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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]]. For the FOH, the signal is reconstructed as a [[Piecewise linear function|piecewise linear]] approximation to the original signal that was sampled. A mathematical model such as the FOH (or, more commonly, the [[zero-order hold]]) is necessary because, in the [[Nyquist–Shannon sampling theorem|sampling and reconstruction theorem]], a sequence of [[dirac delta function|dirac impulses]], ''x''<sub>s</sub>(''t''), representing the discrete samples, ''x''(''nT''), is [[low-pass filter]]ed to recover the original signal that was sampled, ''x''(''t''). However, outputting a sequence of dirac impulses is decidedly impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either the predictive or delayed FOH.
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| Even though this is '''not''' what is physically done, an identical output can be generated by applying the hypothetical sequence of dirac impulses, ''x''<sub>s</sub>(''t''), to a [[LTI system|linear, time-invariant system]], otherwise known as a [[electronic filter|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.
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| ==Basic first-order hold==
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| [[Image:Sampled.signal.svg|thumb|Ideally sampled signal ''x''<sub>s</sub>(''t'').]]
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| The first-order hold is the hypothetical [[filter (signal processing)|filter]] or [[LTI system]] that converts the ideally sampled signal
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| : <math>
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| \begin{align}
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| x_s(t) & {} = x(t) \ T \sum_{n=-\infty}^{\infty} \delta(t - nT) \\
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| & {} = T \sum_{n=-\infty}^{\infty} x(nT) \delta(t - nT)
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| \end{align}
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| </math>
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| [[Image:Firstorderhold.signal.svg|thumb|Piecewise linear signal ''x''<sub>FOH</sub>(''t'').]]
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| to the piecewise linear signal
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| :<math>x_{\mathrm{FOH}}(t)\,= \sum_{n=-\infty}^{\infty} x(nT) \mathrm{tri} \left(\frac{t - nT}{T} \right) \ </math>
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| [[Image:Firstorderhold.impulseresponse.svg|thumb|Impulse response (non-causal) of first-order hold ''h''<sub>FOH</sub>(''t'').]]
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| resulting in an effective [[impulse response]] of
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| : <math>h_{\mathrm{FOH}}(t)\,= \frac{1}{T} \mathrm{tri} \left(\frac{t}{T} \right)
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| = \begin{cases}
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| \frac{1}{T} \left( 1 - \frac{|t|}{T} \right) & \mbox{if } |t| < T \\
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| 0 & \mbox{otherwise}
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| \end{cases} \ </math>
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| : where <math>\mathrm{tri}(x) \ </math> is the [[triangular function]].
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| The effective frequency response is the [[continuous Fourier transform]] of the impulse response.
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| :{|
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| |-
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| |<math>H_{\mathrm{FOH}}(f)\,</math>
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| |<math>= \mathcal{F} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= \left( \frac{e^{i \pi fT} - e^{-i \pi fT}}{i 2 \pi fT} \right)^2 \ </math>
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| |-
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| |<math>= \mathrm{sinc}^2(fT) \ </math>
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| |}
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| : where <math>\mathrm{sinc}(x) \ </math> is the [[sinc function]].
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| The [[Laplace transform]] [[transfer function]] of the FOH is found by substituting ''s'' = ''i'' 2 π ''f'':
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| :{|
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| |<math>H_{\mathrm{FOH}}(s)\,</math>
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| |<math>= \mathcal{L} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= \left( \frac{e^{sT/2} - e^{-sT/2}}{sT} \right)^2 \ </math>
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| |}
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| 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. This acausality is also reflected in the impulse response of the FOH filter beginning to respond before impulse is applied.
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| ==Delayed first-order hold==
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| [[Image:Delayedfirstorderhold.signal.svg|thumb|Delayed piecewise linear signal ''x''<sub>FOH</sub>(''t'').]] | |
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| The '''delayed first-order hold''', sometimes called '''causal first-order hold''', is identical to the FOH above except that its output is delayed by one [[sampling frequency|sample period]] resulting in a delayed piecewise linear output signal
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| :<math>x_{\mathrm{FOH}}(t)\,= \sum_{n=-\infty}^{\infty} x(nT) \mathrm{tri} \left(\frac{t - T - nT}{T} \right) \ </math>
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| [[Image:Delayedfirstorderhold.impulseresponse.svg|thumb|Impulse response of causal first-order hold ''h''<sub>FOH</sub>(''t'').]]
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| resulting in an effective [[impulse response]] of
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| : <math>h_{\mathrm{FOH}}(t)\,= \frac{1}{T} \mathrm{tri} \left(\frac{t-T}{T} \right)
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| = \begin{cases}
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| \frac{1}{T} \left( 1 - \frac{|t-T|}{T} \right) & \mbox{if } |t-T| < T \\
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| 0 & \mbox{otherwise}
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| \end{cases} \ </math>
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| : where <math>\mathrm{tri}(x) \ </math> is the [[triangular function]].
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| The effective frequency response is the [[continuous Fourier transform]] of the impulse response.
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| :{|
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| |-
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| |<math>H_{\mathrm{FOH}}(f)\,</math>
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| |<math>= \mathcal{F} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= \left( \frac{1 - e^{-i 2\pi fT}}{i 2 \pi fT} \right)^2 \ </math>
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| |-
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| |<math>= e^{-i 2 \pi fT} \mathrm{sinc}^2(fT) \ </math>
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| |}
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| : where <math>\mathrm{sinc}(x) \ </math> is the [[sinc function]].
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| The [[Laplace transform]] [[transfer function]] of the delayed FOH is found by substituting ''s'' = ''i'' 2 π ''f'':
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| :{|
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| |-
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| |<math>H_{\mathrm{FOH}}(s)\,</math>
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| |<math>= \mathcal{L} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= \left( \frac{1 - e^{-sT}}{sT} \right)^2 \ </math>
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| |}
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| The delayed output makes this a [[causal system]]. The impulse response of the delayed FOH does not respond before the input impulse.
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| This kind of delayed piecewise linear reconstruction is physically realizable by implementing a [[digital filter]] of gain ''H''(''z'') = 1 − ''z''<sup>−1</sup>, 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 [[integrator|integrating]] (in continuous-time, ''H''(''s'') = 1/(''sT'')) the DAC output.
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| ==Predictive first-order hold==
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| [[Image:Predictivefirstorderhold.signal.svg|thumb|Predictive FOH output signal ''x''<sub>FOH</sub>(''t'').]]
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| Lastly, the '''predictive first-order hold''' is quite different. This is a ''causal'' hypothetical LTI system or filter that converts the ideally sampled signal
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| :{|
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| |-
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| |<math>x_s(t)\,</math>
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| |<math>= x(t) \ T \sum_{n=-\infty}^{\infty} \delta(t - nT) \ </math>
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| |-
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| |<math>= T \sum_{n=-\infty}^{\infty} x(nT) \delta(t - nT) \ </math>
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| |}
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| 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
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| :{| | |
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| |<math>x_{\mathrm{FOH}}(t)\,</math>
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| |<math>= \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) \ </math>
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| |-
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| |<math>= \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) \ </math>
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| |}
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| [[Image:Predictivefirstorderhold.impulseresponse.svg|thumb|Impulse response of predictive first-order hold ''h''<sub>FOH</sub>(''t'').]]
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| resulting in an effective [[impulse response]] of
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| :{|
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| |<math>h_{\mathrm{FOH}}(t)\,</math>
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| |<math>= \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) \ </math>
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| |-
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| |<math>= \begin{cases}
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| \frac{1}{T} \left( 1 + \frac{t}{T} \right) & \mbox{if } 0 \le t < T \\
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| \frac{1}{T} \left( 1 - \frac{t}{T} \right) & \mbox{if } T \le t < 2T \\
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| 0 & \mbox{otherwise}
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| \end{cases} \ </math>
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| |}
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| : where <math>\mathrm{rect}(x) \ </math> is the [[rectangular function]] and <math>\mathrm{tri}(x) \ </math> is the [[triangular function]].
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| The effective frequency response is the [[continuous Fourier transform]] of the impulse response.
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| :{|
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| |-
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| |<math>H_{\mathrm{FOH}}(f)\,</math>
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| |<math>= \mathcal{F} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= (1 + i 2\pi fT) \left( \frac{1 - e^{-i 2\pi fT}}{i 2\pi fT} \right)^2 \ </math>
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| |-
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| |<math>= (1 + i 2\pi fT) e^{-i 2\pi fT} \mathrm{sinc}^2(fT)) \ </math>
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| |}
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| : where <math>\mathrm{sinc}(x) \ </math> is the [[sinc function]].
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| The [[Laplace transform]] [[transfer function]] of the predictive FOH is found by substituting ''s'' = ''i'' 2 π ''f'':
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| :{|
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| |-
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| |<math>H_{\mathrm{FOH}}(s)\,</math>
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| |<math>= \mathcal{L} \{ h_{\mathrm{FOH}}(t) \} \ </math>
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| |-
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| |<math>= (1 + sT) \left( \frac{1 - e^{-sT}}{sT} \right)^2 \ </math>
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| |}
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| This a [[causal system]]. The impulse response of the predictive FOH does not respond before the input impulse.
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| This kind of piecewise linear reconstruction is physically realizable by implementing a [[digital filter]] of gain ''H''(''z'') = 1 − ''z''<sup>−1</sup>, 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'').
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| ==See also==
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| * [[Nyquist–Shannon sampling theorem]]
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| * [[Zero-order hold]]
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| *[[Bilinear interpolation]]
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| ==External links==
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| * [http://www.dsplog.com/2007/03/25/zero-order-hold-and-first-order-hold-based-interpolation/ Zero order hold and first order hold based interpolation]
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| [[Category:Digital signal processing]]
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| [[Category:Electrical engineering]]
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| [[Category:Control theory]]
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| [[Category:Signal processing]]
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