First-order hold

Ideally sampled signal x_s(t).

First-order hold is the hypothetical filter or LTI system that converts the ideally sampled signal

${\displaystyle x_{s}(t)\,}$	${\displaystyle =x(t)\ T\sum _{n=-\infty }^{\infty }\delta (t-nT)\ }$
	${\displaystyle =T\sum _{n=-\infty }^{\infty }x(nT)\delta (t-nT)\ }$

Piecewise linear signal x_FOH(t).

to the piecewise linear signal

${\displaystyle x_{\mathrm {FOH} }(t)\,=\sum _{n=-\infty }^{\infty }x(nT)\mathrm {tri} \left({\frac {t-nT}{T}}\right)\ }$

Impulse response (non-causal) of first-order hold h_FOH(t).

resulting in an effective impulse response of

${\displaystyle h_{\mathrm {FOH} }(t)\,={\frac {1}{T}}\mathrm {tri} \left({\frac {t}{T}}\right)={\begin{cases}{\frac {1}{T}}\left(1-{\frac {|t|}{T}}\right)&{\mbox{if }}|t|<T\\0&{\mbox{otherwise}}\end{cases}}\ }$

where ${\displaystyle \mathrm {tri} (x)\ }$ is the triangular function.

The effective frequency response is the continuous Fourier transform of the impulse response.

${\displaystyle H_{\mathrm {FOH} }(f)\,}$	${\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =\left({\frac {e^{i\pi fT}-e^{-i\pi fT}}{i2\pi fT}}\right)^{2}\ }$
	${\displaystyle =\mathrm {sinc} ^{2}(fT)\ }$

where ${\displaystyle \mathrm {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}\ }$ is the normalized sinc function.

The Laplace transform transfer function of FOH is found by substituting s = i 2 π f:

${\displaystyle H_{\mathrm {FOH} }(s)\,}$	${\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =\left({\frac {e^{sT/2}-e^{-sT/2}}{sT}}\right)^{2}\ }$

This is an acausal system in that the linear interpolation function moves toward the value of the next sample before such sample is applied to the hypothetical FOH filter.

Delayed piecewise linear signal x_FOH(t).

Delayed first-order hold, sometimes called causal first-order hold, is identical to FOH above except that its output is delayed by one sample period resulting in a delayed piecewise linear output signal

${\displaystyle x_{\mathrm {FOH} }(t)\,=\sum _{n=-\infty }^{\infty }x(nT)\mathrm {tri} \left({\frac {t-T-nT}{T}}\right)\ }$

Impulse response of causal first-order hold h_FOH(t).

resulting in an effective impulse response of

${\displaystyle h_{\mathrm {FOH} }(t)\,={\frac {1}{T}}\mathrm {tri} \left({\frac {t-T}{T}}\right)={\begin{cases}{\frac {1}{T}}\left(1-{\frac {|t-T|}{T}}\right)&{\mbox{if }}|t-T|<T\\0&{\mbox{otherwise}}\end{cases}}\ }$

where ${\displaystyle \mathrm {tri} (x)\ }$ is the triangular function.

The effective frequency response is the continuous Fourier transform of the impulse response.

${\displaystyle H_{\mathrm {FOH} }(f)\,}$	${\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =\left({\frac {1-e^{-i2\pi fT}}{i2\pi fT}}\right)^{2}\ }$
	${\displaystyle =e^{-i2\pi fT}\mathrm {sinc} ^{2}(fT)\ }$

where ${\displaystyle \mathrm {sinc} (x)\ }$ is the sinc function.

The Laplace transform transfer function of the delayed FOH is found by substituting s = i 2 π f:

${\displaystyle H_{\mathrm {FOH} }(s)\,}$	${\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =\left({\frac {1-e^{-sT}}{sT}}\right)^{2}\ }$

The delayed output makes this a causal system. The impulse response of the delayed FOH does not respond before the input impulse.

This kind of delayed piecewise linear reconstruction is physically realizable by implementing a digital filter of gain H(z) = 1 − z⁻¹, applying the output of that digital filter (which is simply x[n]−x[n−1]) to an ideal conventional digital-to-analog converter (that has an inherent zero-order hold as its model) and integrating (in continuous-time, H(s) = 1/(sT)) the DAC output.

Predictive FOH output signal x_FOH(t).

Lastly, the predictive first-order hold is quite different. This is a causal hypothetical LTI system or filter that converts the ideally sampled signal

${\displaystyle x_{s}(t)\,}$	${\displaystyle =x(t)\ T\sum _{n=-\infty }^{\infty }\delta (t-nT)\ }$
	${\displaystyle =T\sum _{n=-\infty }^{\infty }x(nT)\delta (t-nT)\ }$

into a piecewise linear output such that the current sample and immediately previous sample are used to linearly extrapolate up to the next sampling instance. The output of such a filter would be

${\displaystyle x_{\mathrm {FOH} }(t)\,}$	${\displaystyle =\sum _{n=-\infty }^{\infty }\left(x(nT)+\left(x(nT)-x((n-1)T)\right){\frac {t-nT}{T}}\right)\mathrm {rect} \left({\frac {t-nT}{T}}-{\frac {1}{2}}\right)\ }$
	${\displaystyle =\sum _{n=-\infty }^{\infty }x(nT)\left(\mathrm {rect} \left({\frac {t-nT}{T}}-{\frac {1}{2}}\right)-\mathrm {rect} \left({\frac {t-nT}{T}}-{\frac {3}{2}}\right)+\mathrm {tri} \left({\frac {t-nT}{T}}-1\right)\right)\ }$

Impulse response of predictive first-order hold h_FOH(t).

resulting in an effective impulse response of

${\displaystyle h_{\mathrm {FOH} }(t)\,}$	${\displaystyle ={\frac {1}{T}}\left(\mathrm {rect} \left({\frac {t}{T}}-{\frac {1}{2}}\right)-\mathrm {rect} \left({\frac {t}{T}}-{\frac {3}{2}}\right)+\mathrm {tri} \left({\frac {t}{T}}-1\right)\right)\ }$
	${\displaystyle ={\begin{cases}{\frac {1}{T}}\left(1+{\frac {t}{T}}\right)&{\mbox{if }}0\leq t<T\\{\frac {1}{T}}\left(1-{\frac {t}{T}}\right)&{\mbox{if }}T\leq t<2T\\0&{\mbox{otherwise}}\end{cases}}\ }$

where ${\displaystyle \mathrm {rect} (x)\ }$ is the rectangular function and ${\displaystyle \mathrm {tri} (x)\ }$ is the triangular function.

The effective frequency response is the continuous Fourier transform of the impulse response.

${\displaystyle H_{\mathrm {FOH} }(f)\,}$	${\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =(1+i2\pi fT)\left({\frac {1-e^{-i2\pi fT}}{i2\pi fT}}\right)^{2}\ }$
	${\displaystyle =(1+i2\pi fT)e^{-i2\pi fT}\mathrm {sinc} ^{2}(fT))\ }$

where ${\displaystyle \mathrm {sinc} (x)\ }$ is the sinc function.

The Laplace transform transfer function of the predictive FOH is found by substituting s = i 2 π f:

${\displaystyle H_{\mathrm {FOH} }(s)\,}$	${\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }$
	${\displaystyle =(1+sT)\left({\frac {1-e^{-sT}}{sT}}\right)^{2}\ }$

This a causal system. The impulse response of the predictive FOH does not respond before the input impulse.

This kind of piecewise linear reconstruction is physically realizable by implementing a digital filter of gain H(z) = 1 − z⁻¹, applying the output of that digital filter (which is simply x[n]−x[n−1]) to an ideal conventional digital-to-analog converter (that has an inherent zero-order hold as its model) and applying that DAC output to an analog filter with transfer function H(s) = (1+sT)/(sT).

• Nyquist–Shannon sampling theorem
• Zero-order hold
• Bilinear interpolation

• Sankar, Krishna (2007). "Zero order hold and first order hold based interpolation". dspLog Signal Processing for Communication.