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
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`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-and-
`hold 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
`1/ 12
`between ±½ LSB, has a mean of zero, and a standard deviation of
` LSB
`(0.29 LSB). For example, passing an analog signal through an 8 bit digitizer
`adds an rms noise of:
`, or about 1/900 of the full scale value. A 12
`0.29 / 256
`0.29 / 4096  1 / 14,000
`bit conversion adds a noise of:
`, while a 16 bit
`0.29 / 65536  1 / 227,000
`conversion adds:
`. 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
`
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`Chapter 3- ADC and DAC
`
`37
`
`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.
`
`a. Original analog signal
`
`3.025
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`3.020
`
`3.015
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`3.010
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`3.005
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`3.000
`
`Amplitude (in volts)
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`0
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`5
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`10
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`15
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`20
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`25
`Time
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`30
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`35
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`40
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`45
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`50
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`analog
`input
`
`digital
`output
`
`S/H
`
`ADC
`
`c. Digitized signal
`
`3025
`
`3020
`
`3015
`
`3010
`
`3005
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`3000
`
`Digital number
`
`b. Sampled analog signal
`
`3.025
`
`3.020
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`3.015
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`3.010
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`3.005
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`3.000
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`Amplitude (in volts)
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`0
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`5
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`10
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`15
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`20
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`25
`Time
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`30
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`50
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`0
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`15
`20
`25
`30
`Sample number
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`35
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`40
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`45
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`50
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`d. Quantization error
`
`pdf
`
`1.0
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`0.5
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`0.0
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`Error (in LSBs)
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`-0.5
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`-1.0
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`0
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`10
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`15
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`25
`30
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`45
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`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
`A 2  B 2  C
`in quadrature:
`. The total noise on the digitized signal is
`0.2552  0.292  0.386
`therefore given by:
` 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 peak-
`to-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
`
`4
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`Chapter 3- ADC and DAC
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`39
`
`b. Dithering noise added
`
`original analog signal
`
`3005
`
`3004
`
`3003
`
`3002
`
`Millivolts
`
`3001
`
`3000
`
`a. Digitization of a small amplitude signal
`
`analog signal
`
`digital signal
`
`3005
`
`3004
`
`3003
`
`3002
`
`3001
`
`3000
`
`Millivolts (or digital number)
`
`with added noise
`
`30
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`35
`
`40
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`45
`
`50
`
`25
`Time
`
`0
`
`5
`
`10
`
`15
`20
`25
`30
`35
`Time (or sample number)
`
`40
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`45
`
`50
`
`0
`
`5
`
`10
`
`15
`
`20
`
`c. Digitization of dithered signal
`
`original analog signal
`
`digital signal
`
`0
`
`5
`
`10
`
`15
`20
`25
`30
`35
`Time (or sample number)
`
`40
`
`45
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`50
`
`3005
`
`3004
`
`3003
`
`3002
`
`3001
`
`3000
`
`Millivolts (or digital number)
`
`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.
`
`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
`continuous 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.
`
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`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.
`
`6
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`

`Chapter 3- ADC and DAC
`
`41
`
`b. Analog frequency = 0.09 of sampling rate
`
`Time (or sample number)
`
`d. Analog frequency = 0.95 of sampling rate
`
`3
`
`2
`
`1
`
`0
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`-1
`
`-2
`
`-3
`
`3
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`2
`
`1
`
`0
`
`-1
`
`-2
`
`-3
`
`Amplitude
`
`Amplitude
`
`a. Analog frequency = 0.0 (i.e., DC)
`
`Time (or sample number)
`
`c. Analog frequency = 0.31 of sampling rate
`
`3
`
`2
`
`1
`
`0
`
`-1
`
`-2
`
`-3
`
`3
`
`2
`
`1
`
`0
`
`-1
`
`-2
`
`-3
`
`Amplitude
`
`Amplitude
`
`Time (or sample number)
`
`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 recreate 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 there 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
`
`7
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`42
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`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,  .
`
`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 180 phase shift. Only two phase
`shifts are possible: 0 (no phase shift) and 180 (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.
`
`8
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`

`

`Chapter 3- ADC and DAC
`
`43
`
`DC
`
`Nyquist
`Frequency
`
`GOOD
`
`ALIASED
`
`0.0
`
`0.5
`1.0
`1.5
`2.0
`Continuous frequency (as a fraction of the sampling rate)
`
`2.5
`
`0.5
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`0.0
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`Digital frequency
`
`270
`
`180
`
`90
`
`0
`
`-90
`
`Digital phase (degrees)
`
`0.0
`
`0.5
`1.0
`1.5
`2.0
`Continuous frequency (as a fraction of the sampling rate)
`
`2.5
`
`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 right-
`hand 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
`
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`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, 2fs, 3fs, 4fs, etc., has received a copy and a left-for-
`right 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 ½fs. 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, fs, 2fs, 3fs, 4fs, 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
`
` along the horizontal axis are spaced closer in (f) than in (d).
`fS , 2fS , 3fS
`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
`
`10
`
`

`

`Chapter 3- ADC and DAC
`
`45
`
`Time Domain
`
`Frequency Domain
`
`b. Original signal's spectrum
`
`100
`
`f
`200
`s
`
`300
`Frequency
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`2f
`400
`s
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`500
`
`3fs
`600
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`3
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`2
`
`Amplitude
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`1
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`0
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`0
`0
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`3
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`a. Original analog signal
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`0
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`1
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`2
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`3
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`4
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`5
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`Time
`
`c. Sampling at 3 times highest frequency
`
`d. Duplicated spectrum from sampling
`
`lower
`sideband
`
`upper
`sideband
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`100
`
`f
`200
`s
`
`300
`Frequency
`
`2f
`400
`s
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`500
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`3fs
`600
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`2
`
`Amplitude
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`1
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`0
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`0
`0
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`3
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`original signal
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`impulse train
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`0
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`1
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`2
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`3
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`4
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`5
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`Time
`
`e. Sampling at 1.5 times highest frequency
`
`f. Overlapping spectra causing aliasing
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`fs
`100
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`2f
`200
`s
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`3f
`300
`s
`Frequency
`
`4f
`400
`s
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`5fs
`500
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`6fs
`600
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`2
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`Amplitude
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`1
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`0
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`0
`0
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`original signal
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`impulse train
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`1
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`Time
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`0
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`Amplitude
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`Amplitude
`
`Amplitude
`
`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.
`
`11
`
`

`

`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
`f  0, H ( f )  1
`represented by
`. For
`.
`fS
`
`H ( f )  
`
`sin(f /fs )
`f /fs
`
`
`
`sin (x) / (x)
`, called the sinc function or sinc(x).
`This is of the general form:
`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 zeroth-
`order 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)
`
`12
`
`

`

`Chapter 3- ADC and DAC
`
`47
`
`Time Domain
`
`Frequency Domain
`
`b. Spectrum of impulse train
`
`0
`0
`
`100
`
`f
`200
`s
`
`300
`Frequency
`
`2f
`400
`s
`
`500
`
`3fs
`600
`
`d. Spectrum multiplied by sinc
`
`"correct" spectrum
`
`sinc
`
`2
`
`1
`
`Amplitude
`
`0
`
`2
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`1
`
`Amplitude
`
`a. Impulse train
`
`0
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`1
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`2
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`3
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`4
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`5
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`Time
`
`c. Zeroth-order hold
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`5
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`0
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`0
`0
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`100
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`500
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`3
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`2
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`1
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`0
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`-1
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`-2
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`-3
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`3
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`2
`
`1
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`0
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`-1
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`-2
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`-3
`
`Amplitude
`
`Amplitude
`
`0
`
`1
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`2
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`3
`
`4
`
`Time
`
`f
`200
`s
`
`300
`Frequency
`
`2f
`400
`s
`
`3fs
`600
`
`e. Ideal reconstruction filter
`
`0
`0
`
`100
`
`f
`200
`s
`
`300
`Frequency
`
`2f
`400
`s
`
`500
`
`3fs
`600
`
`g. Reconstructed spectrum
`
`2
`
`1
`
`Amplitude
`
`0
`
`2
`
`1
`
`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).
`
`f. Reconstructed analog signal
`
`0
`
`1
`
`2
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`3
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`4
`
`5
`
`Time
`
`0
`
`0
`0
`
`100
`
`f
`200
`s
`
`300
`Frequency
`
`2f
`400
`s
`
`500
`
`3fs
`600
`
`3
`
`2
`
`1
`
`0
`
`-1
`
`-2
`
`-3
`
`Amplitude
`
`13
`
`

`

`48
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`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 b