analysis

The fft opcode computes a windowed and overlapped complex-valued Discrete Fourier Transform (DFT) on the a-rate parameter signal `in.` It stores the results in the wavetables `re` and `im`.

The complementary opcode ifft computes a windowed and overlapped Inverse Discrete Fourier Transform on the wavetable pair `re` and `im`, and returns samples of the resulting audio waveform.

These opcodes are designed to be used together to implement sound synthesis algorithms that use spectral modification techniques. If a boxcar window is used for both fft and ifft, an fft-ifft pair has unity gain. See the right panel for the header syntax of fft and ifft.

#### `fft`

The fft opcode is a specialop, that executes at the a-rate and k-rate, but returns a value at the k-rate.

The fft opcode returns a 1 if a new DFT has been calculated since the last k-pass, and 0 otherwise. If a new DFT has been computed, the real components are placed in the wavetable parameter `re`, and the imaginary components are placed in the wavetable parameter `im`.

The optional parameters `len`, `shift`, and `size` control the operation of the fft opcode.

The `len` parameter sets the size of the holding buffer for new audio samples. In most cases, `len` is also the size of the DFT.

The `shift` parameter controls the number of audio samples to add to the holding buffer before computing a new DFT. For a simple, non-overlapped DFT, `shift` is set to the same value as `len`. For an overlapped DFT, `shift` is set to a value smaller than `len`. For example, if `len` is 1024 and `shift` is 128, the opcode computes a new 1024 DFT every 128 samples.

On the first call to fft, a buffer `hbuf` of size `len` is created and zeroed, and the `in` parameter is placed in position `hbuf[len - shift]`.

Subsequent calls fill `hbuf[len - shift + 1], hbuf[len - shift + 2] ...` until the buffer is filled, and then the DFT computation begins. The optional `size` parameter may be used to set the DFT size; if `size` is not used, the `len` parameter is used. The DFT size may be no larger than 8192, and must be a power of 2.

The table `win` may be supplied to window the audio samples prior to computing the DFT. If it is not supplied, a boxcar window is used. When `hbuf` is filled for the first time, a buffer `new` with `size` values is created. Each buffer variable `new[i]` takes the value `win[i]*hbuf[i]`. If `size` is greater than `len`, the extra values of `new[i]` are set to zero.

Once `new` is filled, a DFT is performed on the buffer, and the real and imaginary results placed in the wavetables `re` and `im` respectively, which must be able to hold `size` values. The first position in each table holds the DC DFT value, the `size/2` position holds the Nyquist frequency coefficient value, and the positions after `size/2` hold values that code the reflection of the spectrum above the Nyquist frequency.

The `shift` parameter controls the data overlap between successive DFT calculations. After the first DFT is computed, the `hbuf` buffer is shifted forward by `shift` values. The `shift` spaces at the end of the buffer are the place where future calls to `fft` place `in` values. Once the `hbuf` buffer is refilled, the `new` is refilled, and a new DFT is performed.

The right panel describes the default values and legal ranges for the `fft` parameters `len`, `shift`, `size`, and `win`.

#### `ifft`

The ifft opcode runs at a-rate, and returns audio samples created from the complex DFT values in the `re` and `im` tables. The opcode assumes these tables are in the format created by the fft opcode.

The optional parameters `len`, `shift`, and `size` control the operation of the ifft opcode.

The `len` parameter sets the size of the holding buffer for output audio samples. Since in most cases `len` is also the size of the IDFT, the `size` parameter defaults to `len`. The IDFT size may be no larger than 8192, and must be a power of 2.

During the first call to ifft, the opcode computes the IDFT of the `re` and `im` tables. If `re` and `im` are greater than `size`, only the first `size` elements of the wavetables are used to compute the IDFT.

The first `len` components of the IDFT result are multiplied point-by-point by the windowing table `win`, and placed in an output buffer `out` of length `len`.

The value `out[0]` is returned on this first call, the next call returns `out[1]`, etc. Each sample is scaled by `shift/len`, so that an fft-ifft pair using boxcar windows has unity signal gain.

On the call where `out[shift-1]` is returned, the next IDFT is calculated, in the following way.

The contents of the `out` buffer are shifted forward `shift` elements, and the last `shift` values of `out` are set to zero. A new IDFT is computed, and the first `len` components of the result are multiplied point-by-point with the `win` table, and added into the `out` buffer. Values from `out[0]` to `out[shift-1]` are returned as described above, and the cycle repeats.

The right panel describes the default values and legal ranges for the `ifft` parameters `len`, `shift`, `size`, and `win`.

#### Example

The right panel shows a simple example, using `fft` and `ifft` together in a simple spectral modification algorithm.

## FFT and IFFT

`

    specialop fft(asig in,
                  table re,
                  table im [,
                  ivar len,
                  ivar shift,
                  ivar size,
                  table win])

    See right panel for algorithm
    details. Characteristics of
    parameters described below.

    in: audio input signal that
    is processed by the opcode.

    re: table that holds the real
    portion of the DFT. Must have
    at least size samples.

    im: table holds the imaginary
    portion of the DFT. Must have
    at least size samples.

    len: optional parameter that
    sets the number of samples to
    use. may not be negative. if
    zero or not provided, it is
    the next power of two greater
    than a_rate/k_rate.

    shift: optional parameter that
    sets the shift amount of the
    analysis window. may not be
    negative. if not provided or
    zero, set to len.

    size: optional parameter that
    sets the DFT size. may not
    be negative. if zero, set to
    len. must be a power of 2,
    and no greater than 8192.

    win: windowing table for
    analysis. if not provided,
    a boxcar of length len.
    may not have fewer than
    len samples.

    aopcode ifft(table re,
                 table im [,
                 ivar len,
                 ivar shift,
                 ivar size,
                 table win])

    See right panel for algorithm
    details. Descriptions of
    parameter limits for fft also
    hold for ifft.

`

## Example

`

    // hanning window table

    table win(window, 1024, 2);

    // space for fft

    table re(empty, 1024);
    table im(empty, 1024);
    table re_m(empty, 1024);
    table im_m(empty, 1024);

    // signal new fft done

    ksig flag;

    // signal to process

    asig in;

    flag = fft(in, re, im, 1024, 128,
    	   1024, win);

    if (flag)
     {
      // modify re and im here
      // put results in re_m and im_m
     }

    output(ifft(re_m, im_m, 1024, 128,
    	   1024, win));

`

## Portamento

The core opcode port is a k-rate filter, that converts a step transition of the k-rate parameter `ctrl` into a smooth transition with an exponential trajectory. When applied to a pitch control signal (in Hertz), it confers a portamento effect on pitch changes.

The right panel shows the header syntax and algorithm for the port opcode. A k-rate parameter `htime` sets the time that the output signal traverses one half of its total excursion.

This section concludes our descriptions of the SAOL core opcode library. In the final chapter in this section, we describe how users may write new opcodes in SAOL.

Next section: Part IV/4: User-Defined Opcodes

## `port`

`

    kopcode port(ksig ctrl,
                 ksig htime)

    ctrl: input k-rate signal
           to be filtered.

    htime: half-transition
           time, in seconds.
           one half of the
           time for the return
           value of port to
           reflect a step change
           in ctrl.

    port returns the value:

    o + (n - o)*(1 - 2^(t/htime))

    where o is the old value of
    ctrl and n is the new value
    of ctrl. o and n are updated
    whenever ctrl and n are not
    equal (o = n, n = ctrl).

    t is set to zero at each
    ctrl transition, and incremented
    by the 1/k_rate on each call.

    on first call, both o and n
    are set to ctrl.

`

Copyright 1999 John Lazzaro and John Wawrzynek.