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I have a root-raised cosine pulse signal of bandwidth $500$ MHz. If the signal is modulated at the frequency $6$ GHz, the result should apparently look like this : enter image description here

If I understand well, to modulate the root raised cosine, it needs to be "multiplied" by a signal like $h \left( t \right) = \cos \left( \omega t \right)$ with $\omega = 2 \pi f$ and $f = 6$ GHz. My problem is that, in Matlab, when I multiply my pulse by a cosine this is not the result I get at all. I get something like this : enter image description here

Could someone please explain what I might be doing wrong here.

Thank you in advance for your help!

ZaellixA
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shokmri
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  • It looks like you are plotting 16 different waveforms, as opposed to a single modulated waveform. Assuming that was intented, in addition to the sampling considerations that Dan mentioned, make sure you plot along the appropriate dimensions (i.e., rows vs columns). – Ash Jun 01 '23 at 17:08
  • Hello, thank you, this helped me resolve part of my problem. The problem was that Matlab was doing matrix multiplication instead of point by point multiplication of the vectors, hence why I got many waveforms instead of one. I am facing another problem now, is that, when I plot a cosine wave from -8 ns in time to 8 ns in time, and I choose for example 400 samples (or points) for the plot, I get a certain number of periods of the cosine in my plot, then if i change this number of samples (for example i put 500 samples instead of 400), I get a DIFFERENT number of periods in the plot but why.. – shokmri Jun 16 '23 at 18:04

1 Answers1

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The OP is not sampling high enough to visibly see a carrier. From inspection of the graph, it appears the time spacing is only 1 ns which corresponds to a sampling rate of 1 GHz. To visibly see the carrier cycles at 6 GHz, the sampling rate needs to be sufficiently greater than twice the highest frequency based on how many samples per 6 GHz cycle is desired to be able to see visually; for example if we want 10 samples for every cycle, the sampling rate would need to be 60 GHz or a time interval spacing of 16.67 ps.

Technical details:

This is for visual aesthetic only. In practice, we would not need 60 GHz to directly sample an actual waveform that is at a 6 GHz carrier with 500 MHz BW, nor would we need to sample anywhere near as high as the carrier itself. To meet Nyquist, the minimum sampling rate is 2x the highest bandwidth (I assume this is close to 500 MHz in this case but I don't know if the bandwidth given is one-sided or two-sided and what the actual roll-off factor is for the RC shape), plus additional margin for realizable filtering (typically at least 20% to 30% more but is traded with filter complexity) so in practice if the analog input bandwidth to the Analog to Digital Converter was high enough, we could directly sample a modulated signal at an analog 6 GHz carrier with a much lower sampling rate. This is referred to as "undersampling".

When simulating modulated waveforms, it rarely makes any practical sense to simulate the actual carrier (unless we want it for a pretty picture or other similar use). Any modelling we would do can equivalently be done at a complex baseband (using "I" as in-phase and "Q" as quadrature components). For purpose of simulation, it doesn't matter what the actual carrier frequency is, we can model any channel and components at carrier frequency =0 (complex baseband).

Dan Boschen
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  • Thank you for your reply, I actually haven't thought about upsampling. Just to confirm, this number of upsampling is related to the Root Raised Cosine filter impulse response and not to the cos(wt) where w=2pif , if my understanding is correct? – shokmri Jun 01 '23 at 12:18
  • @shokmri if your goal is to be able to "see" the plot showing the sinusoidal carrier frequency (which I wouldn't recommend for any purpose of analysis as there is really no additional information there), then I am simply saying to be sure you have multiple samples for any one of those cycles, make sense? – Dan Boschen Jun 01 '23 at 14:04
  • Hello, thank you for your answer, I was able to see the plot by increasing the number of samples as you have suggested! I have another problem, it's that, when I plot my cosine wave alone, from -8 ns (in time) to 8 ns with a certain number of samples say 400 samples, i get a certain number of periods in my plot, but when i change this number of samples to 500 samples and keep same time interval, i see a different number of periods in the plot, but I have no idea why this number of periods inside the same time interval changes. Any idea on why this is happening ? Thank you – shokmri Jun 16 '23 at 18:09
  • @shokmri Hi! Glad it helped. If I answered your original question, please select the answer to close this out. Your new question is good to post as a new question, since Stack Exchange discourages continued discussion in the comments, thanks! – Dan Boschen Jun 16 '23 at 22:00