Aufgaben:Exercise 4.18: Non-Coherent FSK Demodulation - Revision history

Guenter at 14:25, 8 October 2022

2022-10-08T14:25:29Z

Hwang at 03:34, 11 September 2022

2022-09-11T03:34:44Z

Guenter at 12:53, 30 August 2022

2022-08-30T12:53:21Z

Guenter at 12:40, 30 August 2022

2022-08-30T12:40:25Z

Guenter at 15:04, 24 August 2022

2022-08-24T15:04:21Z

Hwang at 17:10, 16 August 2022

2022-08-16T17:10:57Z

Hwang at 15:01, 16 August 2022

2022-08-16T15:01:10Z

Guenter at 15:27, 15 August 2022

2022-08-15T15:27:29Z

Guenter: Guenter moved page Aufgaben:Exercise 4.18: Non-coherent FSK Demodulation to Aufgaben:Exercise 4.18: Non-Coherent FSK Demodulation

2022-08-15T15:24:26Z

Guenter moved page Exercise 4.18: Non-coherent FSK Demodulation to Exercise 4.18: Non-Coherent FSK Demodulation

Guenter: Guenter moved page Aufgaben:Aufgabe 4.18: Nichtkohärente FSK–Demodulation to Aufgaben:Exercise 4.18: Non-coherent FSK Demodulation

2022-08-15T15:21:49Z

Guenter moved page Aufgabe 4.18: Nichtkohärente FSK–Demodulation to Exercise 4.18: Non-coherent FSK Demodulation

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}
-[[File:EN_Dig_A_4_18.png|right|frame|Block diagram for non-coherent demodulation '''KORREKTUR: amount -> magnitude''']]
+[[File:EN_Dig_Z_4_18_neu.png|right|frame|Block diagram for non-coherent demodulation]]
 We consider&nbsp; "Frequency Shift Keying"&nbsp; $\rm (FSK)$&nbsp; with&nbsp; $M = 2$ &nbsp; &#8658; &nbsp; binary signaling.&nbsp; The two low-pass basis functions in this case are complex:
 :$$\xi_1(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},$$

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}
-[[File:EN_Dig_A_4_18.png|right|frame|Block diagram for non-coherent demodulation]]
+[[File:EN_Dig_A_4_18.png|right|frame|Block diagram for non-coherent demodulation '''KORREKTUR: amount -> magnitude''']]
 We consider&nbsp; "Frequency Shift Keying"&nbsp; $\rm (FSK)$&nbsp; with&nbsp; $M = 2$ &nbsp; &#8658; &nbsp; binary signaling.&nbsp; The two low-pass basis functions in this case are complex:
 :$$\xi_1(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},$$

@@ Line 72: / Line 72: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Solutions 1, 2 and 4</u> are correct:
+'''(1)'''&nbsp; <u>Solutions 1, 2 and 4</u>&nbsp; are correct:
-*In coherent demodulation, orthogonality is given when the modulation index $h$ is a multiple of $0.5$. Binary FSK with $h = 0.5$ is also called <i>Minimum Shift Keying</i> (MSK).
+*In coherent demodulation,&nbsp; orthogonality is given when the modulation index&nbsp; $h$&nbsp; is a multiple of&nbsp; $0.5$.&nbsp; Binary FSK with&nbsp; $h = 0.5$&nbsp; is also called&nbsp; "Minimum Shift Keying"&nbsp; $\rm (MSK)$.
 *Since phase control compensates for the phase shift (delay) on the AWGN transmission channel &nbsp; &#8658; &nbsp; ${\rm e}^{{\rm j}\phi} \cdot {\rm e}^{-{\rm j}\phi} = 1$, the signals in the equivalent low-pass region are actually: &nbsp; $r(t) = s(t) + n(t)   \hspace{0.05cm}.$
-'''(2)'''&nbsp; Here, only the <u>first solution</u> is correct, i.e., now $h = 1, \ 2, \ 3, \text{ ...}$ must be an integer.
+'''(2)'''&nbsp; Here,&nbsp; only the&nbsp; <u>first solution</u>&nbsp; is correct,&nbsp; i.e., now $h = 1, \ 2, \ 3, \text{ ...}$&nbsp; must be an integer.
 *So a non-coherent demodulation of FSK is not possible.
 *Because of the missing phase control, the following is also true:
 :$$r(t) = s(t) \cdot {\rm e }^{ - {\rm j }\hspace{0.05cm}\phi} + n(t)   \hspace{0.05cm}.$$
@@ Line 87: / Line 89: @@
 :$$p_{\rm S} = {\rm Pr}({\cal{E}}) = {\rm Pr}({\cal{E}}| \hspace{0.05cm}m = m_0) =    {\rm Pr}(\hat{m}= m_1| \hspace{0.05cm}m = m_0)\hspace{0.05cm}.$$
 *This probability is obtained from the given equation with $\gamma = 1$.
 *In this case, ${\it \Gamma} ={\rm Z}$ is always and the decision rule is: &nbsp; Decide on the symbol $m_0$ if $y_1 > y_2$.
 *It follows that:
 :$$p_{\rm S} = \frac{1}{1 + \gamma^2} \cdot
@@ Line 94: / Line 98: @@
   {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)}\hspace{0.05cm}.$$
-*With $E_{\rm S}/N_0 = 10$, we get $p_{\rm S} = 1/2 \cdot {\rm e}^{\rm &ndash;5} \approx \underline {0.337 \cdot \%}$.
+*With&nbsp; $E_{\rm S}/N_0 = 10$,&nbsp; we get&nbsp; $p_{\rm S} = 1/2 \cdot {\rm e}^{\rm &ndash;5} \approx \underline {0.337 \cdot \%}$.
-'''(4)'''&nbsp; For symmetry reasons, this probability results in
+'''(4)'''&nbsp; For symmetry reasons,&nbsp; this probability results in
 :$${\rm Pr}({\it \Gamma} = {\rm Z}\hspace{0.05cm}\cap\hspace{0.05cm} {\rm error}  ) =
   {1}/{2} \cdot {\rm Pr}\{(\hat{m} = m_1)\hspace{0.05cm}\cap\hspace{0.05cm}{\it \Gamma} = {\rm Z} \hspace{0.05cm}| \hspace{0.05cm}m_0 \} + {1}/{2}\cdot {\rm Pr}\{(\hat{m} = m_0)\hspace{0.05cm}\cap\hspace{0.05cm}({\it \Gamma} = {\rm Z}) \hspace{0.05cm}| \hspace{0.05cm}m_1 \} $$
@@ Line 106: / Line 110: @@
-'''(5)'''&nbsp; Since ${\rm U}$ and ${\rm Z}$ give a complete system according to statistics, with the results of subtasks '''(3)''' and '''(4)''':
+'''(5)'''&nbsp; Since&nbsp; ${\rm U}$&nbsp; and&nbsp; ${\rm Z}$&nbsp; give a&nbsp; "complete system"&nbsp; according to statistics,&nbsp; with the results of subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)''':
 :$${\rm Pr}({\it \Gamma} = {\rm U}\hspace{0.05cm}\cap\hspace{0.05cm} {\rm error}  ) ={\rm Pr}( {\rm error}  ) -
   {\rm Pr}({\it \Gamma} = {\rm Z}\hspace{0.05cm}\cap\hspace{0.05cm} {\rm error}  ) = 3.37 \cdot 10^{-3} - 6.7 \cdot 10^{-5} = \underline{3.3 \cdot 10^{-3}}
   \hspace{0.05cm}.$$
-*Thus, the conditional probability we are looking for is ${\rm Pr}({\it \Gamma}= {\rm U}\hspace{0.05cm}|\hspace{0.05cm} {\rm error})$:
+*Thus,&nbsp; the conditional probability,&nbsp; we are looking for,&nbsp; is&nbsp; "${\rm Pr}({\it \Gamma}= {\rm U}\hspace{0.05cm}|\hspace{0.05cm} {\rm error})$":
 :$${\rm Pr}({\it \Gamma}= {\rm U}\hspace{0.05cm}|\hspace{0.05cm} {\rm error}  )  =
-\frac{{\rm Pr}({\it \Gamma} = {\rm U}\hspace{0.05cm}\cap\hspace{0.05cm} {\rm error}  ) }{{\rm Pr}( {\rm error}  )} = \frac{3.3 \cdot 10^{-3}}{3.37 \cdot 10^{-3}}  \hspace{0.15cm} \underline{= 98 \%}
+\frac{{\rm Pr}({\it \Gamma} = {\rm U}\hspace{0.05cm}\cap\hspace{0.05cm} {\rm error}  ) }{{\rm Pr}( {\rm error}  )} = \frac{3.3 \cdot 10^{-3}}{3.37 \cdot 10^{-3}}  \hspace{0.15cm} \underline{\approx 98 \%}
   \hspace{0.05cm}.$$
 {{ML-Fuß}}

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}
-[[File:EN_Dig_A_4_18.png|right|frame|Non-coherent demodulation]]
+[[File:EN_Dig_A_4_18.png|right|frame|Block diagram for non-coherent demodulation]]
-We consider <i>Frequency Shift Keying</i>&nbsp; (FSK) with&nbsp; $M = 2$ &nbsp; &#8658; &nbsp; binary signaling. The two low-pass basis functions in this case are complex and are:
+We consider&nbsp; "Frequency Shift Keying"&nbsp; $\rm (FSK)$&nbsp; with&nbsp; $M = 2$ &nbsp; &#8658; &nbsp; binary signaling.&nbsp; The two low-pass basis functions in this case are complex:
 :$$\xi_1(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},$$
 :$$ \xi_2(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}-{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T \hspace{0.05cm}.$$
-Then, for the two possible waveforms in the low-pass region, with the mean symbol energy&nbsp; $E_{\rm S}$:
+Then,&nbsp; for the two possible waveforms in the low-pass region,&nbsp; with the mean symbol energy&nbsp; $E_{\rm S}$:
 :$$m_0\text{:}\hspace{0.2cm} s_{\rm TP,\hspace{0.05cm}0}  = \sqrt{E_{\rm S}} \cdot \xi_1(t) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\boldsymbol{ s}_{\rm 0} = (\sqrt{E_{\rm S}}, 0)\hspace{0.05cm},$$
 :$$m_1\text{:}\hspace{0.2cm} s_{\rm TP,\hspace{0.05cm}1}  = \sqrt{E_{\rm S}} \cdot \xi_2(t) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\boldsymbol{ s}_{\rm 1} = (0, \sqrt{E_{\rm S}})\hspace{0.05cm}.$$
-Here,&nbsp; $h$&nbsp; indicates the so-called <i>modulation index</i>. This must meet certain criteria to result in orthogonal waveforms. However, these criteria depend on whether a coherent or a non-coherent demodulator is used at the receiver.
+Here,&nbsp; $h$&nbsp; indicates the so-called&nbsp; "modulation index".&nbsp; This must meet certain criteria to result in orthogonal waveforms.&nbsp; However,&nbsp; these criteria depend on whether a coherent or a non-coherent demodulator is used at the receiver.
-The diagram shows the non-coherent demodulator for binary <i>Frequency Shift Keying</i>&nbsp; (FSK) in the lower section. All complex signals are labeled in blue, complex values in green, and real values in red.
+The diagram shows the non-coherent demodulator for&nbsp; "Binary Frequency Shift Keying"&nbsp; $\rm (BFSK)$&nbsp; in the lower section.&nbsp; All complex signals are labeled in blue,&nbsp; complex values in green,&nbsp; and real values in red.
-Compared to the decision process given in the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation| "theory part"]],&nbsp; we now consider a complex decision that outputs a <i>safety flag</i>&nbsp; ${\it \Gamma} = \{{\rm Z}, \ {\rm U}\}$&nbsp; in addition to the estimated value. The logical values&nbsp; ${\rm Z}$&nbsp; and&nbsp; ${\rm U}$&nbsp; stand for a realiable ('''Z'''uverlässige) and an '''U'''nreliable decision, respectively. Thus, there are four possibilities for the decision, controlled by the parameter&nbsp; $\gamma$:
+Compared to the decision process given in the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation| "theory part"]],&nbsp; we now consider a complex decision that outputs a&nbsp; "safety flag"&nbsp; ${\it \Gamma} = \{{\rm Z}, \ {\rm U}\}$&nbsp; in addition to the estimated value.&nbsp; The logical values&nbsp; ${\rm Z}$&nbsp; and&nbsp; ${\rm U}$&nbsp; stand for
-:$$\hat{m} \hspace{-0.1cm} \ = \ \hspace{0.1cm} m_0,\hspace{0.05cm}{\it \Gamma} = {\rm Z}, \hspace{0.5cm}{\rm falls} \hspace{0.15cm}y_1 > \gamma \cdot y_2\hspace{0.05cm},$$
+*realiable decision&nbsp; (German: "zuverlässig" &nbsp; &rArr; &nbsp; subscript:&nbsp; "Z"),
-:$$\hat{m} \hspace{-0.1cm} \ = \ \hspace{0.1cm} m_0,\hspace{0.05cm}{\it \Gamma} = {\rm U}, \hspace{0.5cm}{\rm falls} \hspace{0.15cm}y_2 < y_1 < \gamma \cdot y_2\hspace{0.05cm},$$
+* unreliable decision&nbsp; (German: "unzuverlässig" &nbsp; &rArr; &nbsp; subscript:&nbsp; "U").
-For the probability that the decision erroneously chooses the symbol&nbsp; $m_1$&nbsp; and, moreover, indicates that this decision is to be considered reliable (particularly reprehensible), the following applies
+Thus,&nbsp; there are four possibilities for the decision,&nbsp; controlled by the parameter&nbsp; $\gamma$:
-:$${\rm Pr}\{\hat{m} = m_1,\hspace{0.05cm}{\it \Gamma} = {\rm Z} \hspace{0.05cm}| \hspace{0.05cm}m_0 \} = \frac{1}{1 + \gamma^2} \cdot
+:$$\hat{m} \hspace{-0.1cm} \ = \ \hspace{0.1cm} m_0,\hspace{0.15cm}{\it \Gamma} = {\rm Z}, \hspace{0.5cm}{\rm if} \hspace{0.15cm}y_1 > \gamma \cdot y_2\hspace{0.05cm},$$
   {\rm exp } \left [ - \frac{\gamma^2 \cdot E_{\rm S}}{(1+\gamma^2) \cdot N_{\rm 0}}\right ]   \hspace{0.05cm}.$$
@@ Line 31: / Line 36: @@
-''Note:''
+Note:&nbsp; The exercise belongs to the the chapter&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation|"Carrier Frequency Systems with Non-Coherent Demodulation"]].
@@ Line 39: / Line 43: @@
 ===Questions===
 <quiz display=simple>
-{Which statements are true in the case of coherent demodulation of the FSK?
+{Which statements are true in the case of coherent demodulation of the BFSK?
 |type="[]"}
 + Orthogonality results if&nbsp; $h$&nbsp; is an integer.
 + Orthogonal waveforms also result for&nbsp; $h = 0.5$.&nbsp;
 - Orthogonality cannot be achieved in principle.
-+ For the AWGN channel&nbsp; $r(t) = s(t) + n(t)$ is valid.
++ For the AWGN channel  &nbsp; $r(t) = s(t) + n(t)$ &nbsp; is valid.
-{Which statements are true for the non-coherent demodulation of FSK?
+{Which statements are true for the non-coherent demodulation of the BFSK?
 |type="[]"}
 + Orthogonality results when&nbsp; $h$&nbsp; is an integer.
 - Orthogonal waveforms also result for&nbsp; $h = 0.5$.&nbsp;
 - Orthogonality cannot be achieved in principle.
-- For the AWGN channel&nbsp; $r(t) = s(t) + n(t)$ is valid.
+- For the AWGN channel &nbsp; $r(t) = s(t) + n(t)$ &nbsp; is valid.
-{Let&nbsp; $E_{\rm S}/N_0 = 10$. What is the error probability, i.e., the probability that the estimated value deviates from the transmitted message?
+{Let&nbsp; $E_{\rm S}/N_0 = 10$.&nbsp; What is the error probability,&nbsp; i.e.,&nbsp; the probability that the estimated value deviates from the transmitted message?
 |type="{}"}
 $p_{\rm S} \ = \ $ { 0.337 3% } $\ \%$
-{Let&nbsp; $\gamma = 2$&nbsp; and&nbsp; $E_{\rm S}/N_0 = 10$. What is the probability that the security feature signals a reliable decision despite the errors?
+{Let&nbsp; $\gamma = 2$&nbsp; and&nbsp; $E_{\rm S}/N_0 = 10$.&nbsp; What is the probability that the security feature signals a reliable decision despite the errors?
 |type="{}"}
 ${\rm Pr}({\it \Gamma} = {\rm Z\hspace{0.05cm}  | \hspace{0.05cm}error}) \ = \ $ { 6.7 3% } $\ \cdot 10^{\rm &ndash;5}$
-{Furthermore, let&nbsp; $\gamma = 2$ and&nbsp; $E_{\rm S}/N_0 = 10$. What is the (conditional) probability that the additional information "unreliable" is displayed in case of an error?
+{Furthermore,&nbsp; let&nbsp; $\gamma = 2$&nbsp; and&nbsp; $E_{\rm S}/N_0 = 10$.&nbsp; What is the&nbsp; (conditional)&nbsp; probability that the additional information&nbsp; "unreliable"&nbsp; is displayed in case of an error?
 |type="{}"}
 ${\rm Pr}({\it \Gamma} = {\rm U\hspace{0.05cm}  | \hspace{0.05cm}error}) \ = \ $ { 98 3% } $\ \%$

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}
-[[File:P_ID2080__Dig_A_4_18.png|right|frame|Non-coherent demodulation]]
+[[File:EN_Dig_A_4_18.png|right|frame|Non-coherent demodulation]]
 We consider <i>Frequency Shift Keying</i>&nbsp; (FSK) with&nbsp; $M = 2$ &nbsp; &#8658; &nbsp; binary signaling. The two low-pass basis functions in this case are complex and are:
 :$$\xi_1(t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},$$

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