A Waveform That Continually Repeats Itself After the Same Time Interval

Test Equipment Principles

Morgan Jones , in Building Valve Amplifiers (Second Edition), 2014

Nyquist, and equivalent sample rate

The Nyquist criterion states that a repetitive waveform can be correctly reconstructed provided that the sampling frequency is greater than double the highest frequency to be sampled. Practical considerations usually increase this frequency slightly, so the digital audio on compact disc needed a 44.1   kHz sample rate even though its audio bandwidth was limited to 20   kHz. This simple theory implies that a 100   MHz oscilloscope requires an ADC that samples at 200   MS/s (megasamples per second), yet the first generation HP54600B could only sample at 20   MS/s, whereas its usurper the TDS3012 sampled at 1.25   GS/s. Why did these two 100   MHz oscilloscopes have such wildly different sample rates?

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Test instruments and measurements

John Linsley Hood , in Audio Electronics (Second Edition), 1999

Time-base synchronisation

The ability to 'freeze' a repetitive waveform on the 'scope display is an essential quality in any oscilloscope, and facilities are invariably provided to allow trace synchronisation with any specified signal input, but also from an external 'sync' input. Various types of synchronisation are also usually provided by switch selection, in the hope that some of these may be more effective that others. My choice of words is prompted by the experience, over many years and with many 'scopes, that some 'scopes and some waveforms can prove very difficult to lock on the screen. In fact, one of my personal priorities in the choice of an oscilloscope is ease of synchronisation. Sadly, one only ever discovers whether this is good or poor after the purchase of the instrument – unless one has had previous experience with that make and model.

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The Oscilloscope and Its Use

John Clayton Rawlins M.S. , in Basic AC Circuits (Second Edition), 2000

Triggering Control Functions

Recall that the sweep is the action of the electron beam moving from left to right across the scope face. Each sweep must begin at the same time-base point on a repetitive waveform as shown in Figure 3.39 so that each succeeding sweep overlays precisely on the preceding sweep. The trigger circuits determine when each sweep begins, thus providing a stable CRT display. If the sweep is not stabilized, the waveform appears to run across the screen, creating a random display pattern, or it produces a waveform envelope as shown in Figure 3.40 .

Figure 3.39. Relationship of Trigger Point and Displayed Waveform

(Reproduced from Basic Oscilloscope Operation,Tektronix, Inc., 1978) Copyright © 1978

Figure 3.40. Untriggered 300-Hz Waveform

As stated previously, the sweep trace seen on the face of the oscilloscope is the result of an electron beam moving from left to right (as we view it) across the face of the CRT. Depending on the setting of the time/division control the beam takes a certain amount of time to traverse the ten horizontal divisions on the face of the CRT. When the trace reaches the right side of the CRT, the beam is turned off (turning off the beam by the scope circuits is called blanking) and the voltage on the horizontal deflection plates is reset (resetting the voltages is called retrace) so that the beam, when turned back on, will reappear at the left side of the CRT. If the oscilloscope is in "free run" (non-triggered) mode, or if the oscilloscope display is set so that the trace is not triggered, the start of the sweep begins again as soon as the beam reaches the right side of the screen and enough time elapses for completion of blanking and retrace. This free run operation will typically result in a display of multiple waveforms that seem to randomly superimpose themselves one on top of the other.

The repeating waveform pattern of Figure 3.41 when displayed in free run mode wouldoverlay the series of successive waveform time-base durations (shown in Figure 3.42 ) so that the oscilloscope display would appear similar to that shown in Figure 3.43 . This effect is a random display pattern typically seen in free run mode.

Figure 3.41. Free Run Sweep (Time-Base Duration = 1 Full Sweep)

Figure 3.42. Overlay of Successive Waveforms with Scope in Free Run Mode

Figure 3.43. Random Display of Scope in Free Run Mode

On the other hand, a triggered oscilloscope display utilizes the scope's triggering circuits to determine the exact point at which a given waveform will begin to be displayed on the face of the CRT. For example, if the waveform of Figure 3.41 is displayed using a triggered oscilloscope where each successive display of the waveform begins at exactly the same point on the waveform as determined by the triggering circuits, then the display will appear as a stable waveform as shown in Figure 3.44 .

Figure 3.44. Stable Triggered Presentation of Waveform

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Dynamic Range

Morgan Jones , in Valve Amplifiers (Fourth Edition), 2012

Noise and THD+N

Although using a CCIR468-2 filter to weight distortion is cheap and effective, its rising response with frequency may create another problem. Properly designed circuitry creates very little distortion. To put it another way, the distortion could be of comparable amplitude to the noise that all electronics generates. When we make our THD measurement, using our meter, how do we know that we are not actually measuring the amplitude of the noise?

The best solution is to view the distortion residual on a 20   MHz oscilloscope (or one with the 20 MHz filter engaged). If the waveform appears clean, we are measuring mostly distortion; if a repetitive waveform is difficult to discern, we are probably measuring noise. Thus, all practical measurements made by a meter are actually THD+N, and we have to be certain that the noise is sufficiently small to be ignored. Digital oscilloscopes are excellent for making this decision objectively. The oscilloscope can be set to measure the RMS amplitude of the distortion residual, and this measurement can be compared with the same measurement but with averaging engaged. Averaging multiple waveforms cancels the noise (which is random), but maintains the repetitive distortion residual. If there is negligible (<10%) difference between the two measurements, we are measuring distortion. But if the averaged value drops to 71% of the unaveraged value, we are measuring equal amounts of noise and distortion residual, and it is time to stop recording distortion figures.

By definition, white noise has constant amplitude with frequency, whereas distortion harmonics occur at very specific frequencies. Our meter is a broadband device, which means that it is equally sensitive to all frequencies across the audio bandwidth. Thus, although the noise power in a particular frequency band could be quite low, and possibly significantly less than the amplitude of an adjacent distortion harmonic, when summed, the noise powers could easily swamp the distortion powers. This wouldn't be a problem if it were not for the fact that the ear/brain combination can pick distortion harmonics out of the broadband noise because it works like a spectrum analyser.

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Radio communication techniques

D.I. Crecraft , S. Gergely , in Analog Electronics: Circuits, Systems and Signal Processing, 2002

9.4.2 The spectrum of FM

The spectrum of FM is more complex than that of aM, and is not so easily analysed. Consider a sinusoidal modulating signal, causing a peak frequency deviation of Δ f _c from the unmodulated value, f _co, of the carrier. It may be thought that, because the frequency is swept continuously from (f _co − f _c) to(f _co + Δ f _c), the spectrum will be continuous too, containing all the frequencies between (f _co − Δ/ f _c) and (f _co + Δ / f _c ). However, this is not the case. Consider the waveform of the modulated signal. Since, in this case, the modulation is a repetitive waveform, the modulated waveform is repetitive too, with the same period. Figure 9.14 shows an example of this.

Because of this repetition, or periodicity, the modulated waveform's spectrum must be a line spectrum, with components at multiples (harmonics) of the modulating signal's frequency f _m. These components cluster around the carrier frequency, as shown in Figure 9.16. When frequency modulation was first suggested, it was thought by some that its bandwidth would be simply the range from (f _co − Δ f _c) to (f _co + Δ f _c), or simply 2Δ f _c, and this could be made less than that of AM by keeping the deviation smaller than the maximum modulating frequency. This is also a fallacy. No matter how small the deviation, there must be at least two side frequencies, at (f _co − f _m) and (f _co + f _m), for every frequency component f _m of the baseband signal, and the bandwidth is at least 2f_m , just like AM. Of course, the waveform, and the phases of the side frequencies, are different to those of AM.

Fig. 9.16. Frequency modulation spectra with sine wave modulation and various values of modulation index m.

The modulation index of FM is defined as the ratio of the frequency deviation to the modulating frequency

$m=\frac{\Delta f_{\text{c}}}{f_{\text{m}}}=\frac{\Delta {\omega }_{\text{c}}}{{\omega }_{\text{m}}}$

Narrow-band FM is characterized by m < 1 or so, and has mainly just two side frequencies, like AM. At greater values of m, more pairs of side frequencies are created, as you can see in Figure 9.16.

Wide-band FM is characterized by m > 2 or more, and has many side frequencies.

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Electrical Measurement

MJ Cunningham MSC, PHD, MIEE, CEng , GL Bibby BSc, CEng, MIEE , in Electrical Engineer's Reference Book (Sixteenth Edition), 2003

11.7.2 Digital wattmeters ^7–9

Digital wattmeters employ various techniques for sampling the voltage and current waveforms of the power source to produce a sequence of instantaneous power representations from which the average power is obtained. One new NPL method uses high-precision successive-approximation A/D integrated circuits with the multiplication and averaging of the digitised outputs being completed by computer. ¹⁰ , ¹¹ A different 'feedback time-division multiplier' principle is employed by Yokogawa Electric Limited.^r

The NPL digital wattmeter uses 'sample and hold' circuits to capture the two instantaneous voltage (v _x ) and current (i _x ) values, then it uses a novel double dual-slope multiplication technique to measure the average power from the mean of numerous instantaneous (v _x i _x ) power measurements captured at precise intervals during repetitive waveforms. Referring to the single dual-ramp ( Figure 11.9(a) ) the a.c. voltage v _x (captured at T ₀) is measured as v _x T ₁ = V _r(T ₂ – T ₁). If a second voltage (also captured at T ₀) is proportional to i _x and integrated for T ₂ – T _1, then, if it is reduced to zero (by V _r) during the time T ₃ – T ₂ it follows that i _x (T ₂ – T ₁) = V _r(T ₃ − T ₂).

The instantaneous power (at T ₀) is v _x i _x = K(T ₃ – T ₂) and, by scaling, the mean summation of all counts such as T ₃ – T ₂ equals the average power. The prototype instrument measures power with an uncertainty of about ±0.03% f.s.d. (and ±0.01%, between 50 and 400 Hz, should be possible after further development).

The 'feedback time division multiplier' technique develops rectangular waveforms with the pulse height and width being proportional, respectively, to the instantaneous voltage (v _x) and current (i _x); the average 'area' of all such instantaneous powers (v _x i _x) is given (after 1.p. filtering) as a d.c. voltage measured by a DVM scaled in watts; such precision digital wattmeters, operating from 50 to 500 Hz, have ±0.05% to ±0.08% uncertainty for measurements up to about 6kW.

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Instrumentation and interfacing

B. HOLDSWORTH BSc (Eng), MSc, FIEE , R.C. WOODS MA, DPhil , in Digital Logic Design (Fourth Edition), 2002

10.3 Schmitt input gates

The internal circuitry of a 'Schmitt input gate' is based upon the well-known 'Schmitt trigger' circuit described in section 10.2 above. A 'Schmitt inverter gate' may be regarded as a Schmitt trigger circuit giving outputs at the correct voltage levels to drive subsequent digital logic gates correctly. A 'Schmitt input buffer gate' is similar but has an extra stage of logic inversion prior to the output. In each case, varying analogue voltages, not necessarily corresponding to the logic level specifications for the logic family concerned, may be applied to the Schmitt input. The output will take the appropriate logic level (0 or 1) according to whether the input is above or below the relevant threshold voltage fixed by the manufacturer. Other types of Schmitt input gates, with more complex logic functions, are also available. The threshold voltage is usually fixed between the normal logic level voltages, so that a Schmitt input gate will operate correctly if it is driven by a conventional logic gate rather than an analogue voltage source.

On circuit diagrams, a Schmitt input gate is indicated by the gate symbol for the corresponding conventional logic gate, but with the addition of the special symbol

drawn adjacent to an input or centrally inside the gate symbol as appropriate. This special symbol has a stylised derivation from the letter 'S' and from a diagram indicating hysteresis between input and output voltages. Examples of circuit symbols for typical Schmitt input gates are shown in Figures 10.2(a) and (b).

Figure 10.2. (a) Circuit symbol for a Schmitt inverter (e.g. as in IC type 74LS19) (b) Circuit symbol for a Schmitt NAND gate (e.g. as in IC type 74LS18) (c) Simple digital signal delay generator using a Schmitt input gate

One typical use of such a Schmitt input gate is shown in Figure 10.2(c). In this circuit, an RC network is used to convert an input logic waveform into a waveform with slow edges, that is, a signal where the rising and falling edges are governed by the usual exponential law with time constant τ = RC. It is bad practice to apply such a waveform to a conventional logic gate because these slow edges can potentially cause severe problems such as oscillations at the output or out-of-specification output voltages with gates not specifically designed to handle slow edges. However, Schmitt input gates are expressly designed and intended to be able to handle slow edges, and so the output state changes after a delay time determined by the precise value of the threshold voltage and the time constant τ = RC.

The Schmitt delay circuit can be developed into a simple oscillator circuit for producing a repetitive waveform, as shown in Figure 10.3(a). In this circuit, a logic transition at the output of the inverter is fed back to the input of the inverter, but the RC network connected to the Schmitt input causes a delay before the complementary transition occurs at the output. Because of the non-zero hysteresis at the gate input, the gate input voltage varies exponentially between the two threshold voltages, and the output oscillates between logic low and high levels indefinitely, as shown in Figure 10.3(b). The frequency of the waveform produced depends upon the values of R and C determining the delay time. However, a serious disadvantage of this circuit is that the exact oscillation frequency also depends upon the precise input threshold voltages and the difference between them, which are usually not precisely known.

Figure 10.3. (a) Simple low-precision RC oscillator circuit using a single Schmitt inverter (b) Voltage waveforms in the single-inverter oscillator circuit (c) Simple oscillator circuit using two Schmitt inverters (d) Voltage waveforms in the double-inverter oscillator circuit (e) Gated oscillator circuit (f) Precision oscillator using a resonant quartz crystal

A simple oscillator circuit that overcomes this difficulty to some extent is shown in Figure 10.3(c). Gate G1 must be a Schmitt input gate; G2 need not have a Schmitt input, although in practice it will usually be part of the same IC package as gate G1 and so will also have a Schmitt input. The voltage waveforms in this circuit are shown in Figure 10.3(d). Suppose that the output voltage of gate G2 is currently at logic high level, V _OH; because of the inversion in gate G2, this requires that the output of gate G1 is at logic low level. Therefore, the voltage at point A will fall, according to the usual exponential decay law, until the input voltage to gate G1 falls below the threshold value. For simplicity it will be assumed here that gate G1 has a high input impedance (much greater than R ₁) so that it draws no current from the RC network. It follows that the voltage at its input is the same as the voltage at point A, and the precise value of resistor R ₁ is unimportant. It will also be assumed here for simplicity that the Schmitt threshold voltages, V _T, for rising and falling edges, are exactly half of the logic high voltage level, with zero hysteresis, and also that the logic low voltage level V _OL = 0V. At the instant where the input of gate G1 falls below its threshold voltage, the voltage across the capacitor will be half of the logic high voltage, or ½V _OH. Then the output of gate G1 goes high and so the output of gate G2 goes low. At this point, the voltage at point A is now negative, with value −½V _OH, and it starts rising towards the logic high value with time constant RC. When the voltage at the input to gate G1 rises above its threshold value V _T = ½V _OH, then the output of gate G1 changes to logic low, the output of gate G2 changes to logic high, and the voltage at point A is immediately + V _OH + V _T = 3V _OH/2 (i.e., the output of gate G2 plus the capacitor voltage of ½V _OH). The whole cycle can now repeat indefinitely. It is straightforward to calculate the time taken for the first half of the cycle, during which the voltage at point A rises from −½V _OH to the threshold value + ½V _OH with an aiming voltage of + V _OH and time constant RC; the result is t ₀ = RC ln(3). The time taken for the other half-cycle of the operation is the same, so that the oscillation frequency is given by f = 1/[2RC ln(3)]. The oscillation frequency is now relatively insensitive to the actual threshold voltage values. However, the precise mark-to-space ratio of the signal at the output of gate G2 depends upon the exact threshold voltage values, and so if it is important to have a 1:1 mark-to-space ratio then the output of gate G2 can be taken through a divide-by-2 circuit as shown in Figure 10.3(c).

In practice, a Schmitt input NAND gate might be used in place of one of the inverters, as shown in Figure 10.3(e), to produce a 'gated oscillator'. This only produces a repetitive waveform when the additional input to the first NAND gate is held at logic high level. When the gating input is held low, the oscillator output is held low. Since the precise oscillation frequency still depends to some extent upon the voltage threshold values, this type of RC oscillator is only suitable for applications where the utmost frequency stability and accuracy is not required. A similar circuit, as shown in Figure 10.3(f), using a quartz crystal which resonates at a frequency precisely specified by its manufacturer, will usually be employed in cases where excellent frequency stability or precision is of paramount importance.

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Introduction to Noise Measurement

Art Kay , in Operational Amplifier Noise, 2012

5.2 Equipment for Measuring Noise: Oscilloscope

One disadvantage of using a true RMS meter for measuring noise is that the meter does not help you to know the nature of the noise. For example, the true RMS meter cannot tell the difference between noise pickup at a specific frequency and broadband noise. The oscilloscope, however, allows you to observe the time-domain noise waveform. Note that most different types of noise have a distinct waveform shape, so you can determine what type of noise dominates.

Both digital and analog oscilloscopes can be used to measure noise. Since noise is random in nature, analog oscilloscopes cannot trigger on the noise signal. Analog oscilloscopes can only trigger on repetitive waveforms. Nevertheless, analog oscilloscopes display distinctive patterns when connected to sources of noise. Figure 5.2 shows the result of a broadband measurement using an analog oscilloscope. Note that analog oscilloscopes tend to create an average or "smeared" waveform because of the phosphorescent quality of the display and the inability of the analog oscilloscope to trigger noise. One disadvantage of most standard analog oscilloscopes is that they are not able to capture low-frequency noise (1/f noise).

Figure 5.2. White noise on analog oscilloscope

Digital oscilloscopes have some convenient features that help with measuring noise. Digital oscilloscopes can capture 1/f noise. Digital oscilloscopes also have the ability to mathematically compute RMS. Figure 5.3 shows the same noise source in Figure 5.2 captured using a digital oscilloscope.

Figure 5.3. White noise on digital oscilloscope

There are some general guidelines that should be followed when using an oscilloscope for noise measurements. First of all, it is important to check the noise floor of your oscilloscope before measuring your noise signal. This can be done by connecting BNC shorting cap across the oscilloscope input or shorting the oscilloscope lead to the ground lead, if a 1×probe is being used. This is an important consideration because the measurement range is 10 times lower when using a 1×probe. Most good oscilloscopes have a 1   mV per division range with a 1×oscilloscope probe or direct BNC connection, and a 10   mV per division noise floor with a 10×probe.

Note that a direct BNC connection is preferred over a 1×oscilloscope probe because the ground lead connection can pick up RFI/EMI interference (see Figure 5.4). One way to avoid this issue is to remove the oscilloscope probe ground lead and top cover, and use the ground on the side of the probe (see Figure 5.5). Figure 5.6 shows a BNC shorting cap.

Figure 5.4. The ground lead can pick up RFI/EMI

Figure 5.5. Oscilloscope probe with ground lead removed

Figure 5.6. BNC shorting cap

Most oscilloscopes have a bandwidth limiting feature. To accurately measure the noise of the oscilloscopes, bandwidth must be greater than the noise bandwidth of the circuit that you are measuring. However, for best measurement results, the oscilloscope bandwidth should be limited to some value above the noise bandwidth. For example, assume that an oscilloscope has a full bandwidth of 400   MHz, and a bandwidth of 20   MHz when the limiting feature is turned ON. If you are measuring the noise of a circuit with a noise bandwidth of 100   kHz, then it makes sense to turn on the bandwidth limiting feature. For this example, the noise floor is lower because the RFI/EMI interference outside the bandwidth of interest will be eliminated. Figures 5.7 and 5.8 show the noise floor of a typical digitizing oscilloscope with and without bandwidth limiting. Figure 5.9 shows that the noise floor is substantially higher with a 10×probe.

Figure 5.7. Oscilloscope noise floor with 1×probe and bandwidth limiting

Figure 5.8. Oscilloscope noise floor with 1×probe and without bandwidth limiting

Figure 5.9. Oscilloscope noise floor with 10×probe and without bandwidth limiting

The coupling mode for the oscilloscope must also be considered when making noise measurements. AC coupling should be used with broadband measurements because the noise signal generally rides upon a larger DC voltage. For example, a 1-mVpp noise signal may ride on a 2-V DC signal. Thus, in the AC coupling mode the DC signal is eliminated, allowing for the highest gain. However, note that the AC coupling mode should not be used to measure 1/f noise. This is because the bandwidth in AC coupling mode generally has a lower cutoff frequency of approximately 10   Hz. Of course, this number will vary for different models, but the point is that the lower cut frequency is too high for most 1/f noise measurements. Typically, 1/f characterization is done from 0.1 to 10   Hz. So for 1/f measurements, DC coupling with an external bandpass filter is generally used. Table 5.2 summarizes the general guidelines for noise measurements with oscilloscopes.

Table 5.2. General Guidelines for Noise Measurements With Oscilloscopes

•

Do not use 10×probes for low noise measurements

•

Use direct BNC connection (10 times better noise floor)

•

Use BNC shorting cap to measure noise floor

•

Use bandwidth limiting, if appropriate

•

Use digital oscilloscope in DC coupling for 1/f noise measurements (AC coupling has a 10-Hz high–pass filter)

•

Use AC coupling for broadband measurements, if necessary

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Image Quality

Harry L. Snyder , in Handbook of Human-Computer Interaction, 1988

Temporal Vision

For many display technologies, such as the common cathode-ray tube (CRT) display, there is a need to refresh the display periodically to maintain adequate luminance and to avoid the perception of flicker caused by the the limited persistence of the image. Because the determination of the minimum acceptable refresh rate is often critical to the bandwidth requirements of the device, one must be able to predict the refresh rate at which an image ceases to flicker and is seen as nonflickering or fused.

There is a large volume of literature dealing with research on the sensitivity of the visual system to pulses of light. Much of this research has used a stimulus or display which is sequentially turned on and off in rapid succession with the observer asked to determine the minimum rate at which flicker is no longer perceived. This minimum rate is termed the critical fusion frequency (CFF). A plot of the CFF versus other display parameters, such as mean display luminance or off-axis angle, abounds in the literature. Brown (1965) has written an excellent review of this basic research literature.

Unfortunately, the data in the vision literature are often not of direct use to the display designer because the luminance sources used for the experiments are typically fast-response devices, such as glow modulator tubes. Thus, in these experiments the luminance intensity over time is not characteristic of many display devices which have appreciable persistence and gradual decay of their luminance.

Fortunately, the approach used to characterize the spatial response of the visual system can also be used meaningfully and usefully to characterize the temporal response. In this case, the pertinent function of the visual system is called the temporal CTF.

One can create a repetitive waveform of varying luminance and Fourier analyze it in the same fashion that one does a spatially varying signal. In fact, the Fourier analysis of a temporally varying signal is the more traditional application of this analysis form in that it is often applied to the analysis of auditory waveforms. The results of the analysis are, of course, a series of sine-wave frequencies, each having its own amplitude and phase relationship to the other frequencies.

The utility of the Fourier analysis approach to temporal vision was demonstrated in 1958 by deLange, who conducted an experiment to determine the effect of various waveforms upon the CFF at several luminance levels. His results indicate that the sensitivity of the visual system to temporally varying stimuli can be predicted by the frequency and modulation of the Fourier fundamental, irrespective of the waveform itself. deLange's data for each of the four different waveforms illustrated in Figure 12 fit very close to each of the three different curves, one for each of three retinal illumination levels. Thus, the modulation and frequency combination determined by the Fourier fundamental for each of the waveforms is a total predictor of the threshold curve, irrespective of waveform differences.

Figure 12. DeLange's (1958) results.

As illustrated in Figure 12, deLange also noted that the sensitivity to flicker is increased with increasing luminance and that the typical bandpass response shifts to a higher center frequency with increasing luminance, not unlike the response of the visual system in the spatial domain (Figure 7). deLange's (1958) results cover the frequency range from 1 to over 50 Hz, while the research of Almagor, Farley, and Snyder (1979) extended these data to a lower frequency of 0.01 Hz. They found that the sensitivity continues to decrease (i.e., the threshold modulation continues to increase) monotonically below 1 Hz (Figure 13).

Figure 13. Effect of low temporal frequencies on the CSF, from Almagor, Farley, and Snyder (1979).

Numerous other experiments have been conducted since those of deLange, all of which show with good consistency that the visual system behaves essentially as a Fourier analyzer in the temporal domain. Fortunately, this means that we can take the Fourier fundamental and the first several harmonics of the waveform, compare those coefficients with known threshold curves, and estimate whether the display exhibiting that waveform will be seen as flickering or fused. The only difficult part of this process, other than measuring the waveform, is to know which visual threshold curves to use for the purposes of making the comparison. Ample research data exist to define temporal CTFs for a variety of conditions, including field size, target size, and mean luminance.

Figure 13 is particularly important in the current context as the data were obtained with a wide field display covering approximately 160 degrees of angular vision. Thus, the visual periphery was stimulated as was the central foveal field. The importance of this research lies in the fact that the peripheral field is more sensitive to flicker than is the central field. Accordingly, one needs to calculate the necessary refresh rate for a display to avoid flicker in the peripheral field if the display is itself a bright display with dark characters, whereas one needs to calculate the refresh rate for the foveal field to avoid flicker if the display is one of light characters on a dark background. Because the data of Almagor et al. (1979) amply stimulated both the foveal and peripheral fields, these data can be used (in conjunction with the Fourier transform results of any repetitive waveform to predict the refresh rate necessary to avoid flicker) either in the fovea or in the periphery.

To predict perceived flicker or fusion for a display from these temporal CTF data, one needs to know (or assume) a refresh rate, a peak luminance, a mean luminance, and the decay (or persistence) function of the luminance. From the temporally repetitive luminance waveform of the display, one then uses a discrete Fourier transform to obtain the luminance modulation at the Fourier fundamental frequency. If the modulation is above the CTF at this fundamental frequency, the display will be seen to flicker, while a modulation below threshold will result in a nonflickering perception. This approach has been discussed in the literature by Bryden (1966), Krupka and Fukui (1973), and Snyder (1980).

It should be noted that some intuitive approaches to eliminating flicker are in fact erroneous. For example, a CRT with a P4 phosphor that flickers at 60 Hz may be replaced with a longer persistence phosphor. In many cases, if the same mean display luminance is maintained, the longer persistence phosphor will increase flicker because its decay pattern places more luminance power in the same Fourier fundamental frequency.

As mentioned above in the section on spatial vision, the visual system performs some automatic tradeoffs of benefit to the display user. One of these tradeoffs is the variation in the integration time of the eye. As early as 1885, Bloch showed that the visual threshold is constant in total energy if the stimulus duration is less than some upper time limit, t. That is, that the eye integrates light energy over the stimulus duration, and the threshold is a direct function of this integration rather than of the momentary energy level itself. Several experiments (e.g. Herrick, 1956) have demonstrated that this upper limit for foveal vision is approximately 100 ms; further, it appears that the upper limit for peripheral vision is approximately the same, at least out to about 20 degrees off axis. The results of Barlow (1958) further support the upper limit of temporal integration as 0.1 s with no difference in this limit for fields ranging from 0.111 square degree to 27.6 square degrees.

This integration time has been shown to vary significantly with differences in luminance of the flickering field (Almagor et al., 1979). Increases in luminance result in decreases in integration time as measured by both CFFs and the duration of the temporal "Mach" band, an overshoot perception of continuing luminance change at the end of a ramp function of luminance change. Figure 14 illustrates the stimulus luminance function and the related subjective brightness experience associated with the temporal Mach band.

Figure 14. Overshoot of perceived luminance, from Almagor, Farley, and Snyder (1979).

The importance of this varying integration time lies in the viewing of dynamically noisy displays, such as television displays of the low-light-level sensor type or displays which have an intrinsic amount of computer-inserted noise. Images embedded in dynamic, uncorrelated noise should be viewed in a dim environment and with low display luminance to increase the integration time of the visual system and thereby permit the system to filter out more noise. Because the eye trades off integration time for integration space, the acuity remains reasonably high while the perceived noise is reduced through increased temporal integration. Conversely, images of moving objects should be viewed under higher display luminance conditions to reduce integration time and therefore reduce the image blur induced by the integration time itself. Similary, because spatial vision improves in its resolution capability as a function of display luminance, those applications requiring fine discrimination should be both noise free and of greater luminance; for example, this argument would apply to CAD and CAE displays.

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Instrumentation and Measurements

Sverre Grimnes , Ørjan G Martinsen , in Bioimpedance and Bioelectricity Basics (Third Edition), 2015

The Sum of a Fundamental Sine Wave and Its Harmonic Components: Fourier Series

The only waveform containing just one frequency is the sine wave. A periodic waveform can be created by a sum of sine waves, each being a harmonic component of the sine wave at the fundamental frequency determined by the period. This is illustrated in Figure 8.6(a) , showing the sum of a fundamental and its third and fifth harmonic components. It indicates that uneven harmonic components may lead to a square wave, with a precision determined by the number of harmonic components included.

Figure 8.6. Summation of harmonic sine waves, waveform dependence of phase relationships. Amplitude of fundamental sine wave   =   1. Time domain; (a) in-phase harmonics, (b) phase-shifted harmonics. (c) Amplitude magnitude line frequency spectrum, equal for both cases.

Figure 8.6(c) shows the frequency spectrum of the waveform. It is a line or discrete spectrum, because it contains only the three discrete frequencies. Continuously repetitive waveforms have line spectra, and their periodicity is composed only of the fundamental and its harmonic components.

Fourier formulated the mathematical expression for the sum of the fundamental and its harmonics. The condition is that a fundamental period of a waveform f(t) can be determined, and that the waveform f(t) is extended outside its defined interval so that it is periodic with period 2π.

(8.25) $\mathrm{f}\left(\mathrm{t}\right)=\frac{{\mathrm{a}}_0}{2}+\sum _{\mathrm{n}=1}^{\infty }({\mathrm{a}}_{\mathrm{n}}\cos \mathrm{n}{\mathrm{\omega }}_1\mathrm{t}+{\mathrm{b}}_{\mathrm{n}}\sin \mathrm{n}{\mathrm{\omega }}_1\mathrm{t})$

where a_n and b_n are the amplitudes of each harmonic component n, a₀ is the DC component, and ω₁ the angular fundamental frequency defining the period 2π.

According to the Fourier series Eq. 8.25 , any periodic waveform is the sum of a fundamental sinusoid and a series of its harmonics. Notice that, in general, each harmonic component consists of a sine and cosine component. Of course, either of them may be zero for a given waveform in the time domain. Such a waveform synthesis (summation) is done in the time domain, but each wave is a component in the frequency domain. The frequency spectrum of a periodic function of time f(t) is therefore a line spectrum. The amplitudes of each discrete harmonic frequency component is:

(8.26) ${\mathrm{a}}_{\mathrm{n}}=\frac{1}{\mathrm{\pi }}\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}\mathrm{f}\left(\mathrm{t}\right)\cos \left(\mathrm{n}{\mathrm{\omega }}_1\mathrm{t}\right)d\mathrm{t}$

(8.27) ${\mathrm{b}}_{\mathrm{n}}=\frac{1}{\mathrm{\pi }}\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }}\mathrm{f}\left(\mathrm{t}\right)\sin \left(\mathrm{n}{\mathrm{\omega }}_1\mathrm{t}\right)d\mathrm{t}$

(8.28) ${\mathbf{A}}_{\text{n}}={\text{a}}_{\text{n}}+{\text{jb}}_{\text{n}}$

(8.29) ${\mathrm{A}}_{\mathrm{n}}=\sqrt{{\mathrm{a}}_{\mathrm{n}}^2+{\mathrm{b}}_{\mathrm{n}}^2}$

(8.30) $\mathrm{\phi }=\arctan \left(\frac{{\text{b}}_{\text{n}}}{{\text{a}}_{\text{n}}}\right)$

Because the waveform is periodic, the integration can be limited to the period interval 2π as defined by ω₁. However, the number n of harmonic components may be infinite. The presentation of a signal in the time or frequency domain contains the same information; it is a choice of how data are to be presented and analyzed.

Figure 8.6 illustrates how two rather different waveforms in the time domain may have the same amplitude magnitude A_n frequency spectrum. The amplitude magnitude frequency spectrum does not contain all necessary information; the phase information is lacking. For each harmonic component, both the sine and cosine (Eqs 8.26 and 8.27) or the magnitude and phase (Eqs 8.29 and 8.30), must be given, a magnitude and a phase spectrum. The amplitude A _n is a vector; therefore, amplitude magnitudes A_n cannot just be added as scalars. A given waveform is the sum of only one unique set of sine and cosine harmonics.

An infinite number of harmonics must be added to obtain, for instance, a true square wave. The Fourier series for a periodic square wave of unit amplitude is (see Figure 8.6(a)):

(8.31) $\mathrm{f}\left(\mathrm{t}\right)=\frac{4}{\mathrm{\pi }}\sum _{\mathrm{n}=1,3,5\dots }\frac{1}{\mathrm{n}}\sin \mathrm{n}{\mathrm{\omega }}_1\mathrm{t}$

This square wave can therefore be realized as the sum of only sine components.

Any waveform with sharp ascending or descending parts, like the square wave or sawtooth, contains large amplitudes of higher harmonic components. The triangular pulses contain more of the lower harmonics. No frequencies lower than the fundamental, corresponding to the repetition rate, exist. The waveform may contain a DC component; if it is symmetrical around zero, the DC component is zero. However, if the waveform is started or stopped nonsynchronized with the period, it is no longer periodic. During those nonperiodic intervals, the Fourier series approach is no more valid.

By using nonsinusoids as excitation waveforms, a system is excited at several frequencies simultaneously. If the system is linear, the response of each sine wave can be added. If the system is nonlinear, new frequencies are created influencing the frequency spectrum. The square wave to the left of Figure 8.7 has no DC component. One of the ramps in the middle is used in scanning devices such as polarographs. As drawn, the waveform has a DC component. The pulse to the right has a DC component dependent on the repetition frequency.

Figure 8.7. Square, ramp, and pulse periodic waveforms.

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