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:** Â Signed twos-complement representation of

sampled signals

**Figure 2:** Â 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:** Â 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:** Â 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 as defined by Davis and Jones (we willFigure 5 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:** Â 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 2^{n} quantization levels (some examples

for common data word widths are shown in **Table 2** ).

N Quantization Levels for n-bit data words (N = 2 ^{n}levels) |
---|

2^{8} = 256 |

2^{16} = 65,536 |

2^{20} = 1,048,576 |

2^{24} = 16,777,216 |

2^{32} = 4,294,967,296 |

2^{64} = 18,446,744,073,729,551,616 |

**Table 2:**

An n-bit data word yields 2^{n} 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:** Â 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:** Â 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:** Â 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 :

- The “real world” analog input signal, typically from a

microphone or line-level source - The ADC word size and conversion errors
- DSP finite word length effects such as quantization

errors resulting from truncation and rounding, and filter

coefficient quantization - The DAC word size
- The analog output circuitry connecting to a speaker
- 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:** Â Fixed-point DSP noise floor with a typical

16-bit ADC/DAC at 92 dB

**Figure 10:** Â 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:** Â 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:** Â 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.

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