16Oscillators, Function Generators, Frequency

and Waveform Synthesizers

16.1 Introduction

This chapter covers the more commonly used types of signal source instrumentsintroduced in Chap. 7. It surveys operating principles of these types and providessome guidance in understanding the specifications provided by manufacturersof these instruments.

16.2 Sine-Wave Oscillators

As used here, the term sine-wave oscillators refers to oscillator circuits whichnaturally produce sinusoidal waveforms. These circuits consist of an ac amplifierwith a positive feedback path from output to input. In the feedback path is afilter network: Its input-output (transfer) gain is very low except close to thedesired frequency of oscillation. Hence the ability of the circuit to operate as anoscillator (that is, to operate regeneratively by producing its own amplifier input)is restricted to a narrow band around this frequency.

16.2.1 Radio-frequency (RF) signal generators

This important class of instrument began in the late 1920s when the need wasrecognized for producing radio receiver test signals with accurate frequencies(1 percent) and amplitudes (1 or 2 dB) over wide ranges. The basic block diagramof these instruments remained nearly unchanged until about 1970, whenfrequency synthesizer circuits began to usurp “free-running” oscillators in thewaveform generator section of the instruments.

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Figure 16.1 shows a simplified form of a typical rf sine-wave oscillator usedin a signal generator. An amplifier producing an output current proportional toits input voltage (transconductance amplifier) drives a tunable filter circuitconsisting of a parallel LC resonant circuit with a magnetically coupled outputtap. The tap is connected to the amplifier input with the proper polarity toprovide positive feedback at the resonant frequency of the filter. To assureoscillation, it is necessary to design the circuit so that the gain around the loop—the amplifier gain times the filter transfer function—remains greater than unityover the desired frequency tuning range. But since the output signal amplitudestays constant only when the loop gain is exactly unity, the design must alsoinclude level-sensitive control of the loop gain. Such a level-responding circuitis nonlinear, because it does nothing until the amplifier output signal reachessome predetermined amplitude, rather than responding proportionately, like alinear circuit. In the oscillator shown here, the automatic gain control (AGC)circuit provides the level control function. The AGC monitors the sine-waveamplitude at the amplifier output and acts to smoothly decrease its gain whenthe desired amplitude is reached.

This oscillator circuit produces a low level of harmonics in its output. This isbecause the sine-wave amplitude can be kept within the linear operating rangeof the amplifier by the AGC circuit. Many other oscillator circuits depend onamplifier saturation as the gain-limiting mechanism, and this generates a gooddeal of harmonic energy, which degrades the signal purity.

16.2.2 Audio oscillators

At audio frequencies there is more emphasis on purity of waveform than at rf,so it is natural to use an oscillator circuit which generates a sine wave as itscharacteristic waveform: an amplifier with filtered feedback. For audio

Figure 16.1 Radio-frequency sine-wave os-cillator.

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frequencies, however, the elements of an LC resonant circuit for the filter becomeexpensive and large, with iron or ferrite core construction necessary to get thelarge values of inductance required. Another handicap is that ferrocore inductorshave some nonlinear characteristics which increase harmonic output when theyare used in a resonant filter circuit.

Using resistors and capacitors, it is possible to get a voltage transfer functionwhich resembles that of a resonant circuit in both its phase and amplituderesponse. The four-element RC network in the dashed portion of Fig. 16.2 ischaracterized by an input-output voltage ratio V1/V0 (called a transfer function)which varies with frequency. This transfer function has an amplitude peak of1/3 V0 at the “resonant” frequency 1/(2πRC) and a phase shift passing throughzero at this frequency. The value of Q, a measure of the bandwidth of thetransfer function, is only 1/3, normally a poor value for use in a feedbackoscillator. (Low resonant circuit Q means frequency drift and noise in theoutput signal.) However, if R1 and R2 are adjusted so that their transfer functionV2/V0 is about the same as V1/V0 at resonance (that is, 1/3), then the combinedtransfer function (V1-V2)/V0 exhibits a much sharper (higher Q) character.This topology, with resistors R1 and R2 added to the four-element network, isnow a bridge circuit, and it was called the “Wien bridge” by William R.Hewlett.If a high-gain amplifier is connected as shown, oscillations can be generatedat the resonant frequency, provided the bridge is just slightly unbalanced toprovide net positive feedback for the amplifier.

The frequency of the Wien bridge oscillator is tuned by using ganged variablecapacitors for the two C’s. Since a 10:1 capacitance ratio is readily obtained in amechanical capacitor, this provides continuous tuning over a decade of frequency.This is, in itself, a decided advantage over LC resonant circuit tuning, which

Figure 16.2 Wien bridge oscillator.

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exhibits a square-root variation of resonance vs. capacitance tuning. Decadeband switching is provided by switching the R’s in pairs.

As always with feedback oscillators, once oscillation starts, it is necessary tostabilize the loop gain at unity to provide a constant sine-wave amplitude. Thisproblem was solved by replacing R2 with a nonlinear resistor, such as a smallChristmas-tree light bulb. Since tungsten has a positive temperature coefficient,an increasing sine-wave amplitude causes the bulb resistance to increase, thusincreasing the negative feedback in the amplifier. Because the filament thermaltime constant is a second or more, its resistance does not change over the muchshorter time of one cycle of the sine-wave signal, and so the filament resistancecontributes negligible distortion to the output signal.

16.2.3 Performance and specifications

Since there are no ideal signal sources producing perfectly stable, noiselesswaveforms, it is necessary to describe the actual performance of instruments sothat the user knows what (and what not) to expect. Several parameters, becauseof their utility, have become standardized for describing the quality of waveformsproduced by signal sources in quantitative terms. These parameters are discussedin this and similar sections in this chapter.

Frequency accuracy. The frequency of rf and audio oscillators is usuallycontrolled by mechanically varied capacitors. The relationship between frequencyand capacitance is recorded as a dial calibration and can be read and set tobetter than 1 percent. However, other factors usually decrease this accuracy:Manufacturing tolerances, component aging, and ambient temperature are themain ones.

Amplitude accuracy. The amplitude of the output waveform is usuallyexpressed in volts or dBm (decibels referred to 1 mW of power). Specificationsfor amplitude vary widely; as might be expected, more expensive instrumentsusually come with more precise numbers. Rf signal generators, which oftencost more than audio sources, also usually have better specifications. Somesignal generators have a meter for reading the output amplitude, and a verniercontrol to set it to a standard level. Other generators have a digital readoutand an automatic circuit for keeping the amplitude constant. There is usuallya switchable attenuator to reduce the output level in 10-dB (or smaller) steps.The accuracy of the output level degrades over a wide frequency range (this iscalled “flatness” on data sheets) and with larger values of attenuation. Botheffects are mainly due to parasitic impedances in the attenuator. Output levelsare specified with a standard resistive load placed at the generator output. Thisis usually 50 Ω for rf sources but may be 75 Ω if the source is to be used in videoapplications.

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Typically, audio oscillators also have a step attenuator in the output, butlevels between the steps are set with an uncalibrated potentiometer. Both flatnesswith frequency and attenuator accuracy are usually specified. The standardload is 600 Ω.

Frequency stability. The ideal sine wave is noiseless and absolutely constant inboth frequency and amplitude. Nonideal oscillators, of course, depart from this,and stability is a measure of the degree to which they approximate the ideal.Stability is further classified as long-term and short-term, depending on whethersignificant frequency change occurs on a time scale of minutes or fractions of asecond. An example of long-term frequency stability is “warmup drift,” a changein frequency usually caused by dimensional change in inductors and capacitorsas they are heated over minutes or hours by circuit power dissipation. Suchstability is specified in terms such as “less than 10 kHz change 30 minutes afterpower-on.” On the other hand, short-term stability is affected by physical factorswhich have higher frequency content, such as random noise, power supply ripple,and microphonics. These lead to more rapid frequency variations of the outputsignal. The effects of short-term factors are combined on an rms basis and thenexpressed as frequency modulation (FM) of the oscillator. A typical example ofsuch a specification is “hum and noise less than 10 Hz rms deviation.”

Harmonic distortion. For a signal trying to be a perfect sine wave, one measureof how closely it approximates the ideal is its harmonic distortion. A “harmonic”is a sine wave having a frequency which is an integer multiple of the basicsignal frequency. If a sine wave is accompanied by some of its harmonics, theresultant waveform is no longer sinusoidal; that is, it is distorted. As the numberand the amplitude of harmonic terms increase, so does the distortion. Theamount of harmonic distortion is usually specified in one of two ways. Lesscommon is to give an upper bound for any harmonic term; e.g., “all harmonicterms less than –80 dBc” means every harmonic is no larger than 80 dB belowthe “carrier” or output signal. A weakness in this specification is that there canbe a large number of harmonic terms and hence a lot of total harmonic energy.More commonly, the manufacturer combines the energy of all harmonics andexpresses the rms value as a percentage of the fundamental, such as “harmonicdistortion less than 0.5 percent.” In the latter case, sometimes residual noisein the output is added to the harmonic energy to give a noise-plus-distortionspecification.

16.3 Function Generators

This term is used to describe a class of oscillator-based signal sources in whichthe emphasis is on versatility. Primarily, this means providing a choice of outputwaveforms. It also includes continuous tuning over wide bands with max-minfrequency ratios of 10 or more, sub-Hz to MHz frequencies, flat output amplitude,

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and sometimes modulation capabilities: frequency sweeping, frequencymodulation (FM), and amplitude modulation (AM). In their frequency accuracyand stability, function generators are inferior to sine-wave oscillators, but theirperformance is quite adequate for many applications.

Some frequency synthesizers (Sec. 16.4) are also called “precision” functiongenerators by their manufacturers. What this means is that circuitry has beenadded to the synthesizer to produce other waveforms in addition to sine waves.

16.3.1 Threshold-decision oscillators

To realize the versatility described above—especially the wide frequency tuningrange—function generators nearly always use a basically different oscillatormechanism than filtered feedback around an amplifier. This is the threshold-decision oscillator. Section 7.3.1 has additional discussion on this oscillator.

The threshold-decision oscillator requires three basic elements:

1. A circuit whose state (voltage, current, etc.) changes with time

2. A way to reset this circuit to an initial state

3. A method for deciding when to perform the reset

Figure 16.3a shows a simplified form of such an oscillator. An RC circuit chargesfrom a positive supply. The time-changing state of this circuit is the voltageacross the capacitor. A comparator monitors this voltage and, when this reachesa reference level (the decision criterion), momentarily closes the switch. Theswitch discharges the capacitor, restoring the RC circuit to its initial state,and the cycle restarts. Figure 16.3b shows a couple of cycles of the oscillatorwaveform.

The typical function generator oscillator circuit, shown simplified in Fig. 16.3c,is a little more complex than the circuit of Fig. 16.3a in order to gain someversatility. Two current sources i+ and i– are available to charge and dischargecapacitor C. A switch, under control of a bistable flip-flop, determines whichcurrent source is connected. The voltage across C is monitored by twocomparators. Assume that i+ is initially connected to C. This causes its voltageto rise linearly. This property is important, since many applications require alinear ramp waveform. When the voltage reaches the high reference, the uppercomparator responds, resetting the flip-flop. This actuates the switch, connectingi- to C, which then begins to discharge linearly. When the voltage of C reachesthe low reference, the lower comparator responds, resetting the flip-flop andbeginning a new cycle.

One cycle of oscillation consists of two linear voltage segments across C. Ifthe current sources are the same magnitude, the slopes of these segments willbe of equal magnitude but opposite sign. The voltage across C will therefore bea symmetrical triangle waveform. The frequency of oscillation can be computed

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as the inverse of the period, and the period is the sum of the durations of thepositive and negative segments. For the symmetrical case, this is

(16.1)

The triangle waveform is one of the basic waveforms generated by this oscillator.The other is the square wave at the output of the flip-flop. Narrow pulses arealso available from the comparators, but these are seldom used.

Varying the frequency of this oscillator is simple. Since, from Eq. 16.1, thefrequency of oscillation is proportional to the charging current, both currentsources are designed so that their magnitudes vary linearly with a common

Figure 16.3 Threshold-decision oscillator. (a) Simple circuit; (b) waveform across C in simplecircuit; (c) function generator version; (d) asymmetrical triangle produced in function generator.

where fosc = frequency of oscillation T = time duration of either segment i = charge (and discharge) current C = capacitor, farads Vh = high reference voltage Vl = low reference voltage

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control voltage. This makes the oscillator frequency directly proportional to thisvoltage. In practice, continuous frequency control is obtained by a potentiometerwhich produces the control voltage, while frequency bands are switched bychanging the value of C in decade steps.

The linearity of the triangle waveform segments is directly related to thequality of the current sources: how constant their current remains as the voltageacross C varies.

Varying the duty factor of the output waveforms is just a simple matter ofvarying the ratio of i+ to i–. Duty factors as asymmetrical as 100:1 are readilyachievable. A little design effort is required to maintain a constant waveformfrequency while varying the duty factor: A nonlinear relation is needed betweenthe increase in one current source and the decrease in the other. Figure 16.3dshows the waveform across the capacitor when the magnitude of i– is 10 timesthat of i+.

16.3.2 Producing sine waves with a function generator

From this description of a function generator oscillator, it is evident that trianglewaves and square waves are its natural or intrinsic waveforms, while sine wavesare not. Since a versatile source should include the most basic signal—a sinewave—in its repertoire, it is necessary to create sine waves from the waveformswhich are present.

One possible method would be to select the fundamental from the trianglewaveform, using a low-pass or bandpass filter. However, this is impractical,because such a filter would need to be tunable to track the wide tuning range ofthe oscillator. Instead, what is usually done is to apply a triangle wave to theinput of a nonlinear device whose transfer function approximates a sinusoidalshape. A big advantage of this approach is that such a device is frequency-independent, thus doing away with the need for filtering. Distortion in the outputwaveform is determined by the accuracy of the approximation to a true sinusoidaltransfer function.

A practical method of implementing such a transfer function is shown in Fig.16.4a. This circuit effects a piecewise-linear approximation to a sinusoidaltransfer function. As the voltage of the incoming triangle waveform departsfrom zero toward its positive peak, it turns on successive biased diodes, beginningwith D1. As each diode conducts, it connects its associated resistor across thebottom leg of the voltage divider formed by these resistors and Rin. The value ofthe voltage division is the transfer function. Consequently, as more diodesconduct and connect their resistors in parallel, the resistance of the bottom legof the divider decreases, and so does the value of the transfer function. Thesame action occurs on negative excursions, with being the first diode toconduct. The result of these actions is a transfer function composed of a series ofstraight-line segments. Bias voltages controlling diode conduction points areproduced by the bottom row of resistors. The Ri and the bias voltages are chosento cause the straight-line segments to approximate a sinusoidal waveform, as

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seen in Fig. 16.4b, where the dots indicate the breakpoints, or divisions betweensegments. Only four breakpoints per half waveform are shown here for simplicity;usually six are used. With the proper placement of six breakpoints, distortion ofthe output sine approximation can be as low as 0.25 percent.

Figure 16.4 Sine waves from triangle waves. (a) Converter circuit; (b) converter piecewise-linear transferfunction.

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16.3.3 Modulation

Most function generators provide a terminal where the user can apply a voltageto vary the frequency of the generator. The user’s voltage is added to thefrequency control voltage coming from the front panel potentiometer. This is aconvenient way to produce frequency modulation (FM). The quality of theresultant FM will be directly related to how linearly the oscillator frequencyresponds to the control voltage, since nonlinearity will produce distortion in themodulation.

A special case of FM is frequency sweeping. The ability of function generatorsto sweep frequency makes possible some interesting applications. With anoscilloscope and a function generator, one can make a rudimentary networkanalyzer (Chap. 28), an instrument used to display the frequency transferfunctions of filters and other circuits. One connects the frequency control terminalto the horizontal scanning waveform of an (analog) oscilloscope. This is anasymmetrical triangle wave, much like Fig. 16.3d. By doing this, the scanningwaveform causes the frequency of the function generator to track with thehorizontal position of the scope trace. In turn, this maps a linear frequencyscale onto the X axis of the scope. By then connecting a test circuit between thefunction generator signal and the vertical scope input, one may view (repetitively)the amplitude vs. frequency characteristic of the test circuit.

Amplitude modulation (AM) is a little harder to produce in these oscillatorsthan FM. Simultaneously increasing the switching thresholds and the chargingcurrents will vary the amplitude of the triangle waveform, but there is likely tobe a lot of simultaneous FM unless this is precisely done. A better method is toapply the oscillator output signal to an amplifier whose gain may be varied witha control signal. Because of this additional complexity, AM is usually includedonly in the more expensive instruments.

16.3.4 Specifications

Frequency accuracy of function generators is usually defined in terms offrequency dial accuracy, and 5 to 10 percent of full scale would be representativeof many sources. Residual FM is usually not specified. It is higher than that ofa feedback oscillator, but this fact is not too important for the usual applicationsof function generators. On the other hand, amplitude flatness is quite good.Over the full range of the instrument, it is usually better than 1 dB. However,absolute amplitude accuracy is usually not given, and only an uncalibratedcontrol regulates the output level. For sine waves, distortion is specified, usuallyas an rms percentage of the amplitude of the fundamental.

Rise and fall times of pulse waveforms are nearly always given. Linearity oftriangle waveforms—that is, the deviation of the segment voltages from astraight-line shape—is less frequently specified. However, deviations of lessthan 1 percent are common, especially in the lower-frequency ranges of aninstrument.

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16.4 Frequency Synthesizers

Starting in the early 1960s, frequency synthesis techniques have grownsteadily in instrument applications and have become the dominant technologyin signal sources. Synthesis refers to the use of a fixed-frequency oscillatorcalled the reference oscillator or, sometimes, the clock. The latter term isborrowed from computers and refers to the role of this oscillator in pacing orsynchronizing all the other circuits in the instrument. This is usually aprecision crystal oscillator with an output at some cardinal frequency, such as10 MHz. Various signal processing circuits then operate on the reference signal(or are paced by it) to produce a large choice of output frequencies. Everypossible output frequency is derived from the reference oscillator frequency bymultiplying its frequency by the fraction m/n, where m and n are integers. Theinteger m, and sometimes n, are supplied by the user via the front panelkeyboard or an equivalent input. Suppose the reference frequency is 10 MHzand n is 10,000. Then, by varying m, the user can generate a range of outputfrequencies spaced 1 kHz apart.

16.4.1 Direct synthesis

By assembling a circuit assortment of frequency dividers and multipliers, mixers,and bandpass filters, an output m/n times the reference can be generated. Thereare many possible ways to do this, and the configuration actually used is chosenprimarily to avoid strong spurious signals, which are low-level, non-harmonicallyrelated sinusoids. Figure 16.5 shows one way to produce a 13-MHz output froma 10-MHz reference. The inputs to the mixer are 10 and 3 MHz, the mixerproduces sum and difference frequency outputs, and the bandpass filter on theoutput selects the 13-MHz sum. Notice that another bandpass filter could havebeen used to select the 7-MHz difference, if that were wanted.

The principal advantage of the direct synthesis technique is the speed withwhich the output frequency may be changed. Against this are a variety ofdisadvantages which, together with better alternative techniques (see below),have caused direct synthesis to fall out of favor. It is very hardware-intensive,and therefore expensive. When switching frequencies, phase continuity is lost,unless certain constraints are placed on when to switch, such as only at exactmillisecond intervals in the m/10,000 example given above. It is also very proneto spurious signals in the output.

16.4.2 Indirect synthesis

This name derives from the use of an oscillator other than the reference togenerate the output. However, by placing the oscillator in a phase locked loop,its frequency is controlled so that its output is the desired m/n times the referencefrequency.

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Phase locked loop synthesis. This most popular technique came into wide usewith the availability of programmable divider ICs. These are counters whosecount modulus (the number they reach before starting over) is externallyprogrammable. With such a divider, a phase-locked loop (PLL) becomes avariable-modulus frequency multiplier. In addition, other PLLs act as bandpassfilters, selecting the sum or difference frequencies in a mixer output. Figure16.6 shows how one could synthesize the same output as the circuit of Fig. 16.5by using PLLs. The reference is divided to 1 MHz, which is applied to the loopphase detector. With the variable modulus divider programmed to 13, the loopwill stabilize when the voltage controlled oscillator (VCO) outputs exactly 13MHz. Programming the divider to another number will cause the loop to lock tothat number times 1 MHz. This observation reveals that the basic frequencyresolution—the spacing between available output frequencies—is equal to theloop reference frequency, or 1 MHz in this case.

This fact opens up an interesting possibility: If we want a synthesizerfrequency resolution of, say, 1 Hz, why not divide the reference oscillator to 1Hz and use that frequency as the loop reference? Of course, it would also benecessary to use a very large modulus of 13,000,000 in the loop divider in theabove example, but this could be an LSI part of modest cost.

Figure 16.5 Direct frequency synthesis.

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Two important facts foreclose this possibility. One is that the loop frequencyswitching speed is of the order of 10 times the period of the loop reference, or 10s in this case! The other is that frequency multiplier loops also multiply noise atthe phase detector, which appears as noise sidebands on the oscillator output.This noise would be raised 142 dB in this example!

In practice, the maximum modulus used is a few thousand. When fineresolution is needed, sequences of multiplication, division, and addition areused, involving a number of loops. To introduce the idea of a multiple-loopsynthesizer, Fig. 16.7 shows one scheme for synthesizing 13.1 MHz. Themultiplier PLL is the same as the one in Fig. 16.6, only its modulus is 31,thereby producing a 31-MHz output. This is divided by 10, and the resulting3.1 MHz is added to 10 MHz by the summing PLL at the right. Techniques likethis can be extended to reach any desired resolution. Since multiplicationnumbers are low and the loop reference frequency is high, the output will havelow noise sidebands and be capable of fast switching. All this occurs at a costof greater circuit complexity.

Fractional-N synthesis. Various attempts have been made to achieve finefrequency resolution in one PLL while avoiding the penalties of poor noise andswitching speed. A commercially successful technique which achieves this isknown as “fractional-N” synthesis, after the fact that the PLL locks to a non-integer multiple of the loop reference. This multiple is written as N ·F, where N

Figure 16.6 Indirect frequency synthesis.

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is an integer and F is a rational number less than 1. Using loop referencefrequencies of 100 kHz and higher, fractional-N synthesizers achieve resolutionsin the microhertz region and switching speeds of a millisecond or less.

To introduce this technique by using a simple choice of numbers, assumethat 1.1 is the desired multiplier: that is, the PLL should lock at 1.1 MHz for aloop reference of 1 MHz. Figure 16.8a shows a block diagram of a fractional-Nloop configured for this case. To explain how it works, it is useful to assume thatthe loop is already locked to 1.1 MHz, and then to examine its operation to seethe conditions which make this true.

First, consider the phase detector. The divider is set to 1 (the value of N), soits output pulse rate is the same as its input, the VCO. Since there are 11 cyclesof the 1.1-MHz VCO for every 10 of the 1.0-MHz reference, then, for each cycleof the reference, the VCO phase advances 0.1 cycle, or 36°, relative to thereference. The phase detector samples this phase difference every referencecycle, and so its output voltage increases positively, let’s say, by a stepcorresponding to 36° every microsecond. Ordinarily, this condition would quicklydrive most phase detectors out of range, since their range is, at most, 360°.

At the bottom of Fig. 16.8a is an accumulator. This is a decimal registerclocked by the reference, to whose contents is added an amount F each cycle.The accumulator is scaled so that 0.999…is the maximum value it can hold. In

Figure 16.7 Multiple loop indirect synthesis.

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Figure 16.8 Fractional-N frequency synthesis. (a) Fractional-N circuit;(b) waveforms in fractional-N circuit.

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the example, F has the value 0.1, so, beginning with zero, the accumulator willoverflow on the tenth clock pulse. However, 10 clock pulses also correspond tothe time for the phase detector output to change by a voltage equal to 360°.

Along with the accumulator overflow there is a carry output from the adder.This carry is connected to the divider and momentarily changes the dividermodulus from 1 to 2—in effect skipping one cycle from the VCO. For the phasedetector, the loss of one divider pulse is equivalent to abruptly retarding theVCO phase by 360°. So, instead of advancing another 36°, the phase detectoroutput falls negatively an amount corresponding to 324° (0.9 cycle) at the sametime that the accumulator contents change from 0.9 to 0.0.

Although the periodic pulse removal keeps the phase detector output withinits linear range so that the loop can lock, the sawtooth waveform, shown inFig. 16.8b, in the phase detector output is very undesirable, as it producesphase modulation sidebands on the oscillator. But, when this loop is locked tothe offset frequency of 1.1 MHz, the accumulator contents predict the phasedetector sawtooth which results from the frequency difference between thetwo phase detector inputs. By properly scaling and polarizing the output froma D/A converter connected to the accumulator, the D/A output waveform maybe used to cancel the phase detector sawtooth. (See Chap. 7 for an explanationof D/A converters.) The two waveforms (shown in Fig. 16.8b) are added, sampledat times indicated by the dots, and filtered to provide the oscillator controlvoltage.

In the general case of a fractional-N synthesis loop, the integer part of thedesired multiplier is supplied to the digital divider and the fractional part tothe accumulator. The sudden removal of 360° of oscillator phase (as seen by thephase detector) is accomplished either by changing the modulus momentarilyto N+1 or by “swallowing” one cycle from the oscillator input to the phase detector.The accumulator may be made arbitrarily long, allowing very small frequencyoffsets (e.g., 1 µHz) to be used.

Limitations of indirect synthesis. The most important concern for a user of anindirect synthesizer is usually its behavior when a frequency change iscommanded, that is, its frequency switching speed. Because PLLs are feedbackcontrol circuits, there is a finite response time when an operating change isimposed. For some applications, this is of little importance. For others, such asfrequency hopping, the time required to change frequencies may be intolerablyslow. It is highly advisable to compare carefully the synthesizer specificationswith the needs of the proposed application.

16.4.3 Sampled sine-wave synthesis

Many technical workers are familiar with sampling theory, in particular thecentral precept of this theory: A waveform may be exactly reconstructed from asequence of uniformly spaced samples of its values, provided the sampling rate

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(i.e., frequency) is at least twice that of the highest-frequency component in thewaveform. Applying this to frequency synthesis means generating samples of asine wave and interpolating among the sample values to obtain a smoothwaveform, in accordance with sampling theory. Although the sine wave valuescould be computed as the signal is generated, in practice they are stored in alookup table. Values of the sine-wave phase are used as the argument for thetable lookup. The phase values are determined by an accumulator, similar tothat in the fractional-N circuit, in which the phase increment used is a numberproportional to the desired frequency. The numeric output from the lookup tablemust be converted to an electrical signal and smoothed for output, as the nextsection shows.

Advantages and limitations. There are great advantages to this approach. Themost apparent is instantaneous frequency switching. This is possible becausethe size of the angle increment between table lookups may be changedinstantaneously. For instance, a tabular increment of 36° would generate anoutput frequency of 0.1 times the sample rate. If the increment is changed to72°, the output frequency becomes 0.2 times the sample rate. The waveformphase at which this happens may also be chosen arbitrarily. With enoughcomputing ability to determine the lookup argument, various precise yet complexmodulation formats may be generated, such as FM, PM, and linear sweeps.

Another advantage is that the frequency of a sine wave synthesized in thismanner (point-by-point) is accurate, just as with frequency synthesis alreadydescribed. This is due to two causes: first, the phase increment is usually accurateto 6 or 8 digits, and second, the time intervals at which new phase informationis computed and the sine function evaluated are precise, coming from theprecision source. Since a sine-wave frequency is defined in terms of the derivativeof phase (time rate of change of phase), both of these quantities—time andphase—are accurately controlled in this technique.

The principal disadvantage of this technique is its limited frequency range.A fair amount of digital logic and arithmetic must occur in each sample. Inpractice, 2.5 samples per output cycle is about the minimum sampling rate.With this requirement, and using TTL logic to implement the circuits, themaximum output frequency is probably less than 10 MHz.

Typical block diagram. A typical, although simplified, block diagram of a sampledsine-wave synthesizer is shown in Fig. 16.9a. An accurate reference clock regulatesthe timing of a digital accumulator: that is, a register whose stored numericalvalue increases by an externally supplied constant value during each clock cycle.This constant represents the increase in the phase, during the time of a clockperiod, of the desired synthesized signal. Thus it is labeled “frequency constant”since, in the limit, frequency is the rate of change of the phase of a sine wave. Thecontents of the accumulator therefore correspond to the sample values of thephase of the desired signal, modulo 2π radians (the remainder after dividing the

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Figure 16.9 Sampled sine-wave frequency synthesis. (a) Block diagram of technique; (b) sampled sine waveforms:(1) ideal signal to be synthesized, (2) D/A output, and (3) filtered D/A output.

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phase by 2π). These contents, representing phase, are supplied to the sine lookuptable, which produces a number corresponding to the sine of its input. The sinenumber is then changed into a voltage by the digital-to-analog converter, whoseoutput is a series of steps corresponding to the samples. Some form of outputlevel control, such as an attenuator, is generally provided, although not shown inthe figure.

Constructing the sine wave. Typical waveforms which are important in carryingout this process are shown in Fig. 16.9b. Waveform (1) represents the ideal sinewave to be synthesized. This waveform doesn’t exist within the instrument, butsample values of it are computed and are shown by the dots along the sinewave. Note the uniform time axis spacing of these samples. The sample valuesare supplied to a D/A converter, whose output is a series of voltage steps, shownas a discontinuous waveform (2). The stepped waveform (2) is smoothed by aninterpolator, which is just a low-pass filter whose cutoff frequency is less thanhalf the clock rate. The sine-wave output from this filter is shown as waveform(3). Note that this reconstructed sine wave is delayed, with respect to waveform(1), by one-half the sampling interval; this is characteristic of the sample-and-hold process.

A couple of interesting details about constructing the sine lookup table arenot evident in the simplified diagram. One is that the stored values of the sinefunction need only represent one quadrant (a fourth part of a cycle, or 0 to π/2radians), because of sine-wave symmetry. Simple digital logic, including invertingand complementing functions, is used to generate the data for the other threequadrants.

Another detail concerns the resolution of the stored table. It is desirable torepresent the frequency by six or eight decimal digits. Yet to store the sinefunction, even only one-fourth of it, to this degree of resolution would require ahuge ROM or other storage device. In practice, the ROM input argument is onlytwo or three digits, which are supplied by the most significant digits in theaccumulator. The remaining resolution is computed by relying on the fact thatlinear interpolation of the sine function is quite accurate for small distancesbetween arguments. The lookup circuit therefore uses the least significant digitsfrom the accumulator and linearly interpolates between table entries. Suchinterpolation requires determining the slope of the function, but these data arealready available: the derivative of the sine is the cosine, so a table lookup π/2radians from the current position gives this number!

Limitations of sampled sine-wave synthesis. Being a sampled-data system,this synthesis technique is subject to the inherent limitations of such systems:

Quantization noise. This is caused by finite word-length effects. Usually enoughresolution is provided in these synthesizers to maintain the noise floorcomparable with that of other techniques.

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Aliasing. For output frequencies close to half the sampling rate, the outputlow-pass filter may not adequately remove the sampling image (“alias”), whichwill appear as a spurious component.

Spurious components. A more serious problem in the output waveform is dueto imperfections in the D/A converter. Signal degradation derives from two kindsof imperfections: level inaccuracies due to bit weight errors, and transient energygenerated when the D/A changes levels. The latter problem can be minimizedby adding a sample-and-hold circuit to the output. This is a circuit which assumesthe output voltage of the D/A at a time when the levels are not changing, anddoes not “look” at the D/A during the time when they are changing.

16.4.4 Synthesized function generators

Signal sources with this name are usually indirect frequency synthesizers, andso their basic output signal is a sine wave. However, their designers have madean effort to approach the waveform versatility of classic function generators byadding circuitry to produce pulse and triangle waveforms, and sometimesmodulation capability. The result is a most useful combination: a functiongenerator with the frequency precision of a synthesizer.

A particularly interesting feature is the ability to generate a very preciseramp, which is a triangle wave with one segment of negligible width. This isachieved by supplying a linear phase detector with two accurate signals with asmall frequency offset. For instance, if the two signals are 20.000 and 20.001MHz, the detector output is a 20-MHz pulse train whose pulse widths are linearlyincreasing, returning to zero 1000 times per second. With a low-pass filter, aprecise 1-kHz ramp is recovered whose linearity is superior to the best analogfunction generator. Figure 16.10a shows the block diagram of this technique,Fig. 16.10b shows the phase detector output before entering the low-pass filter,and Fig. 16.10c shows the linear ramp at the output of the low-pass filter.

16.4.5 Synthesizer specification

Since frequency synthesizers are complex instruments, a number of categoriesare required to characterize their performance and signal quality. Followingare the most important of these.

Frequency range and resolution. Although manufacturers naturally herald thetotal frequency range of their products, it is worthwhile to examine thespecifications carefully to see whether this range is covered in a single band ora series of contiguous bands. In the latter case, adjacent bands may have differentcharacteristics such as noise. Likewise, when a frequency transition crosses theboundary between two of these bands, the output transient may be quite a bitlarger than a normal transition which occurs for a transition within a band.

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Figure 16.10 Precision ramp generator. (a) Block diagram of technique; (b) unfiltered phasedetector output; (c) filtered output.

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Resolution is most commonly a constant number, such as 0.1 Hz, over the entirefrequency range. However, some commercial instruments feature a much smallerresolution, such as 1 Hz, over their lowest frequency ranges.

Frequency switching speed. This specification is a measure of the time requiredfor the source to stabilize at a new frequency after the change command isissued. Since the majority of synthesized sources use phase-locked loops, thespecification really reflects the loop transient response characteristics. Becausethese transients die off asymptotically, specifications often use nearness-of-approach language, such as “to within 100 Hz of the final frequency.” Suchnumbers are usually in the order of tens of milliseconds, although some sourcesdesigned for fast switching claim switching times in hundreds of microseconds.As mentioned above, digital sine-wave synthesizers can change frequency nearlyinstantaneously, usually in just a few reference clock cycles.

Signal purity. This specification is most easily understood as applying to thefrequency domain. It describes how well the output signal approximates theideal single spectral line. There are two principal contributors which degradethis ideal: phase noise and spurious signals (“spurs”).

Phase noise. This term is somewhat peculiar to synthesizers. It refers to thesidebands which result from phase modulation of the carrier by noise. AMsidebands from noise modulation exist also, but these are usually insignificant,because of the amplitude-limiting mechanism of oscillators. Phase noise isspecified most simply as the total sideband power, in dB with respect to thecarrier (dBc). In higher-performance instruments, the noise power is specifiedin the frequency domain, using frequency offset from the carrier as the abscissa.The specification is in the form of a loglog plot of noise power in dBc per Hz BW,or its verbal equivalent.

Spurious signals. These are non-harmonically related tones which exist in theoutput along with the desired sine wave. There are several sources of spurs,including unintentional coupling among circuits in the instrument and distortionproducts in signal mixers. An undesirable property of spurs is that they generallymove in frequency at various rates as the signal frequency is changed, oftencrossing over the output. Such “crossover spurs” can’t be removed by filtering.It is customary to specify spurious signals by declaring an upper bound for theiramplitudes in dBc.

16.5 Arbitrary Waveform Synthesizers

By generalizing the sampled sine-wave synthesis technique to allow otherwaveforms to be stored in memory, a versatile signal source instrument can beachieved: the arbitrary waveform synthesizer. In this instrument, the user

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determines the shape of one period of the waveform and its repetition rate. It iseasy to imagine how useful this ability would be in many applications. A goodexample is the testing of patient cardiac monitors by simulating various ECGwaveforms to which the monitors must respond in a prescribed manner.

16.5.1 Principles of operation

The operation of the arbitrary waveform synthesizer is similar to that of thestored sine-wave synthesizer, in that stored digital sample values of the waveformare converted to sequential voltage values of a signal by a digital-to-analog (D/A) converter. However, there are a few distinctions between the two instruments.

Figure 16.11a shows a basic block diagram of an arbitrary waveformsynthesizer. It is very much like that of the sampled sine-wave synthesizer, butthere are a few differences, both in the hardware and in operation. The heartof the instrument is the random-access memory (RAM). This memory has anumber of locations—for instance, 256—in which the user can store sequentialamplitude values of a particular waveform to be generated. Only one period ofthe waveform is stored, as the example in Fig. 16.11b. This particular waveformis a combination of a sine wave plus its third harmonic, typical of a powertransformer’s magnetizing current. The sample values are assumed to be evenlyspaced in time.

Provision must be made for the user to load the sample values into the RAM.In the simplest case, the user can enter the successive values from a keyboard.A more useful method, less prone to numerical mistakes, is for the user to drawthe desired waveform on paper and then to trace over the waveform with adigitizer stylus which inputs the numerical values directly. Some more expensiveinstruments provide a graphical editor by which the user can construct andsmooth a waveform.

Once the sample values are loaded into the RAM, they are ready to bestepped through at a rate which will produce the desired repetition rate(frequency) of the waveform. This is controlled by the size of the “frequencyword” input to the phase accumulator, exactly like the sine-wave synthesizer.However, one difference is immediately apparent in the operation ofinterpolating between adjacent sample entries. For arbitrary functions, theslope of each line segment between entry values, which is needed for linearinterpolation, must be computed rather than looked up as the cosine of thatvalue. This point-by-point process can be made more efficient by enlargingthe RAM, and computing and storing the slope values as the points are entered.Thus the slopes need be computed only once, after which they are looked upas needed.

As the interpolated RAM contents are output in sequence, the sample numbersare supplied to the D/A converter, which produces an output voltage proportionalto each number. Hence the D/A output is not a smooth waveform but rather astepped approximation, similar to Fig. 16.9b, but with many more and finer

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Figure 16.11 Arbitrary waveform synthesis. (a) Block diagram of technique; (b) typical arbitrary waveform,showing a few stored values.

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steps. The steps, which contain high-frequency energy resulting from thesampling process, are removed by the low-pass filter, and the output from thefilter is the desired smooth waveform.

It is apparent that the complexity of the waveform to be synthesized by thisprocess is limited by the number of samples. With 256 samples per period asassumed here, fairly complex waveforms can be generated, but even so it isobvious that any fine waveform structure which might exist between sampleswill not be “seen” by the sampling process. In fact, the situation is actually morerestrictive than this. As a working rule, a minimum of about three to four samplesper cycle of the highest frequency in the waveform should be used. This rule ofthumb derives from the need to filter the sampling image (“alias”) from theoutput. The desired signal and the image approach each other as the signalapproaches one-half the sampling frequency (that is, 2 samples per cycle).Therefore, if there are, say, 256 samples and the sampling (clocking) rate is 1MHz, the highest frequency in the waveform should be about one-third to one-fourth of the sampling rate, or about 300 kHz.

Another way to state this limitation, which is basic and comes from thesampling theorem, is that the waveform to be synthesized must be band-limited.In theory, this limitation would eliminate synthesizing common waveforms,such as square waves, which are not band-limited. However, if the fundamentalfrequency of the waveform is low, enough harmonics can be included to generatea good approximation, usable for most purposes. A 1-kHz square wave, in theexample above, would be represented by about 150 different frequencycomponents and would have a rise time of about 1.5 s.

16.5.2 Arbitrary waveform generator specifications

Because an ARB is a generalized form of the sampled sine-wave frequencysynthesizer, the same limitations and types of specifications apply. In addition,it is important to know (1) the clock rate and (2) the number of stored samples.The ratio of these will determine the highest fundamental frequency of thesynthesized waveform. For the numbers used in the discussion above, this wouldbe 1 MHz/256, or about 4 kHz. It is possible to overcome this rather low frequencylimit if provision is made for the instrument to use fewer samples per cycle.This, however, means synthesizing simpler (less complex and harmonic-rich)waveforms, since the upper frequency limit of about 300 kHz for any componentcannot be exceeded.

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