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Suppose x(t) is a continuous-time signal and x(n) is a discrete time signal obtained by sampling the signal x(t) at a Nyquest sampling frequency Fs. The signal x(t) requires a bandwidth of B1 Hz for faithful transmission through a communication channel, while the signal x(n) requires B2 Hz in the same communication channel.

1-Is there are any relationship between B1 and B2? are the two equal?

2- Is it true that a digital signal utilizes bandwidth more efficiently as compared analog signal (perhaps by multiplexing) ?

Update

Thanks for useful comments. I am a little bit confused. I can get a clear idea if someone can answer my following question:

Digital signals can utilize bandwidth more efficiently as compared to Analog. Is this true or false?

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I think you're confused about the potential benefits of digital modulation when compared to analog methods. In your description, the discrete-time signal $x[n]$ contains the exact same information as the continuous-time signal $x(t)$, so there is no difference between the two. In a theoretical sense, you can perfectly reconstruct $x(t)$ given the samples of $x[n]$, assuming that $x(t)$ is bandlimited and your sample rate is high enough. It is true that digital modulation schemes, like PSK, OFDM, etc. can be more capable, but they are still transmitted using continuous-time signals. – Jason R Jan 29 at 14:46
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Just to add to Jason's comments: information theory (see e.g. en.wikipedia.org/wiki/Noisy-channel_coding_theorem) doesn't care what kind of modulation you use - the amount of information that you can transmit through a noisy channel is the same, regardless. – Paul R Jan 29 at 15:17
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To add to the other comments, particularly with regard to information theory... Signals are information sources. Different information sources may be more more or less efficient in how they represent the information they convey. It is easier to decompose information and then reassemble it so it is conveyed in an efficient way using digital techniques. This is largely due to the advances in digital computing as compared with analog computing / signal processing techniques. – B Z Jan 29 at 17:26
@BruceZenone so can I say that digital techniques are more efficient because of "advances in digital computing as compared to analog computing", AND not just because of digital modulation per se? – gpuguy Jan 30 at 8:12
@gpuguy I believe that is a fair characterization. – B Z Jan 31 at 14:39

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