View source for Aufgaben:Exercise 4.18Z: BER of Coherent and Non-Coherent FSK


{{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}} 

[[File:EN_Dig_Z_4_18.png|right|frame|Bit error probabilities of BPSK and BFSK]]
The diagram shows the bit error probability for&nbsp;  [[Modulation_Methods/Non-Linear_Digital_Modulation#FSK_.E2.80.93_Frequency_Shift_Keying|"binary FSK modulation"]]&nbsp; $\rm (BFSK)$&nbsp; with
* coherent demodulation,&nbsp; or

* incoherent demodulation


in comparison with binary phase modulation&nbsp; $\rm (BPSK)$.&nbsp; Orthogonality is always assumed.&nbsp; 
*For coherent demodulation,&nbsp; the modulation index&nbsp; $h$&nbsp; can be a multiple of&nbsp; $0.5$,&nbsp; so that the purple curve is also valid for&nbsp; "Minimum Shift Keying"&nbsp; $\rm (MSK)$.&nbsp; 

*On the other hand,&nbsp; for non-coherent demodulation of a BFSK,&nbsp; the modulation index&nbsp; $h$&nbsp; must be a multiple of&nbsp; $1$.&nbsp;


This system comparison is based on the AWGN channel,&nbsp; characterized by the ratio&nbsp; $E_{\rm B}/N_0$.&nbsp; The equations for the bit error probabilities are as follows for
* BFSK with&nbsp; <U>coherent</u>&nbsp; demodulation:
:$$p_{\rm B} = {\rm Q } \left ( \sqrt {{E_{\rm B}}/{N_0} }\right ) \hspace{0.05cm}.$$

* BFSK with&nbsp; <U>non-coherent</u>&nbsp; demodulation:
:$$p_{\rm B} = {1}/{2} \cdot {\rm e}^{- E_{\rm B}/{(2N_0) }}\hspace{0.05cm}.$$

* BPSK,&nbsp; only&nbsp; <U>coherent</u>&nbsp; demodulation possible:
:$$p_{\rm B} = {\rm Q } \left ( \sqrt {{2 \cdot E_{\rm B}}/{N_0} }\right ) \hspace{0.05cm}.$$


<u>Remember:</u>
#For&nbsp; BPSK,&nbsp; the log ratio&nbsp; $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$&nbsp; must be at least&nbsp; $9.6 \, \rm dB$&nbsp; so that the bit error probability does not exceed the value&nbsp; $p_{\rm B} = 10^{\rm -5}$.&nbsp; 
#For binary modulation methods,&nbsp; $p_{\rm B}$&nbsp; can also be replaced by&nbsp; $p_{\rm S}$&nbsp; and &nbsp;$E_{\rm B}$&nbsp; by &nbsp;$E_{\rm S}$.&nbsp; 
#Then we speak of the symbol error probability&nbsp; $p_{\rm S}$&nbsp; and the symbol energy&nbsp; $E_{\rm S}$.



<u>Notes:</u>
* The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation| "Carrier Frequency Systems with Non-Coherent Demodulation"]].

* However,&nbsp; reference is also made to the chapter&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation| "Carrier Frequency Systems with Coherent Demodulation"]].

* Further information can be found in the book&nbsp; [[Modulation_Methods|"Modulation Methods"]].
* Use the approximation&nbsp; ${\rm lg}(2) \approx 0.3$.




===Questions===
<quiz display=simple>
{For BFSK and &nbsp;<u>coherent demodulation</u>,&nbsp; which&nbsp; $E_{\rm B}/N_0$&nbsp; is required to satisfy the requirement&nbsp; $p_{\rm B} &#8804; 10^{\rm -5}$?
|type="{}"}
$10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ = \ $ { 12.6 3% } $\ \rm dB$

{Are the following statements correct: &nbsp; The same result as in&nbsp; '''(1)'''&nbsp; is obtained for
|type="[]"}
- the coherent FSK with modulation index&nbsp; $\eta = 0.7$,
+ the coherent FSK with modulation index&nbsp; $\eta = 1$.

{For BFSK with modulation index&nbsp; $h = 1$&nbsp; and&nbsp; <u>non-coherent demodulation</u>,&nbsp; which&nbsp; $E_{\rm B}/N_0$&nbsp; is required for&nbsp; $p_{\rm B} &#8804; 10^{\rm -5}$&nbsp; to be satisfied?
|type="{}"}
$10 \cdot {\rm lg} \, E_{\rm B}/N_0 \ = \ ${ 13.4 3% } $\ \rm dB$

{What is the error probability with&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 = 12.6 \ \rm dB$&nbsp; for BFSK and&nbsp; <u>non-coherent demodulation</u>?
|type="{}"}
$p_{\rm B} \ = \ ${ 0.012 5% } $\ \%$
</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''&nbsp; A comparison of the equations in the information section makes it clear that for BFSK with coherent demodulation,&nbsp;  the AWGN ratio $E_{\rm B}/N_0$&nbsp;  must be doubled to achieve the same error probability as for BPSK.

*In other words: &nbsp; The coherent BFSK curve lies &nbsp;  $10 \cdot {\rm lg} \, (2) \approx 3 \ \rm dB$ &nbsp; to the right of the BPSK curve. 

*To guarantee&nbsp;  $p_{\rm B} &#8804; 10^{\rm &ndash;5}$,&nbsp;  it must hold:
:$$10 \cdot {\rm lg}\hspace{0.05cm} {E_{\rm B}}/ {N_{\rm 0}}\approx
 9.6\,\,{\rm dB} + 3\,\,{\rm dB}\hspace{0.15cm} \underline{=12.6\,\,{\rm dB}}\hspace{0.05cm}.$$


'''(2)'''&nbsp; <u>Solution 2</u>&nbsp;  is correct:
*The given equation is valid not only for the MSK&nbsp;  $($this is a BFSK with&nbsp;  $h = 0.5)$,&nbsp;  but for any form of orthogonal BFSK.

*Such a BFSK exists if the modulation index&nbsp;  $h$&nbsp;  is an integer multiple of&nbsp;  $0.5$,&nbsp;  for example for&nbsp;  $h = 1$.
 
*With&nbsp;  $h = 0.7$&nbsp;  there is no orthogonal FSK.

*It can be shown that for&nbsp;  $h = 0.7$&nbsp;  there is even a smaller error probability than with orthogonal FSK:

*With&nbsp;  $10 \cdot {\rm lg} \, E_{\rm B}/N_0 = 12.6 \ \rm dB$ one even achieves $p_{\rm B} \approx 10^{\rm &ndash;6}$,&nbsp;  here,&nbsp;  i.e.&nbsp;  an improvement by one power of ten.




'''(3)'''&nbsp; From the inverse function of the given equation,&nbsp;  one obtains:
:$$\frac{E_{\rm B}} {2 \cdot N_{\rm 0}}= {\rm ln}\hspace{0.05cm}\frac{1}{2  p_{\rm B}}= {\rm
 ln}(50000)\approx 10.82\hspace{0.3cm} 
\Rightarrow \hspace{0.3cm} {E_{\rm B}}/ {N_{\rm 0}}= 21.64 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 10 \cdot {\rm lg}\hspace{0.09cm} {E_{\rm B}}/ {N_{\rm 0}}\hspace{0.15cm}
 \underline{\approx 13.4\,\,{\rm dB}}\hspace{0.05cm}.$$



'''(4)'''&nbsp; From&nbsp;  $10 \cdot {\rm lg} \, E_{\rm B}/N_0 = 12.6 \ \rm dB$&nbsp;  follows:
:$${E_{\rm B}} /{N_{\rm 0}}= 10^{1.26} \approx 16.8 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 \frac{E_{\rm B}} {2 \cdot N_{\rm 0}}\approx 8.4 \hspace{0.3cm}
\Rightarrow \hspace{0.3cm} p_{\rm B} = {1}/{2} \cdot {\rm e}^{- 8.4}\hspace{0.15cm} \underline{ \approx  0.012 \%= 12 \cdot 10^{\rm -5}}\hspace{0.05cm}.$$

This means: &nbsp; For the same&nbsp;  $E_{\rm B}/N_0$,&nbsp;  the error probability for non-coherent demodulation is increased by a factor of about&nbsp; $12$&nbsp; compared to coherent demodulation according to subtask&nbsp;  '''(1)'''.
{{ML-Fuß}}



[[Category:Digital Signal Transmission: Exercises|^4.5 Non-Coherent Demodulation^]]