# Bandwidth required to transmit a binary sequence

Does a digital binary number have any bandwidth? I explain suppose T want to express the number 10 in binary digits: 1 0 1 0 would that digital number have any BW in frequency domain? I would say the question has no meaning as frequency is the dual in time and a number 5 whether expressed in decimal or analog is not a quantity changing with time. am I right?

• Please improve your question's title. Like this, it says nothing about the content of this question – Marcus Müller Jul 7 '18 at 10:50
• You are right: only signals and systems have bandwidth; sequences of numbers do not. (Even though, mathematically, sequences have a DFT, I think "bandwidth" is meaningful only in when the sequence is given a physical context). – MBaz Jul 7 '18 at 16:28

It's not meaningful, in my opinion, to talk of the "bandwidth" of a constant number. Unless you want to characterize that constant number as DC (because the value does not change in time). And the bandwidth of DC is zero.

But a stream of bits toggling on and off, yes, a bandwidth (which is the rate of change of the bits or "bitrate", if we're considering two-sided bandwidth or half the bitrate if we're considering the single-sided bandwidth) is meaningful.

In fact, a very important fundamental theorem from Claude Shannon relates bandwidth, signal-to-noise ratio, and rate of information together.

• I suppose that a modulation scheme can be designed to communicate over any channel that has non-zero bandwidth. In this sense the available bandwidth is a design constraint imposed by the channel rather than something derived from the information source. – firdes Jul 7 '18 at 11:25
• " a modulation scheme can be designed to communicate over any channel that has non-zero bandwidth", sure @firdes. but no matter what modulation scheme, the greater the noise, the less information you will be able to squeeze through the channel. In the binary case of the OP, this is consistent with the Shannon formula in the case that $\tfrac{S}{N}=1$. then, with 1 bit-per-symbol, you need detect only whether the noise-corrupted signal is positive or negative. – robert bristow-johnson Jul 9 '18 at 5:17

if you mean by right, is something scientifically true? by Popper, you can only show that it is false. The question if 5 has bandwidth needs to be refutable in order to be scientific.

It is refutable if 5 takes on a physical meaning like 3 bits transferred in 30 seconds.

If meaningful, the question of 5 has bandwidth without being refutable is a metaphysical question and truth is in a different context, than a scientific truth

• is physics (or physical science) necessary for the Shannon Channel Capacity Theorem to be true? seems to me to be a purely mathematical thing. if what you mean by "refutable" is what Popper means by "falsifiable", then i am not sure Popper's Demarcation is applicable. when i think of Popperian Demarcation, i am thinking that "falsification" means that reality between the (new) physical theory and the current physical theories must make a material difference to be science. measuring that material difference either falsifies or confirms a scientific claim. – robert bristow-johnson Jul 9 '18 at 5:26