Aufgaben:Exercise 5.6Z: Single-Carrier and Multi-Carrier System: Difference between revisions
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{{quiz-Header|Buchseite= | {{quiz-Header|Buchseite=Modulation_Methods/General_Description_of_OFDM | ||
}} | }} | ||
[[File:P_ID1660__Z_5_6.png| | [[File:P_ID1660__Z_5_6.png|right|frame|Signal space assignments for $\rm SC$ (above), $\rm MC$ (bottom)]] | ||
In | In this exercise, a comparison is to be made between | ||
* | *a single-carrier $\rm (SC)$ system $(N = 1)$, and | ||
* | *a multi-carrier $\rm (MC)$ system with $N = 32$ carriers. | ||
For both transmission systems (see diagram), a data bit rate of $R_{\rm B} = 1 \ \rm Mbit/s$ is required in each case. | |||
=== | Notes: | ||
*The exercise belongs to the chapter [[Modulation_Methods/Allgemeine_Beschreibung_von_OFDM|General Description of OFDM]]. | |||
*Reference is also made to the chapter [[Modulation_Methods/Quadratur%E2%80%93Amplitudenmodulation|Quadrature Amplitude Modulation]]. | |||
===Questions=== | |||
<quiz display=simple> | <quiz display=simple> | ||
{ | {Which mapping does the single-carrier system use? | ||
|type=" | |type="()"} | ||
- ASK, | - ASK, | ||
+ BPSK, | + BPSK, | ||
| Line 28: | Line 30: | ||
- 16-QAM | - 16-QAM | ||
{ | {Which mapping does the multi-carrier system use? | ||
|type=" | |type="()"} | ||
- ASK, | - ASK, | ||
- BPSK, | - BPSK, | ||
| Line 35: | Line 37: | ||
+ 16-QAM | + 16-QAM | ||
{ | {Calculate the symbol duration $T_{\rm SC}$ of the single-carrier system. | ||
|type="{}"} | |type="{}"} | ||
$T_{\rm SC} \ = \ $ { 1 3% } $\ \rm | $T_{\rm SC} \ = \ $ { 1 3% } $\ \rm µ s$ | ||
{ | {Calculate the symbol duration $T_{\rm MC}$ of the multi-carrier system. | ||
|type="{}"} | |type="{}"} | ||
$T_{\rm | $T_{\rm MC} \ = \ $ { 128 3% } $\ \rm µ s$ | ||
{ | {Which of the following statements is true? | ||
|type=" | |type="()"} | ||
- | - The intersymbol interferences are independent of the symbol duration $T$. | ||
+ | + The intersymbol interferences decrease with increasing symbol duration $T$. | ||
- | - The intersymbol interferences increase with increasing symbol duration $T$. | ||
</quiz> | </quiz> | ||
=== | ===Solution=== | ||
{{ML-Kopf}} | {{ML-Kopf}} | ||
'''(1)''' From the diagram on the front page, it is immediately apparent that the single-carrier system is based on "binary phase modulation" $\rm (BPSK)$ ⇒ <u>solution 2</u>. | |||
'''(1)''' | |||
'''(2)''' In contrast, the multi-carrier system is based on $\rm 16–QAM$ ⇒ <u>solution 4</u>. | |||
'''(3)''' | '''(3)''' In general, for an OFDM system with $N$ carriers and $M$ signal space points, the symbol duration is: | ||
:$$T = N \cdot {\rm{log}_2}\hspace{0.04cm}(M) \cdot T_{\rm{B}}.$$ | :$$T = N \cdot {\rm{log}_2}\hspace{0.04cm}(M) \cdot T_{\rm{B}}.$$ | ||
*Because of $R_{\rm{B}} = 1 \ \rm Mbit/s$, the bit duration for BPSK is equal to $T_{\rm{B}} = 1 \ \rm µ s$. | |||
:$$ T_{\rm{SC}} = 1 \cdot {\rm{log}_2}\hspace{0.04cm}(2) \cdot T_{\rm{B}}\hspace{0.15cm}\underline {= 1\,\,{\rm \ | *From this, the symbol duration of the single-carrier system with $N = 1$ and $M = 2$ is: | ||
:$$ T_{\rm{SC}} = 1 \cdot {\rm{log}_2}\hspace{0.04cm}(2) \cdot T_{\rm{B}}\hspace{0.15cm}\underline {= 1\,\,{\rm µ s}}.$$ | |||
'''(4)''' Similarly, for the multi-carrier system with $N = 32$ and $M = 16$, we obtain: | |||
:$$T_{\rm{MC}} = 32 \cdot {\rm{log}_2}\hspace{0.04cm}(16) \cdot T_{\rm{B}}\hspace{0.15cm}\underline {= 128\,\,{\rm µ s}}.$$ | |||
'''(5)''' | '''(5)''' <u>Solution 2</u> is correct because: | ||
*At large symbol duration, the relative fraction extending from the predecessor symbol into the symbol under consideration and thus causing intersymbol interference $\rm (ISI)$ is smaller than at small symbol duration. | |||
{{ML-Fuß}} | {{ML-Fuß}} | ||
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[[Category: | [[Category:Modulation Methods: Exercises|^5.5 General Description of OFDM^]] | ||
[[de:Aufgaben:Aufgabe 5.6Z: Einträger–und Mehrträgersystem]] | |||
Latest revision as of 17:53, 16 March 2026
In this exercise, a comparison is to be made between
- a single-carrier $\rm (SC)$ system $(N = 1)$, and
- a multi-carrier $\rm (MC)$ system with $N = 32$ carriers.
For both transmission systems (see diagram), a data bit rate of $R_{\rm B} = 1 \ \rm Mbit/s$ is required in each case.
Notes:
- The exercise belongs to the chapter General Description of OFDM.
- Reference is also made to the chapter Quadrature Amplitude Modulation.
Questions
Solution
(1) From the diagram on the front page, it is immediately apparent that the single-carrier system is based on "binary phase modulation" $\rm (BPSK)$ ⇒ solution 2.
(2) In contrast, the multi-carrier system is based on $\rm 16–QAM$ ⇒ solution 4.
(3) In general, for an OFDM system with $N$ carriers and $M$ signal space points, the symbol duration is:
- $$T = N \cdot {\rm{log}_2}\hspace{0.04cm}(M) \cdot T_{\rm{B}}.$$
- Because of $R_{\rm{B}} = 1 \ \rm Mbit/s$, the bit duration for BPSK is equal to $T_{\rm{B}} = 1 \ \rm µ s$.
- From this, the symbol duration of the single-carrier system with $N = 1$ and $M = 2$ is:
- $$ T_{\rm{SC}} = 1 \cdot {\rm{log}_2}\hspace{0.04cm}(2) \cdot T_{\rm{B}}\hspace{0.15cm}\underline {= 1\,\,{\rm µ s}}.$$
(4) Similarly, for the multi-carrier system with $N = 32$ and $M = 16$, we obtain:
- $$T_{\rm{MC}} = 32 \cdot {\rm{log}_2}\hspace{0.04cm}(16) \cdot T_{\rm{B}}\hspace{0.15cm}\underline {= 128\,\,{\rm µ s}}.$$
(5) Solution 2 is correct because:
- At large symbol duration, the relative fraction extending from the predecessor symbol into the symbol under consideration and thus causing intersymbol interference $\rm (ISI)$ is smaller than at small symbol duration.
