Aufgaben:Exercise 5.2: Band Spreading and Narrowband Interferer - Revision history

Bene: Text replacement - "power spectral density" to "power-spectral density"

2022-02-17T11:41:31Z

Text replacement - "power spectral density" to "power-spectral density"

Bene: Text replacement - "autocorrelation" to "auto-correlation"

2022-02-17T11:29:04Z

Text replacement - "autocorrelation" to "auto-correlation"

Bene: Text replacement - "power density spectrum" to "power spectral density"

2022-02-17T11:20:34Z

Text replacement - "power density spectrum" to "power spectral density"

Guenter at 13:40, 8 December 2021

2021-12-08T13:40:45Z

Guenter at 13:38, 8 December 2021

2021-12-08T13:38:28Z

Guenter at 13:03, 8 December 2021

2021-12-08T13:03:19Z

Guenter at 12:50, 8 December 2021

2021-12-08T12:50:28Z

Guenter: Guenter moved page Aufgaben:Exercise 5.2: Bandspreading and Narrowband Interferer to Aufgaben:Exercise 5.2: Band Spreading and Narrowband Interferer

2021-12-08T12:30:22Z

Guenter moved page Exercise 5.2: Bandspreading and Narrowband Interferer to Exercise 5.2: Band Spreading and Narrowband Interferer

Hwang at 15:17, 29 November 2021

2021-11-29T15:17:35Z

Hwang at 14:33, 29 November 2021

2021-11-29T14:33:49Z

@@ Line 6: / Line 6: @@
 of band spreading]]
 A spread spectrum system is considered according to the given diagram in the equivalent low-pass range:
-*Let the digital signal &nbsp;$q(t)$&nbsp; possess the power spectral density &nbsp;${\it \Phi}_q(f)$,&nbsp; which is to be approximated as rectangular with bandwidth&nbsp; $B = 1/T = 100\ \rm  kHz$&nbsp;&nbsp; (a rather unrealistic assumption):
+*Let the digital signal &nbsp;$q(t)$&nbsp; possess the power-spectral density &nbsp;${\it \Phi}_q(f)$,&nbsp; which is to be approximated as rectangular with bandwidth&nbsp; $B = 1/T = 100\ \rm  kHz$&nbsp;&nbsp; (a rather unrealistic assumption):
 :$${\it \Phi}_{q}(f) =
 \left\{ \begin{array}{c} {\it \Phi}_{0} \\
@@ Line 35: / Line 35: @@
 <quiz display=simple>
-{What is the power spectral density &nbsp;${\it \Phi}_c(f )$&nbsp; of the spreading signal &nbsp;$c(t)$?&nbsp; What value results at the frequency &nbsp;$f = 0$?
+{What is the power-spectral density &nbsp;${\it \Phi}_c(f )$&nbsp; of the spreading signal &nbsp;$c(t)$?&nbsp; What value results at the frequency &nbsp;$f = 0$?
 |type="{}"}
 ${\it \Phi}_c(f = 0) \ = \ $ { 0.1 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The power spectral density&nbsp; $\rm (PDS)$&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular ACF,&nbsp; which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
+'''(1)'''&nbsp; The power-spectral density&nbsp; $\rm (PDS)$&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular ACF,&nbsp; which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
 :$${\it \varphi}_{c}(\tau) = \frac{1}{T_c} \cdot {\rm rect} \big(\frac{\tau}{T_c} \big ) \star {\rm rect} \big(\frac{\tau}{T_c} \big ) \hspace{0.05cm}.$$
 *From this follows &nbsp;${\it \Phi}_{c}(f) = {1}/{T_c} \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] = T_c \cdot {\rm sinc}^2 \left(f T_c \right ) \hspace{0.05cm}$&nbsp; with maximum value

@@ Line 17: / Line 17: @@
 *Thus,&nbsp; in the low-pass range,&nbsp; the bandwidth&nbsp; (only the components at positive frequencies)&nbsp; is equal to &nbsp;$B/2$&nbsp; and the bandwidth in the band-pass range is &nbsp;$B$.
 *The band spreading is done by multiplication with the PN sequence &nbsp;$c(t)$&nbsp; of the chip duration &nbsp;$T_c = T/100$&nbsp; <br>("PN" stands for "pseudo-noise").
-*To simplify matters,&nbsp; the following applies to the autocorrelation function:
+*To simplify matters,&nbsp; the following applies to the auto-correlation function:
 :$$ {\it \varphi}_{c}(\tau) = \left\{ \begin{array}{c}1 - |\tau|/T_c \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ {\rm{otherwise}} \hspace{0.05cm}. \\ \end{array}\begin{array}{*{20}c} -T_c \le \tau \le T_c \hspace{0.05cm}, \\ \\ \end{array}$$
 *At the receiver,&nbsp; the same spreading sequence &nbsp;$c(t)$&nbsp; is again added phase-synchronously.

@@ Line 6: / Line 6: @@
 of band spreading]]
 A spread spectrum system is considered according to the given diagram in the equivalent low-pass range:
-*Let the digital signal &nbsp;$q(t)$&nbsp; possess the power density spectrum &nbsp;${\it \Phi}_q(f)$,&nbsp; which is to be approximated as rectangular with bandwidth&nbsp; $B = 1/T = 100\ \rm  kHz$&nbsp;&nbsp; (a rather unrealistic assumption):
+*Let the digital signal &nbsp;$q(t)$&nbsp; possess the power spectral density &nbsp;${\it \Phi}_q(f)$,&nbsp; which is to be approximated as rectangular with bandwidth&nbsp; $B = 1/T = 100\ \rm  kHz$&nbsp;&nbsp; (a rather unrealistic assumption):
 :$${\it \Phi}_{q}(f) =
 \left\{ \begin{array}{c} {\it \Phi}_{0} \\
@@ Line 35: / Line 35: @@
 <quiz display=simple>
-{What is the power density spectrum &nbsp;${\it \Phi}_c(f )$&nbsp; of the spreading signal &nbsp;$c(t)$?&nbsp; What value results at the frequency &nbsp;$f = 0$?
+{What is the power spectral density &nbsp;${\it \Phi}_c(f )$&nbsp; of the spreading signal &nbsp;$c(t)$?&nbsp; What value results at the frequency &nbsp;$f = 0$?
 |type="{}"}
 ${\it \Phi}_c(f = 0) \ = \ $ { 0.1 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The power density spectrum&nbsp; $\rm (PDS)$&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular ACF,&nbsp; which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
+'''(1)'''&nbsp; The power spectral density&nbsp; $\rm (PDS)$&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular ACF,&nbsp; which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
 :$${\it \varphi}_{c}(\tau) = \frac{1}{T_c} \cdot {\rm rect} \big(\frac{\tau}{T_c} \big ) \star {\rm rect} \big(\frac{\tau}{T_c} \big ) \hspace{0.05cm}.$$
 *From this follows &nbsp;${\it \Phi}_{c}(f) = {1}/{T_c} \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] = T_c \cdot {\rm sinc}^2 \left(f T_c \right ) \hspace{0.05cm}$&nbsp; with maximum value

@@ Line 68: / Line 68: @@
 [[File:P_ID1869__Mod_A_5_2b.png|right|frame|Power density spectrum of the pseudo-noise spread signal]]
 :$$B_c= \frac{1}{T_c} \cdot \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\it \Phi}_{c}(f)\hspace{0.1cm} {\rm d}f = \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\rm sinc}^2 \left(f T_c \right )\hspace{0.1cm} {\rm d}f $$
-:$$\Rightarrow \hspace{0.3cm} B_c=  = \frac{1}{T_c}\hspace{0.15cm}\underline {= 10\,{\rm MHz}} \hspace{0.05cm}$$
+:$$\Rightarrow \hspace{0.3cm} B_c=  \frac{1}{T_c}\hspace{0.15cm}\underline {= 10\,{\rm MHz}} \hspace{0.05cm}$$
 The graph illustrates,
 *that&nbsp; $B_c$&nbsp; is given by the first zero of the&nbsp; $\rm sinc^2$ function in the equivalent low-pass range,

@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The power density spectrum&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular AKF, which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
+'''(1)'''&nbsp; The power density spectrum&nbsp; $\rm (PDS)$&nbsp; ${\it \Phi}_c(f)$&nbsp; is the Fourier transform of the triangular ACF,&nbsp; which can be represented with rectangles of width&nbsp; $T_c$&nbsp; as follows:
 :$${\it \varphi}_{c}(\tau) = \frac{1}{T_c} \cdot {\rm rect} \big(\frac{\tau}{T_c} \big ) \star {\rm rect} \big(\frac{\tau}{T_c} \big ) \hspace{0.05cm}.$$
-*From this follows &nbsp;${\it \Phi}_{c}(f) = {1}/{T_c} \cdot \big[ T_c \cdot {\rm si} \left(\pi f T_c \right ) \big ] \cdot \big[ T_c \cdot {\rm si} \left(\pi f T_c \right ) \big ] = T_c \cdot {\rm si}^2 \left(\pi f T_c \right ) \hspace{0.05cm}$&nbsp; with maximum value
+*From this follows &nbsp;${\it \Phi}_{c}(f) = {1}/{T_c} \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] \cdot \big[ T_c \cdot {\rm sinc} \left(f T_c \right ) \big ] = T_c \cdot {\rm sinc}^2 \left(f T_c \right ) \hspace{0.05cm}$&nbsp; with maximum value
 :$${\it \Phi}_{c}(f = 0) = T_c = \frac{T}{100}= \frac{1}{100 \cdot B} = \frac{1}{100 \cdot 10^5\,{\rm 1/s}} = 10^{-7}\,{\rm 1/Hz} \hspace{0.15cm}\underline {= 0.1 \cdot 10^{-6}\,{\rm 1/Hz}}\hspace{0.05cm}.$$
+'''(2)'''&nbsp; By definition,&nbsp; with&nbsp; $T_c = T/100 = 0.1\ \rm  &micro; s$:
+[[File:P_ID1869__Mod_A_5_2b.png|right|frame|Power density spectrum of the pseudo-noise spread signal]]
-[[File:P_ID1869__Mod_A_5_2b.png|right|frame|Power density spectrum of the PN spread signal]]
+:$$B_c= \frac{1}{T_c} \cdot \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\it \Phi}_{c}(f)\hspace{0.1cm} {\rm d}f = \hspace{-0.03cm} \int_{-\infty }^{+\infty} \hspace{-0.03cm} {\rm sinc}^2 \left(f T_c \right )\hspace{0.1cm} {\rm d}f $$
 :$$\Rightarrow \hspace{0.3cm} B_c=  = \frac{1}{T_c}\hspace{0.15cm}\underline {= 10\,{\rm MHz}} \hspace{0.05cm}$$
 The graph illustrates,
-*that&nbsp; $B_c$&nbsp; is given by the first zero of the&nbsp; $\rm si^2$ function in the equivalent low-pass range,
+*that&nbsp; $B_c$&nbsp; is given by the first zero of the&nbsp; $\rm sinc^2$ function in the equivalent low-pass range,
-*but at the same time also gives the equivalent&nbsp; (equal area)&nbsp; bandwidth in the bandpass region.
+*but at the same time also gives the equivalent&nbsp; (equal area)&nbsp; bandwidth in the band-pass region.
-'''(3)'''&nbsp; <u>Solutions 2 and 5</u> are correct:
+'''(3)'''&nbsp; <u>Solutions 2 and 5</u>&nbsp; are correct:
-*The LDS&nbsp; ${\it \Phi}_s(f)$&nbsp; results from the convolution of&nbsp; ${\it \Phi}_q(f)$&nbsp; and&nbsp; ${\it \Phi}_c(f)$.&nbsp; This actually gives&nbsp; $B_s = B_c + B$ for the bandwidth of the transmitted signal.
+*The PDS&nbsp; ${\it \Phi}_s(f)$&nbsp; results from the convolution of&nbsp; ${\it \Phi}_q(f)$&nbsp; and&nbsp; ${\it \Phi}_c(f)$.&nbsp; This actually gives&nbsp; $B_s = B_c + B$&nbsp; for the bandwidth of the transmitted signal.
 *Since the spreading signal&nbsp; $c(t) ∈ \{+1, –1\}$&nbsp; multiplied by itself always gives the value&nbsp; $1$,&nbsp; naturally&nbsp; $b(t) ≡ q(t)$&nbsp; and consequently&nbsp; $B_b = B$.
 *Obviously, the bandwidth&nbsp; $B_b$&nbsp; of the band compressed signal is not equal to&nbsp; $2B_c + B$,&nbsp; although the convolution&nbsp; ${\it \Phi}_s(f) ∗ {\it \Phi}_c(f)$&nbsp; suggests this.
 *This is due to the fact that the power density spectra must not be convolved, but the spectral functions&nbsp; (amplitude spectra)&nbsp; $S(f)$&nbsp; and&nbsp; $C(f)$&nbsp; must be assumed, taking into account the phase relations.
-*Only then can the LDS&nbsp; $B(f)$&nbsp; be determined from&nbsp; ${\it \Phi}_b(f)$.&nbsp; Clearly, the following is also true:&nbsp; $C(f) ∗ C(f) = δ(f)$.
+*Only then can the PDS&nbsp; $B(f)$&nbsp; be determined from&nbsp; ${\it \Phi}_b(f)$.&nbsp; Clearly,&nbsp; the following is also true: &nbsp; $C(f) ∗ C(f) = δ(f)$.
-'''(4)'''&nbsp; Only the <u>first solution</u> is correct. The solution shall be clarified by the diagram at the end of the page:
+'''(4)'''&nbsp; Only the&nbsp; <u>first solution</u>&nbsp; is correct.&nbsp; The solution shall be clarified by the diagram at the end of the page:
-*In the upper diagram the LDS&nbsp; ${\it \Phi}_i(f)$&nbsp; of the narrowband interferer is approximated by two Dirac functions at&nbsp; $±f_{\rm T}$&nbsp; with weights&nbsp; $P_{\rm I}/2$.&nbsp;&nbsp; Also plotted is the bandwidth&nbsp; $B = 0.1 \ \rm MHz$&nbsp; (not quite true to scale).
+*In the upper diagram the PDS&nbsp; ${\it \Phi}_i(f)$&nbsp; of the narrowband interferer is approximated by two Dirac delta functions at&nbsp; $±f_{\rm T}$&nbsp; with weights&nbsp; $P_{\rm I}/2$.&nbsp;&nbsp; Also plotted is the bandwidth&nbsp; $B = 0.1 \ \rm MHz$&nbsp; (not quite true to scale).
-*The receiver-side multiplication with&nbsp; $c(t)$&nbsp; – actually with the function of the band compression, at least with respect to the useful part of&nbsp; $r(t)$ – causes a band spreading with respect to the interference signal&nbsp; $i(t)$.&nbsp;Without considering the useful signal,&nbsp; $b(t) = n(t) = i(t) · c(t)$.&nbsp; It follows:
+*The receiver-side multiplication with&nbsp; $c(t)$&nbsp; – actually with the function of the band compression,&nbsp; at least with respect to the useful part of&nbsp; $r(t)$ –&nbsp; causes a band spreading with respect to the interference signal&nbsp; $i(t)$.&nbsp; Without considering the useful signal,&nbsp; $b(t) = n(t) = i(t) · c(t)$.&nbsp; It follows:
-:$${\it \Phi}_{n}(f)  =  {\it \Phi}_{i}(f) \star {\it \Phi}_{c}(f) =  \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm si}^2 \left( \pi \cdot (f - f_{\rm T}) \cdot T_c \right )+ \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm si}^2 \left( \pi \cdot (f + f_{\rm T}) \cdot T_c \right ) \hspace{0.05cm}.$$
+:$${\it \Phi}_{n}(f)  =  {\it \Phi}_{i}(f) \star {\it \Phi}_{c}(f) =  \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm sinc}^2 \left(   (f - f_{\rm T}) \cdot T_c \right )+ \frac{P_{\rm I}\cdot T_c}{2}\cdot {\rm sinc}^2 \left( (f + f_{\rm T}) \cdot T_c \right ) \hspace{0.05cm}.$$
 [[File:P_ID1870__Mod_A_5_2c.png|right|frame|Power density spectra before and after band spreading]]
 *Note that&nbsp; $n(t)$&nbsp; is used here only as an abbreviation and does not denote AWGN noise. &nbsp;
-*In a narrow range around the carrier frequency&nbsp; $f_{\rm T} = 30 \ \rm MHz$,&nbsp; the LDS&nbsp; ${\it \Phi}_n(f)$&nbsp; is almost constant.&nbsp; Thus, the interference power after band spreading is:
+*In a narrow range around the carrier frequency&nbsp; $f_{\rm T} = 30 \ \rm MHz$,&nbsp; the PDS&nbsp; ${\it \Phi}_n(f)$&nbsp; is almost constant.&nbsp; Thus,&nbsp; the interference power after band spreading is:
 :$$ P_{n} = P_{\rm I} \cdot T_c \cdot B = P_{\rm I}\cdot \frac{B}{B_c} = \frac{P_{\rm I}}{J}\hspace{0.05cm}. $$
-*This means: &nbsp; the interference power is reduced by the factor&nbsp; $J = T/T_c$&nbsp; by band spreading, which is why&nbsp; $J$&nbsp; is often called spreading gain.
+*This means: &nbsp; The interference power is reduced by the factor&nbsp; $J = T/T_c$&nbsp; by band spreading,&nbsp; which is why&nbsp; $J$&nbsp; is often called&nbsp; "spreading gain".
-*However, such a spreading gain is only given for a narrowband interferer.
+*However,&nbsp; such a&nbsp; "spreading gain"&nbsp; is only given for a narrowband interferer.

@@ Line 39: / Line 39: @@
 ${\it \Phi}_c(f = 0) \ = \ $ { 0.1 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
-{Calculate the equivalent bandwidth &nbsp;$B_c$&nbsp; of the spread signal as the width of the equal-area LDS rectangle.
+{Calculate the equivalent bandwidth &nbsp;$B_c$&nbsp; of the spread signal as the width of the equal-area&nbsp; $\rm PDS$&nbsp; rectangle.
 |type="{}"}
 $B_c \ = \ $ { 10 3% } $\ \rm MHz$
@@ Line 51: / Line 51: @@
 + $B_b$&nbsp; is exactly equal to &nbsp;$B$.
-{What is the effect of band spreading on a narrowband interferer at the carrier frequency?&nbsp; Let &nbsp;$f_{\rm I} = f_{\rm T}$.
+{What is the effect of band spreading on a&nbsp; "narrowband interferer"&nbsp; at the carrier frequency?&nbsp; Let &nbsp;$f_{\rm I} = f_{\rm T}$.
 |type="[]"}
 + The interfering influence is weakened by band spreading.

@@ Line 19: / Line 19: @@
 *To simplify matters, the following applies to the autocorrelation function:
 :$$ {\it \varphi}_{c}(\tau) = \left\{ \begin{array}{c}1 - |\tau|/T_c \\ 0 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ {\rm{otherwise}} \hspace{0.05cm}. \\ \end{array}\begin{array}{*{20}c} -T_c \le \tau \le T_c \hspace{0.05cm}, \\ \\ \end{array}$$
-*At the receiver, the same spreading sequence &nbsp;$c(t)$&nbsp; is again added in phase synchronism.
+*At the receiver, the same spreading sequence &nbsp;$c(t)$&nbsp; is again added phase-synchronously.
 *The interference signal &nbsp;$i(t)$&nbsp; is to be neglected for the time being.
 *In subtask&nbsp; '''(4)'''&nbsp; &nbsp;$i(t)$&nbsp; denotes a narrowband interferer at carrier frequency &nbsp;$f_{\rm T} = 30 \ \rm MHz = f_{\rm I}$&nbsp; with power &nbsp;$P_{\rm I}$.

@@ Line 5: / Line 5: @@
 [[File:EN_Mod_A_5_2.png|right|frame|Considered model<br>
 of the band spreading]]
-A ''Spread Spectrum System''&nbsp; is considered according to the given diagram in the equivalent low-pass range:
+A spread spectrum system is considered according to the given diagram in the equivalent low-pass range:
 *Let the digital signal &nbsp;$q(t)$&nbsp; possess the power density spectrum &nbsp;${\it \Phi}_q(f)$, which is to be approximated as rectangular with bandwidth &nbsp;$B = 1/T = 100\ \rm  kHz$&nbsp;&nbsp; (a rather unrealistic assumption):
 :$${\it \Phi}_{q}(f) =