Aufgaben:Exercise 4.5: Non-Linear Quantization - Revision history

Guenter at 15:10, 9 April 2022

2022-04-09T15:10:19Z

Guenter at 14:12, 9 April 2022

2022-04-09T14:12:47Z

Guenter at 13:23, 9 April 2022

2022-04-09T13:23:47Z

Noah at 14:43, 7 April 2022

2022-04-07T14:43:14Z

Noah at 17:13, 6 April 2022

2022-04-06T17:13:35Z

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Guenter at 13:53, 16 March 2022

2022-03-16T13:53:34Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.5: Nichtlineare Quantisierung to Aufgaben:Exercise 4.5: Non-Linear Quantization

2022-03-16T13:47:22Z

Guenter moved page Aufgabe 4.5: Nichtlineare Quantisierung to Exercise 4.5: Non-Linear Quantization

Guenter at 16:25, 9 November 2021

2021-11-09T16:25:19Z

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:46:13Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:59:18Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

@@ Line 26: / Line 26: @@
 This means:
 *In the range &nbsp;$|q_{\rm A}| ≤ 1/64$&nbsp; holds &nbsp;$q_{\rm K} = q_{\rm A}$.
-*For &nbsp;$q_{\rm A} > 1/64$,&nbsp; there are the following six additional ranges &nbsp;$(k = 1$, ... , $6)$&nbsp; of the compressor characteristic:<br>&nbsp; &rArr; &nbsp; range $k\hspace{0.3cm}{\rm (if}\hspace{0.3cm} 2^{k-7}< q_{\rm A} \le 2^{k-6}) \hspace{0.05cm}$ &nbsp; &rArr; &nbsp; $q_{\rm K}(q_{\rm A}) = 2^{4-k} \cdot q_{\rm A} + {k}/{8};$
+*For &nbsp;$q_{\rm A} > 1/64$,&nbsp; there are the following six additional ranges &nbsp;$(k = 1$, ... , $6)$&nbsp; of the compressor characteristic:&nbsp; <br>&nbsp; &rArr; &nbsp; range $k\hspace{0.3cm}{\rm (if}\hspace{0.3cm} 2^{k-7}< q_{\rm A} \le 2^{k-6}) \hspace{0.05cm}$ &nbsp; &rArr; &nbsp; $q_{\rm K}(q_{\rm A}) = 2^{4-k} \cdot q_{\rm A} + {k}/{8}.$
 *Another six domains exist for negative &nbsp;$q_{\rm A}$&nbsp; values with &nbsp;$k = -1$, ... , $-6$,&nbsp; which are point-symmetric with respect to the origin.&nbsp; <br>However,&nbsp; these are not considered further in this exercise.

@@ Line 80: / Line 80: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The sample&nbsp; $q_{\rm A} = 0.4$&nbsp; belongs to the segment&nbsp; $k = 5$ covering the range&nbsp; $1/4 < q_{\rm A} ≤ 1/2$&nbsp; From the given equation it follows that with&nbsp; $k = 5$:
+'''(1)'''&nbsp; The sample&nbsp; $q_{\rm A} = 0.4$&nbsp; belongs to the segment&nbsp; $k = 5$&nbsp; covering the range&nbsp; $1/4 < q_{\rm A} ≤ 1/2$.&nbsp; From the given equation it follows that with&nbsp; $k = 5$:
 :$$q_{\rm K}(q_{\rm A}) = 2^{4-k} \cdot q_{\rm A} + {k}/{8}={1}/{2}\cdot 0.4 + {5}/{8} \hspace{0.15cm}\underline {= 0.825}\hspace{0.05cm}.$$
-'''(2)'''&nbsp; The input value of the linear quantizer is now&nbsp; $q_{\rm K} = 0.825$, so the following calculation applies:
+'''(2)'''&nbsp; The input value of the linear quantizer is now&nbsp; $q_{\rm K} = 0.825$,&nbsp; so the following calculation applies:
 :$${105}/{128} < q_{\rm K} = 0.825 \le {106}/{128}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 105 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \mu = 128 + 105\hspace{0.15cm}\underline { = 233} \hspace{0.05cm}.$$
@@ Line 91: / Line 91: @@
 '''(3)'''&nbsp; According to the specification page, the quantization interval&nbsp; $μ = 128 + m$&nbsp; is given by the value&nbsp;
-$q_{\rm Q} = 1/256 + m/128$&nbsp; represented.&nbsp; With&nbsp; $m = 105$&nbsp; it follows:
+$q_{\rm Q} = 1/256 + m/128$.&nbsp; With&nbsp; $m = 105$&nbsp; it follows:
 :$$q_{\rm Q} = \frac{1}{256} + \frac{105}{128} \hspace{0.15cm}\underline {\approx 0.824} \hspace{0.05cm}.$$
-'''(4)'''&nbsp; According to the sample solution to the subtask&nbsp; '''(3)'''&nbsp; with the input value&nbsp; $q_{\rm A} = 0.04$:
+'''(4)'''&nbsp; According to the sample solution to subtask&nbsp; '''(3)'''&nbsp; with the input value&nbsp; $q_{\rm A} = 0.04$:
-:$$ \frac{1}{32} < q_{\rm A} \frac{1}{16}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} k = 2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} q_{\rm K} = 2^2 \cdot 0.04 + \frac{2}{8}= 0.41$$
+:$$ \frac{1}{32} < q_{\rm A} \le \frac{1}{16}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} k = 2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} q_{\rm K} = 2^2 \cdot 0.04 + \frac{2}{8}= 0.41$$
 :$$\Rightarrow \hspace{0.3cm}\frac{52}{128} < q_{\rm K} = 0.41 \le \frac{53}{128}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} m = 52 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \mu = 128 + 52 = 180\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm}q_{\rm Q} = \frac{1}{256} + \frac{52}{128} \hspace{0.15cm}\underline {= 0.41} \hspace{0.05cm}.$$
@@ Line 103: / Line 103: @@
-[[File:Mod_A_4_5_ML_S2a_version2.png|right|frame|characteristics of compressor (blue) and expander (green)]]
+[[File:Mod_A_4_5_ML_S2a_version2.png|right|frame|Characteristics of compressor (blue) & expander (green)]]
 '''(5)'''&nbsp; We are looking for the solution in several steps:
-*In the compressor has&nbsp; $q_{\rm A} = 0.4$&nbsp; led to the initial value&nbsp; $q_{\rm K} = 0.825$&nbsp; and after quantization to the value&nbsp; $q_{\rm Q} = 0. 824$&ndash; see subtasks&nbsp; '''(1)'''&nbsp; and&nbsp; '''(3)'''.&nbsp; Note the red marks in the graph.
+*In the compressor: &nbsp; $q_{\rm A} = 0.4$&nbsp; led to the initial value&nbsp; $q_{\rm K} = 0.825$&nbsp; and after quantization to the value&nbsp; $q_{\rm Q} = 0. 824$ &nbsp; &rArr; &nbsp; see subtasks&nbsp; '''(1)'''&nbsp; and&nbsp; '''(3)'''.&nbsp; &rArr; &nbsp; red marks in the graph.
-*The graph shows that, on the receiver side, this results in&nbsp; $v_{\rm Q} = 0.824$&nbsp; approximately back to&nbsp; $v_{\rm E} ≈ 0.4$&nbsp; &rArr; &nbsp; brown marks in the graph.
+*On the receiver side,&nbsp; this results in&nbsp; $v_{\rm Q} = 0.824$&nbsp; approximately back to&nbsp; $v_{\rm E} ≈ 0.4$ &nbsp; &rArr; &nbsp; brown marks in the graph.
-However, due to quantization, this is only an approximation.&nbsp; Exactly:
+:Although the expander characteristic&nbsp; $v_E(υ_{\rm Q})$&nbsp; is equal to the inverse function of the compressor characteristic&nbsp; $q_K(q_{\rm A})$&nbsp; an error results because the input&nbsp; $v_{\rm Q}$&nbsp; of the expander is discrete in value&nbsp; (influence of quantization).
-'''(6)'''&nbsp; Correct are <u>statements 1 and 4</u>, as can be verified by the following left graph:
+'''(6)'''&nbsp; Correct are the&nbsp; <u>statements 1 and 4</u>,&nbsp; as can be verified by the following left graph:
 [[File:P_ID1622__Mod_A_4_5f.png|right|frame|13-segment characteristic curves: left:&nbsp; $q_{\rm Q}(q_{\rm A})$, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; right:&nbsp; $v_{\rm E}(q_{\rm A})$]]
-*The width of each step is different in each segment.
+*The width of each step is different in each segment.&nbsp; In the outermost segment&nbsp; $(k = 6)$&nbsp; the step width is&nbsp; $0.5/16 = 1/32$,&nbsp; in the next segment&nbsp; $(k = 5)$&nbsp; only more&nbsp; $0.25/16 = 1/64$.
 *The step widths in the further segments are&nbsp; $1/128 \ (k = 4)$,&nbsp; $1/256 \ (k = 3)$,&nbsp; $1/512\ (k = 2)$&nbsp; and&nbsp; $1/1024 \ (k = 1)$.
-*The innermost range from&nbsp; $-1/64$&nbsp; to&nbsp; $+1/64$&nbsp; is divided into&nbsp; $64$&nbsp; steps, resulting in the step width&nbsp; $1/2048$&nbsp;.
+*The innermost range from&nbsp; $-1/64$&nbsp; to&nbsp; $+1/64$&nbsp; is divided into&nbsp; $64$&nbsp; steps,&nbsp; resulting in the step width&nbsp; $1/2048$.
-*The step height, on the other hand, is constantly equal&nbsp; $1/8$&nbsp; divided by&nbsp; $16 = 1/128$&nbsp; in the segments&nbsp; $k ≠ 0$&nbsp; and equal&nbsp; $1/256$ in the middle segment.
+*The step height,&nbsp; on the other hand,&nbsp; is constantly equal&nbsp; $1/8$&nbsp; divided by&nbsp; $16 = 1/128$&nbsp; in the segments&nbsp; $k ≠ 0$&nbsp; and equal&nbsp; $1/256$ in the middle segment.
-'''(7)'''&nbsp; Correct here is <u>only the second statement</u>:
+'''(7)'''&nbsp; Correct here is&nbsp; <u>only the second statement</u>:
-*By the expander, the quantization is now along the bisector of the angle.
+*By the expander,&nbsp; the quantization is now along the bisector of the angle.
-*In each segment, step width and step height are constant.
+*In each segment,&nbsp; step width and step height are constant.
-*As the right graphic shows, however, in the next inner segment the width and the height are only half as large.
+*As the right graphic shows,&nbsp; however,&nbsp; in the next inner segment the width and the height are only half as large.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:EN_Mod_Z_4_5.png|right|frame|PCM system with companding]]
-To investigate&nbsp; ''nonlinear quantization''&nbsp; we start from the outlined system model.
+To investigate&nbsp; "non-linear quantization"&nbsp; we start from the outlined system model.
 *We disregard the influence of the channel and the PCM coding or decoding.
-*Thus, &nbsp;$v_{\rm Q}(ν \cdot T_{\rm A}) = q_{\rm Q}(ν \cdot T_{\rm A})$ always applies, whereby the time specification &nbsp;$ν \cdot T_{\rm A}$&nbsp; is omitted in the following.
+*Thus, &nbsp;$v_{\rm Q}(ν \cdot T_{\rm A}) = q_{\rm Q}(ν \cdot T_{\rm A})$&nbsp; always applies,&nbsp; whereby the time specification &nbsp;$ν \cdot T_{\rm A}$&nbsp; is omitted in the following.
-By comparing one output variable with one input variable at a time, it is possible to determine the influence
 * of the compressor &nbsp; ⇒ &nbsp; &nbsp;$q_{\rm K}(q_{\rm A})$,
 * of the linear quantizer &nbsp; ⇒ &nbsp; &nbsp;$q_{\rm Q}(q_{\rm K})$,
-* of the nonlinear quantizer &nbsp; ⇒ &nbsp; &nbsp;$q_{\rm Q}(q_{\rm A})$,
+* of the non-linear quantizer &nbsp; ⇒ &nbsp; &nbsp;$q_{\rm Q}(q_{\rm A})$,
-* of the expander &nbsp; ⇒ &nbsp; &nbsp;$v_{\rm E}(v_{\rm Q})$, and
+* of the expander &nbsp; ⇒ &nbsp; &nbsp;$v_{\rm E}(v_{\rm Q})$,&nbsp; and
-* of the overall system &nbsp; ⇒ &nbsp; &nbsp;$v_{\rm E}(q_{\rm A})$
+* of the overall system &nbsp; ⇒ &nbsp; &nbsp;$v_{\rm E}(q_{\rm A})$.
-&nbsp; The following assumptions are made:
+The following assumptions are made:
-* All samples &nbsp;$q_{\rm A}$&nbsp; are in the range of values &nbsp;$±1$&nbsp; .
+* All samples &nbsp;$q_{\rm A}$&nbsp; are in the value range &nbsp;$±1$&nbsp; .
-* The (linear) quantizer works with &nbsp;$M = 256$&nbsp; quantization levels, which are marked with &nbsp;$μ = 0$&nbsp; to &nbsp;$μ = 255$&nbsp;.
+* The&nbsp; (linear)&nbsp; quantizer works with &nbsp;$M = 256$&nbsp; quantization levels,&nbsp; which are marked with &nbsp;$μ = 0$&nbsp; to &nbsp;$μ = 255$&nbsp;.
-* For compression, the so-called 13-segment characteristic is used.
+* For compression,&nbsp; the so-called&nbsp; "13-segment"&nbsp; characteristic is used.
 This means:
 *In the range &nbsp;$|q_{\rm A}| ≤ 1/64$&nbsp; holds &nbsp;$q_{\rm K} = q_{\rm A}$.
-*For &nbsp;$q_{\rm A} > 1/64$&nbsp;, &nbsp;$k = 1$, ... , $6$&nbsp; the following six additional ranges of the compressor characteristic:
+*For &nbsp;$q_{\rm A} > 1/64$,&nbsp; there are the following six additional ranges &nbsp;$(k = 1$, ... , $6)$&nbsp; of the compressor characteristic:<br>&nbsp; &rArr; &nbsp; range $k\hspace{0.3cm}{\rm (if}\hspace{0.3cm} 2^{k-7}< q_{\rm A} \le 2^{k-6}) \hspace{0.05cm}$ &nbsp; &rArr; &nbsp; $q_{\rm K}(q_{\rm A}) = 2^{4-k} \cdot q_{\rm A} + {k}/{8};$
-:$$q_{\rm K}(q_{\rm A}) = 2^{4-k} \cdot q_{\rm A} + {k}/{8}\hspace{0.9cm} {\rm im\,\,range}\hspace{0.9cm}2^{k-7}< q_{\rm A} \le 2^{k-6} \hspace{0.05cm}.$$
+*Another six domains exist for negative &nbsp;$q_{\rm A}$&nbsp; values with &nbsp;$k = -1$, ... , $-6$,&nbsp; which are point-symmetric with respect to the origin.&nbsp; <br>However,&nbsp; these are not considered further in this exercise.
@@ Line 39: / Line 34: @@
 Hints:
-*The task belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|Pulse code modulation]].
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|"Pulse Code Modulation]].
-*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Compression_and_expansion|Compression and Expansion]].
+*Reference is made in particular to the section&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Compression_and_expansion|"Compression and Expansion"]].
@@ Line 47: / Line 42: @@
 <quiz display=simple>
-{If &nbsp;$q_{\rm A} = 0.4$.&nbsp; What is the output value &nbsp;$q_{\rm K}$&nbsp; of the compressor?
+{If &nbsp;$q_{\rm A} = 0.4$: &nbsp; What is the output value &nbsp;$q_{\rm K}$&nbsp; of the compressor?
 |type="{}"}
 $q_{\rm K} \ = \ $ { 0.825 3% }
-{ To which quantization interval &nbsp;$μ$&nbsp; does &nbsp;$q_{\rm A} = 0.4$ belong?
+{ To which quantization interval &nbsp;$μ$&nbsp; does &nbsp;$q_{\rm A} = 0.4$&nbsp; belong?
 |type="{}"}
 $\mu \ = \ $ { 233 }
-{which quantization value &nbsp;$q_{\rm Q}$&nbsp; belongs to &nbsp;$q_{\rm A} = 0.4$?
+{Which quantization value &nbsp;$q_{\rm Q}$&nbsp; belongs to &nbsp;$q_{\rm A} = 0.4$?
 |type="{}"}
 $q_{\rm Q} \ = \ $ { 0.824 3% }
-{In contrast, what quantization value &nbsp;$q_{\rm Q}$&nbsp; belongs to &nbsp;$q_{\rm A} = 0.04$?
+{In contrast,&nbsp; what quantization value &nbsp;$q_{\rm Q}$&nbsp; belongs to &nbsp;$q_{\rm A} = 0.04$?
 |type="{}"}
 $q_{\rm Q} \ = \ $ { 0.41 3% }
-{at the receiver, the input value is &nbsp;$v_{\rm Q} = 211/256 ≈ 0.824$&nbsp; What value &nbsp;$v_{\rm E}$&nbsp; does the expander provide?
+{At the receiver,&nbsp; the input value is &nbsp;$v_{\rm Q} = 211/256 ≈ 0.824$.&nbsp; What value &nbsp;$v_{\rm E}$&nbsp; does the expander provide?
 |type="{}"}
 $v_{\rm E} \ = \ $ { 0.398 3% }
-{What are the properties of the characteristic &nbsp;$q_{\rm Q}(q_{\rm A})$&nbsp;?
+{What are the properties of the&nbsp;  "non-linear quantizer characteristic" &nbsp;$q_{\rm Q}(q_{\rm A})$&nbsp;?
 |type="[]"}
 + The characteristic &nbsp;$q_{\rm Q}(q_{\rm A})$&nbsp; approximates the compressor characteristic in steps.
@@ Line 75: / Line 70: @@
 + The step height is equal in all segments&nbsp; $($except for&nbsp; $k = 0)$&nbsp;.
-{What are the properties of the characteristic &nbsp;$v_{\rm E}(q_{\rm A})$&nbsp;?
+{What are the properties of the&nbsp; "overall system characteristic" &nbsp;$v_{\rm E}(q_{\rm A})$&nbsp;?
 |type="[]"}
 - The characteristic &nbsp;$v_{\rm E}(q_{\rm A})$&nbsp; approximates the compressor characteristic in steps.

← Older revision		Revision as of 14:43, 7 April 2022
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−	{{quiz-Header\|Buchseite=~~Modulationsverfahren~~/~~Pulscodemodulation~~	+	{{quiz-Header\|Buchseite=Modulation_Methods/Pulse_Code_Modulation
	}}		}}

← Older revision		Revision as of 13:53, 16 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.1 ~~Pulscodemodulation~~^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1620__Mod_A_4_5.png|right|frame|PCM-System mit Kompandierung]]
+[[File:EN_Mod_Z_4_5.png|right|frame|PCM-System mit Kompandierung]]
 Zur Untersuchung der&nbsp; ''nichtlinearen Quantisierung''&nbsp; gehen wir vom skizzierten Systemmodell aus.
 * Den Einfluss des Kanals und der PCM–Codierung bzw. –Decodierung lassen wir  außer Acht.
 *Somit gilt stets &nbsp;$v_{\rm Q}(ν · T_{\rm A}) = q_{\rm Q}(ν · T_{\rm A})$, wobei im Weiteren auf die Zeitangabe &nbsp;$ν · T_{\rm A}$&nbsp; verzichtet wird.
+P_ID1620__Mod_A_4_5.png
 Durch den Vergleich von jeweils einer Ausgangsgröße mit einer Eingangsgröße kann man den Einfluss

← Older revision		Revision as of 12:46, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^4.1 Pulscodemodulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulscodemodulation^]]

@@ Line 40: / Line 40: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulationsverfahren/Pulscodemodulation|Pulscodemodulation]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Pulscodemodulation|Pulscodemodulation]].
-*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulationsverfahren/Pulscodemodulation#Kompression_und_Expandierung|Kompression und Expandierung]].
+*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulation_Methods/Pulscodemodulation#Kompression_und_Expandierung|Kompression und Expandierung]].