Data Acquisition

Texas Instruments Incorporated

How digital filters affect analog
audio signal levels
By Jorge Arbona, Applications Engineer,and Supriyo Palit, Software Systems Engineer
Introduction
Digital audio processing provides a great amount of flexibility to system designers. Multiple filter structures can becascaded to form equalization (EQ), low-pass, high-pass,
shelf, and many other filter combinations with relatively
low power consumption and little PCB space. Infiniteimpulse-response (IIR) filters can be used to easily simulate filter functions performed by analog counterparts.
Digital audio signals are represented as an array of bitswith a fixed resolution. This means that the signal is discrete in nature, both in amplitude and in time. If the sourceof this data is analog, it is quantized and sampled at fixedintervals (sampling periods) by an analog-to-digital converter (ADC). An audio engineer has to be careful toensure that the signal being recorded is not clipped, while
maintaining the signal as loud as possible to maximize thesignal-to-noise ratio (SNR). An ADC has an amplitude
limit, which may be defined as a full-scale voltage, meaning that any signal above a certain amount of volts at theconverter’s input may result in clipping. Also, the signal isquantized into a number with a certain fixed resolution(which might also be close to the clipping point).
A mastering engineer also has to be careful when working with music in the digital domain. Some EQ can be

added to boost certain frequencies as well as to achieve
other effects. If there is an extravagant amount of headroom, the engineer can boost, boost, and boost. However,the final medium for the music is a CD (which is limited to
16-bit audio), so trade-offs have to be made because therest of the music sounds too “quiet” compared to the SNRof the medium.
Boosting frequency bands in the digital domain cancreate certain problems when the bands are converted into
the analog domain. Digital-to-analog converters (DACs)
can also clip the signal if their digital input is larger thantheir full-scale voltage. Most processors allow a certain
amount of headroom to work with intermediate values, butultimately a DAC expects a certain data width bounded by
maximum and minimum values. If the signal is scaled down
and certain frequencies are then boosted to accommodate
the higher peaks, then the SNR at the flat region(s) will
suffer from a lower SNR.

Understanding the problem
A very important aspect to consider when digital filters areused is how the signal level is affected upon its conversion
from the digital to the analog domain. Suppose that a system provides a digital signal to a processing unit and converts it to analog using an ideal DAC without any proc­ess­
ing being applied, as shown in Figure 1. In this example, a0-dBFS digital signal is provided to the DAC and converted

Amplitude (dB)

–130

0

–130

2 n–1–1
0–2 n–1
1-kHz Time
EQ

Source Data

0

–130

1 k Frequency (Hz)DAC Range

DAC Range

1 k Frequency (Hz)1 k Frequency (Hz)DACOUT (V)0

Amplitude (dB)

Amplitude (dB)Figure 1. Full-scale digital signal represented as a 1-Vpeak analog signal

2 n–1–1
0–2 n–1Time

Processor

1
0
–1
Time

DAC

Amp

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into the analog domain. A relationship between the digitalcode and the analog output amplitude is provided in thespecification of a codec as the full-scale amplitude. If thespecification of the full-scale amplitude is 0.707 VRMS (or1 Vpeak), this means that a full-scale 0-dBFS digital sinusoid will result in a 1-Vpeak sinusoid, as shown in the figure.
If a DAC is bounded by –2n–1 and 2n–1 – 1, amplifying asignal beyond these limits will distort the signal by clipping
it at its output (assuming saturation logic), as shown in
Figure 2. Note that it is typical for most signal processors
to allow some amount of headroom before providing data to
a DAC. It is important for the data within the proc­ess­or’s

memory to remain undistorted. Figure 2 illustrates the
DAC input limits where output clipping may occur if
exceeded.
A solution to this problem would be to ensure that thesignal is not amplified beyond the DAC’s limits (i.e., ensurethat positive gain is not applied to the source signal).
However, there are cases for which the solution is not as
obvious. Performing a boost relative to the full-scale
amplitude of the DAC input at a specific frequency range
will also cause adverse effects. In Figure 3, a 500-Hz signal
is boosted by 6 dB. The distortion observed in the analog
output is due to DAC clipping.

Figure 2. Excessive gain in the digital domain can makethe signal exceed the DAC’s upper and lower limits

DAC’s Positive Limit

2 n–1–1

0–2 n–1

DAC’s Negative Limit

Figure 3. Possible effects of boost on a specific frequency band
Analog Output

6
0

–130

–130
Frequency (Hz)1k

DACOUT (V)

1k

DAC’s
Limits60

Amplitude (dB)

Amplitude (dB)Processor

2 n–1–1
0–2 n–1Time

Frequency (Hz)

1
0
–1
Time

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This concept is also illustrated in Figure 4. Note thatthe noise from the source data is inherited when passed
to the larger bus width of the processor’s memory. As
mentioned previously, data could be scaled down by the
maximum amount of total boost to accommodate theboosted regions. However, as seen in Figure 5, even if theboost reference point is in a good position, the DAC signal
might be affected by the output SNR. If the amount ofboost does not compromise overall system SNR significantly, then sim­ple scaling might be a viable solution.
Some low-power codecs provide 100 dB of SNR, whichallows some amount of scaling without sacrificing the SNRof the original 16-bit source.

Figure 4. Signal content throughout the digital
signal chain

Max

Max
Boost

Max

Min

Quantization and number representation

Min

Low
Data ispassed to frequencies
memory are boosted

16-BitDigitalSignal

Max

Min

Min

In digital processing, a real number is represented as aninteger value with a fixed precision. This is called quanti­za­tion, and the quantized value is an approximation of
the original value. The integer value can be represented
as a fixed-point number or a floating-point number. An
integer value represented as a fixed-point number iscomposed of magnitude bits and fractional bits. An integer value represented as a floating-point number is composed of exponent bits and mantissa bits. This discussion
will henceforth be restricted to fixed-point numbers andfixed-point arithmetic.
A fixed-point number is represented as a twoscomplement integer with a fixed number of digits after theradix point (or the decimal point). These digits make up
the fractional part of the number. The digits before theradix point are the magnitude part and denote the rangeof the number. The magnitude part also contains the signof the number.

Clipping

32-BitProcessor
MemoryDAC
clips thesignalDAC
Input

Digital data coming into an audio processor is considered to be a real number lying between –1 and 1 – 1LSB.
Assuming that the real value is represented as a 16-bit
fixed-point number, the number –1 will be represented as
1000000000000000 in binary (or 0x8000 in hexadecimal).
In twos-complement arithmetic, 0x8000 corresponds to an
integer value equal to –32768. This means that dividing
the integer number by 32768 will result in the quantized

Figure 5. Signal content with the use of scaling

Max

Max

Max

Max

Boost
Ref

Min

Min

Min
Data ispassed tomemory16-BitDigitalSignalMax

Min
Data isscaleddown

32-BitProcessor
MemoryMin
Boost isadded tospecific
frequencies

DAC
Input

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Figure 6. Fixed-point representation of a real number
Magnitude Part
Fractional Part
15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Decimal Point

Signal bits

approximation of the real value. The largest positive number in 16 bits is 0111111111111111 in binary (or 0x7FFF
in hexadecimal). The corresponding integer value is 32767.
Dividing this by the scale factor of 32768 produces thelargest real number that can be represented in this format.
The number is 32767/32768 = 0.999969482421875. The
fixed-point representation is shown in Figure 6.
In this representation, there are 15 fractional bits and 1
magnitude bit, which is also the sign bit. This means that areal number must lie between –1 and 0.999969482421875
before quantization. If the real number is above or below
this range, it cannot be represented in the given format
because the 16-bit register will overflow. To accommodate
larger real numbers, the magnitude part needs to be
increased at the expense of a reduced fractional part. This
format is also known as the 1.15 format (1 = magnitude
bits, and 15 = sign bits). Input to a digital processor isalways represented in 1.n format, where n is the numberof fractional bits (15, 19, 23, or 31). A value of 0 dBFS
corresponds to the RMS value of a full-scale sine wave
whose amplitude is (2n – 1)/2n. The largest real number inthe given format is represented by 2n. The number of bitsthat are used to represent a signal is called the signal bit
width or the data bit width.

where S is a uniformly distributed random signal, NC is
DAC circuit noise, and NQ is quantization noise. Using a
100-dB DAC and a 120-dB signal processor will result inan overall SNR of 99.96 dB.
It should be noted that the overall SNR is also limited by
the source—the input to the digital audio processor. If the
input is provided as a 16-bit number, then the signal-toquantization-noise ratio (SQNR) of the system can, at
best, be 96 dB (assuming a uniformly distributed random
signal, unweighted). So, even a higher-bit internal representation (lower NQ) will not provide much improvementin this case.

Overflow and saturation

Noise bits

Overflow occurs when a processing unit’s computation
results in a value greater in magnitude than the data bit
width. Overflow is typically associated with computation in
the accumulator, where successive numbers of the same
sign are added and stored. Accumulators usually keep
accumulating even after overflow because the final result,if within bounds, will still come out correctly.
The output of the accumulator is saturated before it isstored as a signal value. Saturation is a process where apositive overflow is converted to the maximum positive
number and a negative overflow is converted to the minimum negative number. Saturation is a nonlinear operation
and results in severe harmonic distortion of the output.
Headroom bits are used to prevent saturation.

Earlier it was mentioned that the number of signal bits
de­termines the performance of the digital audio system.
More bits are sometimes needed for filter-response
calculations.
A filter implementation consists of a data path throughwhich the signal flows and is stored as delay elements forthe filter. The signal and delay values are multiplied by thecoefficients associated with the filter taps. Coefficient quan­tization also plays an important role in system perform­
ance. The product of the signal and coefficient values isstored in the accumulator, which usually has a higher bit
width than that of the signal. Subsequent products areadded in the accumulator (at a higher bit width), and thefinal filter output is then stored back in signal precision(at a lower bit width).

Signal and noise bits affect the performance of a system.
The digital audio processor adds quantization noise, andthe overall performance will be the effect of both the analog
circuit noise and the quantization noise. Assuming thatboth noise sources are the outcome of independent random
processes, the overall system noise performance can be
defined as
 S2

,
SNR = 10 log10  2
 N + N2 
Q
 C

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Consider the biquad filter implementation in Figure 7.
In this figure, the input and output signals are represented
by “A bits.” The a and b coefficients are represented by“B bits.” The input signal and its delay elements are multi­
plied by the coefficients and added at the accumulator.
The multiplier and accumulator together are A + B bits
wide. The output signal is then quantized by the Q blockand stored as an A-bit number. This introduces a quantization error, which is a noise source for the digital filter;
therefore extra bits are needed to ensure that the noise
contribution from the digital filter is below the target
SNR. These extra bits are called the noise bits. The effectof noise is more pronounced with IIR filters than withfinite-impulse-response (FIR) filters. The number of noise
bits also depends on the sampling frequency and the cutoff frequency of the digital filter. As the sampling fre­quency increases, the number of noise bits required
increases. As the cutoff frequency decreases, the numberof noise bits required increases. For a 48-kHz operation,
14 to 16 noise bits are enough to maintain the target SNR
for a 40-Hz IIR filter.

Headroom bits
Other than signal and noise bits, additional bits are needed
to prevent overflow. These bits are called headroom bits.
An end-to-end audio-processing chain will usually preserve the signal level. This means that if a 0-dB signal is
input to the signal chain, the output will measure 0 dB or
less. (Usually there is a signal compressor that will limit
the signal swing to a few decibels below zero.) If boost
filters are used to amplify specific signal bands, the remain­ing bands are usually attenuated to prevent the signal fromgoing above 0 dB. For the latter case, when the inputsignal level is at 0 dB (also known as the neutral signal
level), the output signal will be lower than 0 dB, and only
the amplified bands will reach 0 dB at the output. This will
reduce the average volume level of the audio signal.
In spite of the signal level being maintained at 0 dB, thesignal can overflow at intermediate processing points. To
prevent overflow, headroom bits—i.e., bits in addition tosignal and noise bits—are needed.
There can be two sources of overflow:
1. An audio-processing chain can have a filter whose gain
(at some specific frequency values) is greater than 0 dB.
The filter can be part of a cascaded filter chain (e.g.,
low-pass, high-pass, and/or band-pass filters) whose
overall gain is 0 dB, or it can be a frequency-selective
filter that amplifies a specific frequency band relative
to the neutral signal level (e.g., shelf and EQ filters).
Note that if a real number is represented in 1.n format(where n is the number of fractional bits), the magnitude of the number is always less than 1. So, if a filter
with a gain of more than 0 dB (a real number greaterthan 1) is used, then the output value from the filter isgoing to overflow if the input value is 0 dB (a real number equal to 1). To prevent overflow in such cases,
more headroom bits are needed.

Figure 7. Biquad filter implementation
A BitsB BitsA+B Bits

x(n)

b0

+

A Bits
y(n)

Q

z–1

z–1
b1

–a1

z–1

z–1
b2

–a2

Figure 8. Representation of signal withheadroom bits

Headroom
Bits
InputSignal
Bits

OutputSignal
Bits
Noise
Bits

2. A filter with a gain of less than or equal to 0 dB can haveinstantaneous real values greater than 1. To ensure thatthese instantaneous values do not overflow, headroom
bits are needed.
A pictorial representation of the signal in the audio proc­
essor is shown in Figure 8. An important point to note isthat headroom bits are primarily used to accommodate
intermediate signal growth. It is expected that at the endof the final processing block, the output will fit within thesignal bit width. Otherwise, for low-signal amplitudes, theoutput will still be within limits and not distort; but, for
high-signal amplitudes, the output will get saturated andcause distortion. To prevent distortion, it is best to attenuate the signal prior to the final output.

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Figure 9. Scaling the transfer function
1-kHz EQ

Amplitude (dB)

DAC Range

Input
2 n–1–1
0–2 n–1
Time

6
0

–130
1 k Frequency (Hz)Scaling
Scaling can be used to avoid saturation for filters with gain
exceeding 0 dB. A boost filter can be used intentionallythat will gain a particular frequency. Even a multisection
low-pass filter can have a biquad section that actually gains
certain frequencies that are higher than the available head­room (the overall response will still be 0 dB). In such acase, whether or not to use scaling can be determined bymultiplying the input-signal level by the maximum gain of
the total filter response. If the product is greater than theavailable DAC headroom, then scaling could be used to
avoid saturation.
One method of scaling is to attenuate the system’s trans­
fer function by an amount equal to the maximum amplitudeof the filter’s transfer function. The scaling factor can be
defined as
S = max H(e jω ) ,where 0 ≤ ω ≤ π. A second method is to scale the input
signal by S. Figure 9 demonstrates the effect of scaling thetransfer function. A full-scale sinusoid is input to the trans­
fer function, which attenuates the flat frequencies by 6 dB.
Relative to –6 dBFS, the 1-kHz signal is boosted by 6 dB.
In some cases, due to the filter structure and the instantaneous signal sequences, the output of a filter can be
more than 0 dB even though it does not have a gain of
more than 0 dB. An FIR filter can increase the gain of asignal by a sum of the absolute value of the filter taps if
the individual memory elements are at 0 dB with a sign
opposite to that of the taps. The response of the filter may
not exceed 0 dB, so additional headroom may be applied.
Computing additional headroom for IIR filters is complex

because they have feedback elements, and finding a closedform expression to determine the upper limit for instantaneous gain is complex. In fact, one of the reasons signal
processors provide additional headroom (above the DAC
limit) is to allow headroom for instantaneous values. Mea­
surements might be needed to compute additional headroom. In some cases, the SNR may need to be traded off
for distortion that is due to saturation, and then an analog
gain may need to be added to get the signal back to 0 dB.
When scaling is used, it is sometimes desirable to add
additional gain (boost) in the analog output stage to compensate. Special care should be taken to ensure that thesignal of the boosted regions does not saturate the output
amplifiers as well, resulting in a distorted signal. Boost isalso provided at the final output stage of the processor tocompensate for scaling. This is required for multisection,
0-dB filters where scaling has been done to prevent overflow for one or more of the individual sections. For filtersthat gain frequencies above 0 dB (EQ and shelf filters),
the neutral signal level is scaled below 0 dB. In this case,the final-stage boost is not required. The result is a loss of
SNR for the flat regions.
A more elegant solution is to limit the amount of filter
gain based upon the volume gain applied at the digitalproc­essor, which is very well-suited for headphone use. At
higher volume levels, the frequency boost can be lowered
and, ultimately, be flat at full volume.
In some cases, the frequency boost is kept constant,while the signal is compressed when the volume is high.
This is the anti-clipping dynamic-range compressor (DRC)
function: At low volume levels, the original SNR is maintained; but, as volume increases, the scaling is proportionately increased to prevent distortion.

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Figure 10. Headphone volume relative to human hearing
140-dB SPL

Threshold of Pain
Jet Aircraft
Truck–6 dBVAmp

32 Ω

Quiet Room
0-dB SPL

Threshold of Hearing–106 dBV

Regardless of the method used, it is important to consider how humans perceive sound and noise. Human
hearing has an outstanding dynamic range. Headphone
amplifiers trade between noise floor and output power
to best accommodate this range. For instance, theTLV320AIC3254 audio codec can deliver a very highsound-pressure level (SPL) with just 500 mVRMS into a
typical 32-Ω or 16-Ω headphone load, and at the sametime can have a noise floor of 100 dB (A-weighted) below
full scale, which can be below the threshold of hearing(see Figure 10). Sometimes, it is not even necessary to
add additional amplification after scaling is performed,
since the output power could be very well above the
comfortable listening level.

References
1. David Zaucha, “Importance of precision on performancefor digital audio filters,” 112th Audio Engineering
Society (AES) Convention, Munich, Germany, April
2002, Paper 5613.
2. W. Marshall Leach, Jr., Introduction to Electroacoustics
and Audio Amplifier Design, 3rd ed. (Dubuque, IA:
Kendall/Hunt Publishing Co., 2003), p. 7.

Related Web sites
www.ti.com/digitalaudio
www.ti.com/sc/device/TLV320AIC3254

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