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The Relationship of Dynamic Range to Data Word Size in Digital Audio Processing

Since the introduction of the compact disc in the early 1980s,
digital technology has become the standard for the recording
and storage of high-fidelity audio. It is not difficult to see
why. Digital signals are robust. Digital signals can be
transmitted and copied without distortion. Digital signals can
be played back without degrading the carrier. Who would want to
go back to scraping a needle along a vinyl groove now?

Another advantage of digital audio signals is the ease with
which they can be manipulated. Digital Signal Processing (DSP)
technology has advanced to such an extent that almost any audio
product, from a mobile phone to a professional mixing console,
contains a DSP chip. Once again the reasons for the success of
DSP are simple: stability, reliability, enhanced performance,
and programmability. Signal processing functions can be
implemented for a fraction of the cost, and in a fraction of
the space required by analog circuitry, as well as providing
functionality that simply couldn't be done in analog. In fact,
so ubiquitous has it now become that, for many people, the word
“digital” has become synonymous with “high quality”.

The ever-increasing performance and falling cost of DSP
hardware have generated new applications and new markets for
digital audio in both the consumer and professional audio
sectors. Digital Versatile Disk (DVD) and digital surround
sound in the home, digital radio and hands-free cellular phones
in the car are just a few of the DSP-based technologies which
have appeared in the last few years. The demands on the
quality, speed and flexibility of DSP has also increased as
more functionality is added to DSP products: a DSP might now be
required for mixing, equalization, dynamic range compression,
and data decompression, all in one product, implemented on one
chip.

16-bit, 44.1 kHz PCM digital audio continues to be the
standard for high quality audio in most current applications
such as CD, DAT, and high-quality PC audio. Recent
technological developments and improved knowledge of human
hearing, however, have created a demand for greater data word
lengths. Analog-to-digital converters (ADCs) now available
support 18, 20, and 24 bits and are capable of exceeding the
96dB dynamic range available using 16-bit data words. Many
recording studios now routinely master their recordings using
20- or 24-bit recorders. These technological developments are
beginning to make their way into the consumer and “prosumer”
audio applications. The most obvious consumer audio impact is
DVD, which is capable of carrying audio with up to 24-bit
resolution at sample rates well above 48 kHz. Another example
is a 16-channel digital home studio recorder, capable of
sampling at a 96 kHz sample rate with 24-bit resolution. In
fact, three trends can be identified which have influenced the
current generation of digital audio formats which are set to
replace CD digital audio. These can be summarized as
follows:

  • Higher resolution”either 20 or 24 bits per data word
  • Higher sampling frequency”typically 96 kHz and 192
    kHz
  • More audio channels for a more realistic “3D” sound
    experience.

Low-cost, higher-performance DSPs are now appearing on the
market to satisfy the high dynamic range requirements for
processing or synthesizing audio signals. How many bits are
required for processing audio signals? Is it 16, 20, 24, or 32
bits? Does the audio application require fixed-point of
floating-point arithmetic? What undesirable side-effects of
quantization should the audio designer look out for?

The first section in this article briefly reviews desirable
characteristics of a DSP for use in audio applications, and
then discusses the differences in data formats for fixed- and
floating-point processors. Next, the relationship of dynamic
range to data word size in processing audio signals is
examined. This will aid in determining how many bits would be
required for your application, whether it is a lower-cost,
low-fidelity consumer device or high-performance,
high-fidelity professional audio gear. Finally, to design a
system with either CD-quality or professional-quality audio, it
is suggested that for a digital filter routine to operate
transparently, the resolution of the processing system must be
considerably greater than that of the input signal. For the
highest-quality, professional audio systems, a 32-bit DSP is
offered as a suggested solution.


What Are the Benefits of Using a DSP to Process Audio
Signals?

A digital signal processor has one purpose: to operate on
quantized signal data as quickly and efficiently as possible.
Compared to a typical CPU or microcontroller, a
well-architected DSP usually contains the following desirable
characteristics to perform real-time DSP computations on audio
signals:

  • Fast and Flexible Arithmetic
    Single-cycle computation for multiplication with
    accumulation, arbitrary amounts of shifting, and standard
    arithmetic and logical operations.

  • Extended Dynamic Range for Extended Sum-of Product
    Calculations

    Extended sums-of-products, common in DSP algorithms, are
    supported in multiply-accumulate units. Extended precision in
    the multiplier's accumulator provides extra bits for
    protection against overflow in successive additions to ensure
    that no loss of data or range occurs.

  • Single-cycle Fetch of Two Operands For Sum-of-Products
    Calculations

    In extended sums-of-products calculations, two operations are
    needed on each cycle to feed the calculation. The DSP should
    be able to sustain two-operand data throughput, whether the
    data is stored on-chip or off.

  • Hardware Circular Buffer Support For Efficient Storage
    and Retrieval of Samples

    A large class of DSP algorithms, including digital filters,
    requires circular data buffers. A circular buffer is a finite
    segment of the DSP's memory defined by the programmer that is
    used to store samples for processing. Hardware Circular
    Buffering is designed to allow automatic address pointer
    wraparounds to the beginning of the buffer for simplifying
    circular buffer implementations, and thus reducing overhead
    and improving performance. When circular buffering is
    implemented in hardware, the DSP programmer does not have to
    be concerned with the additional overhead of testing and
    resetting the address pointer so that it does not go beyond
    the boundary of the buffer.

  • Efficient Looping and Branching for Repetitive DSP
    Operations

    DSP algorithms are repetitive and are most logically
    expressed as loops. For digital filter routines, a running
    sum of MAC operations is typically executed in fast and efficient loop structures. A DSP's program sequencer, or
    control unit, should allow looping of code with minimal or
    zero overhead. Any loop branching, loop decrementing, and
    termination test operations are built into the DSP control
    unit hardware. Also, no overhead penalties should result for
    conditional branching instructions which branch based on a
    computation unit's status bits.

All of the above architectural features are used for
implementation of DSP-type operations. For example, convolution
is a common signal processing operation involving the
multiplication of two sets of discrete data, an input
multiplied with a shifted version of the impulse response to a
system, and keeping a running sum of the outputs. This is seen
in the following convolution equation :

DSP architectural features are designed to perform these
types of discrete mathematical operations as quickly as
possible, usually within a single instruction cycle. Examining
this equation closely shows elements required for
implementation. The filter coefficients and input samples
required to implement the above equation can be stored in two
memory arrays defined as circular buffers. Both circular
buffers need to be multiplied together and added to the results
of previous iterations. To perform the operation shown above,
the DSP architecture should allow one multiplication to be
executed, along with an addition to a previous result in a
single instruction cycle. Within the same cycle, the
architecture should also contain enough parallelism in the
compute units to enable memory reads of the next sample and
filter coefficient for the next loop iteration. Hardware
looping circuitry included in the architecture would allow
efficient looping through the number of iterations with
zero-overhead. When used in a zero-overhead loop, digital
filter implementations become extremely optimized since no
explicit software decrement, test and jump instructions are
required. Thus, for actual implementation of the convolution
operation, two circular buffers, multipliers, adders, and a
zero-overhead loop construct are required. A digital signal
processor contains the necessary building blocks to accomplish
implementation of discrete-time filter operations.

In performing these types of repetitive DSP calculations,
quantization errors from truncation and rounding can accumulate
over time, degrading the quality of the DSP algorithmic result.
The number of bits of resolution used in the arithmetic
computations, along with a given filter structure realization,
will determine the robustness of a filter algorithm's signal
manipulation. The rest of this article will discuss how many
bits would potentially be required for a particular audio
application, as this is determined by the complexity of the
processing and the desired target signal quality.


DSP Numeric Data Formats: Do I Require Fixed or Floating
Point Arithmetic For My Audio Application?

Depending on the complexity of the application, the audio
system designer must decide on how much computational accuracy
and dynamic range will be needed. The most common native data
types are explained briefly in this section. 16- and 24-bit
fixed-point DSPs are designed to compute integer or fractional
arithmetic. 32-bit DSPs, such as the Analog Devices ADSP-2106x
SHARC family, were traditionally offered as floating-point
devices; however, this popular family of DSPs can equally
perform both floating-point arithmetic and integer or
fractional fixed-point arithmetic.

16-, 24-, and 32-Bit Fixed-Point Arithmetic
DSPs that can perform fixed-point operations typically use a
twos complement binary notation for representing signals. The
representation of the fixed-point format can be signed
(twos-complement) or unsigned integer or fractional notation.
Most DSP operations are optimized for signed fractional
notation.

The numeric format in signed fractional notation makes sense
to use in DSP computations, because in a fractional representation
it would easily correspond to a ratio of the full range of
samples produced from a 5V ADC, as shown in Figure
1
. It is harder to overflow a fractional result, because
multiplying a fraction by a fraction results in a smaller
number, which is then either truncated or rounded. The highest
full-scale positive fractional number would be 0.99999, while
the highest full scale negative number is -1.0. Anything in
between the highest representable signal from the converter
would be a fractional representation of the “loudest” signal.
For example, the midway positive amplitude for a converter
would be 1/2, and this would be interpreted as a fractional
value of 0x4000 by the DSP.

Figure 1:  Signed twos-complement representation of
sampled signals

Figure 2:  Fractional and integer formats for a N-bit
number

In the fractional format, the binary point is assumed to be
to the to the left of the LSB (sign bit). In the integer
format, the binary point is to the right of the LSB (Figure
2
).

Fractional math is more intuitive for signal manipulation,
and it is the least significant bits in a fractional result
that we will examine in this article, since it is these
lower order bits that can suffer from quantization errors due
to finite word length effects. The more bits that are used to
represent a given audio signal, the more accurate the
arithmetic result.

32-/40-bit Floating-Point Arithmetic
Floating-point math offers flexibility in programming because
it is much harder to overflow a result, while the programmer is
less concerned about scaling inputs to prevent overflow. IEEE
754/854 Floating-point data is stored in a format that is 32
bits wide, where 24 bits represent the mantissa and 8 bits
represent the exponent. The 24-bit mantissa is used for
precision while the exponent is for extending the dynamic
range. For 40-bit extended precision, 32 bits are used for the
mantissa while 8 bits are used to represent the exponent
(Figures 3 and 4 ).

Figure 3:  IEEE 754/854 32-bit single precision
floating-point format

A 32-bit floating point number is represented in decimal
as:

Its binary numeric IEEE format representation is stored on
the 32-bit floating point DSP as:

It is important to know that the IEEE standard always refers
to the mantissa in signed-magnitude format, and not in
twos-complement format. The extra hidden bit effectively
improves the precision to 24 bits and also insures any number
ranges from 1 (1.0000….00) to 2 (1.1111….11) since the
hidden bit is always assumed to be a 1.

Figure 4:  40-bit extended precision floating-point
format

Figure 4 shows the 40-bit extended precision format
available that is also supported on the ADSP-2106x family of
DSPs. With extended precision, the mantissa is extended to 32
bits. In all other respects, it is the same format as the IEEE
standard format. 40-bit extended-precision binary numeric
format representation is stored as:

For audio-processing, the dynamic range of floating point
may be unnecessary for some algorithms, but the flexibility in
programming in floating-point is desirable, especially for high-level programming languages
like C. Keep in mind that many of the fixed-point precision
issues discussed in later sections would still apply for a DSP
that supports floating point arithmetic, at least in terms of
truncation and coefficient quantization. The programmer still
has to convert the fixed-point data coming from an ADC to its
floating-point representation, while the floating-point result
has to be converted back to its fixed-point equivalent when
the data is sent to a DAC.

Floating-point arithmetic was traditionally used for
applications that have very high dynamic range requirements,
such as image processing, graphics, and military/space
applications. The dynamic range offered for 32-bit IEEE
floating-point arithmetic is 1530 dB. Typically in the past,
trade-offs were considered with price vs. performance when
deciding on the use of floating-point processors. Until
recently, the higher cost made 32-bit floating point DSPs
unreasonable for use in audio. Today, designers can achieve
high-quality audio using either 32-bit fixed- or floating-point
processing with the introduction of the lower-cost 32-bit
processors, at a cost comparable to 16-bit
and 24-bit DSPs.


The Relationship of Dynamic Range to Data Word Size in
Digital Audio

One of the top considerations when designing an audio system is
determining acceptable signal quality for the application.
Table 1 shows some comparisons of signal quality for some
audio applications, devices and equipment.

Audio
Device/Application
Dynamic
Range
AM Radio 48 dB
Analog Broadcast TV 60 dB
FM Radio 70 dB
Analog Cassette Player 73 dB
Video Camcorder 75 dB
ADI SoundPort Codecs 80 dB
16-bit Audio Converters 90 to 95 dB
Digital Broadcast TV 85 dB
Mini-Disk Player 90 dB
CD Player 92 to 96 dB
18-bit Audio Converters 104 dB
Digital Audio Tape (DAT) 110 dB
20-bit Audio Converters 110 dB
24-bit Audio Converters 110 to 120 dB
Analog Microphone 120 dB

Table 1:

Some dynamic range comparisons


“Recent advancements within the past decade in human
hearing indicate the sensitivity of the human ear is
such that the dynamic range between the quietest
sound detectable and the maximum sound which can be
experienced without pain is approximately 120dB.
Further studies suggest there is critically important
audio information at frequencies up to 40 kHz and
possibly 80 kHz”

Audio equipment retailers and consumers often use the phrase
'CD-quality sound' when referring to high-dynamic-range audio.
Compare sound quality of a CD player to that of an AM radio
broadcast. For higher quality CD audio, noise is not audible,
especially during quiet passages in music. Lower level signals
are heard clearly. But, the AM radio listener can easily hear
the low-level noise at very audible levels to where it can be a
distraction to the listener. With an increase of an audio
signal's dynamic range, the better distinction one can make
for low-level audio signals while the noise floor is lowered
and becomes undetectable to the listener (“noise floor” is a
term used to describe the point where the audio signal cannot
be distinquished from low-level white noise).

To achieve CD-type signal quality, the trend in recent years
has been to design a system that processes audio signals
digitally, using 16-bit ADCs and DACs with signal-to-noise ratio
(SNR) and dynamic range around 90-93 dB. When processing these
signals, the programmer should normally design the algorithm
with computation precision that is usually greater than
16-bits in compact disk signals. CD-quality audio is just one
example. For whatever the application, the audio system
designer must first determine what is an acceptable SNR and
then decide how much precision is required to produce
acceptable results for the intended application.


Click Here for a summary of the terms shown in
Figure 5
as defined by Davis and Jones (we will
be referring to many of these terms frequently
throughout this article).

What Is The SNR and Dynamic Range for a DSP?
In analog and digital terms, SNR (S/N ratio) and dynamic range
are often used synonymously. In pure analog terms, SNR is defined
as the ratio of the largest known signal that exists to the
noise present when no signal exists. In digital terms, SNR and
dynamic range are used synonymously to describe the ratio
between the largest representable number to the quantization
error. A well-designed digital filter should contain a
maximum SNR that is greater than the
converter SNR. Thus, the DSP designer must be sure that the
noise floor of a filter is not larger than the minimum
precision required of the ADC or DAC.

Figure 5:  Audio signal level (dBu) relationship
between dynamic range, SNR, and headroom


“In theoretical terms, there is an increase in the
signal-to-quantization noise or dynamic range by
approximately 6 dB for each bit added to the
word-length of an ADC, DAC or DSP.”

In “real-world” signal processing, quantization is the process
by which a number is approximated by a number of finite
precision. For example, during analog-to-digital conversion, an
infinitely variable signal voltage is represented by a binary
number with a fixed number of bits. The difference between two
consecutive binary values is called the quantization step, or
quantization level. The size of the quantization step defines
the effective noise floor of the quantized signal. The word
length for a given processor determines the number of
quantization levels that are available. For example, an n-bit data word
would yield 2n quantization levels (some examples
for common data word widths are shown in Table 2 ).

N Quantization
Levels for n-bit data words (N = 2n
levels)
28 = 256
216 = 65,536
220 = 1,048,576
224 = 16,777,216
232 = 4,294,967,296
264 = 18,446,744,073,729,551,616

Table 2:

An n-bit data word yields 2n quantization levels

A higher number of bits used to represent a sample will
result in a better approximation of the audio signal and a
reduction in quantization error (noise) that produces an
increase in the SNR. In theoretical terms, there is an increase
in the signal-to-quantization noise or dynamic range by
approximately 6 dB for each bit added to the word length of an
ADC, DAC, or DSP.

Figure 6:  DSP/converter SNR and dynamic range

Note that the “6-dB-Per-Bit-Rule” is an approximation to
calculating the actual dynamic range for a given word width.
The maximum representable signal amplitude to the maximum
quantization error for of an ideal ADC or DSP-based digital
system is actually calculated as:

1.76 dB is based on sinusoidal waveform statistics and
would vary for other waveforms, while n represents the data word
length of the converter or the digital signal processor.

In undithered DSP-based systems, the SNR definition above is
not directly applicable since there is no noise present when
there is no signal. In digital terms, dynamic range and SNR
(Figure 6 ) are often used synonymously to describe
the ratio of the largest representable signal to the
quantization error or noise floor. Therefore, when
referring to SNR or dynamic range in terms of DSP data word
size and quantization errors”both terms mean the same
thing.

Now the question arises, how many bits are required to
design a high quality audio system? In terms of dynamic range
and SNR, what is the best precision one can choose without
sacrificing low cost in a given design? Let's first see
the dynamic range comparisons between DSPs with different
native data-word sizes. Figure 7 shows the dynamic-range
relationship between the three most common DSP fixed-point
processor data-word widths: 16, 24, and 32 bits. The
quantization level comparisons are also given. As stated
earlier, the number of data-word bits used to represent a
signal directly affects the SNR and quantization noise
introduced during the sample conversions and arithmetic
computations.

Figure 7:  Fixed-point DSP dynamic range
comparisons

Precision
(Fixed-Point Binary Representation)
Dynamic Range
(# of bits per data word x 6 db/bit or
resolution)
16-bit 96 dB
24-bit 144 dB
32-bit 192 dB

Table 3:

Dynamic range vs. resolution

Each additional bit of resolution used by the DSP
for calculations will reduce the quantization noise power by
6dB. 16-bit fixed-point numeric precision yields 96 dB [16 x 6
dB per bit], 24-bit fixed-point precision yields 144 dB [24 x 6
dB per bit], while 32-bit fixed-point precision will yield 192
dB [32 x 6 dB per bit]. Note that for native single-precision
math, a 16-bit DSP is not adequate for accurately representing
the full dynamic range required for 'higher-fidelity' audio
signals around 120 dB.

In terms of quantization levels, Figure 8
demonstrates how 32-bit and 24-bit processing can more
accurately represent a processed audio signal as compared to
16-bit processing. 24-bit processing can more accurately
represent a signal 256 times better than 16-bit processing,
while 32-bit processing can more accurately represent signals
65,536 times better than that for 16-bit processing, and 256
times more accurately than that of a 24-bit processor.

Figure 8:  Fixed-point DSP quantization level
comparisons

Using the “6-dB-Per-Bit-Rule,” 32-bit IEEE floating point
dynamic range is determined to be 1530 dB. For floating point
this is calculated by the size of the exponent”6 dB x 255
exponent levels = 1530 dB. (255 levels come from the fact that
there is an 8-bit exponent). For floating-point audio
processing, we can see there is much more dynamic range
available than the 120 dB required for covering the full audio
dynamic range capabilities of the human ear.

Additional Fixed Point MAC Unit Dynamic Range for DSP
Overflow Prevention

Computation overflow/underflow is a hardware limitation that
occurs when the numerical result of the fixed-point computation
exceeds the largest or smallest number that can be represented
by the DSP. Many DSPs include additional bits in the MAC unit
to prevent overflow in intermediate calculations. Extended
sums-of-products, which are common in DSP algorithms, are
achieved in the MAC unit with single-cycle multiply-accumulates
placed in an efficient loop structure. The extra bits of
precision in the accumulator result register provide extended
dynamic range for protection against overflow in successive
multiplies and additions. Thus, no loss of data or range
occurs. Table 4 shows a comparison of the extended
dynamic ranges of 16-bit, 24-bit, and 32-bit DSPs.

Table 4: Comparison of the extended dynamic ranges of fixed-point DSP
multiplier units

Considering Data Word Length Issues When Developing Audio
Algorithms Free From Noise Artifacts


Digital Signal Processing is often discussed as if the signals
to be processed and the filter arithmetic used to process them
are both of infinite precision. However, all implementations of
DSP necessarily use words of finite length to represent each
and every value, be it a digital audio input sample, a filter
coefficient or the result of a multiplication. This finite
precision of representation means that any digital signal
processing performed to generate a desired result introduces
inaccuracy into the result. If a signal goes through several
stages of DSP, then each stage will add more inaccuracy.

The effects of a finite word length can severely effect
signal quality (in other words, lower the system S/N ratio) and
produce unacceptable error when performing DSP calculations.
Undesirable effects of finite precision can result of any of
the following:

  • A/D Conversion Noise
    Finite precision of an input data word sample will introduce
    some inaccuracy for the DSP computation as a result of the
    nonlinearities inherent in the A/D Conversion Process.
    Therefore, the accuracy of the result of an arithmetic
    computation can not be greater than the resolution of the
    quantized sample. In other words, the A/D conversion process
    will establish the noise floor for the DSP (unless the DAC
    has a lower noise floor). The DSP programmer must ensure that
    the noise floor of the processing algorithm does not exceed
    the noise floor of the ADC.

  • Quantization Error of Arithmetic Computations From
    Truncation and Rounding

    DSP Algorithms such as Digital Filters will generate results
    that must be truncated or rounded up (in other words,
    re-quantized). When a processing result need to be stored, it
    must be quantized to the native data-word length of the
    processor, introducing an error. For recursive DSP algorithms
    these re-quantized values are part of a feedback loop,
    causing arithmetic errors that can build up, which then reduces
    the dynamic range of the filter. The smaller the data word of
    the DSP, the more likely these types of errors will show up
    in the D/A converted output analog signal.

    In a n-bit fixed-point system, quantization of results may
    be considered as the addition of noise to the result.
    Consider a multiplication operation in a digital filter,
    including re-quantization of the result. This can be modeled
    as an infinite-precision multiplication followed by an
    addition stage where quantization noise is added to the
    product so that the result is equal to a n-bit number.

    In a digital-signal-processing system, multiplication,
    addition, and shift operations are performed on a sequence of
    n-bit input values. These operations generate results which
    would require more than n bits to be represented accurately.
    The solution to this problem is generally to eliminate the
    low-order bits resulting from an arithmetic operation in
    order to produce a n-bit value which can be stored by the
    system.

    The two most common methods for eliminating the low-order
    bits are truncation and rounding. Truncation is accomplished
    by simply discarding all bits less significant than the least
    significant bit that is retained. Rounding is performed by
    choosing the n-bit number which is closest to the original
    unrounded quantity.

  • Computational Overflow
    Whenever the result of an arithmetic computation is larger
    than the highest positive or negative full-scale value, an
    overflow will occur and the true result will be lost.

  • Coefficient Quantization
    Finite Word Length (n-bit data word size) of a filter
    coefficient can affect pole/zero placement and a digital
    filter's frequency response. This imprecision can cause
    distortion in the frequency response of the filter and, in
    the worst case, instability.

    Errors in the values of a filter's coefficients cause
    alterations in the positions of the transfer-function poles
    and zeros and therefore are manifested as changes to the
    frequency and phase-response characteristics of the filter.
    In a DSP system of finite precision, such deviations cannot
    be avoided. It can, however, be reduced by using greater
    precision for the representation of coefficients. This issue
    is particularly important for poles close to the unit circle
    in the z-plane, where an inaccuracy could make the difference
    between stability and instability.

  • Limit Cycles
    These occur in IIR filters from truncation and rounding of
    multiplication results or addition overflow. These often
    cause periodic oscillations in the output result, even when
    the input is zero.

Other than A/D Conversion Noise, all other effects of having
a finite data-word size are mainly dependent on the precision
of the re-quantization of data and the type of arithmetic
operations used in the DSP algorithm. Any given filter
structure can offer a significantly lower noise floor over
another structure which accomplishes the same task.


“The overall DSP-based audio system dynamic range is
only as good as its weakest link”

In a DSP-based audio system, this means that any one of the
following sources or devices in the audio signal chain will
determine the dynamic range of the overall audio system :

  1. The “real world” analog input signal, typically from a
    microphone or line-level source
  2. The ADC word size and conversion errors
  3. DSP finite word length effects such as quantization
    errors resulting from truncation and rounding, and filter
    coefficient quantization
  4. The DAC word size
  5. The analog output circuitry connecting to a speaker
  6. Another device in the signal path that will further
    process the audio signal.

“For a digital filter routine to operate
transparently, the resolution of the processing
system must be considerably greater than that of the
input signal so that any errors introduced by the
arithmetic computations are smaller than the
precision of the ADC or DAC”

So, the choice of components and the digital filter
implementation will also determine the overall quality of the
processed signal. For example, if we have a 75 dB DAC and a DSP
which can maintain 144 dB dynamic range, the overall 'System'
dynamic range will still only be 75 dB. So the DAC is the
limiting factor. Even though the DSP would compute a given
algorithm and maintain a result that had 122 dB of precision
and dynamic range, the result would have to be truncated in
order for the DAC to properly convert it back to an analog
signal. Now, if the choice is made to use high-quality analog, ADC,
and DAC components, wouldn't one want to be careful to ensure
the signal quality is maintained by the DSP algorithm? Care
must then be taken in a digital system to ensure the DSP is not
the weakest chain in the 'signal chain'.

If a digital-signal-processing algorithm produces
quantization noise artifacts which are above the noise floor of
the input signal, then these artifacts will be audible under
certain circumstances, especially when an input signal is of
low intensity or limited frequency. Therefore, whatever the
dynamic range of a high-quality audio input, be it 16-, 20-, or
24-bit input samples, the digital processing performed
on it should be designed to prevent processing noise from
reaching levels at which it may appear above the noise floor of
the input, and thus become audible content. For a digital filter routine to operate transparently,
the resolution of the processing system must be considerably
greater than that of the input signal so that any errors
introduced by the arithmetic computations are smaller than the
precision of the ADC or DAC. In order for the DSP to maintain
the SNR established by the ADC, all intermediate DSP
calculations require the use of higher precision processing
greater than the input sample word-size.

What are the dynamic ranges that must be maintained for
CD-quality and professional-quality audio designs? Fielder
demonstrated the dynamic range requirements for consumer CD
audio requires 16-bit conversion/processing while the minimum
requirement for professional audio is 20-bits (based on
perceptual tests performed on human auditory capabilities).
Traditional dynamic range application requirements for
high-fidelity audio processing can be categorized into two
groups:

  • 'Consumer CD-Quality' audio systems use 16-bit
    conversion with typical dynamic ranges between 85-93 dB
  • 'Professional-Quality' audio systems use 20- to 24-bit conversion with dynamic ranges between 110-122 dB.


Maintaining 16-Bit 'CD-Quality' Accuracy During DSP
Processing

As we saw in the last section, when using a DSP to process
audio signals, the DSP designer must ensure that any
quantization errors introduced by the arithmetic calculations
executed on the processor are lower than the converter noise
floor. Consider a 'CD-quality' audio system. If the DSP is to
process audio data from a 16-bit ADC (ideal case), a 96 dB SNR
must be maintained through the algorithmic process in order to
maintain a CD-quality audio signal (6×16=96dB). Therefore, it
is important that all intermediate calculations be performed
with higher precision than the 16-bit ADC or DAC resolution. Errors introduced by the arithmetic calculations can be
minimized when using larger data-word width sizes for
processing audio signals. For fractional fixed-point math, we
can visualize the addition of extra 'footroom' bits added to
the right of the least significant bit of the input sample. The
larger word sizes used in the arithmetic operations will ensure
that truncation or round-off errors will be lower than the
noise floor of the DAC, as long as 'optimal' algorithms (better
filter structures) are utilized in conjunction with the larger
word width.

When considering selection of a processor for
implementation, a choice therefore has to be made. Should one
use a lower data-word DSP using double-precision math, or
should a higher data-word DSP be used supporting single-precision math, which is more efficient? It is estimated that
double-precision math operations can take up to 4-5 times the
overhead of single precision math. Double-precision not
only adds computation overhead to a digital filter, it also
doubles the memory storage requirements for the filter
coefficient buffer and the input delay line buffer. Every
application is different, and although some applications may
suffice smaller native data-word width processor, the use of
double-precision computations, coefficients and intermediate
storage comes at the expense of a drastic reduction in
processing throughput.

To visually see the benefits of a larger DSP word size,
let's take a look at the processing of audio signals from a
16-bit ADC that has a dynamic range close to its theoretical
maximum, in this case with a 92 dB signal-to-noise ratio (Figure 9 ). Figure 10 below shows a conceptual
view of a 16-bit data word that is transferred from an ADC to
the DSP's internal memory. Typically, the data transfer would
occur through a serial port interface from the serial ADC, and
the DSP may be configured to automatically perform a direct
memory transfer (DMA) of the sample at the serial port
circuitry to internal memory for processing. Notice that for
the 24-bit and 32-bit processors, there are adequate
'footroom-bits' below the noise floor (to the right) to protect
against quantization errors.

Figure 9:  Fixed-point DSP noise floor with a typical
16-bit ADC/DAC at 92 dB

Figure 10:  16-bit A/D samples at 96 dB SNR

The 16-bit DSP has 4 dB higher SNR than the ADC's 92 dB, so
not much room for error would be allowed in arithmetic
computations. We can easily see that for moderate-to-complex
audio processing using single-precision arithmetic, the 16-bit
DSP data path will not be adequate for precise processing of
16-bit samples as a result of truncation and round-off errors
that can accumulate during the execution of the algorithm. As
shown in Figure 11 , errors resulting from the arithmetic
computations can easily be seen by the output DAC and thus
become audible noise. For example, complex recursive
computations can easily result in the introduction of 18 dB of
quantization noise, and with the 16-bit DSP word width, the
errors are seen by the DAC and hence will be easily heard by
the listener.

Figure 11:  16-bit D/A output samples with finite
length effects

Double-precision math can obviously still be used for the
16-bit DSP if software overhead is available, but the real
performance of the processor will be compromised. A 16-bit DSP
using single-precision processing would only suffice for
low-cost audio applications where processing is not too complex
and SNR requirements are around 75 dB (audio-cassette
quality).

The same algorithm implemented on a 24-bit or 32-bit DSP
would ensure these errors are not seen by the DAC. As can be
seen in the Figure 11 , even though 18 dB of quantization
noise was introduced by the computations in the 24-bit and
32-bit DSP, they remain well below the noise floor of the
16-bit DAC when these two processors run the exact same
algorithm.

The 24-bit DSP has 8 bits below the converter noise floor to
allow for errors. In other words, we have eight digits to the
right of the least significant bit in the 16-bit input sample.
It takes 256 multiplicative processing operations to be
performed before the noise floor of the algorithm goes above
the resolution of the input sample.

A 32-bit DSP has 16-bits below
the noise floor when executing 32-bit fractional math, allowing
for the greatest computation flexibility in developing stable,
noise-free audio algorithms. There are 16 digits to the right
of the least significant bit in the 16-bit input sample. It
would take 65,536 multiplicative processing operations before
the noise floor of the algorithm would go above the resolution
of the 16-bit input. With more room for quantization errors,
filter implementation restrictions seen with 16- or 24-bit DSPs
are now removed.

So, the higher number of bits used to process an audio
signal will result in a reduction in quantization error
(noise). If these errors remain below the noise floor, the
overall 'digital system SNR' established by the converters is
therefore maintained. The DSP should not the limiting factor in
signal quality! When using a 16-bit converter for 'CD-quality'
audio, the general recommendation widely accepted is to use a
higher resolution processor (24- or 32-bit) since additional
bits of precision gives the DSP the ability to maintain the 96
dB SNR of the audio converters.

Is 24-Bit Processing Always Enough for Maintaining 16-Bit
Sample Accuracy?

Now it would appear in some cases, 32-bit processing would be
unnecessary for minimal processing of 16-bit data. In order to
maintain a 96 dB dynamic range, 24 bits would appear to be
sufficient to process a 16-bit signal without any
double-precision math requirement. But the question is then
asked: Is a 24-bit DSP sufficient in all cases to guarantee
that noise introduced in a DSP computation will never go above
a 16-bit noise floor? For moderate and non-recursive DSP
operations, 24-bits should normally be sufficient. However,
research conducted in recent years has clearly shown that for
precise processing of 16-bit signals in recursive audio
processing, a 24-bit DSP may not be sufficient. Recursive
filters are necessary for a wide variety of audio applications
such as graphic equalizers, parametric equalizers, and comb
filters.

In a 1993 AES Journal publication, R. Wilson
demonstrated that even for recursive second-order IIR filter
computations on a 24-bit DSP, the noise floor of the digital
filter can still go above that of the 16-bit sample and hence
become audible. To compensate for this the use of error
feedback schemes (error spectrum shaping) or double-precision
arithmetic were recommended, especially for extremely critical
frequency response designs. The use of double-precision math
can add processor computational overhead by more than a factor
of five in the filter computations, while doubling memory
storage requirements.

Another March 1996 AES Journal publication by W. Chen
came to the same conclusion. In order to maintain the 96-dB
signal-to-noise ratio for 24-bit processing of second-order IIR
filters, a double-precision filter structure was required to
ensure that the digital equalizer output's noise floor was
greater than 96 dB. Chen researched various second-order
realizations to determine the best structure when performing
24-bit processing on 16-bit input. In one test case, he
implemented a single high-pass second-order filter using
direct-form-1 structures, finding these implementations to
yield an SNR between 85 to 88 dB, which is lower than the 96 dB
theoretical maximum of the ideal 16-bit ADC.

Chen's second example consisted of cascading of second-order
structures to implement a sixteenth-order digital equalizer. He
then measured the noise floor of the equalizer using an Audio
Precision System One tester in order to find an adequate
second-order IIR filter structure to meet his target 96-dB
requirement. The results of using the 24-bit DSP on a 16-bit
sample are shown in Table 5 .

Second-Order
Filter Structure
S/N Ratio (dB)
Results for 16th-order Equalizer
Cascaded Form 1 -75 dB
Cascaded Form 2 -63 dB
Cascaded Transposed Form 1 -70 dB
Double Precision Cascaded Form 1 -100 dB
Parallel Form 1 -85 dB
Parallel Transposed Form 1 -79 dB

Table 5: Chen's Results of 24-bit 2nd Order IIR Processing on 16-bit
Data (March 1996 Journal of AES)

Chen's conclusion”in order to maintain a higher
signal-to-noise ratio greater than 96 dB when cascading
multiple second-order stages, double-precision arithmetic was
required. In his optimal implementation of the double-precision
direct-form-1 filter, there was an increase in the number of
instruction cycles (3x increase) and greater memory space (2x
increase) for storing internal filter states.


“When processing of 16-bit samples with a 32-bit
processor versus a 24-bit processor, the 8 additional
bits available below the noise floor and the use of
32-bit filter coefficients will ensure that
double-precision overhead is not necessary when using
any standard second-order IIR filter realization.”

Recall that with a 32-bit DSP, there are 8 extra bits of
precision compared to a 24-bit processor. For a given
second-order filter structure implemented on a 24-bit processor
that is then implemented in a 32-bit fixed-point processor, the
arithmetic result should result in a reduction in the noise
floor by 48 dB. Direct-form 1 filter structures are generally
the best filter structure for use in audio, because of the better
noise performance they provide. For example, we can see
that in Chen's results (Table 5 ), the Parallel Form 1
structure used to construct the equalizer provided the best
result for single-precision 24-bit computation. However, this
is still less than the ideal 96-dB case. The 24-bit processor's
144-dB ideal noise floor is significantly raised by 70 to 80 dB
and, as a result, it is greater than the 16-bit converter's
noise floor. If this same algorithm is implemented on a 32-bit
fixed-point processor, the noise floor of the filter output is
lowered by 48 dB (with the 8 extra 'foot-room' bits) to 133 dB.
This is not only sufficient for remaining lower than a 16-bit
converter's noise floor, but a 32-bit implementation of the
single-precision direct-form 1 structure would be adequate for
even a 24-bit converter's noise floor as well.


Processing 110-120 dB, 20-/24-bit Professional-Quality Audio

When the compact disc was launched in the early 1980s, the
digital format of 16-bit words sampled at 44.1 kHz, was chosen
for a mixture of technical and commercial reasons. The choice
was limited by the quality of available analog-to-digital
converters, by the quality and cost of other digital
components, and by the density at which digital data could be
stored on the medium itself. It was thought that the format
would be sufficient to record audio signals with all the
fidelity required for the full range of human hearing. However,
research since the entrance of CD technology has shown that
this format is imperfect in some respects.

New research conducted within the last decade indicates that
the sensitivity of the human ear is such that the dynamic range
between the quietest sound detectable and the maximum sound
which can be experienced without pain is approximately 120 dB.
Therefore, 16-bit CD-quality audio is no longer thought to be
the highest-quality audio that can be stored and played back.
Also, many audiophiles claimed that CD-quality audio lacked a
certain warmth that a vinyl groove offered. This may have been
due to a combination of the dynamic range limitation of 16-bits
as well as the chosen sample rate of 44.1 kHz. The 16-bit words
used for CD allow a maximum dynamic range of 96 dB although
with the use of dither this is reduced to about 93 dB. Digital
conversion technology has now advanced to the stage where
recordings with a dynamic range of 120dB or greater may be
made, but compact disc is unable to accurately carry them.

Recent technological developments and improved knowledge of
human hearing have created a demand for greater word lengths
and faster sampling rates in the professional and consumer
audio sectors. It has long been assumed that the human ear was
capable of hearing sounds up to a frequency of about 20 kHz and
was completely insensitive to frequencies above this value.
This assumption was a major factor in the selection of a 44.1
kHz sampling rate. New research has suggested that many people
can distinguish the quality of audio at frequencies of up to 25
kHz, and that humans are also sensitive to a degree to
frequencies above even this value. This research is mainly
empirical, but would mean that a substantially higher sampling
frequency is necessary. D. E. Blackmer has suggested that
in order to fully meet the requirements of human auditory
perception, a sound system must be designed to cover the
frequency range to up to 40 kHz (and possibly up to 80 kHz)
with over 120 dB dynamic range to handle transient peaks. This
is beyond the requirements of many of today's digital audio
systems. As a result, 18-, 20-, and even 24-bit ADCs are now
widely available which are capable of exceeding the 96dB
dynamic range available using 16 bits.

The Race Toward The Use of 24-bit A/D and D/A
Conversion

Multibit Sigma-Delta Converters capable of 24-bit conversion
are now in production by various manufacturers, including Analog Devices,
Crystal Semiconductor, and AKM Semiconductor.
The popularity of 24-bit DACs is increasing for both
professional and high-end consumer applications. The reason for
using these higher precision ADCs and DACs for audio processing
is clear: the distortion performance (linearity) of these
higher resolution converters are much better than 16-bit
converters. The other obvious reason is the increase in SNR and
dynamic range that they provide over 16- to 20-bit
technology.


“24-bit ADC and DAC technology is capable of 120-122
dB dynamic range, fully supporting the dynamic range
capability of the human ear up to the threshold of
pain of 120 dB, at sample rates of 96 kHz and 192
kHz”

Many 24-bit converters on the market range from 110 to 120 dB,
which is professional quality and close to the range capable by
the human ear. The higher-end converters range from 117 dB to
122 dB (Conversion errors such as intermodulation distortion
introduced by the 24-bit converters limit the final SNR from
the theoretical 148 dB maximum). These newer 24-bit converters
have up to 120-122 dB dynamic range, easily allowing input
sources such as a 120 dB low-noise condenser microphone.

At many AES conventions in recent years, professional
equipment manufacturers have showcased equipment with 24-bit
conversion and 96 kHz sample rates. New DVD standards are
extending the digital formats to 24-bits at sample rates of 96
kHz and 192 kHz formats. Professional quality audio is emerging
in the consumer audio market sector, traditionally a market with
less stringent audio specifications. The race is on for audio
equipment manufacturers to include 24-bit, 96 kHz converters to
maintain signal quality up to 120 dB.

Comparing 24-Bit and 32-Bit Processing of Audio Signals
with 24-Bit Resolution

For years it has been widely accepted that in most cases 24-bit
DSP processing offers adequate precision for 16-bit samples.
With higher-precision 24-bit converters emerging to support
newer professional and consumer audio standards, what will
become the recommended processor word-width required to
maintain 24-bit precision? For 24-bit conversion, a 24-bit DSP
may no longer be able to adequately process 24-bit samples
without resorting to double-precision math, especially for
recursive second-order IIR algorithms. Newer 24-bit converter
technology is making a strong case for 32-bit processing. The
use of a 32-bit DSP has already become the logical
processor-of-choice for many audio equipment manufacturers when
using a 24-bit signal conversion. Let's examine why this is the
case.

Figure 12 visually demonstrates a typical situation
that can result from moderately complex or recursive processing
of 24-bit samples. Note that the 24-bit sample in this case is
assuming a 1.23 fractional number interpreted from the 24-bit
converters. The extra bits of precision provided by 32-bit fixed-point
processing are to the right of the 24-bit input's LSB. For
example, the parallel combination of second-order IIR filters
can result in significant quantization artifacts from in the
lower order bits of the data word. If both the 24-bit and
32-bit end up producing errors that result in an introduction
of 24 dB of noise (4 bits x 6 dB/bit), the error will show up
on the 24-bit DAC since the 24-bit DSP has the result above the
noise floor. Single-precision computations with 24-bit
processing can limit the result of a processed input to about
15-bit accuracy. Should one use double precision routines on
the 24-bit processor, or should one opt for a 32-bit processor
when using a 24-bit converter? Using a 32-bit processor, the
errors produced during the computations will never be seen by a
120 dB, 24-bit DAC.

Figure 12:  24-bit D/A output samples with finite
length effects

Recall earlier in the article, the analysis of Wilson's and
Chen's research demonstrated that for even second-order IIR
filter designs using a 24-bit processor, one may require the
use of additional error feedback computations or
double-precision math to ensure the noise floor remains lower
that a 16-bit converter. If 24-bit computations can introduce
noise artifacts that can go above a 16-bit noise floor for
complex second order filters, what does that mean? We can
conclude that a 24-bit DSP processing 24-bit samples will
result in the noise floor of the digital filter to always be
greater than the 24-bit converter's noise floor, unless methods
are implemented to reduce the digital filter's noise floor.
These costly methods of implementing error-feedback schemes and
double-precision arithmetic are unavoidable and can add
significant overhead in processing of 24-bit audio data.

With many converter manufacturers introducing 24-bit ADCs
and DACs to meet emerging consumer and professional audio
standards, the audio systems using these higher resolution
converters will require at least 32-bit processing in order to
offer sufficient precision to ensure that a filter algorithm's
quantization noise artifacts will not exceed the 24-bit input
signal. If optimal filter routines are used for complex
processing, any quantization noise introduced in the 32-bit
computations will never be seen by the 24-bit output DAC. In
many cases, the audio designer can choose from a number of
second-order structures because the result will still be
greater than 120 dB. 32-bit processing will guarantee that the
noise artifacts remain below the 120-dB noise floor, and hence
provide a dynamic range of the audio signal up the human ear's
threshold of pain. Therefore, the goal of developing robust
audio algorithms is accomplished, and the only limiting factor
when examining the signal quality (SNR) of the digital audio
system is the precision of the 24-bit ADC and DACs.


Summary of Data Word Size Requirements for Processing Audio
Signals

To maintain high audio-signal quality well above the noise
floor, all intermediate DSP calculations should be done using
higher precision than the bit length of the quantized input
data. High precision storage should also be used between the
DSP's memory and computation units. The use of “optimal” filter
algorithms, higher precision filter coefficients, and higher
precision storage of intermediate samples (available with
extended precision in the MAC unit) will ensure that errors
introduced by the arithmetic computations are much smaller than
the error introduced by the conversion of the results by a DAC.
Therefore, the noise floor of the digital filter algorithm will
be lower than the resolution of the ADCs and DACs.

A 16-bit DSP may suffice for low-cost audio applications
where processing is not complex and SNR requirements are around
75 dB. However, 16-bit DSPs using single-precision computations
will not be adequate for precise processing of 16-bit signals.
When using 16-bit ADCs and DACs in an audio system that will
process 'CD-quality' signals having a dynamic range of 90 to 96
dB, a 16-bit data path may not be adequate as a result of
truncation and rounding errors accumulating during execution of
the DSP algorithm. Double-precision routines can be utilized to
lower the digital filter's noise floor as long as the software
overhead is available.

While complexity for new DSP algorithms increase as audio
standards and requirements are increasing, designers are
looking to 18-bit, 20-bit, and 24-bit converters to increase
the signal quality. A 16-bit DSP will not be adequate due to
these higher resolution converter's dynamic range capabilities
exceeding those of a 16-bit DSP processor. However, a 16-bit DSP may
still be able to interface to these higher precision
converters, but this would then require the use of
double-precision arithmetic. Double-precision operations slow
down the true performance of the processor while increasing
programming complexity. Memory requirements for
double-precision math are doubled. Even if double-precision
math can be used, the interfaces to these higher precision
converters in many cases would require glue logic to move the
data to and from the DSP.

At least 24 bits are required in processing if the quality
of 16 bits is to be preserved. However, even with 24-bit
processing, it has been demonstrated that care would need to be
taken to ensure the noise floor of the digital filter algorithm
is not greater than the established noise floor of the 16-bit
signal, especially for recursive IIR audio filters. Recursive
IIR filters can introduce quantization noise above the noise
floor of a 16-bit converter when using a 24-bit DSP and
therefore 24-bit processing requires software overhead to lower
the digital filter's noise floor. Again, double-precision math
is an option, but this can add overhead by as much as a factor
of five.

Using a 32-bit, fixed-point DSP will offer an additional benefit
of ensuring 16-bit signal quality is not impaired during
arithmetic computations. Thus, the higher resolution of the
32-bit DSP will eliminate quantization noise from showing up in
the DAC output, providing improved Signal-to-Noise (SNR) ratio
over 16- and 24-bit DSPs.

When processing 16-bit audio data, the use of 32-bit
processing is especially useful for complex recursive
processing using IIR filters. For example, parametric and
graphic equalizer implementations using cascaded 2nd-order IIR
filters, and comb/allpass filters for audio are more robust
using 32-bit math. A 32-bit processor operating on 16- or
20-bit data removes the filter structure implementation
restrictions that are present for 24-bit processors. Any filter
structure of choice can then be used without worrying about the
level of the noise floor. Double-precision and error-feedback
schemes are therefore eliminated. With 16-bits below the noise
floor on a 32-bit DSP, quantization errors would have to
accumulate up to 96 dB from the LSB before these errors can be
seen by the 16-bit DAC.

At least 32 bits are required if 24-bit signals are to be
preserved with complex, math-intensive, or recursive
processing. Using 24-bit ADCs and DACs will require a 32-bit
DSP in order to offer sufficient precision to ensure that the
noise floor of the algorithm will not exceed the 24-bit input
signal.

For more information about DSPs, visit the Analog Devices Web site.

1 comment on “The Relationship of Dynamic Range to Data Word Size in Digital Audio Processing

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