Ошибка квантования для шага квантования

Ошибки квантования

В реальных
устройствах цифровой обработки сигналов
необходимо учитывать
эффекты, обусловленные квантованием
входных сигналов
и конечной разрядностью всех регистров.
Источниками ошибок
в процессах обработки сигналов являются
округление (усечение)
результатов арифметических операций,
шум аналого-цифрового квантования
входных аналоговых сигналов, неточность
реализации характеристик цифровых
фильтров из-за округления их коэффициентов
(параметров). В дальнейшем с целью
упрощения анализа предполагается, что
вес источники ошибок независимы и не
коррелируют с входным сигналом (хотя
мы и рассмотрим явление предельных
циклов, обусловленных коррелированным
шумом округления).

Эффект квантования
приводят в конечном итоге к погрешностями выходных сигналах цифровых фильтров
(ЦФ), а в некоторыхслучаяхи к неустойчивым
режимам. Выходную ошибку ЦФ будем
рассчитыватькаксуперпозицию ошибок, обусловленных
каждым независимымисточником.

Квантование
чисел
– нелинейная операция;m-разрядное
двоичное числоА
представляетсяb-разрядным
двоичнымчислом
B=F(A),
причем b
<
m. В
результате квантования число А
представляется
с ошибкой

е
=BА= F(А)
А.

Шаг квантования
Q
=
2b
определяется весом младшего
числовогоразряда.
При квантовании
используется усечение или округление.

Усечение
числаА
состоит в отбрасываниит
b
младших разрядов числа, при этом
ошибка усечения
e
ус=
F
ус(А) –А.

Оценим величину
ошибки в предположении m
» b.
Для положительных чисел при любом
способе кодирования –2b
<еус
0. Для
отрицательных чисел при использовании
прямого и обратного кодов ошибка усечения
неотрицательна: 0еус
< 2b,
а в дополнительном коде эта ошибка
неположительна: 0еус
> –2b.
Таким образом, во всех случаях
абсолютное значение ошибки усечения
не превосходит шага квантования:maxeус
< 2b
=Q.

Округление
m-разрядного
числаA
доb
разрядов (b «
m
)b
разряд остается неизменным или
увеличивается на единицув
зависимости от соотношения (больше –
меньше) между отбрасываемой дробью
0,аb+1…ат
и величиной
,
гдеаii
разряд числаA;
i =
b+1,
…,m.
Округление можно практически выполнить
путемприбавления
единицы к (b+1)-му
разряду и усечения полученного числа
до b разрядов.
В таком случае ошибка округления е
=
fок(А)–
А
при всех способах кодирования лежит в
пределах

–2(b+1)
<
еoк
< 2(b+1)
(1.11)

и, следовательно,
m
ax<2b
= Q/2.
(1.12)

В задачах ЦОС
ошибки квантования чисел рассматриваются
как стационарный
шумоподобный процесс с равномерным
распределением
вероятности по диапазону распределения
ошибок квантования.

(nT)

x(nT)

e(nT)

Рис. 3. Линейная модель квантования
сигналов:

(nT) —дискретный
или m-разрядный
цифровой сигнал (m
> b);

x(nT) —квантованный
b-разрядный
цифровой сигнал;

e(nT)
= x(nT)
f(nT)
— ошибка
квантования.

Квантование
дискретных сигналов состоит в
представлении отсчета
(выборки сигнала) числамиx(nT),
содержащими b
числовых разрядов. Квантование сигналов,
как и квантование
чисел – нелинейная операция. Однако
при анализе процессов в ЦФ целесообразно
использовать линейную модель квантования
сигналов – рис. 3.

Верхнее значение ошибки квантования
определяетсясоотношением
(1.11) или (1.12).

Вероятностные
оценки ошибок квантования основаны на
предположениях о том, что
последовательностье(пТ)являетсястационарным
случайным процессом с равномерным
распределением вероятности по
диапазону ошибок квантования ие(пТ)
не коррелирован
с f(nT).
Математическое
ожидание (среднее значение) e
и дисперсия
ошибки квантованияе
определяются
по формулам:

=E(е)
=,

===E(е2)
,

где ре
— плотность вероятности ошибки. По
этим формуламвычисляются
математическое ожидание и дисперсия
для ошибок
округления и усечения:

=

=

В логарифмическоммасштабе

=

Лекция 2

Ошибка квантования

Ошибка квантования

Шум квантования — ошибки, возникающие при оцифровке аналогового сигнала. В зависимости от типа аналого-цифрового преобразования могут возникать из-за округления (до определённого разряда) сигнала или усечения (отбрасывания младших разрядов) сигнала.

Содержание

  • 1 Математическое описание
    • 1.1 Модель
    • 1.2 Детерминированные оценки
    • 1.3 Вероятностные оценки
  • 2 См. также
  • 3 Ссылки

Математическое описание

Модель

Шум квантования можно представить как аддитивный дискретный сигнал e(nT) !, учитывающий ошибки квантования. Если  d(nT) ! — входной сигнал квантователя, а  F[] ! — его передаточная функция, то имеем следующую линейную модель шума квантования:

 e(nT) = F[d(nT)] - d(nT) !

Линейная модель используется для аналитического исследования свойств шума квантования.

Детерминированные оценки

Детерминированные оценки позволяют определить абсолютные границы шума квантования:

|max[e(nT)]| = frac{1}{m} 2^{-b} = frac{1}{m} Q,

где b ! — число разрядов квантования (сигнала e(nT) !),  Q ! — шаг квантования  m = 2 ! — при округлении  m = 1 ! — при усечении.

Вероятностные оценки

Вероятностные оценки основаны на представлении ошибок квантования (сигнала e(nT)) как случайного шумоподобного процесса. Допущения, вводимые относительно шума квантования:

В таком случае математическое ожидание ! M_e и дисперсия ! D_e шума квантования определяется следующим образом (при квантовании используется дополнительный код):

См. также

  • Отношение сигнал-шум
  • Дизеринг

Ссылки

  • Round-Off Error Variance(англ.)
  • «Цифровая обработка сигналов». Л. М. Гольденберг, Б. Д. Матюшкин — М.: Радио и связь, 1985

Wikimedia Foundation.
2010.

Полезное

Смотреть что такое «Ошибка квантования» в других словарях:

  • ошибка квантования — Ошибка, вызванная несоответствием формы выходного (квантованного) и входного (аналогового) сигналов. Зависит от величины шага квантования и частоты дискретизации. [Л.М. Невдяев. Телекоммуникационные технологии. Англо русский толковый словарь… …   Справочник технического переводчика

  • ошибка квантования — kvantavimo paklaida statusas T sritis automatika atitikmenys: angl. quantization error vok. Quantisierungsfehler, m rus. ошибка квантования, f pranc. erreur de quantification, f …   Automatikos terminų žodynas

  • ошибка округления — погрешность, возникающая при квантовании в результате округления амплитуды сигнала до ближайшего уровня квантования. Причина возникновения шума квантования …   Русский индекс к Англо-русскому словарь по музыкальной терминологии

  • Аналого-цифровой преобразователь — Четырёхканальный аналого цифровой преобразователь Аналого цифровой преобразователь[1][2] …   Википедия

  • АЦП — Четырёхканальный аналого цифровой преобразователь Аналого цифровой преобразователь (АЦП, ADC)  устройство, преобразующее входной аналоговый сигнал в дискретный код (цифровой сигнал). Обратное преобразование осуществляется при помощи ЦАП (DAC)… …   Википедия

  • Цифро-аналоговое преобразование — Четырёхканальный аналого цифровой преобразователь Аналого цифровой преобразователь (АЦП, ADC)  устройство, преобразующее входной аналоговый сигнал в дискретный код (цифровой сигнал). Обратное преобразование осуществляется при помощи ЦАП (DAC)… …   Википедия

  • шум дробления — Разновидность шума квантования, возникающего при аналого цифровом преобразовании сигнала малого уровня, когда ошибка квантования для разных отсчетов становится не равновероятной. Эффективной мерой снижения шума дробления является использование… …   Справочник технического переводчика

  • Quantisierungsfehler — kvantavimo paklaida statusas T sritis automatika atitikmenys: angl. quantization error vok. Quantisierungsfehler, m rus. ошибка квантования, f pranc. erreur de quantification, f …   Automatikos terminų žodynas

  • erreur de quantification — kvantavimo paklaida statusas T sritis automatika atitikmenys: angl. quantization error vok. Quantisierungsfehler, m rus. ошибка квантования, f pranc. erreur de quantification, f …   Automatikos terminų žodynas

  • kvantavimo paklaida — statusas T sritis automatika atitikmenys: angl. quantization error vok. Quantisierungsfehler, m rus. ошибка квантования, f pranc. erreur de quantification, f …   Automatikos terminų žodynas

Sensors

Andrea Colagrossi, … Matteo Battilana, in Modern Spacecraft Guidance, Navigation, and Control, 2023

Quantization errors

Quantization error is a systematic error resulting from the difference between the continuous input value and its quantized output, and it is like round-off and truncation errors. This error is intrinsically associated with the AD conversion that maps the input values from a continuous set to the output values in a countable set, often with a finite number of elements. The quantization error is linked to the resolution of the sensor. Namely, a high-resolution sensor has a small quantization error. Indeed, the maximum quantization error is smaller than the resolution interval of the output, which is associated to the least significant bit representing the smallest variation that can be represented digitally:

LSB=FSR2NBIT

where FSR is the full-scale range of the sensor, and NBIT is the number of bits (i.e., the resolution) used in the AD converter to represent the sensor’s output. Quantization errors are typically not corrected, and the discrete values of the output are directly elaborated by the GNC system, which is designed to operate on digital values.

Fig. 6.9 shows a convenient model block to simulate quantization errors.

Figure 6.9. Quantization error model.

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Digital Filters

Marcio G. Siqueira, Paulo S.R. Diniz, in The Electrical Engineering Handbook, 2005

2.11 Quantization in Digital Filters

Quantization errors in digital filters can be classified as:

Round-off errors derived from internal signals that are quantized before or after more down additions;

Deviations in the filter response due to finite word length representation of multiplier coefficients; and

Errors due to representation of the input signal with a set of discrete levels.

A general, digital filter structure with quantizers before delay elements can be represented as in Figure 2.23, with the quantizers implementing rounding for the granular quantization and saturation arithmetic for the overflow nonlinearity.

Figure 6.9. Quantization error model.

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Digital Filters

Marcio G. Siqueira, Paulo S.R. Diniz, in The Electrical Engineering Handbook, 2005

2.11 Quantization in Digital Filters

Quantization errors in digital filters can be classified as:

Round-off errors derived from internal signals that are quantized before or after more down additions;

Deviations in the filter response due to finite word length representation of multiplier coefficients; and

Errors due to representation of the input signal with a set of discrete levels.

A general, digital filter structure with quantizers before delay elements can be represented as in Figure 2.23, with the quantizers implementing rounding for the granular quantization and saturation arithmetic for the overflow nonlinearity.

FIGURE 2.23. Digital Filter Including Quantizers at the Delay Inputs

The criterion to choose a digital filter structure for a given application entails evaluating known structures with respect to the effects of finite word length arithmetic and choosing the most suitable one.

2.11.1 Coefficient Quantization

Approximations are known to generate digital filter coefficients with high accuracy. After coefficient quantization, the frequency response of the realized digital filter will deviate from the ideal response and eventually fail to meet the prescribed specifications. Because the sensitivity of the filter response to coefficient quantization varies with the structure, the development of low-sensitivity digital filter realizations has raised significant interest (Antoniou, 1993; Diniz et al., 2002).

A common procedure is to design the digital filter with infinite coefficient word length satisfying tighter specifications than required, to quantize the coefficients, and to check if the prescribed specifications are still met.

2.11.2 Quantization Noise

In fixed-point arithmetic, a number with a modulus less than one can be represented as follows:

(2.84)x=b0b1b2b3…bb,

where b0 is the sign bit and where b1b2b3bb represent the modulus using a binary code. For digital filtering, the most widely used binary code is the two’s complement representation, where for positive numbers b0 = 0 and for negative numbers b0 = 1. The fractionary part of the number, called x2 here, is represented as:

(2.85)x2={xif b0=0.2−|x|if b0=1.

The discussion here concentrates in the fixed-point implementation.

A finite word length multiplier can be modeled in terms of an ideal multiplier followed by a single noise source e(n) as shown in Figure 2.24.

FIGURE 2.24. Model for the Noise Generated after a Multiplication

For product quantization performed by rounding and for signal levels throughout the filter much larger than the quantization step q = 2b, it can be shown that the power spectral density of the noise source ei(n) is given by:

(2.86)Pei(z)=q212=2−2b12.

In this case, ei(n) represents a zero mean white noise process. We can consider that in practice, ei(n) and ek(n + l) are statistically independent for any value of n or l (for ik). As a result, the contributions of different noise sources can be taken into consideration separately by using the principle of superposition.

The power spectral density of the output noise, in a fixed-point digital-filter implementation, is given by:

(2.87)Py(z)=σe2Σi=1KGi(z)Gi(z−1),

where Pei(ejw)=σe2, for all i, and each Gi(z) is a transfer function from multiplier output (gi(n)) to the output of the filter as shown in Figure 2.25. The word length, including sign, is b + 1 bits, and K is the number of multipliers of the filter.

FIGURE 2.24. Model for the Noise Generated after a Multiplication

For product quantization performed by rounding and for signal levels throughout the filter much larger than the quantization step q = 2b, it can be shown that the power spectral density of the noise source ei(n) is given by:

(2.86)Pei(z)=q212=2−2b12.

In this case, ei(n) represents a zero mean white noise process. We can consider that in practice, ei(n) and ek(n + l) are statistically independent for any value of n or l (for ik). As a result, the contributions of different noise sources can be taken into consideration separately by using the principle of superposition.

The power spectral density of the output noise, in a fixed-point digital-filter implementation, is given by:

(2.87)Py(z)=σe2Σi=1KGi(z)Gi(z−1),

where Pei(ejw)=σe2, for all i, and each Gi(z) is a transfer function from multiplier output (gi(n)) to the output of the filter as shown in Figure 2.25. The word length, including sign, is b + 1 bits, and K is the number of multipliers of the filter.

FIGURE 2.25. Digital Filter Including Scaling and Noise Transfer Functions.

2.11.3 Overflow Limit Cycles

Overflow nonlinearities influence the most significant bits of the signal and cause severe distortion. An overflow can give rise to self-sustained, high-amplitude oscillations known as overflow limit cycles. Digital filters, which are free of zero-input limit cycles, are also free of overflow oscillations if the overflow nonlinearities are implemented with saturation arithmetic, that is, by replacing the number in overflow by a number with the same sign and with maximum magnitude that fits the available wordlength.

When there is an input signal applied to a digital filter, overflow might occur. As a result, input signal scaling is required to reduce the probability of overflow to an acceptable level. Ideally, signal scaling should be applied to ensure that the probability of overflow is the same at each internal node of the digital filter. This way, the signal-to-noise ratio is maximized in fixed-point implementations.

In two’s complement arithmetic, the addition of more than two numbers will be correct independently of the order in which they are added even if overflow occurs in a partial summation as long as the overall sum is within the available range to represent the numbers. As a result, a simplified scaling technique can be used where only the multiplier inputs require scaling. To perform scaling, a multiplier is used at the input of the filter section as illustrated in Figure 2.25.

It is possible to show that the signal at the multiplier input is given by:

(2.88)xi(n)=12πj∮cXi(z)zn−1dz=12π∫02πFi(ejω)X(ejω)ejωndω,

where c is the convergence region common to Fi(z) and X(z).

The constant λ is usually calculated by using Lp norm of the transfer function from the filter input to the multiplier input Fi(z), depending on the known properties of the input signal. The Lp norm of Fi(z) is defined as:

(2.89)‖Fi(ejω)‖p=[12π∫02π|Fi(ejω)|pdω]1p,

for each p ≥ 1, such that ∫02π|Fi(ejω)|pdω≤∞. In general, the following inequality is valid:

(2.90)|xi(n)| ≤ ‖Fi‖p‖X‖q,  (1p+1q=1),

for p, q = 1, 2 and ∞.

The scaling guarantees that the magnitudes of multiplier inputs are bounded by a number Mmax when |x(n)| ≤ Mmax. Then, to ensure that all multiplier inputs are bounded by Mmax we must choose λ as follows:

(2.91)λ=1Max{‖F1‖p,…,‖F1‖p,…, ‖FK‖p},

which means that:

(2.92)‖F′i(ejω)‖p≤1, for‖X(ejω)‖q ≤  Mmax.

The K is the number of multipliers in the filter.

The norm p is usually chosen to be infinity or 2. The L norm is used for input signals that have some dominating frequency component, whereas the L2 norm is more suitable for a random input signal. Scaling coefficients can be implemented by simple shift operations provided they satisfy the overflow constraints.

In case of modular realizations, such as cascade or parallel realizations of digital filters, optimum scaling is accomplished by applying one scaling multiplier per section.

As an illustration, we present the equation to compute the scaling factor for the cascade realization with direct-form second-order sections:

(2.93)λi=1‖∏j=1i−1Hj(z)Fi(z)‖p,

where:

Fi(z)=1z2+m1iz+m2i.

The noise power spectral density is computed as:

(2.94)Py(z)=σe2[3+3λ12∏i=1mHi(z)Hi(z−1)+5Σj=2m1λj2∏i=jmHi(z)Hi(z−1)],

whereas the output noise variance is given by:

(2.95)σo2=σe2[3+3λ12||∏i=1mHi(ejω)||22+5Σj=2m1λj2||∏i=jmHi(ejω)||22].

As a design rule, the pairing of poles and zeros is performed as explained here: poles closer to the unit circle pair with closer zeros to themselves, such that ||Hi(z)||p is minimized for p = 2 or p = ∞.

For ordering, we define the following:

(2.96)Pi=| |Hi(z)| |∞| |Hi(z)| |2.

For L2 scaling, we order the section such that Pi is decreasing. For L scaling, Pi should be increasing.

2.11.4 Granularity Limit Cycles

The quantization noise signals become highly correlated from sample to sample and from source to source when signal levels in a digital filter become constant or very low, at least for short periods of time. This correlation can cause autonomous oscillations called granularity limit cycles.

In recursive digital filters implemented with rounding, magnitude truncation,72 and other types of quantization, limitcycles oscillations might occur.

In many applications, the presence of limit cycles can be harmful. Therefore, it is desirable to eliminate limit cycles or to keep their amplitude bounds low.

If magnitude truncation is used to quantize particular signals in some filter structures, it can be shown that it is possible to eliminate zero-input limit cycles. As a consequence, these digital filters are free of overflow limit cycles when overflow nonlinearities, such as saturation arithmetic, are used.

In general, the referred methodology can be applied to the following class of structures:

State-space structures: Cascade and parallel realization of second-order state-space structures includes design constraints to control nonlinear oscillations (Diniz and Antoniou, 1986).

Wave digital filters: These filters emulate doubly terminated lossless filters and have inherent stability under linear conditions as well as in the nonlinear case where the signals are subjected to quantization (Fettweis, 1986).

Lattice realization: Modular structures allowing easy limit cycles elimination (Gray and Markel, 1975).

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Biomedical signals and systems

Sri Krishnan, in Biomedical Signal Analysis for Connected Healthcare, 2021

2.2.1 Noise power

The quantization error (e) or noise tends to have a random behavior, and they could be mathematically represented using statistical variables. Power of a random variable with a probability density function of p(e) could be obtained by computing the second-order statistics of variance, and it is denoted by

σ2=∫−q/2q/2e2p(e)de

A good assumption for p(e) is a uniform probability density function which will have a value of 1/q over the range of −q/2 to q/2.

=∫−q/2q/2e2·1qde=q212

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Measurement of high voltages

E. Kuffel, … J. Kuffel, in High Voltage Engineering Fundamentals (Second Edition), 2000

Static errors

The quantization error is present because the analogue value of each sample is transformed into a digital word. This A-to-D conversion entails a quantization of the recorder’s measuring range into a number of bands or code bins, each represented by its central value which corresponds to a particular digital code or level. The number of bands is given by 2N, where N is the resolution of the A-to-D converter. The digital output to analogue input relationship of an ideal digitizer is shown diagrammatically in Fig. 3.49. For any input in the range (iΔVav – 0.5 * ΔVav to iΔVav + 0.5 * ΔVav), where iΔVav is the voltage corresponding to the width of each code bin, or one least significant bit (LSB), and iΔVav is the centre voltage corresponding to the i th code, an ideal digitizer will return a value of Ii. Therefore, the response of an ideal digitizer to a slowly increasing linear ramp would be a stairway such as that shown in Fig. 3.50. A quick study of these figures reveals the character of the quantization error associated with the ideal A-to-D conversion process. The maximum error possible is equivalent to a voltage corresponding to ±(½) of an LSB. For an ideal digital recorder, this quantization would be the only source of error in the recorded samples. For a real digital recorder, this error sets the absolute upper limit on the accuracy of the readings. In the case of an 8-bit machine, this upper limit would be 0.39 per cent of the recorder’s full-scale deflection. The corresponding maximum accuracy (lowest uncertainty) of a 10-bit recorder is 0.10 per cent of its full-scale deflection.

Figure 3.49. Analogue input to digital output relation of an ideal A/D converter

Figure 3.49. Analogue input to digital output relation of an ideal A/D converter

Figure 3.50. Response of an ideal A/D converter to a slowly rising ramp

The error caused by discrete time sampling is most easily demonstrated with reference to the recording of sinusoidal signals. As an example we can look at the discrete time sampling error introduced in the measurement of a single cycle of a pure sine wave of frequency f, which is sampled at a rate of four times its frequency. When the sinusoid and the sampling clock are in phase, as shown in Fig. 3.51, a sample will fall on the peak value of both positive and negative half-cycles. The next closest samples will lie at π/2 radians from the peaks. As the phase of the clock is advanced relative to the input sinusoid the sample points which used to lie at the peak values will move to lower amplitude values giving an error (Δ) in the measurement of the amplitude (A) of

Figure 3.51. Sample points with sinusoid and sampling clock in phase. (Error in peak amplitude = 0)

Δ = A(1 − cos ϕ)

where ϕ is the phase shift in the sample points. This error will increase until ϕ – π/4 (Fig. 3.52). For ϕ > π/4 the point behind the peak value will now be closer to the peak and the error will decrease for a ϕ in the range of π/4 to π/2. The maximum per unit value of the discrete time sampling error is given by eqn 3.93,

Figure 3.51. Sample points with sinusoid and sampling clock in phase. (Error in peak amplitude = 0)

Δ = A(1 − cos ϕ)

where ϕ is the phase shift in the sample points. This error will increase until ϕ – π/4 (Fig. 3.52). For ϕ > π/4 the point behind the peak value will now be closer to the peak and the error will decrease for a ϕ in the range of π/4 to π/2. The maximum per unit value of the discrete time sampling error is given by eqn 3.93,

Figure 3.52. Sample points with sampling clock phase advanced to π/4 with respect to the sinusoid. Error in peak amplitude (Δ) is at a maximum

(3.93)Δmax=I−cos(πfts)

where ts is the recorder’s sampling interval and f the sinewave frequency.

The maximum errors obtained through quantization and sampling when recording a sinusoidal waveform are shown in Fig. 3.53. The plotted quantities were calculated for an 8-bit 200-MHz digitizer.

Figure 3.53. Sampling and quantization errors of an ideal recorder

In a real digital recorder, an additional two categories of errors are introduced. The first includes the instrument’s systematic errors. These are generally due to the digitizer’s analogue input circuitry, and are present to some degree in all recording instruments. They include such errors as gain drift, linearity errors, offset errors, etc. They can be compensated by regular calibration without any net loss in accuracy. The second category contains the digitizer’s dynamic errors. These become important when recording high-frequency or fast transient signals. The dynamic errors are often random in nature, and cannot be dealt with as simply as their systematic counterparts and are discussed below.

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Remaining useful life prediction

Yaguo Lei, in Intelligent Fault Diagnosis and Remaining Useful Life Prediction of Rotating Machinery, 2017

6.3.4.3 RUL Prediction

The constructed indicator WMQE is further input into the RUL prediction module. In this module, a PF-based prediction algorithm is utilized to predict RUL of the rotating machinery whose degradation processes are described using a variant of Paris–Erdogan model. The Paris–Erdogan model is formulated as

(6.96)dxdt=c(Δδ)γ, Δδ=mx

where x represents the semicrack length, t is the number of stress cycles (i.e., the fatigue life), c, γ, and m are material constants which are determined by tests, and Δδ is amplitude of stress intensity factor roughly proportional to the square root of x.

It is seen from Eq. (6.96) that there are several model parameters in the Paris–Erdogan model, that is, c, γ, and m, which are difficult to measure during the operation process of the rotating machinery. For convenient application, the Paris–Erdogan model is transformed into the following format with α=cmγ and β = γ/2.

(6.97)dxdt=αxβ.

Then, the above function is rewritten into the following state space model.

(6.98)xk=xk−1+αk−1xk−1βΔtkαk=αk−1zk=xk+νk,

where αk−1 is a random variable following a normal distribution of Nμα,σα2, β is a constant parameter, Δtk=tk−tk−1, zk is the measured WMQE value at tk and νk is the measurement noise following the normal distribution of N0,σν2. With the transformation of the Paris–Erdogan model, the model parameters are more convenient to estimate according to the measurements. In addition, the state space model inherits the superiority of the Paris–Erdogan model in describing the general degradation processes. Therefore, it is supposed to be a good model for a general degradation process.

After the transformation, the unknown model parameters are changed to be Θ=μα,σα2,β,σν2′, where (·)′ denotes the vector transposition. Then, the measured WMQE values constructed from vibration signals are input into the model, and the model parameters are initialized using MLE. It is assumed that there are a series of measurements z0:M=z0,…,zM′ at ordered times t0,…,tM. According to Eq. (6.98), zk is formulated as follows:

(6.99)zk=xk−1+αkxk−1βΔtk+νk.

The degradation state xk−1 has the following relationship with the measurement zk−1.

(6.100)xk−1=zk−1−νk−1.

The degradation state xk−1 is hard to be acquired in real applications. If the measurement noise νk−1 is small enough compared with the measurement itself, it is negligible and xk−1 is approximated by zk−1. Let T=z0βΔt1,…,zM−1βΔtM′. z1:M=z1,…,zM′ is multivariate normally distributed, which is denoted as follows:

(6.101)z1:M∼Nz0:M−1+μαT,σα2TT′+σν2IM,

where IM is an identity matrix of order M.

Let Δz1:M=z1−z0,…,zM−zM−1′, and the log-likelihood function of the unknown parameters based on the measurements is expressed as

(6.102)ℓΘ|z0:M=−M2ln2π−12lnσα2TT′+σν2IM     −12Δz1:M−μαT′σα2TT′+σν2IM−1Δz1:M−μαT     =−M2ln2π−M2lnσα2−12lnTT′+σ˜ν2IM     −12σα2Δz1:M−μαT′σα2TT′+σν2IM−1Δz1:M−μαT,

with σ~ν2=σν2/σα2. The first partial derivatives of ℓΘ|z0:M with respect to μα and σα2 are calculated and formulated with

(6.103)∂ℓΘ|z0:M∂μα=1σα2T′TT′+σ~ν2IM-1Δz1:M−μαT,

(6.104)∂ℓΘ|z0:M∂σα2=−M2σα2+12σα4Δz1:M−μαT′TT′+σ~ν2IM−1Δz1:M−μαT.

Let ∂ℓΘ|z0:M/∂μα=0 and ∂ℓΘ|z0:M/∂σα2=0. The MLE results of μα and σα2 are

(6.105)μα=T′TT′+σ~ν2IM−1Δz1:MT′TT′+σ~ν2IM−1T,

(6.106)σα2=Δz1:M−μαT′TT′+σ~ν2IM−1Δz1:M−μαTM.

With Eqs. (6.105) and (6.106) substituted into Eq. (6.102), the log-likelihood function is reduced into a two-variable function about β and σ~ν2, which is denoted by

(6.107)ℓΘ|z0:M=−M2ln2π−M2lnσα2−12lnTT′+σ~ν2IM−M2.

The MLE values of β and σ~ν2 are obtained by maximizing the log-likelihood function (6.107) through two-dimensional optimizing. Then the MLE values of β and σ~ν2 are substituted into Eqs. (6.105) and (6.106), and the MLE values of μα and σα2 are acquired. The value of σν2 is calculated with σ~ν2 multiplied by σα2. Finally, all of the unknown parameters Θ=μα,σα2,β,σν2′ are initialized.

After parameter initialization, the model parameters are further updated and the RUL is predicted using a PF-based prediction algorithm. Based on the initialized parameters, a series of initial particles y0nn=1:Ns are sampled from the initial PDF of the system state p(y0n|Θ0)∼N(y0,Q0) with

(6.108)y0=x0μα and Q0=000σα2.

Ns is the number of particles and the weight of each particle is set to be 1/Ns. Then new particles ykni=1:Ns are obtained following

(6.109)ykn=xknμαn=xk−1n+μαnxk−1nβΔtkμαn.

When the new measurement zk at tk is available, each particle weight is updated and normalized by

(6.110)wkn=wk−1npzk|ykn, w~kn=wkn∑n=1Nswkn,

where

(6.111)pzk|ykn=12πσνexp−12zk−xknσν2.

The particles are resampled according to the particle weights and their weights are reset to be 1/Ns. After that, the RUL is predicted based on the resampled particles. The RUL lk at tk is defined as

(6.112)lk=inflk:xlk+tk≥λ|x1:k,

where λ is a prespecified failure threshold. Each particle is transmitted following the transition function of Eq. (6.98) from current state until the state value exceeds the failure threshold, and the RUL lknn=1:Ns predicted using each particle is acquired. Then the PDF of the RUL is approximated by

(6.113)plk|z0:k=∑n=1Nsw~knδlk−lkn.

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Orbit and Attitude Sensors

Enrico Canuto, … Carlos Perez Montenegro, in Spacecraft Dynamics and Control, 2018

Exercise 1

Prove that the quantization error defined by Eq. (8.6) is bounded by |n˜y(i)|≤ρy/2 and under the random assumption has zero mean and variance equal to ρy2/12. □

A typical model of the random error in Eq. (8.4), which includes quantization errors, is the linear continuous-time stochastic state equation

(8.7)x˜˙(t)=A˜x˜+G˜w˜d˜(t)=C˜x˜+D˜w˜E{w˜(t)}=0,E{w˜(t)w˜T(t+τ)}=S˜w2δ(τ)E{x˜(0)}=x˜0,E{(x˜(0)−x˜0)(x˜(0)−x˜0)T}=P˜0≥0E{(x˜(0)−x˜0)w˜T(0)}=0,

which is similar to the DT Eq. (4.159) of Section 4.8.1. Eq. (8.7) being continuous-time, the eigenvalues of the state matrix are assumed to lie on the imaginary axis and when equal to zero may be multiple. The statistics in Eq. (8.7) assumes that is a zero-mean second-order stationary white noise with constant spectral density S˜w2, and impulsive covariance, where δ(τ) denotes a Dirac delta (see Sections 13.2.1 and 13.7.3Section 13.2.1Section 13.7.3). The initial state may be modeled as a random vector with mean value x˜0 and covariance matrix P˜0, but is uncorrelated from any simultaneous white noise as expressed by the last identity in Eq. (8.7). This uncorrelation has been already referred to as the causality constraint. In principle, Eq. (8.7) may be unobservable from the output and uncontrollable by the noise , because the output may include polynomial and trigonometric components (deterministic signals) just driven by the initial state x˜0. For instance, a trigonometric component tuned to the angular frequency ω˜ corresponds to a second-order subsystem with eigenvalues ±jω˜. A first-order polynomial corresponds to a second-order subsystem with a pair of zero eigenvalues and a single eigenvector. The mixed case of stochastic processes and deterministic signals can be simplified by assuming that trigonometric and polynomial components are the free response of the equations driven by , and that Eq. (8.7) is observable and controllable.

The simplest model of the class in Eq. (8.7), which is common to inertial sensors (accelerometers in Section 8.4 and gyroscopes in Section 8.5), is the scalar first-order random drift [32]:

(8.8)x˜˙(t)=w˜x,x˜(0)=x˜0d˜(t)=x˜+w˜dE{x˜(0)}=x˜0,var{x˜}=σ02,E{(x˜−x˜0)w˜T(0)}=0w˜=[w˜xw˜d],E{w˜(t)}=0,E{w˜(t)w˜T(t+τ)}=[S˜wx200S˜wd2]δ(τ),

where, if [unit] denotes the unit of measurement of , we find S˜wx2 in [(unit/s)2Hz−1] and S˜wd2 in [unit2Hz−1]. The initial state x˜0 accounts for a constant bias and is uncorrelated with any simultaneous noise; the scalar input noise w˜x and the output noise w˜d in Eq. (8.8) are uncorrelated with each other. The output process is nonstationary and the autocorrelation is given by

(8.9)R˜d(t,t+τ)=S˜wx2min(t,t+τ)+S˜wd2δ(τ).

Although is nonstationary, the spectral density S˜x2(f) can be defined through the AS equation x˜˙=−εx˜+w˜x, where ε > 0 must be sufficiently smaller than the cutoff frequency f0 to be defined in the next paragraph. We can write the following identities:

(8.10)S˜d2(f)=S˜x2(f)+S˜wd2=S˜wx2(2πf)2+S˜wd2,f>ε2π=fεσ˜x2=limf→∞∫0fS˜x2νdν<∞,

where if, for f < ε, S˜x2(f) is bounded, also the variance σ˜x2 is bounded. The Bode plot of S˜d(f) is approximately flat for f>f0=2π−1S˜wx/S˜wd>fε and has a −20 dB/decade slope for fε < f < f0. The first PSD in the first row of Eq. (8.10) is a first-order random walk, which is known, in the realm of inertial sensors, as the (long-term) bias instability of the sensor. The name is appropriate because it describes the long-term fluctuations—bounded because of ε—around the mean sensor bias x˜0. The square root of the second term S˜wd corresponds to the minimum-valued profile of the overall spectral density. Let us call it, as already anticipated, noise floor, although the name sometimes refers to the whole instrument noise (here referred to as the background noise). In the realm of inertial sensors, it is known as the velocity random walk in the case of linear accelerometers, the unit being [m/(s2Hz)], and the angular random walk (ARW) in the case of gyroscopes, the unit being[rad/(sHz)]. The name, which may cause same confusion, is justified by the fact that when either of the two measurements (linear acceleration and angular rate) is time integrated for generating either velocity or attitude measurements, the integrated noise floor becomes a random walk. By restricting to gyroscopes, the ARW unit [rad/(sHz)] is usually simplified to [rad/s], de facto to the non-SI unit [degree/hour]. Indeed, the simplified unit is at the same time the unit of S˜wd and the unit of the root mean square (RMS) σ¯w(t,Δt) of the random walk increment w¯d(t,Δt)=∫tt+Δtw˜d(τ)dτ, namely:

(8.11)σ¯w(t,Δt)=E{1Δt(∫tt+Δtw˜d(τ)dτ)2}=S˜wd.

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Sampling Theory

Luis Chaparro, in Signals and Systems Using MATLAB (Second Edition), 2015

8.3.2 Quantization and Coding

Amplitude discretization of the sampled signal xs(t) is accomplished by a quantizer consisting of a number of fixed amplitude levels against which the sample amplitudes {x(nTs)} are compared. The output of the quantizer is one of the fixed amplitude levels that best represents {x(nTs)} according to some approximation scheme. The quantizer is a non-linear system.

Independent of how many levels or, equivalently, of how many bits are allocated to represent each level of the quantizer, in general there is a possible error in the representation of each sample. This is called the quantization error. To illustrate this, consider the 2-bit or four-level quantizer shown in Figure 8.12. The input of the quantizer are the samples {x(nTs)}, which are compared with the values in the bins [-2Δ,-Δ],[-Δ,0],[0,Δ], and [Δ,2Δ]. Depending on which of these bins the sample falls in it is replaced by the corresponding levels -2Δ,-Δ,0, or Δ, respectively. The value of the quantization step Δ for the four-level quantizer is

Figure 3.53. Sampling and quantization errors of an ideal recorder

In a real digital recorder, an additional two categories of errors are introduced. The first includes the instrument’s systematic errors. These are generally due to the digitizer’s analogue input circuitry, and are present to some degree in all recording instruments. They include such errors as gain drift, linearity errors, offset errors, etc. They can be compensated by regular calibration without any net loss in accuracy. The second category contains the digitizer’s dynamic errors. These become important when recording high-frequency or fast transient signals. The dynamic errors are often random in nature, and cannot be dealt with as simply as their systematic counterparts and are discussed below.

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Remaining useful life prediction

Yaguo Lei, in Intelligent Fault Diagnosis and Remaining Useful Life Prediction of Rotating Machinery, 2017

6.3.4.3 RUL Prediction

The constructed indicator WMQE is further input into the RUL prediction module. In this module, a PF-based prediction algorithm is utilized to predict RUL of the rotating machinery whose degradation processes are described using a variant of Paris–Erdogan model. The Paris–Erdogan model is formulated as

(6.96)dxdt=c(Δδ)γ, Δδ=mx

where x represents the semicrack length, t is the number of stress cycles (i.e., the fatigue life), c, γ, and m are material constants which are determined by tests, and Δδ is amplitude of stress intensity factor roughly proportional to the square root of x.

It is seen from Eq. (6.96) that there are several model parameters in the Paris–Erdogan model, that is, c, γ, and m, which are difficult to measure during the operation process of the rotating machinery. For convenient application, the Paris–Erdogan model is transformed into the following format with α=cmγ and β = γ/2.

(6.97)dxdt=αxβ.

Then, the above function is rewritten into the following state space model.

(6.98)xk=xk−1+αk−1xk−1βΔtkαk=αk−1zk=xk+νk,

where αk−1 is a random variable following a normal distribution of Nμα,σα2, β is a constant parameter, Δtk=tk−tk−1, zk is the measured WMQE value at tk and νk is the measurement noise following the normal distribution of N0,σν2. With the transformation of the Paris–Erdogan model, the model parameters are more convenient to estimate according to the measurements. In addition, the state space model inherits the superiority of the Paris–Erdogan model in describing the general degradation processes. Therefore, it is supposed to be a good model for a general degradation process.

After the transformation, the unknown model parameters are changed to be Θ=μα,σα2,β,σν2′, where (·)′ denotes the vector transposition. Then, the measured WMQE values constructed from vibration signals are input into the model, and the model parameters are initialized using MLE. It is assumed that there are a series of measurements z0:M=z0,…,zM′ at ordered times t0,…,tM. According to Eq. (6.98), zk is formulated as follows:

(6.99)zk=xk−1+αkxk−1βΔtk+νk.

The degradation state xk−1 has the following relationship with the measurement zk−1.

(6.100)xk−1=zk−1−νk−1.

The degradation state xk−1 is hard to be acquired in real applications. If the measurement noise νk−1 is small enough compared with the measurement itself, it is negligible and xk−1 is approximated by zk−1. Let T=z0βΔt1,…,zM−1βΔtM′. z1:M=z1,…,zM′ is multivariate normally distributed, which is denoted as follows:

(6.101)z1:M∼Nz0:M−1+μαT,σα2TT′+σν2IM,

where IM is an identity matrix of order M.

Let Δz1:M=z1−z0,…,zM−zM−1′, and the log-likelihood function of the unknown parameters based on the measurements is expressed as

(6.102)ℓΘ|z0:M=−M2ln2π−12lnσα2TT′+σν2IM     −12Δz1:M−μαT′σα2TT′+σν2IM−1Δz1:M−μαT     =−M2ln2π−M2lnσα2−12lnTT′+σ˜ν2IM     −12σα2Δz1:M−μαT′σα2TT′+σν2IM−1Δz1:M−μαT,

with σ~ν2=σν2/σα2. The first partial derivatives of ℓΘ|z0:M with respect to μα and σα2 are calculated and formulated with

(6.103)∂ℓΘ|z0:M∂μα=1σα2T′TT′+σ~ν2IM-1Δz1:M−μαT,

(6.104)∂ℓΘ|z0:M∂σα2=−M2σα2+12σα4Δz1:M−μαT′TT′+σ~ν2IM−1Δz1:M−μαT.

Let ∂ℓΘ|z0:M/∂μα=0 and ∂ℓΘ|z0:M/∂σα2=0. The MLE results of μα and σα2 are

(6.105)μα=T′TT′+σ~ν2IM−1Δz1:MT′TT′+σ~ν2IM−1T,

(6.106)σα2=Δz1:M−μαT′TT′+σ~ν2IM−1Δz1:M−μαTM.

With Eqs. (6.105) and (6.106) substituted into Eq. (6.102), the log-likelihood function is reduced into a two-variable function about β and σ~ν2, which is denoted by

(6.107)ℓΘ|z0:M=−M2ln2π−M2lnσα2−12lnTT′+σ~ν2IM−M2.

The MLE values of β and σ~ν2 are obtained by maximizing the log-likelihood function (6.107) through two-dimensional optimizing. Then the MLE values of β and σ~ν2 are substituted into Eqs. (6.105) and (6.106), and the MLE values of μα and σα2 are acquired. The value of σν2 is calculated with σ~ν2 multiplied by σα2. Finally, all of the unknown parameters Θ=μα,σα2,β,σν2′ are initialized.

After parameter initialization, the model parameters are further updated and the RUL is predicted using a PF-based prediction algorithm. Based on the initialized parameters, a series of initial particles y0nn=1:Ns are sampled from the initial PDF of the system state p(y0n|Θ0)∼N(y0,Q0) with

(6.108)y0=x0μα and Q0=000σα2.

Ns is the number of particles and the weight of each particle is set to be 1/Ns. Then new particles ykni=1:Ns are obtained following

(6.109)ykn=xknμαn=xk−1n+μαnxk−1nβΔtkμαn.

When the new measurement zk at tk is available, each particle weight is updated and normalized by

(6.110)wkn=wk−1npzk|ykn, w~kn=wkn∑n=1Nswkn,

where

(6.111)pzk|ykn=12πσνexp−12zk−xknσν2.

The particles are resampled according to the particle weights and their weights are reset to be 1/Ns. After that, the RUL is predicted based on the resampled particles. The RUL lk at tk is defined as

(6.112)lk=inflk:xlk+tk≥λ|x1:k,

where λ is a prespecified failure threshold. Each particle is transmitted following the transition function of Eq. (6.98) from current state until the state value exceeds the failure threshold, and the RUL lknn=1:Ns predicted using each particle is acquired. Then the PDF of the RUL is approximated by

(6.113)plk|z0:k=∑n=1Nsw~knδlk−lkn.

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Orbit and Attitude Sensors

Enrico Canuto, … Carlos Perez Montenegro, in Spacecraft Dynamics and Control, 2018

Exercise 1

Prove that the quantization error defined by Eq. (8.6) is bounded by |n˜y(i)|≤ρy/2 and under the random assumption has zero mean and variance equal to ρy2/12. □

A typical model of the random error in Eq. (8.4), which includes quantization errors, is the linear continuous-time stochastic state equation

(8.7)x˜˙(t)=A˜x˜+G˜w˜d˜(t)=C˜x˜+D˜w˜E{w˜(t)}=0,E{w˜(t)w˜T(t+τ)}=S˜w2δ(τ)E{x˜(0)}=x˜0,E{(x˜(0)−x˜0)(x˜(0)−x˜0)T}=P˜0≥0E{(x˜(0)−x˜0)w˜T(0)}=0,

which is similar to the DT Eq. (4.159) of Section 4.8.1. Eq. (8.7) being continuous-time, the eigenvalues of the state matrix are assumed to lie on the imaginary axis and when equal to zero may be multiple. The statistics in Eq. (8.7) assumes that is a zero-mean second-order stationary white noise with constant spectral density S˜w2, and impulsive covariance, where δ(τ) denotes a Dirac delta (see Sections 13.2.1 and 13.7.3Section 13.2.1Section 13.7.3). The initial state may be modeled as a random vector with mean value x˜0 and covariance matrix P˜0, but is uncorrelated from any simultaneous white noise as expressed by the last identity in Eq. (8.7). This uncorrelation has been already referred to as the causality constraint. In principle, Eq. (8.7) may be unobservable from the output and uncontrollable by the noise , because the output may include polynomial and trigonometric components (deterministic signals) just driven by the initial state x˜0. For instance, a trigonometric component tuned to the angular frequency ω˜ corresponds to a second-order subsystem with eigenvalues ±jω˜. A first-order polynomial corresponds to a second-order subsystem with a pair of zero eigenvalues and a single eigenvector. The mixed case of stochastic processes and deterministic signals can be simplified by assuming that trigonometric and polynomial components are the free response of the equations driven by , and that Eq. (8.7) is observable and controllable.

The simplest model of the class in Eq. (8.7), which is common to inertial sensors (accelerometers in Section 8.4 and gyroscopes in Section 8.5), is the scalar first-order random drift [32]:

(8.8)x˜˙(t)=w˜x,x˜(0)=x˜0d˜(t)=x˜+w˜dE{x˜(0)}=x˜0,var{x˜}=σ02,E{(x˜−x˜0)w˜T(0)}=0w˜=[w˜xw˜d],E{w˜(t)}=0,E{w˜(t)w˜T(t+τ)}=[S˜wx200S˜wd2]δ(τ),

where, if [unit] denotes the unit of measurement of , we find S˜wx2 in [(unit/s)2Hz−1] and S˜wd2 in [unit2Hz−1]. The initial state x˜0 accounts for a constant bias and is uncorrelated with any simultaneous noise; the scalar input noise w˜x and the output noise w˜d in Eq. (8.8) are uncorrelated with each other. The output process is nonstationary and the autocorrelation is given by

(8.9)R˜d(t,t+τ)=S˜wx2min(t,t+τ)+S˜wd2δ(τ).

Although is nonstationary, the spectral density S˜x2(f) can be defined through the AS equation x˜˙=−εx˜+w˜x, where ε > 0 must be sufficiently smaller than the cutoff frequency f0 to be defined in the next paragraph. We can write the following identities:

(8.10)S˜d2(f)=S˜x2(f)+S˜wd2=S˜wx2(2πf)2+S˜wd2,f>ε2π=fεσ˜x2=limf→∞∫0fS˜x2νdν<∞,

where if, for f < ε, S˜x2(f) is bounded, also the variance σ˜x2 is bounded. The Bode plot of S˜d(f) is approximately flat for f>f0=2π−1S˜wx/S˜wd>fε and has a −20 dB/decade slope for fε < f < f0. The first PSD in the first row of Eq. (8.10) is a first-order random walk, which is known, in the realm of inertial sensors, as the (long-term) bias instability of the sensor. The name is appropriate because it describes the long-term fluctuations—bounded because of ε—around the mean sensor bias x˜0. The square root of the second term S˜wd corresponds to the minimum-valued profile of the overall spectral density. Let us call it, as already anticipated, noise floor, although the name sometimes refers to the whole instrument noise (here referred to as the background noise). In the realm of inertial sensors, it is known as the velocity random walk in the case of linear accelerometers, the unit being [m/(s2Hz)], and the angular random walk (ARW) in the case of gyroscopes, the unit being[rad/(sHz)]. The name, which may cause same confusion, is justified by the fact that when either of the two measurements (linear acceleration and angular rate) is time integrated for generating either velocity or attitude measurements, the integrated noise floor becomes a random walk. By restricting to gyroscopes, the ARW unit [rad/(sHz)] is usually simplified to [rad/s], de facto to the non-SI unit [degree/hour]. Indeed, the simplified unit is at the same time the unit of S˜wd and the unit of the root mean square (RMS) σ¯w(t,Δt) of the random walk increment w¯d(t,Δt)=∫tt+Δtw˜d(τ)dτ, namely:

(8.11)σ¯w(t,Δt)=E{1Δt(∫tt+Δtw˜d(τ)dτ)2}=S˜wd.

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Sampling Theory

Luis Chaparro, in Signals and Systems Using MATLAB (Second Edition), 2015

8.3.2 Quantization and Coding

Amplitude discretization of the sampled signal xs(t) is accomplished by a quantizer consisting of a number of fixed amplitude levels against which the sample amplitudes {x(nTs)} are compared. The output of the quantizer is one of the fixed amplitude levels that best represents {x(nTs)} according to some approximation scheme. The quantizer is a non-linear system.

Independent of how many levels or, equivalently, of how many bits are allocated to represent each level of the quantizer, in general there is a possible error in the representation of each sample. This is called the quantization error. To illustrate this, consider the 2-bit or four-level quantizer shown in Figure 8.12. The input of the quantizer are the samples {x(nTs)}, which are compared with the values in the bins [-2Δ,-Δ],[-Δ,0],[0,Δ], and [Δ,2Δ]. Depending on which of these bins the sample falls in it is replaced by the corresponding levels -2Δ,-Δ,0, or Δ, respectively. The value of the quantization step Δ for the four-level quantizer is

Figure 8.12. Four-level quantizer and coder.

(8.23)Δ=dynamic range of signal2b=2max|x(t)|22

where b = 2 is number of bits of the code assigned to each level. The bits assigned to each of the levels uniquely represents the different levels [-2Δ,-Δ,0,Δ]. As to how to approximate the given sample to one of these levels, it can be done by rounding or by truncating. The quantizer shown in Figure 8.12 approximates by truncation, i.e., if the sample kΔ≤x(nTs)<(k+1)Δ, for k = −2, −1,0,1, then it is approximated by the level .

To see how quantization and coding are done, and how to obtain the quantization error, let the sampled signal be

x(nTs)=x(t)|t=nTS

The given four-level quantizer is such that if the sample x(nTs) is such that

(8.24)kΔ≤x(nTs)<(k+1)Δ⇒xˆ(nTs)=kΔk=-2,-1,0,1

The sampled signal x(nTs) is the input of the quantizer and the quantized signal xˆ(nTs) is its output. So that whenever

-2Δ≤x(nTs)<-Δ⇒xˆ(nTs)=-2Δ-Δ≤x(nTs)<0⇒xˆ(nTs)=-Δ0≤x(nTs)<Δ⇒xˆ(nTs)=0Δ≤x(nTs)<2Δ⇒xˆ(nTs)=Δ

To transform the quantized values into unique binary 2-bit values, one could use a code such as

xˆ(nTs)⇒binary code-2Δ10-Δ110Δ00Δ01

which assigns a unique 2 bit binary number to each of the 4 quantization levels. Notice that the first bit of this code can be considered a sign bit, “1” for negative levels and “0” for positive levels.

If we define the quantization error as

ε(nTs)=x(nTs)-xˆ(nTs)

and use the characterization of the quantizer given by Equation (8.24) as

xˆ(nTs)≤x(nTs)≤xˆ(nTs)+Δ

by subtracting xˆ(nTs) from each of the terms gives that the quantization error is bounded as follows

(8.25)0≤ε(nTs)≤Δ

i.e., the quantization error for the four-level quantizer being considered is between 0 and Δ. This expression for the quantization error indicates that one way to decrease the quantization error is to make the quantization step Δsmaller. Increasing the number of bits of the A/D converter makes Δ smaller (see Equation (8.23) where the denominator is 2 raised to the number of bits) which in turn makes smaller the quantization error, and improves the quality of the A/D converter.

In practice, the quantization error is random and so it needs to be characterized probabilistically. This characterization becomes meaningful when the number of bits is large, and when the input signal is not a deterministic signal. Otherwise, the error is predictable and thus not random. Comparing the energy of the input signal to the energy of the error, by means of the so-called signal to noise ratio (SNR), it is possible to determine the number of bits that are needed in a quantizer to get a reasonable quantization error.

Example 8.5

Suppose we are trying to decide between an 8 and a 9 bit A/D converter for a certain application where the signals in this application are known to have frequencies that do not exceed 5 kHz. The dynamic range of the signals is 10 volts, so that the signal is bounded as −5 ≤ x(t) ≤ 5. Determine an appropriate sampling period and compare the percentage of error for the two A/Ds of interest.

Solution

The first consideration in choosing the A/D converter is the sampling period, so we need to get an A/D converter capable of sampling at fs = 1/Ts > 2 fmax samples/second. Choosing fs = 4 fmax = 20 k samples/second then Ts = 1/20 msec/sample or 50 microseconds/sample. Suppose then we look at the 8-bit A/D converter, the quantizer has 28 = 256 levels so that the quantization step is Δ=10/256 volts and if we use a truncation quantizer the quantization error would be

0≤ε(nTs)≤10/256

If we find that objectionable we can then consider the 9-bit A/D converter, with a quantizer of 29 = 512 levels and the quantization step Δ=10/512 or half that of the 8-bit A/D converter, and

0≤ε(nTs)≤10/512

So that by increasing one bit we cut the quantization error in half from the previous quantizer. Inputting a signal of constant amplitude 5 into the 9-bit A/D gives a quantization error of [(10/512)/5] × 100% = (100/256)% ≈ 0.4% in representing the input signal. For the 8-bit A/D it would correspond to 0.8% error. ▪

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Compression

StéphaneMallat , in A Wavelet Tour of Signal Processing (Third Edition), 2009

Weighted Quantization and Regions of Interest

Visual distortions introduced by quantization errors of wavelet coefficients depend on the scale 2j. Errors at large scales are more visible than at fine scales [481]. This can be taken into account by quantizing the wavelet coefficients with intervals Δj=Δwj that depend on the scale 2j. For R¯≤1 bit/pixel, wj = 2j is appropriate for the three finest scales. The distortion in (10.34) shows that choosing such weights is equivalent to minimizing a weighted mean-square error.

Such a weighted quantization is implemented like in (10.35) by quantizing weighted wavelet coefficients fB[m]/wj with a uniform quantizer. The weights are inverted during the decoding process. JPEG-2000 supports a general weighting scheme that codes weighted coefficients w[m]fB[m] where w[m] can be designed to emphasize some region of interest Ω ⊂ [0, 1]2 in the image. The weights are set to w[m] = w > 1 for the wavelet coefficients fB[m]=〈f,ψj,p,q1〉 where the support of ψj,p,q1 intersects Ω. As a result, the wavelet coefficients inside Ω are given a higher priority during the coding stage, and the region Ω is coded first within the compressed stream. This provides a mechanism to more precisely code regions of interest in images—for example, a face in a crowd.

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Signal and Image Representation in Combined Spaces

Zoran. Cvetković, Martin. Vetterli, in Wavelet Analysis and Its Applications, 1998

6.1 Two lemmas on frames of complex exponentials

Estimates of bounds on the quantization error in Subsection 4.3 are derived from the next two lemmas [5].

Lemma 1

Letejλnωbe a frame in L2[− σ, σ]. If M is any constant and {μn} is a sequence satisfying |μn − λn| ≤ M, for all n, then there is a number C = C(M, σ, {λn}) such that

(6.1.1)∑n|fμn|2∑n|fλn|2≤C

for every cr-bandlimited signal f(x).

Lemma 2

Letejλnωbe a frame in L2[− σ, σ], with bounds 0 < A ≤ B < ∞ and δ a given positive number. If a sequence { μn } satisfies | λn − μn  < δ for all n, then for every σ-bandlimited signal f(x)

(6.1.2)A1−C2||f||2≤∑n|fμn|2≤B(1+C)2||f||2,

where

(6.1.3)C=BAeγδ−12

Remark 1

If δ in the statement of Lemma 2 is chosen small enough, so that C is less then 1, then ejμnω is also a frame in L2[− σ, σ]. Moreover, there exists some δ 1/4 ({λn},σ), such that whenever δ < δ 1/4 ({λn }, σ), ejμnω is a frame with frame bounds A/A and 9B/4.

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Live HDR Video Broadcast Production

I.G. Olaizola, … J. Gorostegui, in High Dynamic Range Video, 2017

1.4.1 Banding

Banding effects are directly related to quantization errors. As the dynamic range increases, the quantization effects become more noticeable and banding artifacts arise. To avoid this, smaller quantization steps must be taken, but this requires a higher bitdepth (from 8 to 10 or 12 in order to have backwards compatibility with existing infrastructures, and ideally 14 or 16 bits) and nonlinear transform functions based on the HVS that minimize the observable banding effects. Nowadays, this is partially solved by the electro-optical transfer function (EOTF) and opto-electronic transfer function (OETF) mapping curves that will be introduced in a further section.

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