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| − | + | [[Category:ECE440]] [[Category:ECE]] [[Category:Lab]] [[Category:communication systems]] | |
[[Category:discussion]] | [[Category:discussion]] | ||
| − | + | [[ECE 440|ECE 440]] > [[ECE440_Lab1|Lab 1]] > [[ECE440_Lab1_discussion|Discussion page]] | |
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| + | =Discussion page for Lab 1= | ||
This discussion board is used to clarify issues on the lab and pre-lab. | This discussion board is used to clarify issues on the lab and pre-lab. | ||
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A: One can show that the RMS voltage of each spectral component is equal to the corresponding coefficient of the complex Fourier series. To show this, plug in the spectral component | A: One can show that the RMS voltage of each spectral component is equal to the corresponding coefficient of the complex Fourier series. To show this, plug in the spectral component | ||
| − | + | <math> x_k (t) = a_k e^{j k w_0 t} </math> | |
from the complex Fourier series into the formula for RMS | from the complex Fourier series into the formula for RMS | ||
| − | + | <math> x_{k,RMS} = \sqrt{ \frac{1}{T} \int_T | x_k (t)|^2 dt } </math>. | |
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| − | <math> x_{RMS} = \sqrt{ \frac{1}{T} \int_T | | + | |
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A: To combine the RMS voltages to get the net RMS of the first nine spectral components, use the following: | A: To combine the RMS voltages to get the net RMS of the first nine spectral components, use the following: | ||
| − | + | <math>V_{rms,net} = \sqrt{ v_0^2 + v_1^2 + ... }</math>, | |
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| − | <math>V_{rms,net} = \sqrt{ v_0^2 + v_1^2 + ... | + | |
where v_i is the RMS of the i-th spectral component. By Parseval's theorem, Vrms,net should converge to the RMS voltage of the overall signal as the number of spectral components included in the sum approaches infinity. | where v_i is the RMS of the i-th spectral component. By Parseval's theorem, Vrms,net should converge to the RMS voltage of the overall signal as the number of spectral components included in the sum approaches infinity. | ||
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A: A reference resistance is a resistance used in an instrument (e.g., Wavetek RMS Voltmeter) to measure the power of a signal. The power, P in watts, and voltage, V in volts, of the signal are related according to | A: A reference resistance is a resistance used in an instrument (e.g., Wavetek RMS Voltmeter) to measure the power of a signal. The power, P in watts, and voltage, V in volts, of the signal are related according to | ||
| − | <math> | + | <math>P_{avg}=\frac{V_{rms}^2}{R_{ref}}</math>, |
| − | where <math>R_{ref}</math> is the reference resistance in ohms. Now, from the above relationship we can get the dBV of a signal from a dBm reading. Dividing each side by <math>10^{-3}</math> and taking the <math>10\log </math> of each side, we get: | + | where <math>R_{ref}</math> is the reference resistance in ohms. Now, from the above relationship we can get the dBV of a signal from a dBm reading. Dividing each side by <math>10^{-3} W</math> and taking the <math>10\log </math> of each side, we get: |
| − | <math>10\log\left(\frac{P}{10^{-3}}\right)=10\log\left(\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3})}\right)^2\right)=20\log\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3}}\right)</math> | + | <math>10\log\left(\frac{P}{10^{-3}W}\right)=10\log\left(\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3}W)}\right)^2\right)=20\log\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3} W})\right)</math> |
| − | Any <math>R_{ref}</math> can do, however, it is | + | Any <math>R_{ref}</math> can do, however, it is easiest if we pick <math>R_{ref}=1000 \Omega</math> in which case we get: |
<math> \text{Power reading in dBm}=\text{Voltage of signal in dBV} </math>. | <math> \text{Power reading in dBm}=\text{Voltage of signal in dBV} </math>. | ||
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Latest revision as of 09:08, 15 January 2014
ECE 440 > Lab 1 > Discussion page
Discussion page for Lab 1
This discussion board is used to clarify issues on the lab and pre-lab.
Q: How do I get the RMS voltage of spectral components on the 5th question?
A: One can show that the RMS voltage of each spectral component is equal to the corresponding coefficient of the complex Fourier series. To show this, plug in the spectral component
$ x_k (t) = a_k e^{j k w_0 t} $
from the complex Fourier series into the formula for RMS
$ x_{k,RMS} = \sqrt{ \frac{1}{T} \int_T | x_k (t)|^2 dt } $.
Q: Do the RMS voltage of the spectral components need to sum up to the total RMS voltage?
A: To combine the RMS voltages to get the net RMS of the first nine spectral components, use the following:
$ V_{rms,net} = \sqrt{ v_0^2 + v_1^2 + ... } $,
where v_i is the RMS of the i-th spectral component. By Parseval's theorem, Vrms,net should converge to the RMS voltage of the overall signal as the number of spectral components included in the sum approaches infinity.
Q: What is reference resistance? How to get the dBV of a signal from a dBm reading?
A: A reference resistance is a resistance used in an instrument (e.g., Wavetek RMS Voltmeter) to measure the power of a signal. The power, P in watts, and voltage, V in volts, of the signal are related according to
$ P_{avg}=\frac{V_{rms}^2}{R_{ref}} $,
where $ R_{ref} $ is the reference resistance in ohms. Now, from the above relationship we can get the dBV of a signal from a dBm reading. Dividing each side by $ 10^{-3} W $ and taking the $ 10\log $ of each side, we get:
$ 10\log\left(\frac{P}{10^{-3}W}\right)=10\log\left(\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3}W)}\right)^2\right)=20\log\left(\frac{V}{\sqrt(R_{ref}\times 10^{-3} W})\right) $
Any $ R_{ref} $ can do, however, it is easiest if we pick $ R_{ref}=1000 \Omega $ in which case we get:
$ \text{Power reading in dBm}=\text{Voltage of signal in dBV} $.
