bandwidth is defined as the difference between the highest frequency and lowest frequency.

amplitude is the distance between the, let's call it a crest and a trough, in a wave.

cycle is the time per wavelength -- (a crest and a trough)

period is the time in seconds to complete a cycle.

frequency is the number of periods in 1 second, measured in hertz.


based on this, i have a few questions i'd like to focus in on.

a signal sent from source to destination can be either analog from a-z, digital a-z, or can start digital, and then is modem'd (modulated/demodulated) at a device (modem) at the source, sent across the medium, and then vice-versa'd at the destination.

my question is how is the digital converted to analog.

my downstream rate, for example, is 32,339,000 bps. so according to the text i'm reading to convert this to analog we need a frequency of N/2, which is ~16,000,000 Hz. I'm just typing this out as I try to figure it out, i'm totally lost at this point. Is this the first harmonic? then they suggest to make the shape of the analog signal look more like that of the digital signal, we need to increase the bandwidth to 3N/2, 5N/2, or 7N/2. are these sub harmonics? (i just made that up) 3rd harmonics, 5th harmonics, and 7th?

conversion from analog to digital

bit rate conversions

so, according to the book, the vertical lines represent infinite frequency, while the horizontal lines represent zero frequency. this really confuses me. the conversion, when anaolog looks to me like an amplitude.

any help is appreciated



The discussion of harmonic in this context is to show with a simplest approach the maximum bandwidth needed to contain a digital bit stream.

The highest frequency for a digital bit stream occurs when we transmit a sequence that changes on every bit:

1 0 1 0 1 0 ...

with the analog representation of this being a square wave. So if we were to use a simplest D/A converter, the analog output in this case would be a square wave IF we had infinite bandwidth in the frequency domain.

What is useful at this point is to draw on the Fourier Series Expansion, where we have learned that a square wave can be decomposed into an infinite sum of frequencies, starting with a base frequency which is a sine wave that exactly fits into the 0 1 0 1 0 1 0 pattern, which we call the "first harmonic". For a general Fourier Series Expansion, the theory continues that all continuous periodic functions are decomposed into such a first harmonic that is at frequency 1/T where T is the time duration of the base function (in this case 0 1 as a square wave) AND all the higher integer multiples of this frequency (the 2nd, 3rd, 4th.. etc harmonics) some of which may be zero, but if the pattern is repeating (periodic) there will be no frequency components anywhere else. In the case of a square wave it is specifically all the odd harmonics 1, 3, 5... that if you add up more and more of these (each with an amplitude that goes down at 1/f or 1, 1/3, 1/5, etc) the result will get closer and closer to a square wave. You need very high frequencies to represent sharp (fast) edges in a time domain signal!

For the purpose of transmitting information, the pattern 0 1 0 1 0 1 CAN be transmitted with a simple sine wave, as we can use a threshold detector at the other end to reliably convert it back to the 0 1 0 1 pattern and thus recover the information. As we increase bandwidth to include more of the higher harmonics, the signal will get closer and closer to a square wave.

So they are just making the point here that you can use this logic to determine what is the minimum bandwidth required for a digital bit stream. 0 1 0 1 is the highest frequency pattern and we showed above that we can send this with just the first harmonic. Any other patterns such as 0 0 1 1 0 0 1 1 0 0 contain lower frequency components; so as long as our maximum bandwidth exceeds this, these can be transmitted without error as well. In the end our random data stream can be viewed as multiple patterns that are summed together, so in frequency instead of seeing discrete tones, the energy will be spread evenly across that bandwidth (if our data is truly random with one bit completely independent of the next).

I hope this helped!


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