First-order hold (FOH ) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional
to recover the original signal that was sampled,
x (
t ). However, outputting a sequence of Dirac impulses is impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either predictive or delayed FOH.
Even though this is not what is physically done, an identical output can be generated by applying the hypothetical sequence of Dirac impulses, x s (t ), to a
linear time-invariant system, otherwise known as a
linear 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 piecewise linear function in the output.
Basic first-order hold
Ideally sampled signal x s (t ).
First-order hold is the hypothetical
LTI system
that converts the ideally sampled signal
x
s
(
t
)
{\displaystyle x_{s}(t)\,}
=
x
(
t
)
T
∑
n
=
−
∞
∞
δ
(
t
−
n
T
)
{\displaystyle =x(t)\ T\sum _{n=-\infty }^{\infty }\delta (t-nT)\ }
=
T
∑
n
=
−
∞
∞
x
(
n
T
)
δ
(
t
−
n
T
)
{\displaystyle =T\sum _{n=-\infty }^{\infty }x(nT)\delta (t-nT)\ }
Piecewise linear signal x FOH (t ).
to the piecewise linear signal
x
F
O
H
(
t
)
=
∑
n
=
−
∞
∞
x
(
n
T
)
t
r
i
(
t
−
n
T
T
)
{\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
h
F
O
H
(
t
)
=
1
T
t
r
i
(
t
T
)
=
{
1
T
(
1
−
|
t
|
T
)
if
|
t
|
<
T
0
otherwise
{\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
t
r
i
(
x
)
{\displaystyle \mathrm {tri} (x)\ }
is the triangular function .
The effective frequency response is the
continuous Fourier transform
of the impulse response.
H
F
O
H
(
f
)
{\displaystyle H_{\mathrm {FOH} }(f)\,}
=
F
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
e
i
π
f
T
−
e
−
i
π
f
T
i
2
π
f
T
)
2
{\displaystyle =\left({\frac {e^{i\pi fT}-e^{-i\pi fT}}{i2\pi fT}}\right)^{2}\ }
=
s
i
n
c
2
(
f
T
)
{\displaystyle =\mathrm {sinc} ^{2}(fT)\ }
where
s
i
n
c
(
x
)
=
sin
(
π
x
)
π
x
{\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 :
H
F
O
H
(
s
)
{\displaystyle H_{\mathrm {FOH} }(s)\,}
=
L
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
e
s
T
/
2
−
e
−
s
T
/
2
s
T
)
2
{\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 first-order hold
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
x
F
O
H
(
t
)
=
∑
n
=
−
∞
∞
x
(
n
T
)
t
r
i
(
t
−
T
−
n
T
T
)
{\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
h
F
O
H
(
t
)
=
1
T
t
r
i
(
t
−
T
T
)
=
{
1
T
(
1
−
|
t
−
T
|
T
)
if
|
t
−
T
|
<
T
0
otherwise
{\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
t
r
i
(
x
)
{\displaystyle \mathrm {tri} (x)\ }
is the triangular function .
The effective frequency response is the
continuous Fourier transform
of the impulse response.
H
F
O
H
(
f
)
{\displaystyle H_{\mathrm {FOH} }(f)\,}
=
F
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
1
−
e
−
i
2
π
f
T
i
2
π
f
T
)
2
{\displaystyle =\left({\frac {1-e^{-i2\pi fT}}{i2\pi fT}}\right)^{2}\ }
=
e
−
i
2
π
f
T
s
i
n
c
2
(
f
T
)
{\displaystyle =e^{-i2\pi fT}\mathrm {sinc} ^{2}(fT)\ }
where
s
i
n
c
(
x
)
{\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 :
H
F
O
H
(
s
)
{\displaystyle H_{\mathrm {FOH} }(s)\,}
=
L
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
1
−
e
−
s
T
s
T
)
2
{\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 first-order hold
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
x
s
(
t
)
{\displaystyle x_{s}(t)\,}
=
x
(
t
)
T
∑
n
=
−
∞
∞
δ
(
t
−
n
T
)
{\displaystyle =x(t)\ T\sum _{n=-\infty }^{\infty }\delta (t-nT)\ }
=
T
∑
n
=
−
∞
∞
x
(
n
T
)
δ
(
t
−
n
T
)
{\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
x
F
O
H
(
t
)
{\displaystyle x_{\mathrm {FOH} }(t)\,}
=
∑
n
=
−
∞
∞
(
x
(
n
T
)
+
(
x
(
n
T
)
−
x
(
(
n
−
1
)
T
)
)
t
−
n
T
T
)
r
e
c
t
(
t
−
n
T
T
−
1
2
)
{\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)\ }
=
∑
n
=
−
∞
∞
x
(
n
T
)
(
r
e
c
t
(
t
−
n
T
T
−
1
2
)
−
r
e
c
t
(
t
−
n
T
T
−
3
2
)
+
t
r
i
(
t
−
n
T
T
−
1
)
)
{\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
h
F
O
H
(
t
)
{\displaystyle h_{\mathrm {FOH} }(t)\,}
=
1
T
(
r
e
c
t
(
t
T
−
1
2
)
−
r
e
c
t
(
t
T
−
3
2
)
+
t
r
i
(
t
T
−
1
)
)
{\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)\ }
=
{
1
T
(
1
+
t
T
)
if
0
≤
t
<
T
1
T
(
1
−
t
T
)
if
T
≤
t
<
2
T
0
otherwise
{\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
r
e
c
t
(
x
)
{\displaystyle \mathrm {rect} (x)\ }
is the rectangular function and
t
r
i
(
x
)
{\displaystyle \mathrm {tri} (x)\ }
is the triangular function .
The effective frequency response is the
continuous Fourier transform
of the impulse response.
H
F
O
H
(
f
)
{\displaystyle H_{\mathrm {FOH} }(f)\,}
=
F
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {F}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
1
+
i
2
π
f
T
)
(
1
−
e
−
i
2
π
f
T
i
2
π
f
T
)
2
{\displaystyle =(1+i2\pi fT)\left({\frac {1-e^{-i2\pi fT}}{i2\pi fT}}\right)^{2}\ }
=
(
1
+
i
2
π
f
T
)
e
−
i
2
π
f
T
s
i
n
c
2
(
f
T
)
)
{\displaystyle =(1+i2\pi fT)e^{-i2\pi fT}\mathrm {sinc} ^{2}(fT))\ }
where
s
i
n
c
(
x
)
{\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 :
H
F
O
H
(
s
)
{\displaystyle H_{\mathrm {FOH} }(s)\,}
=
L
{
h
F
O
H
(
t
)
}
{\displaystyle ={\mathcal {L}}\{h_{\mathrm {FOH} }(t)\}\ }
=
(
1
+
s
T
)
(
1
−
e
−
s
T
s
T
)
2
{\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 ).
See also
External links