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48-419: While originally developed for use in quality control studies in the realm of manufacturing, SPRT has been formulated for use in the computerized testing of human examinees as a termination criterion. As in classical hypothesis testing , SPRT starts with a pair of hypotheses, say H 0 {\displaystyle H_{0}} and H 1 {\displaystyle H_{1}} for

96-526: A ≈ log ⁡ β 1 − α {\displaystyle a\approx \log {\frac {\beta }{1-\alpha }}} and b ≈ log ⁡ 1 − β α {\displaystyle b\approx \log {\frac {1-\beta }{\alpha }}} In other words, α {\displaystyle \alpha } and β {\displaystyle \beta } must be decided beforehand in order to set

144-553: A bandpass signal is sampled slower than its Nyquist rate , the samples are indistinguishable from samples of a low-frequency alias of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion , because the bandpass signal is still uniquely represented and recoverable. Such undersampling is also known as bandpass sampling , harmonic sampling , IF sampling , and direct IF to digital conversion. Oversampling

192-457: A moiré pattern . The process of volume rendering samples a 3D grid of voxels to produce 3D renderings of sliced (tomographic) data. The 3D grid is assumed to represent a continuous region of 3D space. Volume rendering is common in medical imaging, X-ray computed tomography (CT/CAT), magnetic resonance imaging (MRI), positron emission tomography (PET) are some examples. It is also used for seismic tomography and other applications. When

240-413: A Nyquist rate of B {\displaystyle B} , because all of its non-zero frequency content is shifted into the interval [ − B / 2 , B / 2 ] {\displaystyle [-B/2,B/2]} . Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance,

288-419: A continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every T {\displaystyle T} seconds, which is called the sampling interval or sampling period . Then the sampled function is given by the sequence: The sampling frequency or sampling rate , f s {\displaystyle f_{s}} ,

336-406: A legally defensible method, such as a modified Angoff procedure . Again, the indifference region represents the region of scores that the test designer is OK with going either way (pass or fail). The upper parameter p 2 is conceptually the highest level that the test designer is willing to accept for a Fail (because everyone below it has a good chance of failing), and the lower parameter p 1

384-427: A much lower rate. For most phonemes , almost all of the energy is contained in the 100 Hz – 4 kHz range, allowing a sampling rate of 8 kHz. This is the sampling rate used by nearly all telephony systems, which use the G.711 sampling and quantization specifications. Standard-definition television (SDTV) uses either 720 by 480 pixels (US NTSC 525-line) or 720 by 576 pixels (UK PAL 625-line) for

432-438: A proposed nonlinear function . Digital audio uses pulse-code modulation (PCM) and digital signals for sound reproduction. This includes analog-to-digital conversion (ADC), digital-to-analog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discrete-time, discrete-level analog of a previous electrical analog. While modern systems can be quite subtle in their methods,

480-606: Is parameter estimation of a probability distribution function . Consider the exponential distribution : The hypotheses are Then the log-likelihood function (LLF) for one sample is The cumulative sum of the LLFs for all x is Accordingly, the stopping rule is: After re-arranging we finally find The thresholds are simply two parallel lines with slope log ⁡ ( θ 1 / θ 0 ) {\displaystyle \log(\theta _{1}/\theta _{0})} . Sampling should stop when

528-530: Is a consequence of the Nyquist theorem . Sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early professional audio equipment manufacturers chose sampling rates in the region of 40 to 50 kHz for this reason. There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz Even though ultrasonic frequencies are inaudible to humans, recording and mixing at higher sampling rates

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576-430: Is a major component of quality control, where physical product is examined visually (or the end results of a service are analyzed). Product inspectors will be provided with lists and descriptions of unacceptable product defects such as cracks or surface blemishes for example. Early stone tools such as anvils had no holes and were not designed as interchangeable parts . Mass production established processes for

624-458: Is a simple thresholding scheme: where a {\displaystyle a} and b {\displaystyle b} ( a < 0 < b < ∞ {\displaystyle a<0<b<\infty } ) depend on the desired type I and type II errors , α {\displaystyle \alpha } and β {\displaystyle \beta } . They may be chosen as follows:

672-502: Is a tendency for individual consultants and organizations to name their own unique approaches to quality control—a few of these have ended up in widespread use: In project management , quality control requires the project manager and/or the project team to inspect the accomplished work to ensure its alignment with the project scope. In practice, projects typically have a dedicated quality control team which focuses on this area. Sampling frequency In signal processing , sampling

720-502: Is converted to digital video , a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along scan lines . A common pixel sampling rate is: Spatial sampling in the other direction is determined by the spacing of scan lines in the raster . The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height. Spatial aliasing of high-frequency luma or chroma video components shows up as

768-442: Is effective in eliminating the distortion that can be caused by foldback aliasing . Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum ( intermodulation distortion ), degrading the fidelity. One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs , but with modern oversampling delta-sigma-converters this advantage

816-435: Is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations. Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of low-pass filtering . The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function mapping with

864-431: Is less important. The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for CD and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering . Both Lavry Engineering and J. Robert Stuart state that the ideal sampling rate would be about 60 kHz, but since this

912-469: Is less than 2 sample intervals (see Aliasing ). The corresponding frequency limit, in cycles per second ( hertz ), is 0.5 {\displaystyle 0.5} cycle/sample × f s {\displaystyle f_{s}} samples/second = f s / 2 {\displaystyle f_{s}/2} , known as the Nyquist frequency of the sampler. Therefore, s ( t ) {\displaystyle s(t)}

960-402: Is mathematically equivalent to an ideal low-pass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant ( T ) {\displaystyle (T)} , the sequence of delta functions is called a Dirac comb . Mathematically, the modulated Dirac comb is equivalent to

1008-435: Is not a standard frequency, recommend 88.2 or 96 kHz for recording purposes. A more complete list of common audio sample rates is: Audio is typically recorded at 8-, 16-, and 24-bit depth, which yield a theoretical maximum signal-to-quantization-noise ratio (SQNR) for a pure sine wave of, approximately, 49.93  dB , 98.09 dB and 122.17 dB. CD quality audio uses 16-bit samples. Thermal noise limits

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1056-404: Is set at 70% for a test. We could select p 1 = 0.65 and p 2 = 0.75 . The test then evaluates the likelihood that an examinee's true score on that metric is equal to one of those two points. If the examinee is determined to be at 75%, they pass, and they fail if they are determined to be at 65%. These points are not specified completely arbitrarily. A cutscore should always be set with

1104-467: Is the Hilbert transform of the other waveform, s ( t ) {\displaystyle s(t)} , the complex-valued function, s a ( t ) ≜ s ( t ) + i ⋅ s ^ ( t ) {\displaystyle s_{a}(t)\triangleq s(t)+i\cdot {\hat {s}}(t)} , is called an analytic signal , whose Fourier transform

1152-551: Is the allowance of a composite, one-sided alternative hypothesis, and the introduction of an upper stopping boundary. The method has been used in several medical research studies. Quality control Quality control ( QC ) is a process by which entities review the quality of all factors involved in production . ISO 9000 defines quality control as "a part of quality management focused on fulfilling quality requirements". This approach places emphasis on three aspects (enshrined in standards such as ISO 9001): Inspection

1200-413: Is the average number of samples obtained in one second, thus f s = 1 / T {\displaystyle f_{s}=1/T} , with the unit samples per second , sometimes referred to as hertz , for example 48 kHz is 48,000 samples per second . Reconstructing a continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation formula

1248-449: Is the lowest level that the test designer is willing to accept for a pass (because everyone above it has a decent chance of passing). While this definition may seem to be a relatively small burden, consider the high-stakes case of a licensing test for medical doctors: at just what point should we consider somebody to be at one of these two levels? While the SPRT was first applied to testing in

1296-482: Is the reduction of a continuous-time signal to a discrete-time signal . A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics , which refers to a set of such values. A sampler is a subsystem or operation that extracts samples from a continuous signal . A theoretical ideal sampler produces samples equivalent to

1344-610: Is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters , such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula . Complex sampling (or I/Q sampling ) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers . When one waveform, s ^ ( t ) {\displaystyle {\hat {s}}(t)} ,

1392-483: Is usually the output of a low-pass filter , functionally known as an anti-aliasing filter . Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process. In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. This results in deviations from

1440-519: Is zero for all negative values of frequency. In that case, the Nyquist rate for a waveform with no frequencies ≥  B can be reduced to just B (complex samples/sec), instead of 2 B {\displaystyle 2B} (real samples/sec). More apparently, the equivalent baseband waveform , s a ( t ) ⋅ e − i 2 π B 2 t {\displaystyle s_{a}(t)\cdot e^{-i2\pi {\frac {B}{2}}t}} , also has

1488-472: The null hypothesis and alternative hypothesis respectively. They must be specified as follows: The next step is to calculate the cumulative sum of the log- likelihood ratio , log ⁡ Λ i {\displaystyle \log \Lambda _{i}} , as new data arrive: with S 0 = 0 {\displaystyle S_{0}=0} , then, for i {\displaystyle i} =1,2,..., The stopping rule

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1536-444: The creation of parts and system with identical dimensions and design, but these processes are not uniform and hence some customers were unsatisfied with the result. Quality control separates the act of testing products to uncover defects from the decision to allow or deny product release, which may be determined by fiscal constraints. For contract work, particularly work awarded by government agencies, quality control issues are among

1584-483: The days of classical test theory , as is applied in the previous paragraph, Reckase (1983) suggested that item response theory be used to determine the p 1 and p 2 parameters. The cutscore and indifference region are defined on the latent ability (theta) metric, and translated onto the proportion metric for computation. Research on CCT since then has applied this methodology for several reasons: Spiegelhalter et al. have shown that SPRT can be used to monitor

1632-445: The equivalent baseband waveform can be created without explicitly computing s ^ ( t ) {\displaystyle {\hat {s}}(t)} , by processing the product sequence, [ s ( n T ) ⋅ e − i 2 π B 2 T n ] {\displaystyle \left[s(nT)\cdot e^{-i2\pi {\frac {B}{2}}Tn}\right]} , through

1680-455: The instantaneous value of the continuous signal at the desired points. The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit , by passing the sequence of samples through a reconstruction filter . Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions. For functions that vary with time, let s ( t ) {\displaystyle s(t)} be

1728-400: The integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time: Video digital-to-analog converters operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early games consoles, to 250 MHz or more for the highest-resolution VGA output). When analog video

1776-479: The limits. Quality was thus precisely defined using devices such as plug gauges and ring gauges . However, this did not address the problem of defective items; recycling or disposing of the waste adds to the cost of production, as does trying to reduce the defect rate. Various methods have been proposed to prioritize quality control issues and determine whether to leave them unaddressed or use quality assurance techniques to improve and stabilize production. There

1824-409: The lot is ideal or should be rejected. The SPRT is currently the predominant method of classifying examinees in a variable-length computerized classification test (CCT). The two parameters are p 1 and p 2 are specified by determining a cutscore (threshold) for examinees on the proportion correct metric, and selecting a point above and below that cutscore. For instance, suppose the cutscore

1872-495: The lot to have 3% or less defective widgets, but 1% or less is the ideal lot that would pass with flying colors. In this example, p 1 = 0.01 and p 2 = 0.03 and the region between them is the IR because management considers these lots to be marginal and is OK with them being classified either way. Widgets would be sampled one at a time from the lot (sequential analysis) until the test determines, within an acceptable error level, that

1920-445: The performance of doctors, surgeons and other medical practitioners in such a way as to give early warning of potentially anomalous results. In their 2003 paper, they showed how it could have helped identify Harold Shipman as a murderer well before he was actually identified. More recently, in 2011, an extension of the SPRT method called Maximized Sequential Probability Ratio Test (MaxSPRT) was introduced. The salient feature of MaxSPRT

1968-443: The primary usefulness of a digital system is the ability to store, retrieve and transmit signals without any loss of quality. When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz ( CD ), 48 kHz, 88.2 kHz, or 96 kHz. The approximately double-rate requirement

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2016-496: The product of the comb function with s ( t ) {\displaystyle s(t)} . That mathematical abstraction is sometimes referred to as impulse sampling . Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when s ( t ) {\displaystyle s(t)} contains frequency components whose cycle length (period)

2064-415: The sum of the samples makes an excursion outside the continue-sampling region . The test is done on the proportion metric, and tests that a variable p is equal to one of two desired points, p 1 or p 2 . The region between these two points is known as the indifference region (IR). For example, suppose you are performing a quality control study on a factory lot of widgets. Management would like

2112-729: The theoretically perfect reconstruction, collectively referred to as distortion . Various types of distortion can occur, including: Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling

2160-409: The thresholds appropriately. The numerical value will depend on the application. The reason for being only an approximation is that, in the discrete case, the signal may cross the threshold between samples. Thus, depending on the penalty of making an error and the sampling frequency , one might set the thresholds more aggressively. The exact bounds are correct in the continuous case. A textbook example

2208-440: The top reasons for not renewing a contract. The simplest form of quality control was a sketch of the desired item. If the sketch did not match the item, it was rejected, in a simple Go/no go procedure. However, manufacturers soon found it was difficult and costly to make parts be exactly like their depiction; hence around 1840 tolerance limits were introduced, wherein a design would function if its parts were measured to be within

2256-455: The true number of bits that can be used in quantization. Few analog systems have signal to noise ratios (SNR) exceeding 120 dB. However, digital signal processing operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution. Speech signals, i.e., signals intended to carry only human speech , can usually be sampled at

2304-435: The visible picture area. High-definition television (HDTV) uses 720p (progressive), 1080i (interlaced), and 1080p (progressive, also known as Full-HD). In digital video , the temporal sampling rate is defined as the frame rate  – or rather the field rate  – rather than the notional pixel clock . The image sampling frequency is the repetition rate of the sensor integration period. Since

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