3 ADC and DAC

CHAPTER
3
ADC and DAC
Most of the signals directly encountered in science and engineering are continuous: light intensity
that changes with distance; voltage that varies over time; a chemical reaction rate that depends
on temperature, etc. Analog-to-Digital Conversion (ADC) and Digital-to-Analog Conversion
(DAC) are the processes that allow digital computers to interact with these everyday signals.
Digital information is different from its continuous counterpart in two important respects: it is
sampled, and it is quantized. Both of these restrict how much information a digital signal can
contain. This chapter is about information management: understanding what information you
need to retain, and what information you can afford to lose. In turn, this dictates the selection
of the sampling frequency, number of bits, and type of analog filtering needed for converting
between the analog and digital realms.
Quantization
First, a bit of trivia. As you know, it is a digital computer, not a digit
computer. The information processed is called digital data, not digit data.
Why then, is analog-to-digital conversion generally called: digitize and
digitization, rather than digitalize and digitalization? The answer is nothing
you would expect. When electronics got around to inventing digital techniques,
the preferred names had already been snatched up by the medical community
nearly a century before. Digitalize and digitalization mean to administer the
heart stimulant digitalis.
Figure 3-1 shows the electronic waveforms of a typical analog-to-digital
conversion. Figure (a) is the analog signal to be digitized. As shown by the
labels on the graph, this signal is a voltage that varies over time. To make
the numbers easier, we will assume that the voltage can vary from 0 to 4.095
volts, corresponding to the digital numbers between 0 and 4095 that will be
produced by a 12 bit digitizer. Notice that the block diagram is broken into
two sections, the sample-and-hold (S/H), and the analog-to-digital converter
(ADC). As you probably learned in electronics classes, the sample-and-hold
is required to keep the voltage entering the ADC constant while the
35
36
The Scientist and Engineer's Guide to Digital Signal Processing
conversion is taking place. However, this is not the reason it is shown here;
breaking the digitization into these two stages is an important theoretical model
for understanding digitization. The fact that it happens to look like common
electronics is just a fortunate bonus.
As shown by the difference between (a) and (b), the output of the sample-andhold is allowed to change only at periodic intervals, at which time it is made
identical to the instantaneous value of the input signal. Changes in the input
signal that occur between these sampling times are completely ignored. That
is, sampling converts the independent variable (time in this example) from
continuous to discrete.
As shown by the difference between (b) and (c), the ADC produces an integer
value between 0 and 4095 for each of the flat regions in (b). This introduces
an error, since each plateau can be any voltage between 0 and 4.095 volts. For
example, both 2.56000 volts and 2.56001 volts will be converted into digital
number 2560. In other words, quantization converts the dependent variable
(voltage in this example) from continuous to discrete.
Notice that we carefully avoid comparing (a) and (c), as this would lump the
sampling and quantization together. It is important that we analyze them
separately because they degrade the signal in different ways, as well as being
controlled by different parameters in the electronics. There are also cases
where one is used without the other. For instance, sampling without
quantization is used in switched capacitor filters.
First we will look at the effects of quantization. Any one sample in the
digitized signal can have a maximum error of ±½ LSB (Least Significant
Bit, jargon for the distance between adjacent quantization levels). Figure (d)
shows the quantization error for this particular example, found by subtracting
(b) from (c), with the appropriate conversions. In other words, the digital
output (c), is equivalent to the continuous input (b), plus a quantization error
(d). An important feature of this analysis is that the quantization error appears
very much like random noise.
This sets the stage for an important model of quantization error. In most cases,
quantization results in nothing more than the addition of a specific amount
of random noise to the signal. The additive noise is uniformly distributed
between ±½ LSB, has a mean of zero, and a standard deviation of 1/ 12 LSB
(-0.29 LSB). For example, passing an analog signal through an 8 bit digitizer
adds an rms noise of: 0.29 /256 , or about 1/900 of the full scale value. A 12
bit conversion adds a noise of: 0.29 /4096 . 1 /14,000 , while a 16 bit
conversion adds: 0.29 /65536 . 1 /227,000 . Since quantization error is a
random noise, the number of bits determines the precision of the data. For
example, you might make the statement: "We increased the precision of the
measurement from 8 to 12 bits."
This model is extremely powerful, because the random noise generated by
quantization will simply add to whatever noise is already present in the
Chapter 3- ADC and DAC
3.025
FIGURE 3-1
Waveforms illustrating the digitization process. The
conversion is broken into two stages to allow the
effects of sampling to be separated from the effects of
quantization. The first stage is the sample-and-hold
(S/H), where the only information retained is the
instantaneous value of the signal when the periodic
sampling takes place. In the second stage, the ADC
converts the voltage to the nearest integer number.
This results in each sample in the digitized signal
having an error of up to ±½ LSB, as shown in (d). As
a result, quantization can usually be modeled as
simply adding noise to the signal.
Amplitude (in volts)
a. Original analog signal
3.020
3.015
3.010
3.005
3.000
0
5
10
15
20
25
Time
30
35
40
45
37
50
digital
output
analog
input
ADC
S/H
3.025
3025
c. Digitized signal
3020
Digital number
3.020
3.015
3.010
3.005
3015
3010
3005
3.000
3000
0
5
10
15
20
25
Time
30
35
40
45
50
0
5
10
15
20
25
d. Quantization error
pdf
0.5
0.0
-0.5
-1.0
0
5
10
15
20
25
30
Sample number
35
40
45
50
30
Sample number
1.0
Error (in LSBs)
Amplitude (in volts)
b. Sampled analog signal
35
40
45
50
38
The Scientist and Engineer's Guide to Digital Signal Processing
analog signal. For example, imagine an analog signal with a maximum
amplitude of 1.0 volt, and a random noise of 1.0 millivolt rms. Digitizing this
signal to 8 bits results in 1.0 volt becoming digital number 255, and 1.0
millivolt becoming 0.255 LSB. As discussed in the last chapter, random noise
signals are combined by adding their variances. That is, the signals are added
in quadrature: A 2 %B 2 ' C . The total noise on the digitized signal is
therefore given by: 0.2552 % 0.292 ' 0.386 LSB. This is an increase of about
50% over the noise already in the analog signal. Digitizing this same signal
to 12 bits would produce virtually no increase in the noise, and nothing would
be lost due to quantization. When faced with the decision of how many bits
are needed in a system, ask two questions: (1) How much noise is already
present in the analog signal? (2) How much noise can be tolerated in the
digital signal?
When isn't this model of quantization valid? Only when the quantization
error cannot be treated as random. The only common occurrence of this
is when the analog signal remains at about the same value for many
consecutive samples, as is illustrated in Fig. 3-2a. The output remains
stuck on the same digital number for many samples in a row, even though
the analog signal may be changing up to ±½ LSB. Instead of being an
additive random noise, the quantization error now looks like a thresholding
effect or weird distortion.
Dithering is a common technique for improving the digitization of these
slowly varying signals. As shown in Fig. 3-2b, a small amount of random
noise is added to the analog signal. In this example, the added noise is
normally distributed with a standard deviation of 2/3 LSB, resulting in a peakto-peak amplitude of about 3 LSB. Figure (c) shows how the addition of this
dithering noise has affected the digitized signal. Even when the original analog
signal is changing by less than ±½ LSB, the added noise causes the digital
output to randomly toggle between adjacent levels.
To understand how this improves the situation, imagine that the input signal
is a constant analog voltage of 3.0001 volts, making it one-tenth of the way
between the digital levels 3000 and 3001. Without dithering, taking
10,000 samples of this signal would produce 10,000 identical numbers, all
having the value of 3000. Next, repeat the thought experiment with a small
amount of dithering noise added. The 10,000 values will now oscillate
between two (or more) levels, with about 90% having a value of 3000, and
10% having a value of 3001. Taking the average of all 10,000 values
results in something close to 3000.1. Even though a single measurement
has the inherent ±½ LSB limitation, the statistics of a large number of the
samples can do much better. This is quite a strange situation: adding
noise provides more information.
Circuits for dithering can be quite sophisticated, such as using a computer
to generate random numbers, and then passing them through a DAC to
produce the added noise. After digitization, the computer can subtract
Chapter 3- ADC and DAC
3005
a. Digitization of a small amplitude signal
b. Dithering noise added
3004
3004
3003
Millivolts
original analog signal
analog signal
3002
3003
3002
3001
3001
with added noise
digital signal
3000
0
5
10
15
20
25
30
35
Time (or sample number)
3000
40
45
FIGURE 3-2
Illustration of dithering. Figure (a) shows how
an analog signal that varies less than ±½ LSB can
become stuck on the same quantization level
during digitization. Dithering improves this
situation by adding a small amount of random
noise to the analog signal, such as shown in (b).
In this example, the added noise is normally
distributed with a standard deviation of 2/3 LSB.
As shown in (c), the added noise causes the
digitized signal to toggle between adjacent
quantization levels, providing more information
about the original signal.
50
0
5
10
15
20
25
Time
30
35
40
45
50
40
45
50
3005
Millivolts (or digital number)
Millivolts (or digital number)
3005
39
c. Digitization of dithered signal
3004
original analog signal
3003
3002
3001
digital signal
3000
0
5
10
15
20
25
30
35
Time (or sample number)
the random numbers from the digital signal using floating point arithmetic.
This elegant technique is called subtractive dither, but is only used in the
most elaborate systems. The simplest method, although not always possible,
is to use the noise already present in the analog signal for dithering.
The Sampling Theorem
The definition of proper sampling is quite simple. Suppose you sample a
continuous signal in some manner. If you can exactly reconstruct the analog
signal from the samples, you must have done the sampling properly. Even if
the sampled data appears confusing or incomplete, the key information has been
captured if you can reverse the process.
Figure 3-3 shows several sinusoids before and after digitization. The
continious line represents the analog signal entering the ADC, while the square
markers are the digital signal leaving the ADC. In (a), the analog signal is a
constant DC value, a cosine wave of zero frequency. Since the analog signal
is a series of straight lines between each of the samples, all of the information
needed to reconstruct the analog signal is contained in the digital data.
According to our definition, this is proper sampling.
40
The Scientist and Engineer's Guide to Digital Signal Processing
The sine wave shown in (b) has a frequency of 0.09 of the sampling rate. This
might represent, for example, a 90 cycle/second sine wave being sampled at
1000 samples/second. Expressed in another way, there are 11.1 samples taken
over each complete cycle of the sinusoid. This situation is more complicated
than the previous case, because the analog signal cannot be reconstructed by
simply drawing straight lines between the data points. Do these samples
properly represent the analog signal? The answer is yes, because no other
sinusoid, or combination of sinusoids, will produce this pattern of samples
(within the reasonable constraints listed below). These samples correspond to
only one analog signal, and therefore the analog signal can be exactly
reconstructed. Again, an instance of proper sampling.
In (c), the situation is made more difficult by increasing the sine wave's
frequency to 0.31 of the sampling rate. This results in only 3.2 samples per
sine wave cycle. Here the samples are so sparse that they don't even appear
to follow the general trend of the analog signal. Do these samples properly
represent the analog waveform? Again, the answer is yes, and for exactly the
same reason. The samples are a unique representation of the analog signal.
All of the information needed to reconstruct the continuous waveform is
contained in the digital data. How you go about doing this will be discussed
later in this chapter. Obviously, it must be more sophisticated than just
drawing straight lines between the data points. As strange as it seems, this is
proper sampling according to our definition.
In (d), the analog frequency is pushed even higher to 0.95 of the sampling rate,
with a mere 1.05 samples per sine wave cycle. Do these samples properly
represent the data? No, they don't! The samples represent a different sine wave
from the one contained in the analog signal. In particular, the original sine
wave of 0.95 frequency misrepresents itself as a sine wave of 0.05 frequency
in the digital signal. This phenomenon of sinusoids changing frequency during
sampling is called aliasing. Just as a criminal might take on an assumed name
or identity (an alias), the sinusoid assumes another frequency that is not its
own. Since the digital data is no longer uniquely related to a particular analog
signal, an unambiguous reconstruction is impossible. There is nothing in the
sampled data to suggest that the original analog signal had a frequency of 0.95
rather than 0.05. The sine wave has hidden its true identity completely; the
perfect crime has been committed! According to our definition, this is an
example of improper sampling.
This line of reasoning leads to a milestone in DSP, the sampling theorem.
Frequently this is called the Shannon sampling theorem, or the Nyquist
sampling theorem, after the authors of 1940s papers on the topic. The sampling
theorem indicates that a continuous signal can be properly sampled, only if it
does not contain frequency components above one-half of the sampling rate.
For instance, a sampling rate of 2,000 samples/second requires the analog
signal to be composed of frequencies below 1000 cycles/second. If frequencies
above this limit are present in the signal, they will be aliased to frequencies
between 0 and 1000 cycles/second, combining with whatever information that
was legitimately there.
Chapter 3- ADC and DAC
3
3
b. Analog frequency = 0.09 of sampling rate
2
2
1
1
Amplitude
Amplitude
a. Analog frequency = 0.0 (i.e., DC)
0
0
-1
-1
-2
-2
-3
-3
Time (or sample number)
3
d. Analog frequency = 0.95 of sampling rate
2
2
1
1
Amplitude
Amplitude
Time (or sample number)
3
c. Analog frequency = 0.31 of sampling rate
0
0
-1
-1
-2
-2
-3
41
Time (or sample number)
-3
Time (or sample number)
FIGURE 3-3
Illustration of proper and improper sampling. A continuous signal is sampled properly if the samples contain all the
information needed to recreated the original waveform. Figures (a), (b), and (c) illustrate proper sampling of three
sinusoidal waves. This is certainly not obvious, since the samples in (c) do not even appear to capture the shape of the
waveform. Nevertheless, each of these continuous signals forms a unique one-to-one pair with its pattern of samples.
This guarantees that reconstruction can take place. In (d), the frequency of the analog sine wave is greater than the
Nyquist frequency (one-half of the sampling rate). This results in aliasing, where the frequency of the sampled data is
different from the frequency of the continuous signal. Since aliasing has corrupted the information, the original signal
cannot be reconstructed from the samples.
Two terms are widely used when discussing the sampling theorem: the
Nyquist frequency and the Nyquist rate. Unfortunately, their meaning is
not standardized. To understand this, consider an analog signal composed of
frequencies between DC and 3 kHz. To properly digitize this signal it must
be sampled at 6,000 samples/sec (6 kHz) or higher. Suppose we choose to
sample at 8,000 samples/sec (8 kHz), allowing frequencies between DC and 4
kHz to be properly represented. In this situation their are four important
frequencies: (1) the highest frequency in the signal, 3 kHz; (2) twice this
frequency, 6 kHz; (3) the sampling rate, 8 kHz; and (4) one-half the sampling
rate, 4 kHz. Which of these four is the Nyquist frequency and which is the
Nyquist rate? It depends who you ask! All of the possible combinations are
42
The Scientist and Engineer's Guide to Digital Signal Processing
used. Fortunately, most authors are careful to define how they are using the
terms. In this book, they are both used to mean one-half the sampling rate.
Figure 3-4 shows how frequencies are changed during aliasing. The key
point to remember is that a digital signal cannot contain frequencies above
one-half the sampling rate (i.e., the Nyquist frequency/rate). When the
frequency of the continuous wave is below the Nyquist rate, the frequency
of the sampled data is a match. However, when the continuous signal's
frequency is above the Nyquist rate, aliasing changes the frequency into
something that can be represented in the sampled data. As shown by the
zigzagging line in Fig. 3-4, every continuous frequency above the Nyquist
rate has a corresponding digital frequency between zero and one-half the
sampling rate. If there happens to be a sinusoid already at this lower
frequency, the aliased signal will add to it, resulting in a loss of
information. Aliasing is a double curse; information can be lost about the
higher and the lower frequency. Suppose you are given a digital signal
containing a frequency of 0.2 of the sampling rate. If this signal were
obtained by proper sampling, the original analog signal must have had a
frequency of 0.2. If aliasing took place during sampling, the digital
frequency of 0.2 could have come from any one of an infinite number of
frequencies in the analog signal: 0.2, 0.8, 1.2, 1.8, 2.2, þ .
Just as aliasing can change the frequency during sampling, it can also change
the phase. For example, look back at the aliased signal in Fig. 3-3d. The
aliased digital signal is inverted from the original analog signal; one is a sine
wave while the other is a negative sine wave. In other words, aliasing has
changed the frequency and introduced a 180E phase shift. Only two phase
shifts are possible: 0E (no phase shift) and 180E (inversion). The zero phase
shift occurs for analog frequencies of 0 to 0.5, 1.0 to 1.5, 2.0 to 2.5, etc. An
inverted phase occurs for analog frequencies of 0.5 to 1.0, 1.5 to 2.0, 2.5 to
3.0, and so on.
Now we will dive into a more detailed analysis of sampling and how aliasing
occurs. Our overall goal is to understand what happens to the information
when a signal is converted from a continuous to a discrete form. The problem
is, these are very different things; one is a continuous waveform while the
other is an array of numbers. This "apples-to-oranges" comparison makes the
analysis very difficult. The solution is to introduce a theoretical concept called
the impulse train.
Figure 3-5a shows an example analog signal. Figure (c) shows the signal
sampled by using an impulse train. The impulse train is a continuous signal
consisting of a series of narrow spikes (impulses) that match the original signal
at the sampling instants. Each impulse is infinitesimally narrow, a concept that
will be discussed in Chapter 13. Between these sampling times the value of the
waveform is zero. Keep in mind that the impulse train is a theoretical concept,
not a waveform that can exist in an electronic circuit. Since both the original
analog signal and the impulse train are continuous waveforms, we can make an
"apples-apples" comparison between the two.
Chapter 3- ADC and DAC
DC
43
Nyquist
Frequency
GOOD
ALIASED
Digital frequency
0.5
0.4
0.3
0.2
0.1
0.0
Digital phase (degrees)
0.0
0.5
1.0
1.5
2.0
2.5
0.5
1.0
1.5
2.0
2.5
Continuous frequency (as a fraction of the sampling rate)
270
180
90
0
-90
0.0
Continuous frequency (as a fraction of the sampling rate)
FIGURE 3-4
Conversion of analog frequency into digital frequency during sampling. Continuous signals with
a frequency less than one-half of the sampling rate are directly converted into the corresponding
digital frequency. Above one-half of the sampling rate, aliasing takes place, resulting in the frequency
being misrepresented in the digital data. Aliasing always changes a higher frequency into a lower
frequency between 0 and 0.5. In addition, aliasing may also change the phase of the signal by 180
degrees.
Now we need to examine the relationship between the impulse train and the
discrete signal (an array of numbers). This one is easy; in terms of information
content, they are identical. If one is known, it is trivial to calculate the other.
Think of these as different ends of a bridge crossing between the analog and
digital worlds. This means we have achieved our overall goal once we
understand the consequences of changing the waveform in Fig. 3-5a into the
waveform in Fig. 3.5c.
Three continuous waveforms are shown in the left-hand column in Fig. 3-5. The
corresponding frequency spectra of these signals are displayed in the righthand column. This should be a familiar concept from your knowledge of
electronics; every waveform can be viewed as being composed of sinusoids of
varying amplitude and frequency. Later chapters will discuss the frequency
domain in detail. (You may want to revisit this discussion after becoming more
familiar with frequency spectra).
Figure (a) shows an analog signal we wish to sample. As indicated by its
frequency spectrum in (b), it is composed only of frequency components
between 0 and about 0.33 fs, where fs is the sampling frequency we intend to
44
The Scientist and Engineer's Guide to Digital Signal Processing
use. For example, this might be a speech signal that has been filtered to
remove all frequencies above 3.3 kHz. Correspondingly, fs would be 10 kHz
(10,000 samples/second), our intended sampling rate.
Sampling the signal in (a) by using an impulse train produces the signal
shown in (c), and its frequency spectrum shown in (d). This spectrum is a
duplication of the spectrum of the original signal. Each multiple of the
sampling frequency, fs, 2f s, 3f s, 4f s, etc., has received a copy and a left-forright flipped copy of the original frequency spectrum. The copy is called
the upper sideband, while the flipped copy is called the lower sideband.
Sampling has generated new frequencies. Is this proper sampling? The
answer is yes, because the signal in (c) can be transformed back into the
signal in (a) by eliminating all frequencies above ½f s. That is, an analog
low-pass filter will convert the impulse train, (b), back into the original
analog signal, (a).
If you are already familiar with the basics of DSP, here is a more technical
explanation of why this spectral duplication occurs. (Ignore this paragraph
if you are new to DSP). In the time domain, sampling is achieved by
multiplying the original signal by an impulse train of unity amplitude
spikes. The frequency spectrum of this unity amplitude impulse train is
also a unity amplitude impulse train, with the spikes occurring at multiples
of the sampling frequency, f s, 2f s, 3f s, 4f s, etc. When two time domain
signals are multiplied, their frequency spectra are convolved. This results
in the original spectrum being duplicated to the location of each spike in
the impulse train's spectrum. Viewing the original signal as composed of
both positive and negative frequencies accounts for the upper and lower
sidebands, respectively. This is the same as amplitude modulation,
discussed in Chapter 10.
Figure (e) shows an example of improper sampling, resulting from too low
of sampling rate. The analog signal still contains frequencies up to 3.3
kHz, but the sampling rate has been lowered to 5 kHz. Notice that
fS , 2f S , 3fS þ along the horizontal axis are spaced closer in (f) than in (d).
The frequency spectrum, (f), shows the problem: the duplicated portions of
the spectrum have invaded the band between zero and one-half of the
sampling frequency. Although (f) shows these overlapping frequencies as
retaining their separate identity, in actual practice they add together forming
a single confused mess. Since there is no way to separate the overlapping
frequencies, information is lost, and the original signal cannot be
reconstructed. This overlap occurs when the analog signal contains
frequencies greater than one-half the sampling rate, that is, we have proven
the sampling theorem.
Digital-to-Analog Conversion
In theory, the simplest method for digital-to-analog conversion is to pull the
samples from memory and convert them into an impulse train. This is
Chapter 3- ADC and DAC
45
Time Domain
Frequency Domain
3
3
a. Original analog signal
b. Original signal's spectrum
2
1
Amplitude
Amplitude
2
0
1
-1
-2
0
-3
0
1
2
3
4
0
5
100
fs
200
Time
2fs
400
500
3fs
600
3
3
c. Sampling at 3 times highest frequency
2
d. Duplicated spectrum from sampling
lower
upper
sideband sideband
original signal
1
2
impulse train
Amplitude
Amplitude
300
Frequency
0
1
-1
-2
0
-3
0
1
2
Time
3
4
5
0
100
fs
200
300
Frequency
2fs
400
500
3fs
600
3
3
e. Sampling at 1.5 times highest frequency
2
f. Overlapping spectra causing aliasing
2
impulse train
Amplitude
Amplitude
original signal
1
0
1
-1
-2
0
-3
0
1
2
Time
3
4
5
0
fs
100
2fs
200
3fs
300
Frequency
4fs
400
5fs
500
6fs
600
FIGURE 3-5
The sampling theorem in the time and frequency domains. Figures (a) and (b) show an analog signal composed
of frequency components between zero and 0.33 of the sampling frequency, fs. In (c), the analog signal is
sampled by converting it to an impulse train. In the frequency domain, (d), this results in the spectrum being
duplicated into an infinite number of upper and lower sidebands. Since the original frequencies in (b) exist
undistorted in (d), proper sampling has taken place. In comparison, the analog signal in (e) is sampled at 0.66
of the sampling frequency, a value exceeding the Nyquist rate. This results in aliasing, indicated by the
sidebands in (f) overlapping.
46
The Scientist and Engineer's Guide to Digital Signal Processing
illustrated in Fig. 3-6a, with the corresponding frequency spectrum in (b). As
just described, the original analog signal can be perfectly reconstructed by
passing this impulse train through a low-pass filter, with the cutoff frequency
equal to one-half of the sampling rate. In other words, the original signal and
the impulse train have identical frequency spectra below the Nyquist frequency
(one-half the sampling rate). At higher frequencies, the impulse train contains
a duplication of this information, while the original analog signal contains
nothing (assuming aliasing did not occur).
While this method is mathematically pure, it is difficult to generate the required
narrow pulses in electronics. To get around this, nearly all DACs operate by
holding the last value until another sample is received. This is called a
zeroth-order hold, the DAC equivalent of the sample-and-hold used during
ADC. (A first-order hold is straight lines between the points, a second-order
hold uses parabolas, etc.). The zeroth-order hold produces the staircase
appearance shown in (c).
In the frequency domain, the zeroth-order hold results in the spectrum of the
impulse train being multiplied by the dark curve shown in (d), given by the
equation:
EQUATION 3-1
High frequency amplitude reduction due to
the zeroth-order hold. This curve is plotted
in Fig. 3-6d. The sampling frequency is
represented by fS . For f ' 0, H ( f ) ' 1 .
H( f ) ' /
00
sin(Bf /f s )
Bf /f s
/0
0
This is of the general form: sin (Bx) /(Bx) , called the sinc function or sinc(x).
The sinc function is very common in DSP, and will be discussed in more detail
in later chapters. If you already have a background in this material, the zerothorder hold can be understood as the convolution of the impulse train with a
rectangular pulse, having a width equal to the sampling period. This results in
the frequency domain being multiplied by the Fourier transform of the
rectangular pulse, i.e., the sinc function. In Fig. (d), the light line shows the
frequency spectrum of the impulse train (the "correct" spectrum), while the dark
line shows the sinc. The frequency spectrum of the zeroth order hold signal is
equal to the product of these two curves.
The analog filter used to convert the zeroth-order hold signal, (c), into the
reconstructed signal, (f), needs to do two things: (1) remove all frequencies
above one-half of the sampling rate, and (2) boost the frequencies by the
reciprocal of the zeroth-order hold's effect, i.e., 1/sinc(x). This amounts to an
amplification of about 36% at one-half of the sampling frequency. Figure (e)
shows the ideal frequency response of this analog filter.
This 1/sinc(x) frequency boost can be handled in four ways: (1) ignore it and
accept the consequences, (2) design an analog filter to include the 1/sinc(x)
Chapter 3- ADC and DAC
47
Time Domain
Frequency Domain
2
3
a. Impulse train
b. Spectrum of impulse train
1
Amplitude
Amplitude
2
0
-1
1
-2
0
-3
0
1
2
Time
3
4
5
0
100
fs
200
300
Frequency
2fs
400
500
3fs
600
500
3fs
600
500
3fs
600
500
3fs
600
2
3
c. Zeroth-order hold
2
d. Spectrum multiplied by sinc
Amplitude
Amplitude
"correct" spectrum
1
0
-1
sinc
1
-2
0
-3
0
1
2
3
4
0
5
100
fs
200
Time
e. Ideal reconstruction filter
1
0
0
100
fs
200
300
Frequency
2fs
400
2
3
f. Reconstructed analog signal
2
g. Reconstructed spectrum
1
Amplitude
Amplitude
2fs
400
2
Amplitude
FIGURE 3-6
Analysis of digital-to-analog conversion. In (a), the digital
data are converted into an impulse train, with the spectrum
in (b). This is changed into the reconstructed signal, (f), by
using an electronic low-pass filter to remove frequencies
above one-half the sampling rate [compare (b) and (g)].
However, most electronic DACs create a zeroth-order hold
waveform, (c), instead of an impulse train. The spectrum
of the zeroth-order hold is equal to the spectrum of the
impulse train multiplied by the sinc function shown in (d).
To convert the zeroth-order hold into the reconstructed
signal, the analog filter must remove all frequencies above
the Nyquist rate, and correct for the sinc, as shown in (e).
300
Frequency
0
-1
1
-2
0
-3
0
1
2
3
Time
4
5
0
100
fs
200
300
Frequency
2fs
400
48
The Scientist and Engineer's Guide to Digital Signal Processing
response, (3) use a fancy multirate technique described later in this chapter,
or (4) make the correction in software before the DAC (see Chapter 24).
Before leaving this section on sampling, we need to dispel a common myth
about analog versus digital signals. As this chapter has shown, the amount of
information carried in a digital signal is limited in two ways: First, the number
of bits per sample limits the resolution of the dependent variable. That is,
small changes in the signal's amplitude may be lost in the quantization noise.
Second, the sampling rate limits the resolution of the independent variable, i.e.,
closely spaced events in the analog signal may be lost between the samples.
This is another way of saying that frequencies above one-half the sampling rate
are lost.
Here is the myth: "Since analog signals use continuous parameters, they have
infinitely good resolution in both the independent and the dependent variables."
Not true! Analog signals are limited by the same two problems as digital
signals: noise and bandwidth (the highest frequency allowed in the signal). The
noise in an analog signal limits the measurement of the waveform's amplitude,
just as quantization noise does in a digital signal. Likewise, the ability to
separate closely spaced events in an analog signal depends on the highest
frequency allowed in the waveform. To understand this, imagine an analog
signal containing two closely spaced pulses. If we place the signal through a
low-pass filter (removing the high frequencies), the pulses will blur into a
single blob. For instance, an analog signal formed from frequencies between
DC and 10 kHz will have exactly the same resolution as a digital signal
sampled at 20 kHz. It must, since the sampling theorem guarantees that the
two contain the same information.
Analog Filters for Data Conversion
Figure 3-7 shows a block diagram of a DSP system, as the sampling theorem
dictates it should be. Before encountering the analog-to-digital converter,
antialias filter
Analog
Filter
Analog
Input
reconstruction filter
Digital
Processing
ADC
Filtered
Analog
Input
Digitized
Input
Analog
Filter
DAC
Digitized
Output
S/H
Analog
Output
Analog
Output
FIGURE 3-7
Analog electronic filters used to comply with the sampling theorem. The electronic filter placed before an ADC is
called an antialias filter. It is used to remove frequency components above one-half of the sampling rate that would
alias during the sampling. The electronic filter placed after a DAC is called a reconstruction filter. It also eliminates
frequencies above the Nyquist rate, and may include a correction for the zeroth-order hold.
Chapter 3- ADC and DAC
49
the input signal is processed with an electronic low-pass filter to remove all
frequencies above the Nyquist frequency (one-half the sampling rate). This is
done to prevent aliasing during sampling, and is correspondingly called an
antialias filter. On the other end, the digitized signal is passed through a
digital-to-analog converter and another low-pass filter set to the Nyquist
frequency. This output filter is called a reconstruction filter, and may include
the previously described zeroth-order-hold frequency boost. Unfortunately,
there is a serious problem with this simple model: the limitations of electronic
filters can be as bad as the problems they are trying to prevent.
If your main interest is in software, you are probably thinking that you don't
need to read this section. Wrong! Even if you have vowed never to touch an
oscilloscope, an understanding of the properties of analog filters is important
for successful DSP. First, the characteristics of every digitized signal you
encounter will depend on what type of antialias filter was used when it was
acquired. If you don't understand the nature of the antialias filter, you cannot
understand the nature of the digital signal. Second, the future of DSP is to
replace hardware with software. For example, the multirate techniques
presented later in this chapter reduce the need for antialias and reconstruction
filters by fancy software tricks. If you don't understand the hardware, you
cannot design software to replace it. Third, much of DSP is related to digital
filter design. A common strategy is to start with an equivalent analog filter,
and convert it into software. Later chapters assume you have a basic
knowledge of analog filter techniques.
Three types of analog filters are commonly used: Chebyshev, Butterworth,
and Bessel (also called a Thompson filter). Each of these is designed to
optimize a different performance parameter. The complexity of each filter
can be adjusted by selecting the number of poles and zeros, mathematical
terms that will be discussed in later chapters. The more poles in a filter,
the more electronics it requires, and the better it performs. Each of these
names describe what the filter does, not a particular arrangement of
resistors and capacitors. For example, a six pole Bessel filter can be
implemented by many different types of circuits, all of which have the same
overall characteristics. For DSP purposes, the characteristics of these
filters are more important than how they are constructed. Nevertheless, we
will start with a short segment on the electronic design of these filters to
provide an overall framework.
Figure 3-8 shows a common building block for analog filter design, the
modified Sallen-Key circuit. This is named after the authors of a 1950s paper
describing the technique. The circuit shown is a two pole low-pass filter that
can be configured as any of the three basic types. Table 3-1 provides the
necessary information to select the appropriate resistors and capacitors. For
example, to design a 1 kHz, 2 pole Butterworth filter, Table 3-1 provides the
parameters: k1 = 0.1592 and k2 = 0.586. Arbitrarily selecting R1 = 10K and
C = 0.01uF (common values for op amp circuits), R and Rf can be calculated
as 15.95K and 5.86K, respectively. Rounding these last two values to the
nearest 1% standard resistors, results in R = 15.8K and Rf = 5.90K All of the
components should be 1% precision or better.
50
The Scientist and Engineer's Guide to Digital Signal Processing
C
FIGURE 3-8
The modified Sallen-Key circuit, a building
block for active filter design. The circuit
shown implements a 2 pole low-pass filter.
Higher order filters (more poles) can be
formed by cascading stages. Find k1 and k2
from Table 3-1, arbitrarily select R1 and C
(try 10K and 0.01µF), and then calculate R
and Rf from the equations in the figure. The
parameter, fc, is the cutoff frequency of the
filter, in hertz.
R
R
C
Rf
R '
k1
R1
C fc
R f ' R1 k2
TABLE 3-1
Parameters for designing Bessel, Butterworth, and Chebyshev (6% ripple) filters.
Bessel
k1
k2
# poles
Butterworth
k1
k2
Chebyshev
k1
k2
2
stage 1
0.1251
0.268
0.1592
0.586
0.1293
0.842
4
stage 1
stage 2
0.1111
0.0991
0.084
0.759
0.1592
0.1592
0.152
1.235
0.2666
0.1544
0.582
1.660
6
stage 1
stage 2
stage 3
0.0990
0.0941
0.0834
0.040
0.364
1.023
0.1592
0.1592
0.1592
0.068
0.586
1.483
0.4019
0.2072
0.1574
0.537
1.448
1.846
8
stage 1
stage 2
stage 3
stage 4
0.0894
0.0867
0.0814
0.0726
0.024
0.213
0.593
1.184
0.1592
0.1592
0.1592
0.1592
0.038
0.337
0.889
1.610
0.5359
0.2657
0.1848
0.1582
0.522
1.379
1.711
1.913
The particular op amp used isn't critical, as long as the unity gain frequency is
more than 30 to 100 times higher than the filter's cutoff frequency. This is an
easy requirement as long as the filter's cutoff frequency is below about 100
kHz.
Four, six, and eight pole filters are formed by cascading 2,3, and 4 of these
circuits, respectively. For example, Fig. 3-9 shows the schematic of a 6 pole
0.01µF
0.01µF
0.01µF
10K
10K
9.53K
9.53K
8.25K
0.01µF
402S
3.65K
stage 1
k1 = 0.0990
k2 = 0.040
8.25K
0.01µF
0.01µF
10.2K
10K
stage 2
k1 = 0.0941
k2 = 0.364
10K
stage 3
k1 = 0.0834
k2 = 1.023
FIGURE 3-9
A six pole Bessel filter formed by cascading three Sallen-Key circuits. This is a low-pass filter with
a cutoff frequency of 1 kHz.
10K
Chapter 3- ADC and DAC
51
Bessel filter created by cascading three stages. Each stage has different values
for k1 and k2 as provided by Table 3-1, resulting in different resistors and
capacitors being used. Need a high-pass filter? Simply swap the R and C
components in the circuits (leaving Rf and R1 alone).
This type of circuit is very common for small quantity manufacturing and R&D
applications; however, serious production requires the filter to be made as an
integrated circuit. The problem is, it is difficult to make resistors directly in
silicon. The answer is the switched capacitor filter. Figure 3-10 illustrates
its operation by comparing it to a simple RC network. If a step function is fed
into an RC low-pass filter, the output rises exponentially until it matches the
input. The voltage on the capacitor doesn't change instantaneously, because the
resistor restricts the flow of electrical charge.
The switched capacitor filter operates by replacing the basic resistorcapacitor network with two capacitors and an electronic switch. The newly
added capacitor is much smaller in value than the already existing
capacitor, say, 1% of its value. The switch alternately connects the small
capacitor between the input and the output at a very high frequency,
typically 100 times faster than the cutoff frequency of the filter. When the
switch is connected to the input, the small capacitor rapidly charges to
whatever voltage is presently on the input. When the switch is connected
to the output, the charge on the small capacitor is transferred to the large
capacitor. In a resistor, the rate of charge transfer is determined by its
resistance. In a switched capacitor circuit, the rate of charge transfer is
determined by the value of the small capacitor and by the switching
frequency. This results in a very useful feature of switched capacitor
voltage
voltage
Resistor-Capacitor
R
C
time
low R
high R
time
Switched Capacitor
time
voltage
voltage
f
C/100
C
high f
low f
time
FIGURE 3-10
Switched capacitor filter operation. Switched capacitor filters use switches and capacitors to mimic
resistors. As shown by the equivalent step responses, two capacitors and one switch can perform the
same function as a resistor-capacitor network.
52
The Scientist and Engineer's Guide to Digital Signal Processing
filters: the cutoff frequency of the filter is directly proportional to the clock
frequency used to drive the switches. This makes the switched capacitor filter
ideal for data acquisition systems that operate with more than one sampling
rate. These are easy-to-use devices; pay ten bucks and have the performance
of an eight pole filter inside a single 8 pin IC.
Now for the important part: the characteristics of the three classic filter types.
The first performance parameter we want to explore is cutoff frequency
sharpness. A low-pass filter is designed to block all frequencies above the
cutoff frequency (the stopband), while passing all frequencies below (the
passband). Figure 3-11 shows the frequency response of these three filters on
a logarithmic (dB) scale. These graphs are shown for filters with a one hertz
cutoff frequency, but they can be directly scaled to whatever cutoff frequency
you need to use. How do these filters rate? The Chebyshev is clearly the best,
the Butterworth is worse, and the Bessel is absolutely ghastly! As you
probably surmised, this is what the Chebyshev is designed to do, roll-off (drop
in amplitude) as rapidly as possible.
Unfortunately, even an 8 pole Chebyshev isn't as good as you would like for
an antialias filter. For example, imagine a 12 bit system sampling at 10,000
samples per second. The sampling theorem dictates that any frequency above
5 kHz will be aliased, something you want to avoid. With a little guess work,
you decide that all frequencies above 5 kHz must be reduced in amplitude by
a factor of 100, insuring that any aliased frequencies will have an amplitude of
less than one percent. Looking at Fig. 3-11c, you find that an 8 pole
Chebyshev filter, with a cutoff frequency of 1 hertz, doesn't reach an
attenuation (signal reduction) of 100 until about 1.35 hertz. Scaling this to the
example, the filter's cutoff frequency must be set to 3.7 kHz so that everything
above 5 kHz will have the required attenuation. This results in the frequency
band between 3.7 kHz and 5 kHz being wasted on the inadequate roll-off of the
analog filter.
A subtle point: the attenuation factor of 100 in this example is probably
sufficient even though there are 4096 steps in 12 bits. From Fig. 3-4, 5100
hertz will alias to 4900 hertz, 6000 hertz will alias to 4000 hertz, etc. You
don't care what the amplitudes of the signals between 5000 and 6300 hertz are,
because they alias into the unusable region between 3700 hertz and 5000 hertz.
In order for a frequency to alias into the filter's passband (0 to 3.7 kHz), it
must be greater than 6300 hertz, or 1.7 times the filter's cutoff frequency of
3700 hertz. As shown in Fig. 3-11c, the attenuation provided by an 8 pole
Chebyshev filter at 1.7 times the cutoff frequency is about 1300, much more
adequate than the 100 we started the analysis with. The moral to this story: In
most systems, the frequency band between about 0.4 and 0.5 of the sampling
frequency is an unusable wasteland of filter roll-off and aliased signals. This
is a direct result of the limitations of analog filters.
The frequency response of the perfect low-pass filter is flat across the entire
passband. All of the filters look great in this respect in Fig. 3-11, but only
because the vertical axis is displayed on a logarithmic scale. Another story is
told when the graphs are converted to a linear vertical scale, as is shown
Chapter 3- ADC and DAC
53
Log scale
Linear scale
10
1.6
a. Bessel
1.2
2 pole
0.1
Amplitude
Amplitude
a. Bessel
1.4
1
4
ideal
8
0.01
1
ideal
0.8
0.6
0.4
0.001
2 pole
0.2
0
1
2
3
Frequency (hertz)
4
0
5
10
1
2
4
5
3
4
5
4
5
b. Butterworth
1.4
1
1.2
0.1
Amplitude
2 pole
4
0.01
8
1
0.8
0.6
0.4
0.001
2 pole
0.2
0
1
2
3
Frequency (hertz)
4
5
0
10
4
8
0
0.0001
1
2
Frequency (hertz)
1.6
c. Chebyshev (6% ripple)
c. Chebyshev (6% ripple)
1.4
1
1.2
2 pole
Amplitude
Amplitude
3
Frequency (hertz)
1.6
b. Butterworth
Amplitude
8
0
0.0001
4
0.1
4
0.01
1
0.8
0.6
8
0.4
0.001
2 pole
0.2
0.0001
0
8
0
1
2
3
Frequency (hertz)
4
FIGURE 3-11
Frequency response of the three filters on a
logarithmic scale. The Chebyshev filter has
the sharpest roll-off.
5
0
1
4
2
3
Frequency (hertz)
FIGURE 3-12
Frequency response of the three filters on a
linear scale. The Butterworth filter provides
the flattest passband.
in Fig. 3-12. Passband ripple can now be seen in the Chebyshev filter
(wavy variations in the amplitude of the passed frequencies). In fact, the
Chebyshev filter obtains its excellent roll-off by allowing this passband
ripple. When more passband ripple is allowed in a filter, a faster roll-off
54
The Scientist and Engineer's Guide to Digital Signal Processing
can be achieved. All the Chebyshev filters designed by using Table 3-1 have
a passband ripple of about 6% (0.5 dB), a good compromise, and a common
choice. A similar design, the elliptic filter, allows ripple in both the passband
and the stopband. Although harder to design, elliptic filters can achieve an
even better tradeoff between roll-off and passband ripple.
In comparison, the Butterworth filter is optimized to provide the sharpest rolloff possible without allowing ripple in the passband. It is commonly called the
maximally flat filter, and is identical to a Chebyshev designed for zero
passband ripple. The Bessel filter has no ripple in the passband, but the rolloff far worse than the Butterworth.
The last parameter to evaluate is the step response, how the filter responds
when the input rapidly changes from one value to another. Figure 3-13 shows
the step response of each of the three filters. The horizontal axis is shown for
filters with a 1 hertz cutoff frequency, but can be scaled (inversely) for higher
cutoff frequencies. For example, a 1000 hertz cutoff frequency would show a
step response in milliseconds, rather than seconds. The Butterworth and
Chebyshev filters overshoot and show ringing (oscillations that slowly
decreasing in amplitude). In comparison, the Bessel filter has neither of these
nasty problems.
1.6
1.6
1.4
1.4
a. Bessel
b. Butterworth
1.2
Amplitude
1.0
2
0.8
4
0.6
8 pole
0.4
1.0
2
0.8
4
8 pole
0.6
0.4
0.2
0.2
0.0
0.0
0
1
2
Time (seconds)
3
4
0
1
2
Time (seconds)
3
4
3
4
1.6
1.4
FIGURE 3-13
Step response of the three filters. The times
shown on the horizontal axis correspond to a
one hertz cutoff frequency. The Bessel is the
optimum filter when overshoot and ringing
must be minimized.
c. Chebyshev (6% ripple)
1.2
Amplitude
Amplitude
1.2
1.0
2
0.8
4
0.6
8 pole
0.4
0.2
0.0
0
1
2
Time (seconds)
Chapter 3- ADC and DAC
1.5
1.5
a. Pulse waveform
b. After Bessel filter
1.0
Amplitude
1.0
Amplitude
55
0.5
0.0
0.5
0.0
-0.5
-0.5
100
200
Time
300
400
500
FIGURE 3-14
Pulse response of the Bessel and Chebyshev
filters. A key property of the Bessel filter is that
the rising and falling edges in the filter's output
looking similar. In the jargon of the field, this is
called linear phase. Figure (b) shows the result
of passing the pulse waveform in (a) through a 4
pole Bessel filter. Both edges are smoothed in a
similar manner. Figure (c) shows the result of
passing (a) through a 4 pole Chebyshev filter.
The left edge overshoots on the top, while the
right edge overshoots on the bottom. Many
applications cannot tolerate this distortion.
0
100
200
Time
300
400
500
400
500
1.5
c. After Chebyshev filter
1.0
Amplitude
0
0.5
0.0
-0.5
0
100
200
Time
300
Figure 3-14 further illustrates this very favorable characteristic of the Bessel
filter. Figure (a) shows a pulse waveform, which can be viewed as a rising
step followed by a falling step. Figures (b) and (c) show how this waveform
would appear after Bessel and Chebyshev filters, respectively. If this were a
video signal, for instance, the distortion introduced by the Chebyshev filter
would be devastating! The overshoot would change the brightness of the edges
of objects compared to their centers. Worse yet, the left side of objects would
look bright, while the right side of objects would look dark. Many applications
cannot tolerate poor performance in the step response. This is where the Bessel
filter shines; no overshoot and symmetrical edges.
Selecting The Antialias Filter
Table 3-2 summarizes the characteristics of these three filters, showing how
each optimizes a particular parameter at the expense of everything else. The
Chebyshev optimizes the roll-off, the Butterworth optimizes the passband
flatness, and the Bessel optimizes the step response.
The selection of the antialias filter depends almost entirely on one issue: how
information is represented in the signals you intend to process. While
56
The Scientist and Engineer's Guide to Digital Signal Processing
Step Response
Frequency Response
Voltage gain
at DC
Overshoot
Time to
settle to 1%
Time to
settle to
0.1%
Ripple in
passband
Frequency
for x100
attenuation
Frequency
for x1000
attenuation
Bessel
2 pole
4 pole
6 pole
8 pole
1.27
1.91
2.87
4.32
0.4%
0.9%
0.7%
0.4%
0.60
0.66
0.74
0.80
1.12
1.20
1.18
1.16
0%
0%
0%
0%
12.74
4.74
3.65
3.35
40.4
8.45
5.43
4.53
Butterworth
2 pole
4 pole
6 pole
8 pole
1.59
2.58
4.21
6.84
4.3%
10.9%
14.3%
16.4%
1.06
1.68
2.74
3.50
1.66
2.74
3.92
5.12
0%
0%
0%
0%
10.0
3.17
2.16
1.78
31.6
5.62
3.17
2.38
Chebyshev
2 pole
4 pole
6 pole
8 pole
1.84
4.21
10.71
28.58
10.8%
18.2%
21.3%
23.0%
1.10
3.04
5.86
8.34
1.62
5.42
10.4
16.4
6%
6%
6%
6%
12.33
2.59
1.63
1.34
38.9
4.47
2.26
1.66
TABLE 3-2
Characteristics of the three classic filters. The Bessel filter provides the best step response, making it the choice for
time domain encoded signals. The Chebyshev and Butterworth filters are used to eliminate frequencies in the
stopband, making them ideal for frequency domain encoded signals. Values in this table are in the units of seconds
and hertz, for a one hertz cutoff frequency.
there are many ways for information to be encoded in an analog waveform,
only two methods are common, time domain encoding, and frequency
domain encoding. The difference between these two is critical in DSP, and
will be a reoccurring theme throughout this book.
In frequency domain encoding, the information is contained in sinusoidal
waves that combine to form the signal. Audio signals are an excellent example
of this. When a person hears speech or music, the perceived sound depends on
the frequencies present, and not on the particular shape of the waveform. This
can be shown by passing an audio signal through a circuit that changes the
phase of the various sinusoids, but retains their frequency and amplitude. The
resulting signal looks completely different on an oscilloscope, but sounds
identical. The pertinent information has been left intact, even though the
waveform has been significantly altered. Since aliasing misplaces and overlaps
frequency components, it directly destroys information encoded in the frequency
domain. Consequently, digitization of these signals usually involves an
antialias filter with a sharp cutoff, such as a Chebyshev, Elliptic, or
Butterworth. What about the nasty step response of these filters? It doesn't
matter; the encoded information isn't affected by this type of distortion.
In contrast, time domain encoding uses the shape of the waveform to store
information. For example, physicians can monitor the electrical activity of a
Chapter 3- ADC and DAC
57
person's heart by attaching electrodes to their chest and arms (an
electrocardiogram or EKG). The shape of the EKG waveform provides the
information being sought, such as when the various chambers contract during
a heartbeat. Images are another example of this type of signal. Rather than a
waveform that varies over time, images encode information in the shape of a
waveform that varies over distance. Pictures are formed from regions of
brightness and color, and how they relate to other regions of brightness and
color. You don't look at the Mona Lisa and say, "My, what an interesting
collection of sinusoids."
Here's the problem: The sampling theorem is an analysis of what happens in
the frequency domain during digitization. This makes it ideal to under-stand
the analog-to-digital conversion of signals having their information encoded in
the frequency domain. However, the sampling theorem is little help in
understanding how time domain encoded signals should be digitized. Let's take
a closer look.
Figure 3-15 illustrates the choices for digitizing a time domain encoded signal.
Figure (a) is an example analog signal to be digitized. In this case, the
information we want to capture is the shape of the rectangular pulses. A short
burst of a high frequency sine wave is also included in this example signal.
This represents wideband noise, interference, and similar junk that always
appears on analog signals. The other figures show how the digitized signal
would appear with different antialias filter options: a Chebyshev filter, a Bessel
filter, and no filter.
It is important to understand that none of these options will allow the original
signal to be reconstructed from the sampled data. This is because the original
signal inherently contains frequency components greater than one-half of the
sampling rate. Since these frequencies cannot exist in the digitized signal, the
reconstructed signal cannot contain them either. These high frequencies result
from two sources: (1) noise and interference, which you would like to
eliminate, and (2) sharp edges in the waveform, which probably contain
information you want to retain.
The Chebyshev filter, shown in (b), attacks the problem by aggressively
removing all high frequency components. This results in a filtered analog
signal that can be sampled and later perfectly reconstructed. However, the
reconstructed analog signal is identical to the filtered signal, not the original
signal. Although nothing is lost in sampling, the waveform has been severely
distorted by the antialias filter. As shown in (b), the cure is worse than the
disease! Don't do it!
The Bessel filter, (c), is designed for just this problem. Its output closely
resembles the original waveform, with only a gentle rounding of the edges.
By adjusting the filter's cutoff frequency, the smoothness of the edges can
be traded for elimination of high frequency components in the signal.
Using more poles in the filter allows a better tradeoff between these two
parameters. A common guideline is to set the cutoff frequency at about
one-quarter of the sampling frequency. This results in about two samples
58
The Scientist and Engineer's Guide to Digital Signal Processing
along the rising portion of each edge. Notice that both the Bessel and the
Chebyshev filter have removed the burst of high frequency noise present in
the original signal.
The last choice is to use no antialias filter at all, as is shown in (d). This
has the strong advantage that the value of each sample is identical to the
value of the original analog signal. In other words, it has perfect edge
sharpness; a change in the original signal is immediately mirrored in the
digital data. The disadvantage is that aliasing can distort the signal. This
takes two different forms. First, high frequency interference and noise,
such as the example sinusoidal burst, will turn into meaningless samples,
as shown in (d). That is, any high frequency noise present in the analog
signal will appear as aliased noise in the digital signal. In a more general
sense, this is not a problem of the sampling, but a problem of the upstream
analog electronics. It is not the ADC's purpose to reduce noise and
interference; this is the responsibility of the analog electronics before the
digitization takes place. It may turn out that a Bessel filter should be
placed before the digitizer to control this problem. However, this means the
filter should be viewed as part of the analog processing, not something that
is being done for the sake of the digitizer.
The second manifestation of aliasing is more subtle. When an event occurs
in the analog signal (such as an edge), the digital signal in (d) detects the
change on the next sample. There is no information in the digital data to
indicate what happens between samples. Now, compare using no filter with
using a Bessel filter for this problem. For example, imagine drawing
straight lines between the samples in (c). The time when this constructed
line crosses one-half the amplitude of the step provides a subsample
estimate of when the edge occurred in the analog signal. When no filter is
used, this subsample information is completely lost. You don't need a fancy
theorem to evaluate how this will affect your particular situation, just a
good understanding of what you plan to do with the data once is it acquired.
Multirate Data Conversion
There is a strong trend in electronics to replace analog circuitry with
digital algorithms. Data conversion is an excellent example of this.
Consider the design of a digital voice recorder, a system that will digitize
a voice signal, store the data in digital form, and later reconstruct the
signal for playback. To recreate intelligible speech, the system must
capture the frequencies between about 100 and 3000 hertz. However, the
analog signal produced by the microphone also contains much higher
frequencies, say to 40 kHz. The brute force approach is to pass the analog
signal through an eight pole low-pass Chebyshev filter at 3 kHz, and then
sample at 8 kHz. On the other end, the DAC reconstructs the analog signal
at 8 kHz with a zeroth order hold. Another Chebyshev filter at 3 kHz is
used to produce the final voice signal.
Chapter 3- ADC and DAC
3
3
a. Analog waveform
b. With Chebyshev filter
waveform to
be captured
2
Amplitude
Amplitude
2
1
0
1
0
high-frequency
noise to be rejected
-1
0
100
200
300
Time
400
-1
500
600
0
3
10
20
30
40
50
60
30
40
50
60
Sample number
3
c. With Bessel filter
d. No analog filter
2
Amplitude
2
Amplitude
59
1
0
1
0
-1
-1
0
10
20
30
Sample number
40
50
60
0
10
20
Sample number
FIGURE 3-15
Three antialias filter options for time domain encoded signals. The goal is to eliminate high frequencies (that will alias
during sampling), while simultaneously retaining edge sharpness (that carries information). Figure (a) shows an example
analog signal containing both sharp edges and a high frequency noise burst. Figure (b) shows the digitized signal using
a Chebyshev filter. While the high frequencies have been effectively removed, the edges have been grossly distorted.
This is usually a terrible solution. The Bessel filter, shown in (c), provides a gentle edge smoothing while removing the
high frequencies. Figure (d) shows the digitized signal using no antialias filter. In this case, the edges have retained
perfect sharpness; however, the high frequency burst has aliased into several meaningless samples.
There are many useful benefits in sampling faster than this direct analysis. For
example, imagine redesigning the digital voice recorder using a 64 kHz
sampling rate. The antialias filter now has an easier task: pass all freq-uencies
below 3 kHz, while rejecting all frequencies above 32 kHz. A similar
simplification occurs for the reconstruction filter. In short, the higher sampling
rate allows the eight pole filters to be replaced with simple resistor-capacitor
(RC) networks. The problem is, the digital system is now swamped with data
from the higher sampling rate.
The next level of sophistication involves multirate techniques, using more
than one sampling rate in the same system. It works like this for the digital
voice recorder example. First, pass the voice signal through a simple RC low-
60
The Scientist and Engineer's Guide to Digital Signal Processing
pass filter and sample the data at 64 kHz. The resulting digital data contains
the desired voice band between 100 and 3000 hertz, but also has an unusable
band between 3 kHz and 32 kHz. Second, remove these unusable frequencies
in software, by using a digital low-pass filter at 3 kHz. Third, resample the
digital signal from 64 kHz to 8 kHz by simply discarding every seven out of
eight samples, a procedure called decimation. The resulting digital data is
equivalent to that produced by aggressive analog filtering and direct 8 kHz
sampling.
Multirate techniques can also be used in the output portion of our example
system. The 8 kHz data is pulled from memory and converted to a 64 kHz
sampling rate, a procedure called interpolation. This involves placing seven
samples, with a value of zero, between each of the samples obtained from
memory. The resulting signal is a digital impulse train, containing the desired
voice band between 100 and 3000 hertz, plus spectral duplications between 3
kHz and 32 kHz. Refer back to Figs. 3-6 a&b to understand why this it true.
Everything above 3 kHz is then removed with a digital low-pass filter. After
conversion to an analog signal through a DAC, a simple RC network is all that
is required to produce the final voice signal.
Multirate data conversion is valuable for two reasons: (1) it replaces
analog components with software, a clear economic advantage in massproduced products, and (2) it can achieve higher levels of performance in
critical applications. For example, compact disc audio systems use
techniques of this type to achieve the best possible sound quality. This
increased performance is a result of replacing analog components (1%
precision), with digital algorithms (0.0001% precision from round-off
error). As discussed in upcoming chapters, digital filters outperform analog
filters by hundreds of times in key areas.
Single Bit Data Conversion
A popular technique in telecommunications and high fidelity music reproduction
is single bit ADC and DAC. These are multirate techniques where a higher
sampling rate is traded for a lower number of bits. In the extreme, only a
single bit is needed for each sample. While there are many different circuit
configurations, most are based on the use of delta modulation. Three
example circuits will be presented to give you a flavor of the field. All of
these circuits are implemented in IC's, so don't worry where all of the
individual transistors and op amps should go. No one is going to ask you to
build one of these circuits from basic components.
Figure 3-16 shows the block diagram of a typical delta modulator. The
analog input is a voice signal with an amplitude of a few volts, while the
output signal is a stream of digital ones and zeros. A comparator decides
which has the greater voltage, the incoming analog signal, or the voltage
stored on the capacitor. This decision, in the form of a digital one or zero,
is applied to the input of the latch. At each clock pulse, typically at a few
hundred kilohertz, the latch transfers whatever digital state appears on its
Chapter 3- ADC and DAC
61
clock
analog
input
digital
latch
comparator
positive
charge
injector
clock
negative
charge
injector
clock
delta
modulated
output
FIGURE 3-16
Block diagram of a delta modulation circuit. The input voltage is compared with the voltage
stored on the capacitor, resulting in a digital zero or one being applied to the input of the latch.
The output of the latch is updated in synchronization with the clock, and used in a feedback
loop to cause the capacitor voltage to track the input voltage.
input, to its output. This latch insures that the output is synchronized with the
clock, thereby defining the sampling rate, i.e., the rate at which the 1 bit output
can update itself.
A feedback loop is formed by taking the digital output and using it to drive an
electronic switch. If the output is a digital one, the switch connects the
capacitor to a positive charge injector. This is a very loose term for a circuit
that increases the voltage on the capacitor by a fixed amount, say 1 millivolt
per clock cycle. This may be nothing more than a resistor connected to a large
positive voltage. If the output is a digital zero, the switch is connected to a
negative charge injector. This decreases the voltage on the capacitor by the
same fixed amount.
Figure 3-17 illustrates the signals produced by this circuit. At time equal
zero, the analog input and the voltage on the capacitor both start with a
voltage of zero. As shown in (a), the input signal suddenly increases to 9.5
volts on the eighth clock cycle. Since the input signal is now more positive
than the voltage on the capacitor, the digital output changes to a one, as
shown in (b). This results in the switch being connected to the positive
charge injector, and the voltage on the capacitor increasing by a small
amount on each clock cycle. Although an increment of 1 volt per clock
cycle is shown in (a), this is only for illustration, and a value of 1 millivolt
is more typical. This staircase increase in the capacitor voltage continues
until it exceeds the voltage of the input signal. Here the system reached an
equilibrium with the output oscillating between a digital one and zero,
causing the voltage on the capacitor to oscillate between 9 volts and 10
62
The Scientist and Engineer's Guide to Digital Signal Processing
volts. In this manner, the feedback of the circuit forces the capacitor
voltage to track the voltage of the input signal. If the input signal changes
very rapidly, the voltage on the capacitor changes at a constant rate until a
match is obtained. This constant rate of change is called the slew rate, just
as in other electronic devices such as op amps.
Now, consider the characteristics of the delta modulated output signal. If the
analog input is increasing in value, the output signal will consist of more ones
than zeros. Likewise, if the analog input is decreasing in value, the output will
consist of more zeros than ones. If the analog input is constant, the digital
output will alternate between zero and one with an equal number of each. Put
in more general terms, the relative number of ones versus zeros is directly
proportional to the slope (derivative) of the analog input.
This circuit is a cheap method of transforming an analog signal into a serial
stream of ones and zeros for transmission or digital storage. An especially
attractive feature is that all the bits have the same meaning, unlike the
conventional serial format: start bit, LSB, @ @ @ ,MSB, stop bit. The circuit at
the receiver is identical to the feedback portion of the transmitting circuit. Just
as the voltage on the capacitor in the transmitting circuit follows the analog
input, so does the voltage on the capacitor in the receiving circuit. That is, the
capacitor voltage shown in (a) also represents how the reconstructed signal
would appear.
A critical limitation of this circuit is the unavoidable tradeoff between (1)
maximum slew rate, (2) quantization size, and (3) data rate. In particular, if
the maximum slew rate and quantization size are adjusted to acceptable values
for voice communication, the data rate ends up in the MHz range. This is too
high to be of commercial value. For instance, conventional sampling of a voice
signal requires only about 64,000 bits per second.
A solution to this problem is shown in Fig. 3-18, the Continuously Variable
Slope Delta (CVSD) modulator, a technique implemented in the Motorola
MC3518 family. In this approach, the clock rate and the quantization size are
set to something acceptable, say 30 kHz, and 2000 levels. This results in a
terrible slew rate, which you correct with additional circuitry. In operation, a
shift resister continually looks at the last four bits that the system has produced.
If the circuit is in a slew rate limited condition, the last four bits will be all
ones (positive slope) or all zeros (negative slope). A logic circuit detects this
situation and produces an analog signal that increase the level of charge
produced by the charge injectors. This boosts the slew rate by increasing the
size of the voltage steps being applied to the capacitor.
An analog filter is usually placed between the logic circuitry and the charge
injectors. This allows the step size to depend on how long the circuit has
been in a slew limited condition. As long as the circuit is slew limited, the
step size keeps getting larger and larger. This is often called a syllabic
filter, since its characteristics depend on the average length of the syllables
making up speech. With proper optimization (from the chip manufacturer's
Chapter 3- ADC and DAC
63
15
a. Analog signals
capacitor voltage
Amplitude (volts)
10
5
signal voltage
0
-5
-10
0
10
20
30
40
50
60
70
80
90
100
110
120
50
60
70
80
90
100
110
120
Time (clock cycles)
3
b. Digital output
Digital logic state
2
slew rate limited
1
0
-1
0
10
20
30
40
Time (clock cycles)
FIGURE 3-17
Example of signals produced by the delta modulator in Fig. 3-16. Figure (a) shows the analog
input signal, and the corresponding voltage on the capacitor. Figure (b) shows the delta
modulated output, a digital stream of ones and zeros.
spec sheet, not your own work), data rates of 16 to 32 kHz produce acceptable
quality speech. The continually changing step size makes the digital data
difficult to understand, but fortunately, you don't need to. At the receiver, the
analog signal is reconstructed by incorporating a syllabic filter that is identical
to the one in the transmission circuit. If the two filters are matched, little
distortion results from the CVSD modulation. CVSD is probably the easiest
way to digitally transmit a voice signal.
While CVSD modulation is great for encoding voice signals, it cannot be used
for general purpose analog-to-digital conversion. Even if you get around the
fact that the digital data is related to the derivative of the input signal, the
changing step size will confuse things beyond repair. In addition, the DC level
of the analog signal is usually not captured in the digital data.
The delta-sigma converter, shown in Fig. 3-19, eliminates these problems
by cleverly combining analog electronics with DSP algorithms. Notice that
the voltage on the capacitor is now being compared with ground potential.
The feedback loop has also been modified so that the voltage on the
64
The Scientist and Engineer's Guide to Digital Signal Processing
clock
analog
input
digital
latch
comparator
1
delta
modulated
output
positive
clock
charge
injector
syllabic
filter
logic
negative
0
charge
injector
clock
all 1's,
all 0's
detect
4 bit
shift
register
CVSD Modifications
FIGURE 3-18
CVSD modulation block diagram. A logic circuit is added to the basic delta modulator to
improve the slew rate.
capacitor is decreased when the circuit's output is a digital one, and
increased when it is a digital zero. As the input signal increases and
decreases in voltage, it tries to raise and lower the voltage on the capacitor.
This change in voltage is detected by the comparator, resulting in the charge
injectors producing a counteracting charge to keep the capacitor at zero
volts.
If the input voltage is positive, the digital output will be composed of more
ones than zeros. The excess number of ones being needed to generate the
negative charge that cancels with the positive input signal. Likewise, if the
input voltage is negative, the digital output will be composed of more zeros
than ones, providing a net positive charge injection. If the input signal is equal
to zero volts, an equal number of ones and zeros will be generated in the
output, providing an overall charge injection of zero.
The relative number of ones and zeros in the output is now related to the level
of the input voltage, not the slope as in the previous circuit. This is much
simpler. For instance, you could form a 12 bit ADC by feeding the digital
output into a counter, and counting the number of ones over 4096 clock cycles.
A digital number of 4095 would correspond to the maximum positive input
voltage. Likewise, digital number 0 would correspond to the maximum
negative input voltage, and 2048 would correspond to an input voltage of zero.
This also shows the origin of the name, delta-sigma: delta modulation followed
by summation (sigma).
The ones and zeros produced by this type of delta modulator are very easy to
transform back into an analog signal. All that is required is an analog lowpass filter, which might be as simple as a single RC network. The high
Chapter 3- ADC and DAC
65
clock
analog
input
Digital
Latch
comparator
0
1
positive
charge
injector
clock
negative
clock
charge
injector
delta
modulated
signal
digital
output
Counter
Latch
RESET to
LATCH to
begin ADC
cycle
end ADC
cycle
Digital
low-pass
filter
digital
output
Decimate
FIGURE 3-19
Block diagram of a delta-sigma analog-to-digital converter. In the simplest case, the pulses from a delta
modulator are counted for a predetermined number of clock cycles. The output of the counter is then latched
to complete the conversion. In a more sophisticated circuit, the pulses are passed through a digital low-pass
filter and then resampled (decimated) to a lower sampling rate.
and low voltages corresponding to the digital ones and zeros average out to
form the correct analog voltage. For example, suppose that the ones and zeros
are represented by 5 volts and 0 volts, respectively. If 80% of the bits in the
data stream are ones, and 20% are zeros, the output of the low-pass filter will
be 4 volts.
This method of transforming the single bit data stream back into the original
waveform is important for several reasons. First, it describes a slick way to
replace the counter in the delta-sigma ADC circuit. Instead of simply counting
the pulses from the delta modulator, the binary signal is passed through a
digital low-pass filter, and then decimated to reduce the sampling rate. For
example, this procedure might start by changing each of the ones and zeros in
the digital stream into a 12 bit sample; ones become a value of 4095, while
zeros become a value of 0. Using a digital low-pass filter on this signal
produces a digitized version of the original waveform, just as an analog lowpass filter would form an analog recreation. Decimation then reduces the
sampling rate by discarding most of the samples. This results in a digital
signal that is equivalent to direct sampling of the original waveform.
This approach is used in many commercial ADC's for digitizing voice and other
audio signals. An example is the National Semiconductor ADC16071, which
provides 16 bit analog-to-digital conversion at sampling rates up to 192 kHz.
At a sampling rate of 100 kHz, the delta modulator operates with a clock
frequency of 6.4 MHz. The low-pass digital filter is a 246 point FIR, such as
described in Chapter 16. This removes all frequencies in the digital data
above 50 kHz, ½ of the eventual sampling rate. Conceptually, this can be
66
The Scientist and Engineer's Guide to Digital Signal Processing
viewed as forming a digital signal at 6.4 MHz, with each sample represented
by 16 bits. The signal is then decimated from 6.4 MHz to 100 kHz,
accomplished by deleting every 63 out of 64 samples. In actual operation,
much more goes on inside of this device than described by this simple
discussion.
Delta-sigma converters can also be used for digital-to-analog conversion of
voice and audio signals. The digital signal is retrieved from memory, and
converted into a delta modulated stream of ones and zeros. As mentioned
above, this single bit signal can easily be changed into the reconstructed analog
signal with a simple low-pass analog filter. As with the antialias filter, usually
only a single RC network is required. This is because the majority of the
filtration is handled by the high-performance digital filters.
Delta-sigma ADC's have several quirks that limit their use to specific
applications. For example, it is difficult to multiplex their inputs. When the
input is switched from one signal to another, proper operation is not established
until the digital filter can clear itself of data from the previous signal. Deltasigma converters are also limited in another respect: you don't know exactly
when each sample was taken. Each acquired sample is a composite of the one
bit information taken over a segment of the input signal. This is not a problem
for signals encoded in the frequency domain, such as audio, but it is a
significant limitation for time domain encoded signals. To understand the shape
of a signal's waveform, you often need to know the precise instant each sample
was taken. Lastly, most of these devices are specifically designed for audio
applications, and their performance specifications are quoted accordingly. For
example, a 16 bit ADC used for voice signals does not necessarily mean that
each sample has 16 bits of precision. Much more likely, the manufacturer is
stating that voice signals can be digitized to 16 bits of dynamic range. Don't
expect to get a full 16 bits of useful information from this device for general
purpose data acquisition.
While these explanations and examples provide an introduction to single bit
ADC and DAC, it must be emphasized that they are simplified descriptions of
sophisticated DSP and integrated circuit technology. You wouldn't expect the
manufacturer to tell their competitors all the internal workings of their chips,
so don't expect them to tell you.