3.3 3.3.1
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3.3.2 Problem An average reading power meter is connected to output of transmitter. Transmitter
output is fed into \(75\Omega \) resistive load and the wattmeter read \(67W\)
(a) What is power in dBm units?
(b) What is power in dBk units?
(c) What is the value in dBmV units?
3.3.2.1 \begin{align*} P_{dbm} & =10\log _{10}P_{m}\\ & =10\log _{10}\left ( 67000\right ) \\ & =\fbox {$48.2607$}\ dbm \end{align*}
(b)
\begin{align*} P_{dbk} & =10\log _{10}P_{k}\\ & =10\log _{10}\left ( 0.067\right ) \\ & =\fbox {$-11.7393$}\ dbk \end{align*}
(c)
\[ P=\frac {V^{2}}{R}\]
Hence
\[ 10\log _{10}P=20\log _{10}V-10\log _{10}R \]
Hence
\[ 20\log _{10}V=10\log _{10}P+10\log _{10}R \]
so
\begin{align*} 20\log _{10}V & =10\log _{10}67000+10\log _{10}75000\\ & =\fbox {$97.0114$ dbmV}\end{align*}
3.3.3
Assume that a waveform with known rms value \(V_{rms}\) is applied across a \(50\Omega \) load. Derive a formula that
can be used to computer the \(dbm\) value from \(V_{rms}\)
\[ P\left ( watt\right ) =\frac {V_{rms}^{2}\left ( V\right ) }{R\left ( \Omega \right ) }\]
Hence
\begin{align*} P_{dbm} & =10\log _{10}\left ( 10^{3}\times P_{watt}\right ) \\ & =10\log _{10}\frac {10^{3}\times V_{rms}^{2}\left ( V\right ) }{R\left ( \Omega \right ) }\\ & =10\left ( \log _{10}10^{3}V_{rms}^{2}-\log _{10}R\right ) \\ & =10\left ( \log _{10}10^{3}+\log _{10}V_{rms}^{2}-\log _{10}R\right ) \\ & =10\left ( 3+2\log _{10}V_{rms}-\log _{10}R\right ) \end{align*}
Hence
\[ \fbox {$P_{dbm}=30+20\log _{10}V_{rms}-10\log _{10}R$}\]
When \(R=50\Omega \) , we obtain
\begin{align*} P_{dbm} & =30+20\log _{10}V_{rms}-10\log _{10}50\\ & =30+20\log _{10}V_{rms}-16.9897\\ & =13.0103+20\log _{10}V_{rms}\end{align*}
3.3.4
\begin{align*} Gain(db) & =10\log _{10}\frac {P_{L}}{P_{i}}\\ & =10\log _{10}\frac {\left ( \frac {V_{rms}^{2}}{R_{L}}\right ) }{I_{rms}^{2}R_{in}}\\ & =10\log _{10}\frac {\left ( \frac {10^{2}}{50}\right ) }{\left ( 0.5\times 10^{-3}\right ) ^{2}\times 2000}\\ & =10\log _{10}\frac {10^{5}}{25}\\ & =10\left ( \log _{10}10^{5}-\log _{10}25\right ) \\ & =10\left ( 5-1.39794\right ) \\ & =36.\,\allowbreak 021 \end{align*}
3.3.5
Using the convolution property find the spectrum for \(w\left ( t\right ) =\sin 2\pi f_{1}t\ \cos 2\pi f_{2}t\)
Solution:
\begin{equation} \digamma \left ( w\left ( t\right ) \right ) =\digamma \left ( \sin 2\pi f_{1}t\right ) \otimes \digamma \left ( \cos 2\pi f_{2}t\right ) \tag {1}\end{equation}
\begin{align*} \digamma \left ( \sin 2\pi f_{1}t\right ) & =\frac {1}{2j}\left ( \delta \left ( f-f_{1}\right ) -\delta \left ( f+f_{1}\right ) \right ) \\ \digamma \left ( \cos 2\pi f_{2}t\right ) & =\frac {1}{2}\left ( \delta \left ( f-f_{2}\right ) +\delta \left ( f+f_{2}\right ) \right ) \end{align*}
Hence (1) becomes
\begin{align} \digamma \left ( w\left ( t\right ) \right ) & =\left \{ \frac {1}{2j}\left ( \delta \left ( f-f_{1}\right ) -\delta \left ( f+f_{1}\right ) \right ) \right \} \otimes \left \{ \frac {1}{2}\left ( \delta \left ( f-f_{2}\right ) +\delta \left ( f+f_{2}\right ) \right ) \right \} \nonumber \\ & =\frac {1}{4j}\left \{ \delta \left ( f-f_{1}\right ) -\delta \left ( f+f_{1}\right ) \right \} \otimes \left \{ \delta \left ( f-f_{2}\right ) +\delta \left ( f+f_{2}\right ) \right \} \tag {2}\end{align}
Applying the distributed property of convolution, i.e. \(a\otimes \left ( b+c\right ) =a\otimes b+a\otimes c\) on equation (2) we obtain
\begin{equation} 4j\ \digamma \left ( w\left ( t\right ) \right ) =\delta \left ( f-f_{1}\right ) \otimes \delta \left ( f-f_{2}\right ) +\delta \left ( f-f_{1}\right ) \otimes \delta \left ( f+f_{2}\right ) -\delta \left ( f+f_{1}\right ) \otimes \delta \left ( f-f_{2}\right ) -\delta \left ( f+f_{1}\right ) \otimes \delta \left ( f+f_{2}\right ) \tag {3}\end{equation}
\begin{align} \delta \left ( f-f_{1}\right ) \otimes \delta \left ( f-f_{2}\right ) & ={\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( \lambda -f_{1}\right ) \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f+f_{2}-f_{1}\right ) {\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f+f_{2}-f_{1}\right ) \tag {4}\end{align}
And
\begin{align} \delta \left ( f-f_{1}\right ) \otimes \delta \left ( f+f_{2}\right ) & ={\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( \lambda -f_{1}\right ) \delta \left ( f-\left ( \lambda +f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f-f_{2}-f_{1}\right ) {\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f-f_{2}-f_{1}\right ) \tag {5}\end{align}
And
\begin{align} \delta \left ( f+f_{1}\right ) \otimes \delta \left ( f-f_{2}\right ) & ={\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( \lambda +f_{1}\right ) \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f+f_{2}+f_{1}\right ) {\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f+f_{2}+f_{1}\right ) \tag {6}\end{align}
And
\begin{align} \delta \left ( f+f_{1}\right ) \otimes \delta \left ( f+f_{2}\right ) & ={\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( \lambda +f_{1}\right ) \delta \left ( f-\left ( \lambda +f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f-f_{2}+f_{1}\right ) {\displaystyle \int \limits _{-\infty }^{\infty }} \delta \left ( f-\left ( \lambda -f_{2}\right ) \right ) d\lambda \nonumber \\ & =\delta \left ( f-f_{2}+f_{1}\right ) \tag {7}\end{align}
Substitute (4,5,6,7) into (3) we obtain
\[ \fbox {\ $\digamma \left ( w\left ( t\right ) \right ) =\frac {1}{4j}\left [ \delta \left ( f+f_{2}-f_{1}\right ) +\delta \left ( f-f_{2}-f_{1}\right ) -\delta \left ( f+f_{2}+f_{1}\right ) -\delta \left ( f-f_{2}+f_{1}\right ) \right ] $}\]
or
\begin{equation} \fbox {\ $\digamma \left ( w\left ( t\right ) \right ) =\frac {1}{4j}\left [ \delta \left ( f+\left ( f_{2}-f_{1}\right ) \right ) +\delta \left ( f-\left ( f_{2}+f_{1}\right ) \right ) -\delta \left ( f+\left ( f_{2}+f_{1}\right ) \right ) -\delta \left ( f-\left ( f_{2}-f_{1}\right ) \right ) \right ] $} \tag {8}\end{equation}
\[ w\left ( t\right ) =\sin 2\pi f_{1}t\ \cos 2\pi f_{2}t \]
Using \(\sin \alpha \cos \beta =\frac {1}{2}\left ( \sin \left ( \alpha -\beta \right ) +\sin \left ( \alpha +\beta \right ) \right ) \) , hence
\begin{align} w\left ( t\right ) & =\frac {1}{2}\left ( \sin \left ( 2\pi f_{1}t-2\pi f_{2}t\right ) +\sin \left ( 2\pi f_{1}t+2\pi f_{2}t\right ) \right ) \nonumber \\ & =\frac {1}{2}\left ( \sin \left ( 2\pi \left ( f_{1}-f_{2}\right ) t\right ) +\sin \left ( 2\pi \left ( f_{1}+f_{2}\right ) t\right ) \right ) \nonumber \\ & =\frac {1}{2}\left ( \frac {1}{2j}\left ( \delta \left ( f-\left ( f_{1}-f_{2}\right ) \right ) -\delta \left ( f+\left ( f_{1}-f_{2}\right ) \right ) \right ) +\frac {1}{2j}\left ( \delta \left ( f-\left ( f_{1}+f_{2}\right ) \right ) -\delta \left ( f+\left ( f_{1}+f_{2}\right ) \right ) \right ) \right ) \nonumber \\ & =\frac {1}{4j}\left \{ \delta \left ( f-\left ( f_{1}-f_{2}\right ) \right ) -\delta \left ( f+\left ( f_{1}-f_{2}\right ) \right ) +\delta \left ( f-\left ( f_{1}+f_{2}\right ) \right ) -\delta \left ( f+\left ( f_{1}+f_{2}\right ) \right ) \right \} \nonumber \\ & =\frac {1}{4j}\left \{ \delta \left ( f+\left ( f_{2}-f_{1}\right ) \right ) +\delta \left ( f-\left ( f_{2}+f_{1}\right ) \right ) -\delta \left ( f+\left ( f_{2}+f_{1}\right ) \right ) -\delta \left ( f-\left ( f_{2}-f_{1}\right ) \right ) \right \} \tag {9}\end{align}
Compare (8) and (9) we see they are the same.
3.3.6
\[ w\left ( t\right ) =4\ rect\left ( \frac {t}{4}\right ) -2\ rect\left ( \frac {t}{2}\right ) \]
By linearity of Fourier Transform
\begin{equation} \digamma \left ( w\left ( t\right ) \right ) =4\times \digamma \left ( rect\left ( \frac {t}{4}\right ) \right ) -2\times \digamma \left ( rect\left ( \frac {t}{2}\right ) \right ) \tag {1}\end{equation}
\[ \digamma \left ( rect\left ( \frac {t}{4}\right ) \right ) =4\operatorname {sinc}\left ( 4f\right ) \]
and
\[ \digamma \left ( rect\left ( \frac {t}{2}\right ) \right ) =2\operatorname {sinc}\left ( 2f\right ) \]
Then (1) becomes
\begin{align*} \digamma \left ( w\left ( t\right ) \right ) & =4\times 4\operatorname {sinc}\left ( 4f\right ) -2\times 2\operatorname {sinc}\left ( 2f\right ) \\ & =\fbox {$16\operatorname {sinc}\left ( 4f\right ) -4\operatorname {sinc}\left ( 2f\right ) $}\end{align*}
Or in terms of just the \(\sin \) function, the above becomes
\begin{align*} \digamma \left ( w\left ( t\right ) \right ) & =16\frac {\sin \left ( 4\pi f\right ) }{4\pi f}-4\frac {\sin \left ( 2\pi f\right ) }{2\pi f}\\ & =4\frac {\sin \left ( 4\pi f\right ) }{\pi f}-2\frac {\sin \left ( 2\pi f\right ) }{\pi f}\\ & =\fbox {$\frac {4\sin \left ( 4\pi f\right ) -2\sin \left ( 2\pi f\right ) }{\pi f}$}\end{align*}
3.3.7 If \(w\left ( t\right ) \) has the Fourier Transform \(W\left ( f\right ) =\frac {j2\pi f}{1+j2\pi f}\) find \(X\left ( f\right ) \) for the following waveforms
(a) \(x\left ( t\right ) =w\left ( 2t+2\right ) \)
(b)\(\ x\left ( t\right ) =w\left ( t-1\right ) e^{-jt}\)
(c) \(x\left ( t\right ) =w\left ( 1-t\right ) \)
Answer:
3.3.7.1 \[ w\left ( t\right ) \Leftrightarrow \frac {j2\pi f}{1+j2\pi f}\]
Then
\begin{align*} w\left ( 2t\right ) & \Leftrightarrow \frac {1}{2}X\left ( \frac {f}{2}\right ) \\ w\left ( 2t+2\right ) & \Leftrightarrow \frac {1}{2}X\left ( \frac {f}{2}\right ) e^{j2\pi \frac {f}{2}\left ( 2\right ) }\end{align*}
Hence
\begin{align*} w\left ( 2t+2\right ) & \Leftrightarrow \frac {1}{2}\left ( \frac {j2\pi \frac {f}{2}}{1+j2\pi \frac {f}{2}}\right ) e^{j2\pi f}\\ & \Leftrightarrow \frac {1}{2}\left ( \frac {j\pi f}{1+j\pi f}\right ) e^{j2\pi f}\end{align*}
This can be simplified to
\[ \fbox {$w\left ( 2t+2\right ) \Leftrightarrow \frac {\pi f\ }{2\left ( \pi f-j\right ) }e^{j2\pi f}$}\]
3.3.7.2 \[ w\left ( t\right ) \Leftrightarrow \frac {j2\pi f}{1+j2\pi f}\]
\begin{align*} w\left ( t-1\right ) & \Leftrightarrow X\left ( f\right ) e^{-j2\pi f\left ( -1\right ) }\\ w\left ( t-1\right ) & \Leftrightarrow X\left ( f\right ) e^{j2\pi f}\end{align*}
Now Let \(e^{-jt}=e^{-j2\pi f_{0}t}\) , hence \(2\pi f_{0}=1\) or \(f_{0}=\frac {1}{2\pi }\) , then
\[ w\left ( t-1\right ) e^{-j2\pi f_{0}t}\Leftrightarrow X\left ( f+f_{0}\right ) e^{j2\pi \left ( f+f_{0}\right ) }\]
Hence
\begin{align*} w\left ( t-1\right ) e^{-jt} & \Leftrightarrow \frac {j2\pi \left ( f+f_{0}\right ) }{1+j2\pi \left ( f+f_{0}\right ) }e^{j2\pi \left ( f+f_{0}\right ) }\\ w\left ( t-1\right ) e^{-jt} & \Leftrightarrow \frac {j2\pi \left ( f+\frac {1}{2\pi }\right ) }{1+j2\pi \left ( f+\frac {1}{2\pi }\right ) }e^{j2\pi \left ( f+\frac {1}{2\pi }\right ) }\\ w\left ( t-1\right ) e^{-jt} & \Leftrightarrow \frac {j2\pi \left ( 2\pi f+1\right ) }{2\pi +j2\pi \left ( 2\pi f+1\right ) }e^{j\left ( 2\pi f+1\right ) }\\ w\left ( t-1\right ) e^{-jt} & \Leftrightarrow \frac {j4\pi ^{2}f+j2\pi }{2\pi +j4\pi ^{2}f+j2\pi }e^{j2\pi f}e^{j}\\ w\left ( t-1\right ) e^{-jt} & \Leftrightarrow \frac {2\pi f+1}{-j+2\pi f+1}e^{j2\pi f}e^{j}\end{align*}
Hence
\[ \fbox {$w\left ( t-1\right ) e^{-jt}\Leftrightarrow \frac {2\pi f+1}{1-j+2\pi f}e^{j\left ( 2\pi f+1\right ) }$}\]
3.3.7.3 \[ w\left ( t\right ) \Leftrightarrow \frac {j2\pi f}{1+j2\pi f}\]
\[ w\left ( -t\right ) \Leftrightarrow X\left ( -f\right ) \]
Then
\begin{align*} w\left ( -t+1\right ) & \Leftrightarrow X\left ( -f\right ) e^{j2\pi f\left ( 1\right ) }\\ w\left ( 1-t\right ) & \Leftrightarrow \frac {-j2\pi f}{1-j2\pi f}e^{j2\pi f}\end{align*}
3.3.8
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