Difference between revisions of "Aufgaben:Exercise 2.2Z: Average Code Word Length"
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}} | }} | ||
| − | [[File:P_ID2417__Inf_Z_2_2.png|right|frame| | + | [[File:P_ID2417__Inf_Z_2_2.png|right|frame|Three source coding tables]] |
The aim of data compression is to represent the message of a source with as few binary characters as possible. | The aim of data compression is to represent the message of a source with as few binary characters as possible. | ||
| − | We consider here a discrete-value message source with the symbol set $\rm \{ A, \ B, \ C, \ D\}$ ⇒ symbol | + | We consider here a discrete-value message source with the symbol set $\rm \{ A, \ B, \ C, \ D\}$ ⇒ symbol set size $M = 4$ and the symbol probabilities |
| − | :*$p_{\rm A} = p_{\rm B} = p_{\rm C} = p_{\rm D} = 1/4$ (subtask 1), | + | :*$p_{\rm A} = p_{\rm B} = p_{\rm C} = p_{\rm D} = 1/4$ $($subtask $1)$, |
| − | :* $p_{\rm A} = 1/2, \, p_{\rm B} = 1/4, \, p_{\rm C} = p_{\rm D} = 1/8$ ( | + | :* $p_{\rm A} = 1/2, \, p_{\rm B} = 1/4, \, p_{\rm C} = p_{\rm D} = 1/8$ $($subtask $2$ to $5)$. |
| − | It is assumed that there are no statistical | + | It is assumed that there are no statistical Dependencies between the individual source symbols. |
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Three assignments are given. To be noted: | Three assignments are given. To be noted: | ||
* Each of these binary codes $\rm C1$, $\rm C2$ and $\rm C3$ is designed for a specific source statistic. | * Each of these binary codes $\rm C1$, $\rm C2$ and $\rm C3$ is designed for a specific source statistic. | ||
* All codes are prefix-free and thus immediately decodable without further specification. | * All codes are prefix-free and thus immediately decodable without further specification. | ||
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| + | A measure for the quality of a compression method is the average codeword length $L_{\rm M}$ with the additional unit "bit/source symbol". | ||
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''Hint:'' | ''Hint:'' | ||
| − | *The | + | *The exercise belongs to the chapter [[Information_Theory/Allgemeine_Beschreibung|General Description of Source Coding]]. |
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<quiz display=simple> | <quiz display=simple> | ||
| − | {Determine the | + | {Determine the average codeword length $L_{\rm M}$ for $p_{\rm A} = p_{\rm B} = p_{\rm C} = p_{\rm D} = 1/4$. |
|type="{}"} | |type="{}"} | ||
| − | $\text{C1:}\ \ L_{\rm M} \ = \ $ { 2 1% } $\ \rm bit/source symbol$ | + | $\text{C1:}\ \ L_{\rm M} \ = \ $ { 2 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
| − | $\text{C2:}\ \ L_{\rm M} \ = \ $ { 2.25 1% } $\ \rm bit/source symbol$ | + | $\text{C2:}\ \ L_{\rm M} \ = \ $ { 2.25 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
| − | $\text{C3:}\ \ L_{\rm M} \ = \ $ { 2.25 1% } $\ \rm bit/source symbol$ | + | $\text{C3:}\ \ L_{\rm M} \ = \ $ { 2.25 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
{Which values result for $p_{\rm A} = 1/2, \, p_{\rm B} = 1/4, \, p_{\rm C} = p_{\rm D} = 1/8$? | {Which values result for $p_{\rm A} = 1/2, \, p_{\rm B} = 1/4, \, p_{\rm C} = p_{\rm D} = 1/8$? | ||
|type="{}"} | |type="{}"} | ||
| − | $\text{C1:}\ \ L_{\rm M} \ = \ $ { 2 1% } $\ \rm bit/source symbol$ | + | $\text{C1:}\ \ L_{\rm M} \ = \ $ { 2 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
| − | $\text{C2:}\ \ L_{\rm M} \ = \ $ { 1.75 1% } $\ \rm bit/source symbol$ | + | $\text{C2:}\ \ L_{\rm M} \ = \ $ { 1.75 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
| − | $\text{C3:}\ \ L_{\rm M} \ = \ $ { 2.5 1% } $\ \rm bit/source symbol$ | + | $\text{C3:}\ \ L_{\rm M} \ = \ $ { 2.5 1% } $\ \rm bit/source\hspace{0.15cm} symbol$ |
{How can you recognise prefix-free codes? | {How can you recognise prefix-free codes? | ||
|type="[]"} | |type="[]"} | ||
| − | + No | + | + No codeword is the beginning of another codeword. |
- All codewords have the same length. | - All codewords have the same length. | ||
| − | {For the special source symbol sequence $\rm ADBDCBCBADCA$ , the code symbol sequence $\rm 001101111001100100111000$ results. | + | {For the special source symbol sequence $\rm ADBDCBCBADCA$ , the code symbol sequence $\rm 001101111001100100111000$ results. |
<br>Which code was used? | <br>Which code was used? | ||
|type="()"} | |type="()"} | ||
| − | + | + | + The code $\rm C1$, |
- the code $\rm C2$. | - the code $\rm C2$. | ||
| − | {After coding with $\rm C3$ | + | {After coding with $\rm C3$, you get $\rm 001101111001100100111000$. What is the corresponding source symbol sequence? |
|type="()"} | |type="()"} | ||
- $\rm AACDBACABADAAA$ ... | - $\rm AACDBACABADAAA$ ... | ||
Revision as of 17:04, 7 July 2021
Three source coding tables
The aim of data compression is to represent the message of a source with as few binary characters as possible.
We consider here a discrete-value message source with the symbol set $\rm \{ A, \ B, \ C, \ D\}$ ⇒ symbol set size $M = 4$ and the symbol probabilities
- $p_{\rm A} = p_{\rm B} = p_{\rm C} = p_{\rm D} = 1/4$ $($subtask $1)$,
- $p_{\rm A} = 1/2, \, p_{\rm B} = 1/4, \, p_{\rm C} = p_{\rm D} = 1/8$ $($subtask $2$ to $5)$.
It is assumed that there are no statistical Dependencies between the individual source symbols.
Three assignments are given. To be noted:
- Each of these binary codes $\rm C1$, $\rm C2$ and $\rm C3$ is designed for a specific source statistic.
- All codes are prefix-free and thus immediately decodable without further specification.
A measure for the quality of a compression method is the average codeword length $L_{\rm M}$ with the additional unit "bit/source symbol".
Hint:
- The exercise belongs to the chapter General Description of Source Coding.
Questions
Solution
(1) The average codeword length is generally given by
- $$L_{\rm M} = p_{\rm A} \cdot L_{\rm A} + p_{\rm B} \cdot L_{\rm B}+ p_{\rm C} \cdot L_{\rm C} + p_{\rm D} \cdot L_{\rm D} \hspace{0.05cm}.$$
If the four source symbols are equally probable $($all probabilities exactly $1/4)$, then for this we can also write:
- $$L_{\rm M} = 1/4 \cdot ( L_{\rm A} + L_{\rm B}+ L_{\rm C} + L_{\rm D}) \hspace{0.05cm}.$$
- $\text{Code C1:}$ $L_{\rm M} \hspace{0.15cm}\underline{= 2.00}\ \rm bit/source symbol$,
- $\text{Code C2:}$ $L_{\rm M} \hspace{0.15cm}\underline{= 2.25}\ \rm bit/source symbol$
- $\text{Code C3:}$ $L_{\rm M} \hspace{0.15cm}\underline{= 2.25}\ \rm bit/source symbol$.
(2) With the code table $\text{C1}$ , the average code word length $L_{\rm M} \hspace{0.15cm}\underline{= 2}\ \rm bit/source symbol$ is always obtained, independent of the symbol probabilities.
For the other two codes one obtains:
- $\text{Code C2:}$ $L_{\rm M} = 1/2 \cdot 1 + 1/4 \cdot 2 + 1/8 \cdot 3 + 1/8 \cdot 3 \hspace{0.15cm}\underline{= 1.75}\ \rm bit/source symbol$,
- $\text{Code C3:}$ $L_{\rm M} = 1/2 \cdot 3 + 1/4 \cdot 2 + 1/8 \cdot 1 + 1/8 \cdot 3 \hspace{0.15cm}\underline{= 2.50}\ \rm bit/source symbol$.
You can see the principle from the example:
- Probable symbols are represented by a few binary symbols, improbable ones by more.
- In the case of equally probable symbols, it is best to choose the same codeword lengths.
(3) Solution suggestion 1 is correct:
- The code $\text{C1}$ with uniform length of all codewords is prefix-free,
- but other codes can also be prefix-free, for example the codes $\text{C2}$ and $\text{C3}$.
(4) Solution suggestion 1 is correct:
- Already from "00" at the beginning one can see that the code $\text{C2}$ is out of the question here, because otherwise the source symbol sequence would have to begin with "AA".
- In fact, the code $\text{C1}$ was used.
(5) Solution suggestion 2 is correct:
- The first suggested solution, on the other hand, gives the source symbol sequence for code $\text{C2}$ if the code symbol sequence would be $\rm 001101111001100100111000$ .
