Applets:Digital Filters - Revision history

Noah at 16:44, 26 April 2023

2023-04-26T16:44:13Z

Guenter at 15:02, 24 April 2023

2023-04-24T15:02:42Z

Guenter at 13:55, 21 April 2023

2023-04-21T13:55:34Z

Guenter at 13:31, 21 April 2023

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Guenter at 13:22, 21 April 2023

2023-04-21T13:22:37Z

Guenter at 13:15, 21 April 2023

2023-04-21T13:15:04Z

Guenter at 15:58, 20 April 2023

2023-04-20T15:58:40Z

Noah at 19:46, 16 April 2023

2023-04-16T19:46:41Z

Noah at 19:44, 16 April 2023

2023-04-16T19:44:11Z

Noah at 19:43, 16 April 2023

2023-04-16T19:43:07Z

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 ==Applet Manual==
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Auswahl des Eingangssignals&nbsp; $($Dirac,&nbsp; Sprung oder Rechteck$)$
+&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Input signal selection&nbsp; $($Dirac delta,&nbsp; unit step or rectangular$)$
-&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Einstellungen für Abszisse, Ordinate  und Geschwindigkeit
+&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Settings for abscissa, ordinate and velocity.
-&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Bedienfeld&nbsp; $($Start, Einzelschritt, Gesamt, Pause, Reset$)$
+&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Control panel&nbsp; $($Start, Single step, Total, Pause, Reset$)$
-&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Blockschaltbild mit schrittweiser Anpassung aller Werte
+&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Block diagram with stepwise adjustment of all values.
-&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Grafikfeld für Ausgabe der Ausgangsfolge
+&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Graphic area for output of the output sequence
-&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Aufgabenauswahl
+&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Exercise selection
-&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Fragen und Lösungen
+&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Questions and solutions
 <br clear=all>
 ==About the authors==

← Older revision		Revision as of 15:02, 24 April 2023
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−	[[File:~~Exercise_impuls_v2~~.png\|right \|frame\| Screenshot ~~'''Korrektur''' auch in Deutsch~~]]	+	[[File:EN_DIG_Fil_Mannt.png\|right \|frame\| Screenshot]]

	==Applet Manual==		==Applet Manual==

← Older revision		Revision as of 13:55, 21 April 2023
Line 247:		Line 247:


		+	[[File:Exercise_impuls_v2.png\|right \|frame\| Screenshot '''Korrektur''' auch in Deutsch]]

	==Applet Manual==		==Applet Manual==
−
−	~~[[File:Exercise_impuls_v2.png\|right \|frame\| Screenshot]]~~
−

	'''(A)'''     Auswahl des Eingangssignals  $($Dirac,  Sprung oder Rechteck$)$		'''(A)'''     Auswahl des Eingangssignals  $($Dirac,  Sprung oder Rechteck$)$

@@ Line 257: / Line 257: @@
 &nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Einstellungen für Abszisse, Ordinate  und Geschwindigkeit
-&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Einstellungen für den Programmablauf&nbsp; $($
+&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Bedienfeld&nbsp; $($Start, Einzelschritt, Gesamt, Pause, Reset$)$
-&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Berechnung der&nbsp; "Running Digital Sum"
+&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Blockschaltbild mit schrittweiser Anpassung aller Werte
-&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Blockweise Bitänderung
+&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Grafikfeld für Ausgabe der Ausgangsfolge
-&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Grafikbereich für das Quellensignal&nbsp; $q(t)$
+&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Aufgabenauswahl
-&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Grafikbereich für das Codersignal&nbsp; $c(t)$
+&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Fragen und Lösungen
+<br clear=all>
 ==About the authors==
 <br>

@@ Line 249: / Line 249: @@
 ==Applet Manual==
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Auswahl der Quellensymbolfolge:&nbsp; $\rm A$,&nbsp; $\rm B$&nbsp; oder&nbsp; $\rm C$
+[[File:Exercise_impuls_v2.png|right |frame| Screenshot]]
-&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; "MS43"&nbsp; oder&nbsp; "MMS43"
+&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Auswahl des Eingangssignals&nbsp; $($Dirac,&nbsp; Sprung oder Rechteck$)$
 &nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Berechnung der&nbsp; "Running Digital Sum"

@@ Line 250: / Line 250: @@
 ==Applet Manual==
 <br>
-'''Korrektur'''
+[[File:Exercise_impuls_v2.png|right |frame| Screenshot '''Korrektur''']]
 ==About the authors==
 <br>

@@ Line 80: / Line 80: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$ Consider a non-recursive filter with the filter coefficients $a_0  =  1,\hspace{0.5cm} a_1  = 2,\hspace{0.5cm} a_2  =  1.$
-[[File:P_ID608__Sto_Z_5_3.png|right|frame|Nichtrekursives Filter]]
+[[File:P_ID608__Sto_Z_5_3.png|right|frame|Non-recursive filter]]
 '''(1)''' The conventional impulse response is: $h(t) = \delta (t) + 2 \cdot \delta ( {t - T_{\rm A} } ) + \delta ( {t - 2T_{\rm A} } ).$ <br>&nbsp; &nbsp; &nbsp; &nbsp; &rArr; &nbsp; discrete-time impulse response: $〈h_\mu〉= 〈1,\ 2,\  1〉 .$
@@ Line 129: / Line 129: @@
 {{GraueBox|TEXT=
-[[File:Sto_T_5_2_S3_version2.png |frame| Discrete-time impulse response | rechts]]
+[[File:Sto_T_5_2_S3_version2.png |frame| Discrete-time impulse response]]
 $\text{Example 3:}$&nbsp; The graphic opposite shows the discrete-time impulse response $〈\hspace{0.05cm}h_\mu\hspace{0.05cm}〉$ of a recursive filter of the first order with the parameters $a_0 = 1$ and $b_1 = 0.6$.
 *The (discrete-time) course is exponentially falling and extends to infinity.
-*The ratio of the weights of two successive Diracs is $b_1 = 0.6$.
+*The ratio of the weights of two successive Dirac deltas is $b_1 = 0.6$.
 }}
@@ Line 219: / Line 219: @@
 '''(6)'''&nbsp; Consider and interpret the impulse response and the step response for the filter coefficients&nbsp; $a_0=1$,&nbsp; $b_1=1$,&nbsp; $a_1=a_2= b_2=0$.&nbsp; }}
-:*&nbsp; '''The system is unstable''': &nbsp; A discrete-time Dirac delta function at input $($at time&nbsp; $t=0)$&nbsp; causes an infinite number of Diracs of the same height in the output signal.
+:*&nbsp; '''The system is unstable''': &nbsp; A discrete-time Dirac delta function at input $($at time&nbsp; $t=0)$&nbsp; causes an infinite number of Dirac deltas of the same height in the output signal.
-:*&nbsp; A discrete-time step function at the input causes an infinite number of Diracs with monotonically increasing weights (to infinity) in the output signal.
+:*&nbsp; A discrete-time step function at the input causes an infinite number of Dirac deltas with monotonically increasing weights (to infinity) in the output signal.
 {{BlaueBox|TEXT=

@@ Line 26: / Line 26: @@
 Each signal $x(t)$ can only be represented on a computer by the sequence $〈x_ν〉$ of its samples, where $x_ν$ stands for $x(ν · T_{\rm A})$.
 [[File:P_ID552__Sto_T_5_2_S1_neu.png|right |frame| Block diagram of a digital (IIR&ndash;) filter $M$&ndash;order]]
-*The time interval $T_{\rm A}$ between two samples is limited by the [[https://en.lntwww.de/Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|"sampling theorem"]].
+*The time interval $T_{\rm A}$ between two samples is limited by the [https://en.lntwww.de/Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|"sampling theorem"].
 *We limit ourselves here to causal signals and systems, which means that $x_ν \equiv 0$ for $ν \le 0$.

@@ Line 15: / Line 15: @@
 '''(1)''' the '''discrete-time impulse response''' $〈h_ν〉$ if the “discrete-time Dirac delta function” is present at the input: &nbsp; &nbsp; &nbsp; &nbsp; $〈x_ν〉= 〈1,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0, \text{...}〉,$
-'''(2)''' the '''time-discrete step response''' $〈\sigma_ν〉$ if the “time-discrete step function” is present at the input: &nbsp; &nbsp; &nbsp; &nbsp; $〈x_ν〉= 〈1,\ 1,\ 1,\ 1,\ 1,\ 1,\ 1, \text{...}〉,$
+'''(2)''' the '''discrete-time step response''' $〈\sigma_ν〉$ if the “discrete-time step function” is present at the input: &nbsp; &nbsp; &nbsp; &nbsp; $〈x_ν〉= 〈1,\ 1,\ 1,\ 1,\ 1,\ 1,\ 1, \text{...}〉,$
 '''(3)''' the '''discrete-time rectangle response''' $〈\rho_ν^{(2, 4)}〉$ if the “discrete-time rectangle function” is present at the input: &nbsp; &nbsp; $〈x_ν〉= 〈0,\ 0,\ 1,\ 1,\ 1,\ 0,\ 0, \text{...}〉;$<br>&nbsp; &nbsp; &nbsp; &nbsp; In quotation marks are the beginning of the ones $(2)$ and the position of the last ones $(4)$.
@@ Line 26: / Line 26: @@
 Each signal $x(t)$ can only be represented on a computer by the sequence $〈x_ν〉$ of its samples, where $x_ν$ stands for $x(ν · T_{\rm A})$.
 [[File:P_ID552__Sto_T_5_2_S1_neu.png|right |frame| Block diagram of a digital (IIR&ndash;) filter $M$&ndash;order]]
-*The time interval $T_{\rm A}$ between two samples is limited by the [[Signal_Representation/Time_Discrete_Signal_Representation#Das_Abtasttheorem|sampling theorem]].
+*The time interval $T_{\rm A}$ between two samples is limited by the [[https://en.lntwww.de/Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|"sampling theorem"]].
 *We limit ourselves here to causal signals and systems, which means that $x_ν \equiv 0$ for $ν \le 0$.
-*In order to determine the influence of a linear filter with frequency response $H(f)$ on the time-discrete input signal $〈x_ν〉$, it is advisable to describe the filter discrete-time. In the time domain, this happens with the discrete-time impulse response $〈h_ν〉$.
+*In order to determine the influence of a linear filter with frequency response $H(f)$ on the discrete-time input signal $〈x_ν〉$, it is advisable to describe the filter discrete-time. In the time domain, this happens with the discrete-time impulse response $〈h_ν〉$.
 *On the right you can see the corresponding block diagram. The following therefore applies to the samples of the output signal $〈y_ν〉$ thus holds:
 :$$y_\nu   = \sum\limits_{\mu  = 0}^M {a_\mu  }  \cdot x_{\nu  - \mu }  + \sum\limits_{\mu  = 1}^M {b_\mu  }  \cdot y_{\nu  - \mu } .$$
@@ Line 46: / Line 46: @@
 '''(1)'''&nbsp; The output sequence $〈y_ν〉$ is called the '''discrete-time impulse response''' $〈h_ν〉$ if the “discrete-time Dirac delta function” is present at the input:
 :$$〈x_ν〉= 〈1,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0, \text{...}〉  .$$
-'''(2)'''&nbsp; The output sequence $〈y_ν〉$ is called the '''time-discrete step response''' $〈\sigma_ν〉$ if the “time-discrete step function” is present at the input:
+'''(2)'''&nbsp; The output sequence $〈y_ν〉$ is called the '''discrete-time step response''' $〈\sigma_ν〉$ if the “discrete-time step function” is present at the input:
 :$$〈x_ν〉= 〈1,\ 1,\ 1,\ 1,\ 1,\ 1,\ 1, \text{...}〉  .$$
 '''(3)'''&nbsp; The output sequence $〈y_ν〉$ is called the '''discrete-time rectangle response'''&nbsp; $〈\rho_ν^{(2, 4)}〉$ if the “discrete-time rectangular function” is present at the input:
@@ Line 64: / Line 64: @@
 *The output value $y_ν$ depends only on the current and the previous $M$ input values:
 :$$y_\nu   = \sum\limits_{\mu  = 0}^M {a_\mu   \cdot x_{\mu  - \nu } } .$$
-*Time-discrete impulse response with $〈x_ν〉= 〈1,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0, \text{...}〉$:
+*Discrete-time impulse response with $〈x_ν〉= 〈1,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0, \text{...}〉$:
 :$$〈h_\mu〉= 〈a_0,\ a_1,\  \text{...},\ a_M〉 .$$}}
@@ Line 87: / Line 87: @@
 :$$H(f) = 2\big [ {1 + \cos ( {2{\rm{\pi }\cdot  }f \cdot T_{\rm A} } )} \big ] \cdot {\rm{e} }^{ - {\rm{j} }2{\rm{\pi } }fT_{\rm A} }\hspace{0.5cm}\Rightarrow \hspace{0.5cm}H(f = 0) = 4.$$
-'''(3)''' &nbsp; It follows that the '''time-discrete step response''' $〈\sigma_ν〉$ tends to become $4$ for large $\nu$.
+'''(3)''' &nbsp; It follows that the '''discrete-time step response''' $〈\sigma_ν〉$ tends to become $4$ for large $\nu$.
 '''(4)''' &nbsp; The discrete-time convolution of the input sequence $\left\langle \hspace{0.05cm}{x_\nu  } \hspace{0.05cm}\right\rangle  =  \left\langle {\;1,\;0,\;0,\;0,\;1,\;0,\;0,\;0,\;\text{...} } \hspace{0.05cm} \right\rangle$&nbsp; with $\left\langle \hspace{0.05cm}{h_\nu  } \hspace{0.05cm}\right\rangle = \left\langle \hspace{0.05cm}{1, \ 2,\ 1  } \hspace{0.05cm}\right\rangle$&nbsp; results
@@ Line 121: / Line 121: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; With a recursive filter, the (time-discrete) impulse response extends to infinity with $M = 1$:
+$\text{Conclusion:}$&nbsp; With a recursive filter, the (discrete-time) impulse response extends to infinity with $M = 1$:
 *For reasons of stability, $b_1 < 1$ must apply.
 *With $b_1 = 1$ the impulse response $h(t)$ would extend to infinity and with $b_1 > 1$ the variable $h(t)$ would even continue to infinity.
@@ Line 131: / Line 131: @@
 [[File:Sto_T_5_2_S3_version2.png |frame| Discrete-time impulse response | rechts]]
 $\text{Example 3:}$&nbsp; The graphic opposite shows the discrete-time impulse response $〈\hspace{0.05cm}h_\mu\hspace{0.05cm}〉$ of a recursive filter of the first order with the parameters $a_0 = 1$ and $b_1 = 0.6$.
-*The (time-discrete) course is exponentially falling and extends to infinity.
+*The (discrete-time) course is exponentially falling and extends to infinity.
 *The ratio of the weights of two successive Diracs is $b_1 = 0.6$.
 }}
@@ Line 140: / Line 140: @@
 [[File:EN_Sto_A_5_4_version2.png|right|frame|Proposed filter structure]]
-The graphic shows a second-order digital filter that is suitable for generating a time-discrete sine function on a digital signal processor (DSP) if the input sequence $\left\langle \hspace{0.05cm} {x_\nu  } \hspace{0.05cm}\right\rangle$&nbsp; a (time-discrete) Dirac delta function is:
+The graphic shows a second-order digital filter that is suitable for generating a discrete-time sine function on a digital signal processor (DSP) if the input sequence $\left\langle \hspace{0.05cm} {x_\nu  } \hspace{0.05cm}\right\rangle$&nbsp; a (discrete-time) Dirac delta function is:
 :$$\left\langle \hspace{0.05cm}{y_\nu  }\hspace{0.05cm} \right\rangle  = \left\langle {\, \sin ( {\nu \cdot T_{\rm A} \cdot \omega _0  } )\, }\right\rangle .$$
 The five filter coefficients result from the:
-[https://de.wikipedia.org/wiki/Z-Transformation $Z$ transformation]:
+[https://en.wikipedia.org/wiki/Z-transform "$Z$-transform"]:
 :$$Z \big \{ {\sin ( {\nu T{\rm A}\cdot \omega _0 } )} \big \} = \frac{{z \cdot \sin \left( {\omega _0 \cdot T_{\rm A}} \right)}}{{z^2  - 2 \cdot z \cdot \cos \left( {\omega _0  \cdot T_{\rm A}} \right) + 1}}.$$
 After implementing this equation using a second-order recursive filter, the following filter coefficients are obtained:
@@ Line 181: / Line 181: @@
 {{BlaueBox|TEXT=
-'''(1)'''&nbsp; The filter coefficients are&nbsp; $a_0=0.25$,&nbsp; $a_1=0.5$,&nbsp; $a_2=0.25$,&nbsp; $b_1=b_2=0$.&nbsp; Which filter is it?<br>&nbsp; &nbsp; &nbsp; &nbsp; Interpret the impulse response&nbsp; $〈h_ν〉$,&nbsp; the step response&nbsp; $〈\sigma_ν〉$&nbsp; and the rectangular response&nbsp; $〈\rho_ν^{(2, 8)}〉$, each in a time-discrete representation.}}
+'''(1)'''&nbsp; The filter coefficients are&nbsp; $a_0=0.25$,&nbsp; $a_1=0.5$,&nbsp; $a_2=0.25$,&nbsp; $b_1=b_2=0$.&nbsp; Which filter is it?<br>&nbsp; &nbsp; &nbsp; &nbsp; Interpret the impulse response&nbsp; $〈h_ν〉$,&nbsp; the step response&nbsp; $〈\sigma_ν〉$&nbsp; and the rectangular response&nbsp; $〈\rho_ν^{(2, 8)}〉$, each in a discrete-time representation.}}
 :*&nbsp; Due to the missing&nbsp; $b$ coefficients, it is a non-recursive digital filter &rArr; &nbsp; '''FIR filter''' (''Finite Impulse Response'').
@@ Line 205: / Line 205: @@
 :*&nbsp; The step response is the integral over the impulse response: &nbsp; $\sigma(t) = T \cdot (1-{\rm e}^{-t/T}) ]$ &nbsp;  &rArr; &nbsp; $\sigma_\nu=  10 \cdot (1-{\rm e}^{-\nu/10})$ &nbsp; &rArr; &nbsp; $\sigma_0=1$,&nbsp; $\sigma_1=1.9$,&nbsp; $\sigma_2=2.71$, ...
-:*&nbsp; For large&nbsp; $\nu$&nbsp; values, the (time-discrete) step response tends to the DC signal transmission factor&nbsp; $H(f=0)= 10$:&nbsp; $\sigma_{40}=9.867$,&nbsp; $\sigma_{50}=9.954$,&nbsp;  $\sigma_\infty=10$.
+:*&nbsp; For large&nbsp; $\nu$&nbsp; values, the (discrete-time) step response tends to the DC signal transmission factor&nbsp; $H(f=0)= 10$:&nbsp; $\sigma_{40}=9.867$,&nbsp; $\sigma_{50}=9.954$,&nbsp;  $\sigma_\infty=10$.
 :*&nbsp;The rectangular response&nbsp; $〈\rho_ν^{(2, 8)}〉$&nbsp; increases with a delay of&nbsp; $2$&nbsp; in the same way as&nbsp; $〈\sigma_ν〉$.&nbsp; In the area&nbsp; $\nu \ge 8$&nbsp; the&nbsp; $\rho_ν$&nbsp; values decrease exponentially.
@@ Line 219: / Line 219: @@
 '''(6)'''&nbsp; Consider and interpret the impulse response and the step response for the filter coefficients&nbsp; $a_0=1$,&nbsp; $b_1=1$,&nbsp; $a_1=a_2= b_2=0$.&nbsp; }}
-:*&nbsp; '''The system is unstable''': &nbsp; A time-discrete Dirac delta function at input $($at time&nbsp; $t=0)$&nbsp; causes an infinite number of Diracs of the same height in the output signal.
+:*&nbsp; '''The system is unstable''': &nbsp; A discrete-time Dirac delta function at input $($at time&nbsp; $t=0)$&nbsp; causes an infinite number of Diracs of the same height in the output signal.
 :*&nbsp; A discrete-time step function at the input causes an infinite number of Diracs with monotonically increasing weights (to infinity) in the output signal.
@@ Line 241: / Line 241: @@
 {{BlaueBox|TEXT=
 '''(10)'''&nbsp; We continue with the&nbsp; "sine generator".&nbsp; What modifications do you have to make to generate a&nbsp; "cosine"?}}
-:*&nbsp; With&nbsp; $a_1=0.3826$,&nbsp; $b_1=1.8478$,&nbsp; $b_2=-1$&nbsp; and&nbsp; $〈x_ν〉=〈1, 1, 1, \text{...}〉$&nbsp; is the output sequence&nbsp; $〈y_ν〉$&nbsp; the time-discrete analogon of the step response&nbsp; $\sigma(t)$.
+:*&nbsp; With&nbsp; $a_1=0.3826$,&nbsp; $b_1=1.8478$,&nbsp; $b_2=-1$&nbsp; and&nbsp; $〈x_ν〉=〈1, 1, 1, \text{...}〉$&nbsp; is the output sequence&nbsp; $〈y_ν〉$&nbsp; the discrete-time analogon of the step response&nbsp; $\sigma(t)$.
 :*&nbsp; The step response is the integral over &nbsp; $\sin(\pi\cdot\tau/8)$ &nbsp; within the limits of &nbsp; $\tau=0$ &nbsp; to &nbsp; $\tau=t$ &nbsp; &rArr; &nbsp; $\sigma(t)=-8/\pi\cdot\cos(\pi\cdot\tau/8)+1$.
 :*&nbsp; If you change &nbsp; $a_1=0.3826$ &nbsp; on &nbsp; $a_1=-0.3826\cdot\pi/8=-0.1502$, then &nbsp; $\sigma(t)=\cos(\pi\cdot\tau/8)-1$ &nbsp; &rArr; &nbsp; Values between&nbsp; $0$&nbsp; and&nbsp; $-2$.
-:*&nbsp; Would you still in the block diagram &nbsp; $z_\nu=y_\nu+1$ &nbsp; add, then &nbsp; $z_\nu$ &nbsp; a time-discrete cosine curve with &nbsp; $T_0/T_{\rm A}=16$ &nbsp; and &nbsp; $A=1$.
+:*&nbsp; Would you still in the block diagram &nbsp; $z_\nu=y_\nu+1$ &nbsp; add, then &nbsp; $z_\nu$ &nbsp; a discrete-time cosine curve with &nbsp; $T_0/T_{\rm A}=16$ &nbsp; and &nbsp; $A=1$.