Digital Sampling Error

Digital sampling uncertainty estimation involves defining a representative "signal" to be sampled and specifying a sampling rate, a sampling aperture time, a quantization precision (sampling full scale and significant bits), an impulse response and hysteresis, a sampling noise level, and a sensor bias uncertainty. It also involves deciding on a model or methodology for eventual conversion of digitized data back to analog form. This multi-faceted analysis process is difficult without a structured template approach.

UncertaintyAnalyzer contains specially designed screens to evaluate digital sampling uncer-tainty. Both the analog-to-digital and digital-to-analog conversion processes are handled.

Sampling full scale, significant bits, and aperture time are entered in the Digital Sampling Uncertainty Worksheet. The following drill-down analysis screens and worksheets can be accessed from the Digital Sampling Worksheet:

• Input Signal Characteristics Worksheet

• Sampling Impulse Response Worksheet

• Sampling Noise Worksheet

• Sampling Sensor Worksheet

3.5.1 Input Signal Characteristics Worksheet. The worksheet is used to estimate uncertainty due to sampling rate error. Because signal activity occurs between samples, the rate at which

samples are collected can introduce error into the sampling process. The magnitude of this error depends on the signal being sampled, the conversion of the continuous analog signal to discrete digitized values (A-D), and on the conversion from digital to analog (D-A) following digital signal processing.

The input signal amplitude, amplitude units (e.g. V), and signal frequency units (e.g., Hz) are entered in this worksheet. The sampling rate or sampling frequency is also entered, along with the sampling units (e.g. kHz).

The input is defined from up to 12 components of the representative signal. Each component is expressed in terms of a signal amplitude, frequency, and relative phase that is expressed in degrees or radians. The signal components can be harmonics of a base frequency. A linear or quadratic spline model can be selected to represent the level of digital to analog conversion. A

plot of the signal and the digital sampling points can also be displayed.

3.5.2 Sampling Impulse Response Worksheet. This worksheet is used to estimate the

uncertainty due to impulse response error. Impulse response error is the discrepancy between the instantaneous signal value and the sampled value resulting from the finite time required for the sampling sensor to respond to input stimuli.

The impulse response a(t) is modeled using an exponential response function

a(t)=A₀+(A−A₀)(1−e^−t/tc) (3-1)

where

A0 = impulse response at time t = 0

A = instantaneous signal value tc = response time constant

The response time constant is the time required for the sensor to achieve the value

(3-2)

At any given time t, within the aperture time, the error in the sensed value is A

t

a( )− . (3-3)

The shorter the sensor response time constant, the closer the value of a(t) gets to the instanta-neous signal value A. Consequently, the faster the sensor responds to signal changes, the smaller the impulse response error will be. Conversely, as the sensor response time constant approaches the aperture time, the impulse response error increases significantly.

3.5.2.1 Sampling Impulse Response Error. Assume that the response of a system to a sensed value V is exponential:

) (3-4) with error

(3-5) Ve t

t ^λ

ε( )= ⁻

The unknowns here are the exact time at which the response function is applied, the response parameter λ, and the amplitude of the value V. Since we're focusing on hysteresis uncertainty, we'll consider only the uncertainty due to the sampled time. We'll assume that V can be

represented by the average signal value and that the uncertainty in λ is negligible. The impulse response uncertainty is, accordingly, given by

t

. (3-7)

t t

r t Vλσ e ^λ

σ ( )= ⁻

The uncertainty σt is the uncertainty in the location of the sampled point. If this point can be established with confidence limits, then the standard deviation of σt can be determined as usual.

If the location of the sampled point is unknown, the sampling error is taken as an average over the aperture τ according to

∫

Determining σr in these cases involves the usual definition of the variance

[ ]

The parameter λ is obtained from the time required for the response to reach 1/e the input value:

τe

λ= ¹ (3-10)

3.5.2.2 Hysteresis Error. Hysteresis error arises from residual values left over from previous samples. Represent this quantity by the variable Vh. If the sampling rate is νs, then

The hysteresis error is the sampled value of this residual amount. Thus the hysteresis error is given by

( )

(

)

h t V e ^λ

ε = 1− ⁻ (3-13)

If the sample point t is unknown, we instead use the average error

( )

The hysteresis uncertainty can be obtained from the hysteresis error in the same way as the impulse response uncertainty was obtained. If the sample point is known, then

(3-15)

whereas, if the sample point is unknown,

( ) ( )

3.5.3 Sampling Noise Worksheet. This worksheet is used to estimate the uncertainty in the sensed value of a sampled point due to noise. In electronic devices, noise may include thermal noise, shot noise, stray emf signals and so on. In mechanical and dimensional devices, noise may include vibration, temperature fluctuations, etc.

The Sampling Noise Worksheet allows the user to input information about electronic noise and other noise sources. The electronic noise portion of the worksheet provides a means for computing Thermal Noise and Shot Noise uncertainty estimates. Thermal noise is the noise due to random motion of current carriers (e.g., electrons) in a sampling sensor. Shot noise is the noise due to random fluctuations in the number of carriers in a semiconductor device.

The uncertainty in the sampled value due to thermal noise is estimated from user input values for the Bandwidth, Operating Temperature, and Output Resistance. The uncertainty in the sampled value due to shot noise is estimated from the RMS current and the number of sensor p-n junctions, along with the bandwidth and output resistance. The electronic signal to noise ratio is also computed as supplemental information. It can also be input by the user, if desired.

This other noise portion of the Sampling Noise Worksheet allows the user to input error limits and associated confidence levels for other sources of signal noise. The error limits for a given noise source should be input in amplitude units associated with the sample signal that is defined in the Input Signal Characteristics Worksheet. If the error limits are input in tolerance units or units from a different measurement area, then the appropriate conversion factor must be entered into the Coefficient data field. The coefficient data field can also be used to input a multiplying factor if the noise source undergoes attenuation or amplification by some external factor.

3.5.4 Sampling Sensor Worksheet. This worksheet is used to estimate the uncertainty in the sampled value due to the error in the sampling sensor. This worksheet allows the user to input sensor tolerance limits and associated confidence level. If a confidence level of 100% is entered, then the sensor error is assumed to be uniformly distributed. Otherwise, a normal distribution is assumed for the sensor error.