# What are Discrete and Continue Frequencies?

When it says "Periodic Digital Signal is composed of periodic composite analogue signal with infinite bandwidth with discrete frequencies" and "Non-periodic Digital Signal is composed of non-periodic composite analogue signal with infinite bandwidth with continuous frequencies" in the book, I get very confused.

I understand that digital signals are composed of many composite analogue signals that form the whole digital wave. But I still find this statement rises a lot of questions.

1. Does "Discrete Frequencies" mean the frequency values are in integer numbers only and not any real numbers?
2. Does "Continuous frequencies" mean the frequency values can be in integer or real numbers?
3. If the above 2 questions are false, then why periodic digital signal has discrete frequencies and non-periodic digital signal has continuous frequencies? Periodic signals can be "continuous" too, isn't it?

I could have just memorise this statement but it wouldn't really make sense unless I know exactly what this whole thing mean.

Thanks for any help!

• possible duplicate of Why Periodic Signals has only integer frequencies but non-periodic does not? – stevenvh Aug 28 '11 at 17:24
• @stevenvh I have been too busy this week to have noticed the prior question, hence why I answered here. But I do agree, this looks close enough to be a duplicate. – Kellenjb Aug 28 '11 at 17:41
• @xEnOn If you didn't get an answer to the previous question that you were happy with then you should have provided some clarification to what you meant. – Kellenjb Aug 28 '11 at 17:42
• So sorry. I think I wasn't making myself clear in this question, which sounded like the the other question. I've understood the prior question. I'm actually trying to ask since periodic and non-periodic can have integer or real values as its frequencies, then why are there still there categorisation of discrete and continuous frequencies? Both can simply be just continuous frequencies because they do not need to be in integers, couldn't they? – xenon Aug 28 '11 at 17:56
• @xEnOn - I explained in my answer to your other question that the base frequency of a set of harmonics is the GCD of the harmonic frequencies. So as long as they're relatively rational there's a periodic signal, however low in frequency. Only if they comprise all frequencies (continuous spectrum) the signal isn't repetitive or periodic. – stevenvh Aug 28 '11 at 18:06

## 1 Answer

Lets start by breaking down the first statement of "periodic digital signal is composed of periodic composite analogue signal with infinite bandwidth with discrete frequencies"

• periodic digital signal - A specific example of this would be a simple square wave with a 50% duty cycle.
• composed of periodic composite analogue signal with infinite bandwidth with discrete frequencies - in order to reproduce a square wave fully, you need an infinite amount of sine waves that are at discrete frequencies. Specifically here is what it looks like mathematically: *From wikipedia

So in this mathematical representation you can see that the sumation goes from k=1 to infinity (hence the infinite bandwidth). f can be any number you want it to be, but you always have to have an integer (k) multiple of f which is why they say "Discrete Frequencies". And sine waves are periodic (hence the periodic composite analogue signal).

You can get into some more complex digital signals, but as long as they are periodic you will be able to represent it in some mathematical form like this.

Now for the statement of "Non-periodic Digital Signal is composed of non-periodic composite analogue signal with infinite bandwidth with continuous frequencies"

• Non-periodic Digital Signal - This is a little bit harder to describe mathematically since it is not periodic. Basically this would be something like a digital bit stream that you are sending to someone.
• composed of non-periodic composite analogue signal with infinite bandwidth with continuous frequencies - Because the signal is not periodic you can't reproduce it with only discreet frequencies, instead you have to use other frequencies. This is why they say continuous frequencies.
• Thanks! So is true to say that the non-periodic digital signal is composed of non-periodic composite analogue signal because in order for the composite analogue signal waves to map to the non-periodic digital signal, the harmonic frequencies that compose the mapping composite analogue signal is itself non-periodic. Therefore, the whole composite analogue signal is non-periodic too? – xenon Aug 28 '11 at 19:40