Information about Zero Order Hold
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval.
A zero-order hold reconstructs the following continuous-time waveform from a sample sequence x[n], assuming one sample per time interval T:
The function
is depicted in Figure 1, and 
is the piecewise-constant signal depicted in Figure 2.
In this method, a sequence of dirac impulses, xs(t), representing the discrete samples, x[n], is low-pass filtered to recover a continuous-time signal, x(t).
Even though this is not what a DAC does in reality, the DAC output can be modeled by applying the hypothetical sequence of dirac impulses, xs(t), to a linear, time-invariant filter with such characteristics (which, for an LTI system, are fully described by the impulse response) so that each input impulse results in the correct constant pulse in the output.
Begin by defining a continuous-time signal from the sample values, as above but using delta functions instead of rect functions:
The scaling by T, which arises naturally by time-scaling the delta function, has the result that the mean value of xs(t) is equal to the mean value of the samples, so that the lowpass filter needed will have a DC gain of 1. Some authors use this scaling[1], while many others omit the time-scaling and the T, resulting in a low-pass filter model with a DC gain of T, and hence dependent on the units of measurement of time.
The zero-order hold is the hypothetical filter or LTI system that converts the sequence of modulated Dirac impulses xs(t)to the piecewise-constant signal (shown in Figure 2):
resulting in an effective impulse response (shown in Figure 4) of:
The effective frequency response is the continuous Fourier transform of the impulse response.
The Laplace transform transfer function of the ZOH is found by substituting s = i 2 π f:
The fact that practical digital-to-analog converters (DAC) do not output a sequence of dirac impulses, xs(t) (that, if ideally low-pass filtered, would result in the unique underlying bandlimited signal before sampling), but instead output a sequence of rectangular pulses, xZOH(t) (a piecewise constant function), means that there is an inherent effect of the ZOH on the effective frequency response of the DAC, resulting in a mild roll-off of gain at the higher frequencies (a 3.9224 dB loss at the Nyquist frequency, corresponding to a gain of sinc(1/2) = 2/π). This droop is a consequence of the hold property of a conventional DAC, and is not due to the sample and hold that might precede a conventional analog-to-digital converter (ADC).
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Time-domain model
)
Figure 1. The time-shifted and time-scaled rect function used in the time-domain analysis of the ZOH.
)
Figure 2. Piecewise-constant signal xZOH(t).
)
Figure 3. A modulated Dirac comb xs(t).
- where
is the rectangular function.
The function
is depicted in Figure 1, and is the piecewise-constant signal depicted in Figure 2.
Frequency-domain model
The equation above for the output of the ZOH can also be modeled as the output of a linear time-invariant filter with impulse response equal to a rect function, and with input being a sequence of dirac impulses scaled to the sample values. The filter can then be analyzed in the frequency domain, for comparison with other reconstruction methods such as the Whittaker–Shannon interpolation formula suggested by the Nyquist–Shannon sampling theorem, or such as the first-order hold or linear interpolation between sample values.In this method, a sequence of dirac impulses, xs(t), representing the discrete samples, x[n], is low-pass filtered to recover a continuous-time signal, x(t).
Even though this is not what a DAC does in reality, the DAC output can be modeled by applying the hypothetical sequence of dirac impulses, xs(t), to a linear, time-invariant filter with such characteristics (which, for an LTI system, are fully described by the impulse response) so that each input impulse results in the correct constant pulse in the output.
Begin by defining a continuous-time signal from the sample values, as above but using delta functions instead of rect functions:
The scaling by T, which arises naturally by time-scaling the delta function, has the result that the mean value of xs(t) is equal to the mean value of the samples, so that the lowpass filter needed will have a DC gain of 1. Some authors use this scaling[1], while many others omit the time-scaling and the T, resulting in a low-pass filter model with a DC gain of T, and hence dependent on the units of measurement of time.
Figure 4. Impulse response of zero-order hold hZOH(t). It is identical to the rect function of Figure 1, except now scaled to have an area of 1 so the filter will have a DC gain of 1.
The zero-order hold is the hypothetical filter or LTI system that converts the sequence of modulated Dirac impulses xs(t)to the piecewise-constant signal (shown in Figure 2):
resulting in an effective impulse response (shown in Figure 4) of:
The effective frequency response is the continuous Fourier transform of the impulse response.
- where
is the (normalized) sinc function used in digital signal processing.
The Laplace transform transfer function of the ZOH is found by substituting s = i 2 π f:
The fact that practical digital-to-analog converters (DAC) do not output a sequence of dirac impulses, xs(t) (that, if ideally low-pass filtered, would result in the unique underlying bandlimited signal before sampling), but instead output a sequence of rectangular pulses, xZOH(t) (a piecewise constant function), means that there is an inherent effect of the ZOH on the effective frequency response of the DAC, resulting in a mild roll-off of gain at the higher frequencies (a 3.9224 dB loss at the Nyquist frequency, corresponding to a gain of sinc(1/2) = 2/π). This droop is a consequence of the hold property of a conventional DAC, and is not due to the sample and hold that might precede a conventional analog-to-digital converter (ADC).
References
See also
- Nyquist–Shannon sampling theorem
- First-order hold
In signal processing, reconstruction usually means the determination of an original continuous signal from a sequence of equally spaced samples.
This article takes a generalized abstract mathematical approach to signal sampling and reconstruction.
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This article takes a generalized abstract mathematical approach to signal sampling and reconstruction.
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digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analog signal (current, voltage or electric charge).
An analog-to-digital converter (ADC) performs the reverse operation.
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An analog-to-digital converter (ADC) performs the reverse operation.
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discrete signal or discrete-time signal is a time series, perhaps a signal that has been sampled from a continuous-time signal. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous-time argument, but is a sequence of quantities; that is,
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A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain, usually time. The function of time need not be continuous.
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The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized boxcar function) is defined as:
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step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
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Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x
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The first-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator.
..... Click the link for more information.
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Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x
..... Click the link for more information.
..... Click the link for more information.
A low-pass filter is a filter that passes low frequency signals but attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter.
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A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain, usually time. The function of time need not be continuous.
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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
..... Click the link for more information.
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impulse response of a system is its output when presented with a very brief signal, an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely short in time while
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Electronic filters are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signal components and/or enhance wanted ones.
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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
..... Click the link for more information.
..... Click the link for more information.
impulse response of a system is its output when presented with a very brief signal, an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely short in time while
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Fourier transform, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components
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In mathematics, the sinc function, denoted by , has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function:
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- In digital signal processing and information theory, the normalized sinc function
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In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (
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transfer function is a mathematical representation of the relation between the input and output of a (linear time-invariant) system.
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Explanation
The transfer function is commonly used in the analysis of single-input single-output analog electronic circuits, for instance...... Click the link for more information.
digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analog signal (current, voltage or electric charge).
An analog-to-digital converter (ADC) performs the reverse operation.
..... Click the link for more information.
An analog-to-digital converter (ADC) performs the reverse operation.
..... Click the link for more information.
Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x
..... Click the link for more information.
..... Click the link for more information.
step function if it can be written as a finite linear combination of indicator functions of half-open intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
..... Click the link for more information.
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Not to be confused with Nyquist rate.
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...... Click the link for more information.
In electronics, a sample and hold circuit is used to interface real-world, changing analogue signals to a subsequent system such as an analog-to-digital converter. The purpose of this circuit is to hold the analogue value steady for a short time while the converter or other
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analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit, which converts continuous signals to discrete digital numbers. The reverse operation is performed by a digital-to-analog converter (DAC).
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The first-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator.
..... Click the link for more information.
..... Click the link for more information.
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