Aufgaben:Exercise 5.4: Sine Wave Generator - Revision history

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	[[Category:Theory of Stochastic Signals: Exercises\|^5.2 Digital Filters^]]		[[Category:Theory of Stochastic Signals: Exercises\|^5.2 Digital Filters^]]
	[[de:Aufgaben:5.~~4_Sinusgenerator~~]]		[[de:Aufgaben:Aufgabe 5.4: Sinusgenerator]]

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	[[Category:Theory of Stochastic Signals: Exercises\|^5.2 Digital Filters^]]		[[Category:Theory of Stochastic Signals: Exercises\|^5.2 Digital Filters^]]
			[[de:Aufgaben:5.4_Sinusgenerator]]

Guenter at 17:53, 10 February 2022

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	*Checking the coefficient  $b_1$  confirms the calculation:		*Checking the coefficient  $b_1$  confirms the calculation:
	:$$b_1 = 2 \cdot \cos \left( {{{\rm{\pi }}}/{6}} \right) = 2 \cdot c{\sqrt 3 }/{2} = \sqrt 3 .$$		:$$b_1 = 2 \cdot \cos \left( {{{\rm{\pi }}}/{6}} \right) = 2 \cdot {\sqrt 3 }/{2} = \sqrt 3 .$$

Guenter at 17:44, 10 February 2022

2022-02-10T17:44:21Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The "$1$" at the input affects  $($because of  $a_0= 0)$  at the output only at the time  $\nu = 1$:		'''(1)'''  The  "$1$"  at the input affects  $($because of  $a_0= 0)$  at the output only at the time  $\nu = 1$:
	:$$y_0 \hspace{0.15cm} \underline{= 0},\quad y_1 \hspace{0.15cm} \underline{ = 0.5}.$$		:$$y_0 \hspace{0.15cm} \underline{= 0},\quad y_1 \hspace{0.15cm} \underline{ = 0.5}.$$

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	'''(2)'''  For  $\nu \ge 2$,  the filter is purely recursive:		'''(2)'''  For  $\nu \ge 2$,   the filter is purely recursive:
	:$$y_\nu = b_1 \cdot y_{\nu - 1} - y_{\nu - 2} .$$		:$$y_\nu = b_1 \cdot y_{\nu - 1} - y_{\nu - 2} .$$

	*In particular, one obtains		*In particular,  one obtains
	:$$y_3 = \sqrt 3 \cdot y_2 - y_1 = \sqrt 3 \cdot {\sqrt 3 }/{2} - {1}/{2} = 1;$$		:$$y_3 = \sqrt 3 \cdot y_2 - y_1 = \sqrt 3 \cdot {\sqrt 3 }/{2} - {1}/{2} = 1;$$
	:$$y_4 = \sqrt 3 \cdot y_3 - y_2 = \sqrt 3 \cdot 1 - {\sqrt 3 }/{2} = {\sqrt 3 }/{2};$$		:$$y_4 = \sqrt 3 \cdot y_3 - y_2 = \sqrt 3 \cdot 1 - {\sqrt 3 }/{2} = {\sqrt 3 }/{2};$$
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	'''(4)'''  From  $f_0 = 10 \hspace{0.~~05cm~~} \rm kHz$  follows  $T_0 = 100 \hspace{0.05cm} \rm µ s$  or  $T_0/T = 100$ . This gives:		'''(4)'''  From  $f_0 = 10 \hspace{0.15cm} \rm kHz$  follows  $T_0 = 100 \hspace{0.05cm} \rm µ s$  or  $T_0/T = 100$ . This gives:
	:$$a_1 = \sin \left( {2{\rm{\pi }}\cdot{T}/{T_0 }} \right) = \sin \left( {3.6^ \circ } \right) \hspace{0.15cm} \underline{\approx 0.062},$$		:$$a_1 = \sin \left( {2{\rm{\pi }}\cdot{T}/{T_0 }} \right) = \sin \left( {3.6^ \circ } \right) \hspace{0.15cm} \underline{\approx 0.062},$$
	:$$b_1 = 2 \cdot \cos \left( {2{\rm{\pi }}\cdot{T}/{T_0 }} \right) = 2 \cdot \cos \left( {3.6^ \circ } \right) \hspace{0.15cm} \underline{\approx 1.996}.$$		:$$b_1 = 2 \cdot \cos \left( {2{\rm{\pi }}\cdot{T}/{T_0 }} \right) = 2 \cdot \cos \left( {3.6^ \circ } \right) \hspace{0.15cm} \underline{\approx 1.996}.$$

Guenter at 17:38, 10 February 2022

2022-02-10T17:38:51Z

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			Notes:


	''Notes:''
	*The exercise belongs to the chapter   [[Theory_of_Stochastic_Signals/Digital_Filters\|Digital Filters]]  in this book.		*The exercise belongs to the chapter   [[Theory_of_Stochastic_Signals/Digital_Filters\|Digital Filters]]  in this book.
	*The HTML5/JavaScript applet  [[Applets:Digital_Filters\|"Digital Filters"]]  illustrates the subject matter of this chapter.		*The HTML5/JavaScript applet  [[Applets:Digital_Filters\|"Digital Filters"]]  illustrates the subject matter of this chapter.
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	{Now  $T = 1 \hspace{0.05cm} \rm µ s$  is valid. How must the coefficients  $a_1$  and  $b_1$  be selected so that a  $\text{10 kHz}$ sine wave is generated?		{Now  $T = 1 \hspace{0.05cm} \rm µ s$  is valid.  How must the coefficients  $a_1$  and  $b_1$  be selected so that a  $\text{10 kHz}$ sine wave is generated?
	\|type="{}"}		\|type="{}"}
	$a_1 \ = \ $ { 0.062 3% }		$a_1 \ = \ $ { 0.062 3% }

Guenter at 17:34, 10 February 2022

2022-02-10T17:34:48Z

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	[[File:P_ID622__Sto_A_5_4.png\|right\|frame\|Proposed filter structure]]		[[File:P_ID622__Sto_A_5_4.png\|right\|frame\|Proposed filter structure]]
	The diagram shows a second-order digital filter suitable for generating a discrete-time sinusoidal function on a digital signal processor (DSP), for example:		The diagram shows a second-order digital filter suitable for generating a discrete-time sinusoidal function on a digital signal processor  $\rm (DSP)$,  for example:
	:$$\left\langle \hspace{0.05cm}{y_\nu }\hspace{0.05cm} \right\rangle = \left\langle {\, \sin ( {\nu T \cdot \omega _0 } )\, }\right\rangle .$$		:$$\left\langle \hspace{0.05cm}{y_\nu }\hspace{0.05cm} \right\rangle = \left\langle {\, \sin ( {\nu T \cdot \omega _0 } )\, }\right\rangle .$$
	*It is assumed that the input sequence  $\left\langle \hspace{0.05cm} {x_\nu } \hspace{0.05cm}\right\rangle$  describes a (discrete-time) Dirac function.  Thus, all output values  $y_\nu$  are simultaneously identical to zero for times  $\nu\lt 0$.		*It is assumed that the input sequence  $\left\langle \hspace{0.05cm} {x_\nu } \hspace{0.05cm}\right\rangle$  describes a  (discrete-time)  Dirac delta function.  Thus,  all output values  $y_\nu$  are simultaneously identical to zero for times  $\nu\lt 0$.
	*The total of five filter coefficients result from the  [https://en.wikipedia.org/wiki/Z-transform $Z$-transform]:		*The total of five filter coefficients result from the  [https://en.wikipedia.org/wiki/Z-transform $Z$-transform]:
	:$$Z \{ {\sin ( {\nu T \omega _0 } )} \} = \frac{{z \cdot \sin \left( {\omega _0 T} \right)}}{{z^2 - 2 \cdot z \cdot \cos \left( {\omega _0 T} \right) + 1}}.$$		:$$Z \{ {\sin ( {\nu T \omega _0 } )} \} = \frac{{z \cdot \sin \left( {\omega _0 T} \right)}}{{z^2 - 2 \cdot z \cdot \cos \left( {\omega _0 T} \right) + 1}}.$$

Guenter at 17:18, 10 February 2022

2022-02-10T17:18:07Z

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	''Notes:''		''Notes:''
	*The exercise belongs to the chapter   [[Theory_of_Stochastic_Signals/Digital_Filters\|Digital Filters]]  in this book.		*The exercise belongs to the chapter   [[Theory_of_Stochastic_Signals/Digital_Filters\|Digital Filters]]  in this book.
			*The HTML5/JavaScript applet  [[Applets:Digital_Filters\|"Digital Filters"]]  illustrates the subject matter of this chapter.
	*For the subtasks  '''(1)'''  to  '''(3)'''  the following apply:		*For the subtasks  '''(1)'''  to  '''(3)'''  the following apply:
	:$$a_1 = 0.5,\quad b_1 = \sqrt 3 .$$		:$$a_1 = 0.5,\quad b_1 = \sqrt 3 .$$

Hwang at 10:52, 17 January 2022

2022-01-17T10:52:01Z

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Guenter at 10:36, 12 January 2022

2022-01-12T10:36:03Z

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	[[Category:Theory of Stochastic Signals: Exercises\|^5.2 ~~Digitale Filter~~^]]		[[Category:Theory of Stochastic Signals: Exercises\|^5.2 Digital Filters^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 5.4: Sinusgenerator to Aufgaben:Exercise 5.4: Sine Wave Generator

2022-01-12T08:59:32Z

Guenter moved page Aufgaben:Aufgabe 5.4: Sinusgenerator to Aufgaben:Exercise 5.4: Sine Wave Generator

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