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E302-AC4 Instrumentation 4. Sampled Measurements

CP Imperial College Autumn 2006 4-1

4. Sampled measurements By the end of this section you will be able to: • Describe the function of a finite aperture sampler • Describe the operation of A/D and D/A converters. • Discuss the bandwidth and quantisation noise of A/D converters • Describe capabilities and limitations of Digital Oscilloscopes.

4.1. Discrete time measurements Although this is not necessary, we usually sample a signal to convert a measurement to a digital representation. A discrete time measurement then consists of the following operations: • Signal conditioning to conform to the sampling theorem imposed restrictions • Sampling i.e. recording an instantaneous value of the signal • Quantisation namely approximating the signal value by a finite resolution digital

representation.

In the following we will assume that sampling is performed repetitively at intervals Ts , or a frequency fs by a sample and hold circuit, which, therefore records the value of the signal at

discrete times n ss

nt nTf

= = . By sampling a signal we therefore introduce a mapping between time

and the index n of the measurement. The Sampling theorem states that in order not to lose any information through the sampling process the signal must satisfy some periodicity constraints: 4.1.1. Low-pass sampling Let a signal be given as a generic Fourier series:

( ) ( )0

cos 2N

i i ii

V t A f tπ θ=

= +∑ (1)

This is certainly true for a finite duration signal, where all the frequencies fi can be thought to be harmonics of the signal duration. The theorem states that sampling is unique and reconstruction of the original signal is possible the frequencies fi must satisfy: ( )max / 2B si

f f f= < (2) The sampling theorem essentially says that the amplitude and phase of an infinite duration sinusoidal waveform of frequency f can be recovered only if we record more than two values of the signal during each period. It is not enough to record exactly two values during a period. To see this let’s assume that by coincidence we sample exactly at the zero crossings! We then get only the phase information, up toπ , but no amplitude information whatsoever. If, on the other hand, we obtain infinitesimally more than 2 samples per period we can calculate both the amplitude and the phase of the waveform. Indeed, sampling occurs at a slightly different phase during each period. Then, the amplitude is:

{ }max iA v=



Once the amplitude is determined the phase is obtained from the zero crossing positions. This argument does not clarify why the sampling frequency has to be more than twice the greatest frequency in the signal spectrum. However, the largest frequency specifies the smallest period during which we must obtain at least two samples! From the low-pass sampling data the original signal can be recovered by using the following interpolation formula:

( ) ( ) ( ) ( ) ( ), sincs sn

x t x n g t nT g t f tπ∞

=−∞

= − =∑ (3)

Please note both the inequality in eq. (2) and the infinite number of samples required in eq. (3) . Such considerations make the interpolation formula in eq. (3) of little practical interest. The critical bandwidth / 2N B sf f f= = is called the Nyquist frequency. Strictly speaking, signals containing the Nyquist frequency in their spectrum cannot be reconstructed from the samples at fs. Likewise, a finite duration signal requires a higher than the Nyquist frequency to reconstruct. Just how much higher frequency than the Nyquist rate is required is easy to estimate. During the entire duration

t∆ we require at least one sample more than twice the number of periods to unambiguously resolve both magnitude and phase. So we can write:

1* 2 1 2samp samp upper samp upperN f t f t f ft

= ∆ > ∆ + ⇒ > +∆

All this assumes the sampling clock is “clean”, i.e. perfect. Real clocks have jitter i.e. the nth sampling event happens at:

nt nT tδ= + . The most naïve interpretation would suggest that the maximum possible sampling period must satisfy the Nyquist criterion. Unfortunately, this is not possible, since the jitter uncertainty is usually gaussian, and the probability a sampling event occurs a time tδ away from its ideal position is given by:

This means that any time displacement away from the ideal sampling instances is possible , especially as the sample length is long. Sampling is of course triggered by an oscillator. It can be shown that for many oscillators jitter is a random walk process, i.e. diffusive. The rapidity of the diffusion process is determined by a correlation time tξ∆ expressed as a multiple ξ of the sampling period. The longer this time is the more rapidly the sampling instants diffuse away from their ideal locations. In that case,

( )2

2 212

sampf tsamp t

samp

fP t e

f t

δξξτ δ

πξ

−∆∝ ⇒ =

∆

The naïve counting argument can then be applied with this description of jitter, to estimate a lower bound for the sampling frequency which allows complete reconstruction. This is not an entirely satisfactory approach, as it does not adequately account for whether it is possible to reconstruct the signal by not knowing the sampling instances. Indeed, it can be better done by signal-to-noise ratio arguments, i.e. by asking for the minimum sampling rate which will lead to a required signal-to-noise ration.

( )2

2212

t

P t eδτδ

πτ

−=



4.1.2. Band-pass sampling A practical radio signal has frequency components in some finite range of frequencies:

, ,2

L HL i H C C H L

f ff f f f f f f+≤ ≤ = = − (4)

The sampling signal is, however, a series of narrow impulses with power in all the harmonics of the sampling frequency. The sampling process itself is signal multiplication, i.e. mixing. The mixing process maps the input signal into a signal of mixing images. The downconverted image of the signal band mixed with the mth sampling frequency image lies at:

L S i S H Sf mf f mf f mf− ≤ − ≤ − . Similarly, the negative frequency components must satisfy:

L S i S H Sf mf f mf f mf− + ≥ − + ≥ + + As long as these images do not overlap the signal can be recovered. Solving the overlap problem one can show that the minimum requirement for signal recovery is that the even order mixed down bands do not overlap. The image overlap problem is solved graphically in Figure 1. The general solution for large order mixing products m is:

( )2 2 1 2H S L S H L B Sf mf f m f f f f f− < − − ⇒ − = < It can be shown that in general a band-pass signal can be uniquely sampled if the Nyquist sampling rate satisfies: 2 4B N Bf f f< < (5) The best case condition (lowest sampling rate) condition occurs when the carrier frequency cf and the bandwidth Bf satisfy: 2c B Bf f nf+ = (6) In the limit of small fractional bandwidth we get that the Nyquist rate for band-pass signals satisfies: lim 2

c Bf f N Bf f→∞ = (7) If the band pass signal is sampled at a frequency fs then the reconstruction formula becomes:

( ) ( ) ( ) ( ) ( ) ( ), sinc coss s Cn

x t x n g t nT g t f t f tπ π∞

=−∞

= − =∑ (8)

As was the case with low pass sampling, the minimum sampling rate is one which guarantees two samples per period for all frequency components in the signal. Except that now the bandpass character of the signal guarantees that the signal changes very slowly over a number of periods so that samples obtained during different consecutive periods are as good as if they were obtained during one single period! Once again, a signal of finite duration needs a higher sampling rate than an infinite duration signal to be completely reconstructed. Unfortunately, jitter is a much more severe restriction than is in the lowpass case. The issue with jitter is that if the sampling instance is uncertain then the signal phase is uncertain at the time the sample was taken. In bandpas sampling, though, 2 consecutive samples may have been obtained many periods apart (“many” is in fact of the



order of the ratio of /U sampleN f f= ). And a rather insignificant time jitter may mean that the actual sampling instance is several periods apart from the ideal sample time! In more precise terms we say that all mixing products of the noise power of the sampler circuit are aliased into the signal band. The noise power is therefore amplified by the same /U sampleN f f= factor which is our benefit in terms of lower sampling frequency. The signal to noise ratio in a band-pass signal sampled at samplef Is a factor N lower than the same signal sampled with the low-pass criterion, and the sam equipment. Band-pass sampling is used in extremely high frequency applications, such as sampling oscilloscopes and radar receivers, where the fractional bandwidth (bandwidth to carrier ratio) is extremely small.

Figure 1: Allowed rates for band-pass sampling as a function of the maximum band frequency. In principle frequencies twice the Bandwith are sufficient. 4.1.3. Interpolation The interpolation formulas in eqs. (3) and (8) are not practical for two reasons. Firstly, they require an infinite number of samples. Second, and most important, the interpolation function g(t) is not physically realisable, as it represents a non causal filter (note that g(t) is defined for both positive and negative values of time). A sample and hold or zero order hold approximates the signal by assigning to it, during the interval between two sampling events, the last sampled value. This is indeed an accurate description of the sample and hold circuit preceding an A/D converter. The reconstruction formula suggests the frequency domain response of the sample-and-hold is ( ) ( )sincg f T fTπ= (9) Subsequent low pass filtering removes the discontinuities introduced by the sampling process. A first order hold approximates the signal by its linear interpolation between samples:



( ) ( )( ) ( ) ( )( ) ( )1

1x nT x n T

x t x n T t nTT

− −= − + − (10)

The interpolator is equivalent to a linear filter with a frequency response: ( ) ( )2sincH f T fTπ= (11) Once again low pass filtering can improve the interpolation by removing the derivative discontinuities.

4.2. Signal conditioning The sampling theorem dictates than any signal to be sampled must satisfy the Nyquist bandwidth criterion. The filter used for conditioning is called an anti alias filter. Ideally we would require infinite attenuation at frequencies exceeding the Nyquist frequency. In practice the finite attenuation in the filter stop band provides a contribution to the measurement noise floor. As a rule, we wish to keep the out of band signal which will be aliased into the band to less than ½ LSB. We calculate then the break frequency and order of the antialiasing filter so that the aliased components are less than ½ LSB, as shown in the illustration.

Figure 2: Illustration of the design of an anti-aliasing filter

4.3. D/A conversion We discuss the D/A conversion first because it is more straightforward to implement than A/D conversion. The basic D/A converter structure uses binary weighted current sources which are switched in and out of the circuit to represent a binary number. In Figure 3 we show two such structures, the binary weighted resistive ladder and the R-2R ladder. Both devices are followed by a transimpedance amplifier which sums the current and converts it into a voltage. In the binary weighted converter of N bits, if the 0th bit is LSB and N-1 bit the MSB, the resistance values are given by 1

12N nn NR R− −

−= so that the LSB resistor is 12N − times bigger than the MSB. This introduces a major limitation of this type of converter, as the error arising from component tolerance must be kept below ½ LSB. If η is the fractional component tolerance, the constraint on the maximum number of bits is: ( )1

22 1 logN Nη η−< ⇒ < + (12) This is a severe restriction, as even 1% component tolerance would restrict the length of a converter to about 6 bits. The R-2R ladder alleviates this problem somewhat, in that only two values of components are used, and in general identical components can be manufactured to closer tolerances, especially on ICs. The analysis of the R-2R ladder is an exercise in deriving the Thevenin equivalent circuit by superposition, alternatively turning on and off the voltage sources representing the bits of the input digital data.

FS

Filter Filter Alias

SNRMIN

Pass band



(a) (b)

Figure 3: Simple D/A converters. (a) binary weighted ladder. (b) R-2R ladder

Both converters use an op-amp as a transimpedance amplifier, and are consequently limited by the op-amp’s frequency response and slew rate. A much faster converter can be made by directly switching binary weighted currents, as shown in Figure 4. Scaled current sources are easy to implement on an IC, as they represent a number of transistors connected in parallel. The operation of a weighted current source D/A converter is limited by the larger gate current drive required by the higher bits, and at high speeds by the so-called “injected” charge, i.e. the gate current appearing in the channel of the device and contributing to the converter output. The limitations of this converter are alleviated in the current steering DAC (Figure 5) where switches are used to direct the scaled current.

x8 x4 x2 x1

B3 B2 B1 B0Vcc

Vout

Figure 4: A binary weighted current source DAC.



x8 x4 x2 x1

B3

Vcc

Vout

B2 B1 B0

Figure 5: Current steering DAC

A common characteristic of the converters presented so far is the large DC power dissipation. In high speed and low power applications the resistors of the R-2R ladder can be replaced by capacitors, (C-C/2 , respectively)and the transimpedance amplifier by an integrator. It can be shown that if the integrator is ideal the circuit operation is identical to that of the R-2R ladder. Operation of the capacitive R-2R ladder is limited at lower frequencies by noise currents.



4.4. Quantisation The final step in a discrete measurement is the conversion of the measurement to a digital representation. Clearly not all values of the input signal can be represented digitally. The discrepancy between the signal and its digital representation called quantisation noise. Quantisation is performed by A/D converters which we examine later. 4.4.1. Quantisation Noise

The quantisation noise is assumed to be Gaussian and have a uniform PDF between ± ½ LSB, also denoted ± q/2, i.e.

p(eq) = 1/q (-q/2 < eq < q/2) p(eq) = 0 elsewhere

The ideal average (mean square) quantisation noise power is assumed to be white between –fs and fs and its total power is:

/ 2 2 2

2 2

/ 2

( ) ( )12

qq

q q q q qq

e qE e e p e de deq

∞

−∞ −

= = =∫ ∫ (13)

The quantisation noise dictates the minimum signal to noise ratio achievable in a sampled data system. Since the power of a sinusoidal signal of amplitude A is just 2 2P A= , and since such a signal can be made to fit exactly in the range of an N bit converter by setting the amplitude to half the converter’s range: 2 2NA q= , we can calculate that the maximum S/N ratio, usually called the signal to quantisation noise ratio SNQR is: ( ) ( ) ( )2 2 2 110log 6 10log 3 2 1.76 6.02 dBNSQNR A q N−= = = +i (14) As we shall see later, the argument can be inverted. A converter operating at a particular SNQR is said to be N-bit by inverting this formula. Furthermore, we talk of the effective number of bits ENOB of the converter as the number of bits of an ideal converter which has SNQR equal to the converter’s SNR after all sources of noise and uncertainty have been accounted for.

Figure 6: Flash converter



4.4.2. Flash converters The simplest conceptually and also the fastest A/D converter is the Flash converter, shown in Figure 6. The input signal is compared to all possible values in the conversion range and a decoder selects and outputs the code. Flash converter word length is limited by component tolerances, and they also have a high power dissipation due to the big number of comparators (although the latter can easily be implemented with CMOS gates). 4.4.3. Feedback converters Feedback converters compare the output of an internal D/A converter to the input. The comparison is used to provide a suitable input to the D/A converter. The simplest is the single slope ramp converter, shown in Figure 7, where a counter increments the D/A input until it exceeds the signal input. Such a converter is slow and has a code dependent conversion time.

Figure 7: Single slope ramp converter

A dual slope ramp converter (Figure 8) integrates the input signal for a time t1 and then subtracts from it the integral of a fixed voltage until the output reaches again zero, which takes a time t2. If an intermediate output Vint is reached after t1 , if τ is the integrator time constant then:

1 2 2int

1in ref in ref

t t tV V V V Vtτ τ

= = ⇒ = (15)

The logic times t2 and provides a suitable input to the D/A converter. This type of converter is not only faster, but also exhibits a smaller code-dependent variation of the conversion time. Furthermore, any nonlinearities of the integrator cancel, at least to the lowest order.

Figure 8: Dual slope ramp converter



A very popular (and much faster in multibit applications) converter is the successive approximation converter (Figure 9). An N bit successive approximation converter has a fixed conversion time, of N+1 clock cycles. This allows construction of 16 bit converters with less than 20 µs conversion time.

Figure 9: A successive approximation A/D converter



4.5. Oversampling If we sample a signal at a much higher than the Nyquist rate we should in principle be able to use the extra samples to obtain a higher resolution than the underlying converter. By straightforward oversampling we can in principle gain 0.5 bit of resolution for every doubling of the sampling rate. To see this, we have to compute the power spectral density of the SQNR. The signal power occupies frequencies B sig Bf f f− < < , and as a result, the signal power spectral density is the total signal power divided by (twice) the signal bandwidth:

2

4sigB

APf

=

The quantization noise power spectral density is the total quantization noise power divided by (twice) the sampling frequency, as the quantisation noise occupies frequencies s sf f f− < < :

2

24Ns

qPf

=

The SQNR is then: 2

2

6s s

N B

PSD fASQNRPSD q f

= =

but we have already assumed that the signal fits in the converter range exactly: 2

2 22 2 nA

q−=

So that SQNR is given, in terms of the number of bits n, and the oversampling ratio

2 / 2ks BM f f= =

is given by: 2

2 1 22

6 3 2 3 2n n ks s

N B

PSD fASQNR MPSD q f

− += = = ⋅ = ⋅

we can then write the SQNR in terms of an effective number of bits ENOB:

/ 2ENOB n k= + +1/2

Since: 2 13 2 ENOBSQNR −= ⋅

And clearly the effective number of bits increases by ½ bit for each bit of oversampling. We have averaged M successive measurements to average out the quantisation noise. 4.5.1. Dither

From Ken Pohlmann’s "Principles of Digital Audio," 4th edition, page 46:

"...one of the earliest uses of dither came in World War II. Airplane bombers used mechanical computers to perform navigation and bomb trajectory calculations. Curiously, these computers (boxes filled with hundreds of gears and cogs) performed more accurately when flying on board the aircraft, and less well on ground. Engineers realized that the vibration from the aircraft reduced the error from sticky moving parts. Instead of moving in short jerks, they moved more continuously. Small vibrating motors were built into the computers, and their vibration was called 'dither' from



the Middle English verb 'didderen,' meaning 'to tremble.' Today, when you tap a mechanical meter to increase its accuracy, you are applying dither, and modern dictionaries define 'dither' as 'a highly nervous, confused, or agitated state.' In minute quantities, dither successfully makes a digitization system a little more analog in the good sense of the word." To perform the averaging effectively we need to add some noise, to make sure that the signal and noise sum crosses frequently the converter’s decision threshold. Such intentional noise is called dither. The required dither amplitude typically exceeds the converter’s quantisation step. More precisely, we choose to represent the signal x by a random variable xy x e= + obtained by adding to the signal xe , a random variable representing the dither noise. After N measurements, the ratio of the signal and dither noise averages is:

2 2 2

2 2x

x N x Ne Ne

= = (16)

So that

( )/RMSy x O e N→ + (17)

This is the same result describing averaged measurements in the presence of noise. An important engineering problem remains, though: How can we generate, in hardware, this dither noise component so that it is of the correct magnitude? Is it also possible to make the averaging process converge more rapidly by giving the dither noise some appropriate spectral characteristic? The answer to both is in a very old engineering trick, used originally to scramble space communications to make them more robust to interference!

4.6. ∆-Σ converters

The Delta modulator shown in Figure 10 is a signal to PWM converter. As a PWM signal contains copies of the spectrum both at baseband, it is very easy to decode by retiming and low-pass filtering, also shown in Figure 10

Figure 10: The Delta modulator (left) and demodulator (right)



The sigma delta converter further reduces the in-band quantisation noise in oversampling by nesting the converter inside a feedback loop. It turns out that this arrangement also automatically generates the dither noise required for the enhanced resolution! The converter consists of the sigma delta modulator, (Figure 11) , the output of which is a (possibly multiple bit) pulse width modulated waveform. A subsequent digital filter interprets the output of the modulator.

Figure 11: A sigma delta modulator

The magic in the operation of the sigma delta modulator lies in that the input signal and the quantisation noise have different transfer functions. Assuming the gains of the A/D and D/A converters are both unity, the signal transfer function is:

( ) ( )( )1s

H sG s

H s=

+ (18)

while the quantisation noise transfer function is:

( ) ( )1

1EG sH s

=+

(19)

The total power of the quantisation noise is given by eq. (14), and its power spectral density is:

( ) 2 2( ) / 24q q s sP e E e f q f= = If we require the signal to quantisation noise ratio at much a lower frequency fN (since, after all we are sampling at a frequency that is much greater than the Nyquist frequency), the in-band signal to quantisation noise ratio will approximately be:

( ) ( )( )

22 1

23 2 N

NNN

s s

H fS ffE f H f

−⎛ ⎞= ⎜ ⎟

⎝ ⎠

i (20)

To give a concrete example, consider that the converters are 1 bit wide (i.e the A/D converter is a comparator, and the D/A a switch) and that the filter is an ideal integrator. In terms of the oversampling ratio 2k

s NM f f= = the SQNR is: 3 3 26 1.5 2 kSQNR M += = i (21) Which is the same as the SNQR of a 1.5k+1 bit modulator. So the simplest modulator gains 1.5 bit resolution for every 1 bit oversampling. Using higher order filters and multiple loops we can make much bigger resolution gains with oversampling ratio. Sigma delta converters are very popular in digital audio, and with a bandpass filter in the loop in consumer radio IF stages.

Filter H(s)

A/D Converter

D/A Converter

+

- Vin (analog)

Vout (PWM)

Clock



4.7. Instruments using sampled measurements 4.7.1. The digital storage oscilloscope A digital oscilloscope consists, as shown on Figure 12, of a sampler/A/D converter, and a digital timebase generator. For many applications the digitising oscilloscope has important advantages over its analog counterpart. Most importantly it can trigger at the end of a waveform making easier the observation of transients and one-off events. It can perform computations on waveforms with its built-in processing capability, for instance it can average a number of waveforms and compute the (fast) Fourier transform. It also allows storage in memory of waveforms and comparison to subsequent observations. Since a sampler is central to the operation of the oscilloscope, an important issue is observation of the signal bandwidth versus sampling rate restriction imposed by the Nyquist sampling theorem. Failure to do so results into aliasing and the observation of artefacts. A digital oscilloscope normally operates in Real-time sampling mode: all samples are collected sequentially in a single period as the waveform is received. To somewhat relax the sampling theorem constraints equivalent-time sampling (or coherent sampling) may be employed, as shown in Figure 13: the waveform is reconstructed from samples acquired over a number of cycles of the waveform. The ordering of consecutive samples in the time domain may be sequential or random, and clearly, equivalent time sampling is effective only on periodic waveforms.

Figure 12: Block diagram of a digital oscilloscope



Sampling Techniques

Figure 13: Real versus equivalent time sampling. (a) Sequential sampling. (b) Random sampling

There is an interrelation between the sampling rate fS, the sweep time TX and the memory record length M: S Xf T M= (22) The record length does not need to be equal, of course, to the total available memory. Clearly a large memory allows more flexibility in choosing the displayed sweep time. The sampled data will appear as a set of dots on the screen. This opens the possibility to visual aliasing, i.e. the implicit interpolation performed by the human eye may interpret the signal as being at a different frequency. Often explicit interpolation will be employed, i.e. to oscilloscope will ‘join the dots’ With no interpolation, about 25 samples per period are required to reconstruct a sinusoidal waveform. With a linear interpolator this is reduced to about 10 data points per cycle, and with a sine interpolator the waveform can be accurately reconstructed with as few as 2.5 samples per period (close to the Nyquist rate). Interpolation may introduce diffraction effects, e.g. ringing when observing steep discontinuities. Digital filters can be used to minimise such artefacts at the cost of little additional sampling.



Figure 14: Display, interpolation signal distortion and artefacts.



When the interpolation method is accounted for, the useable storage bandwidth is defined for digital storage single event capture as: USB = Maximum sample rate x (1/C) C depends on the number of samples per cycle, which depends on the method of interpolation. For a dot display (no interpolation) C=25 Linear interpolation C=10 Sine interpolation C=2.5 For repetitive signals, USB = full scope bandwidth (since equivalent time sampling can be used). The Useful Rise Time of a digital scope is approximately TR=1.6 sample periods, as illustrated in Figure 15. Actual bandwidth and rise time of a DSO will therefore change with the timebase setting (sample rate). But USB and TR give an indication of the fastest signals which can be captured.

Figure 15: Useful risetime of a digital scope.

A digital scope allows arithmetic operations on the data acquired. Averaging consecutive sweeps is a common way to enhance the signal to noise ratio, and hence the effective number of bits. All the same, the effective number of bits of the A/D converter front end can be defined as the width of a converter whose quantisation error equals the actual noise floor of the converter when used to digitise a sine wave NA.

The effective number of bits (ENOB) combines various factors into a single measure of performance, which measures the digitise accuracy versus frequency. The difference between the actual and effective width of the converter is called the number of lost bits. If EQ is the quantisation noise power density,

2log A

Q

NLBE

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ (23)



As the noise floor and distortion often rises rapidly with frequency, the effective number of bits correspondingly reduces sharply at higher frequencies. Ensemble averaging can be used to increase n oscilloscope's resolution. If K waveforms are averaged, the signal to noise (in power!) ratio will increase by a factor of K, so the effective resolution will increase by 21 2logN Kδ = bits. To minimise the obvious memory requirement to store many waveforms, subsequent waveforms are added to a running average of previous ones, effectively implementing an IIR filter. 4.7.2. Sampling Oscilloscopes At microwave frequencies real time, or even equivalent time sampling becomes impractical. Yet oscilloscopes that can display signals to frequencies up to 40GHz exist. These exploit aliasing to sample the signal. The waveform to be observed is effectively bandpass sampled, so the bandpass sampling criteria now apply, i.e. the signal needs to have a restricted bandwidth. As the underlying sampling rate may be quite large, this is not a serious restriction. More serious restrictions arise from the need to operate the sampling gate on very high frequency signals. Both the aperture (capture time) and the timing jitter (uncertainty in time position) of the sampler need to be small compared to the highest frequency observed. Finally, it is necessary to use preamplifiers (instead of attenuators in conventional scopes) which can severely restrict the instrument's dynamic range.

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