Phase noise

I'm sorry, but your naive - if not selfish - argument neglects to take into account how others choose to operate their newsreaders. Allowing for this could be referred to as 'consideration' or perhaps 'gentlemanly behaviour'. I'm sure you've heard of these terms.

There is no doubt that interleaving your replies between the trimmed points that you are referring to suits the larger group of people.

And don't forget that some of us have to pay to download your poorly-trimmed and poorly-presented postings.

Try the gentlemanly approach, for once? Take the wider view, perhaps?

Reply Part Three.....

This is one area where lecturers and book-authors repeat parrot-fashion what they have learnt. When a student who is intent on understanding what he is being taught, the best sort, you in this case, the uselessness of the lecturer/author shows up!

Now, I'm not going to answer the exact question you asked, but explain what IMHO is some necessary background. I am prepared to do this because I experienced the similar doubts to yourself when I was going through the mill.

I have to assume that you have some complex variable theory under your belt.....

Remember that e^(jwt) = cos(wt) + j * sin(wt)

Remember that e^(-jwt) = cos(wt) - j * sin(wt)

Add the two together and remember that cos(wt) = 1/2 * ( e^(jwt) + e^(-jwt) )

Now, this expression for cos(wt) is the _REAL reason for phasor representation, and it is not necessarily just an accident caused by someone's preference over a banana.

(You're still free, however, to choose your own model to help your understanding. All that is important is that whatever model you choose gets you the right answer. As an example, I use the wind-of-passing of an express train rushing past a station platform to help me understand the Magnetic Vector Potential, but that's another story!)

If we plot the first term of cos(wt) , e^(jwt), in the complex plane, we get the rotating vector that I discussed previously (and from it, the instantaneous frozen version, the Phasor).

This vector is rotating anti-clockwise starting from the 3 o'clock position.

However, to get the complete representation for cos(wt), we need also to plot the second term, e^(-jwt).

This produced a vector rotating clockwise starting from 3 o'clock.

IT IS GOING BACKWARDS IN TIME!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Because of the factor _MINUS_ t.

When we do the vector sum of the two rotations on the graph at any instant of time, "the imaginary" vertical component vectors s cancel out, and we are left with TWICE the horizontal component vector (hence the factor of 1/2)

Now, here is the thinking behind the rotating vector.

Whatever value we get for e^(-jwt), whether before or after passing through our circuits, it is always the complex conjugate of the value for e^(jwt), SO WHY WASTE TIME PLOTTING IT OR EVALUATING IT?

We end up just using the rotating vector e^(jwt).

We don't need to bother with the factor 1/2 because there is no doubling up.

We can just throw away the "imaginary" part because it would get cancelled out.

Coming back to your question, the e^(-jwt) is going backwards in time.

It won't help you with a smarmy comment at this time, but you'll use this time reversal when evaluating spectra using Fourier's Analysis.

I think your difficulty stems from over use of the rotating disk model- this is leading to a muddling of the angular (phase) approach and the time (linear plot) approach.

The rotating disk helps your visualise the cyclic nature of sine wave but can also confuse things. In particular, in this case, the idea of phase noise.

In the rotating disk model, phase noise manifests itself as uncertainty (due to noise) in the 'position' of the disk relative to were it would be in the noise free case. In the linear plot approach, the uncertainty is in the zero crossing (or other instantaneous reference point) relative to where it would be in the noise free case.

Equally, it is easy to mis-interpret the maths as Gareth has done in:

cos(wt) = 1/2 * ( e^(jwt) +e^(-jwt) )

The term e^(-jwt) isn't some magical time machine relating to "minus time", e^(-jwt) is simply another way of writing 1/(e^jwt) which is a value that decreases as t increasing. (You can check this a basic maths book covering indices. I suspect Gareth needs a refresher in basic maths- this is taught Key Stage 3, maths in the UK so he needs to find a school that will let him sit in with some lessons for the under 15s. )

Why should someone who's unemployed/unemployable be in a hurry?

tox

Normally I would ignore the rather silly outbursts of Mr.Reay because, as is obvious from his outburst below, he seems to be motivated by spite and a desire to seek attention by shouting insulting remarks. He is one of a number of birdbrains-of

-a-feather who twitter together, and who are best ignored.

(I am sorry that my attempts to help with educational matter have sullied the NG with his and others' spitefulness.)

It is necessary to write again because Mr.Reay is talking dangerous nonsense below, the sort of nonsense that perhaps would arise from someone who gave up their education too early, possibly at the "Key Stage 3" that he vaunts below?

-----OOOOO-----

If the discussion were about normal indices, not involving complex notions, then, yes, Mr.Reay would be correct in what he says; that the presence of a power term as the divisor of a quotient would result in progressive decreasing in size as time increased.

However, the discussion is about complex algebra and not real number algebra

The factor e^(jwt) whether as the dividend or the divisor of a quotient causes cyclic variation and not monotonic variation as Mr.Reay suggests in his sentence, "1/(e^jwt) which is a value that decreases as t increasing.".

-----OOOOO-----

Mr.Reay's desperate attempt to score some childish point has just served to make him look silly.

Shame on you, Mr.Reay!

Silly boy!

In addition, considering that he regularly boasts of having _TWO DEGREES_, one in electronics and the other in mathematics, perhaps Mr.Reay could explain the meaning of, "negative frequency" which is expressed using e^(-jwt) (or e^(-st) after Laplacian transformation or as just "1/z" in the further Z transformation so important in today's world of DSP)?

Frequency, as we all know, is the inverse of Time.

If frequency is negative, then it follows that time must also be negative.

Perhaps, with hindsight, Mr.Reay has been over-eager to exhibit behaviour that should have been left behind in the school playground?

Let us keep such behaviour out of the NG, and develop our knowledge through adult discussion! Sure, some of us may be wrong, but that is the nature of discussion; we relearn and move on.

I cannot see that outbursts of the nature of Mr.Reay's are ever of any use or value.

Oh dear, here we go again. You let Gareth out of your killfile and he misbehaves again.

Gareth, if you can't undertand the algebra rules then try another way........

You have used some standard trigometric identities to express a cosine in a exponential form. The cosine is of an ANGLE- in this case an angle given by wt.

w has the dimensions of radians per second

t has the dimension of seconds

The product has no time dimenion, just an value in radians. ie it doesn't represent time, so the idea of it being a 'negative time' is nonsense. It is an ANGLE.

(The j is dimensionless so does not change the above.)

Petty insults and equivocation are not the way forward, OM.

I suggest that you give up while you're still behind.

"Negative frequency" is an artifact of using complex sinusoids to represent signals in the DSP field- no more no less. The signals are represented by a pair of contra-rotating phasors. As they are contra-rotating only one can be in the +ve direction and thus the other is negative. However, this doesn't map to the existance of 'negative frequency'.

Such artifacts are common when you apply maths to the real world and are often put aside after a little thought. The common example is:

"My age is exactly equal to the square root of my grandfather. He is 64. How old am I?"

Actually, most of us know that Frequency is the inverse of the PERIOD of a waveform, usually depicted by T (note T not t).

A look back at your identities (the Euler ones, not your socket puppets) will show that T (ie the period) doesn't appear- the time term is t.

Keep this up and its back to the kill file for you.

-- 73 Brian G8OSN

for FREE training material for all UK amateur radio licences - a RADIO club specifically for those wishing to learn more about amateur radio

Judging by your childish outbursts this morning, or by some other measure?

It is more than that, much more.

It is not just used in DSP, but is a fundamental concept in the derivation of the complex Fourier Transform en route to the development of the Laplace Transform and the Z Transform.

It also comes into play when discussing direct conversion reception techniques, and their converses when transmitting. These techniques are being used to a greater degree in today's GSM phones.

It is important an important part of the phase-cancellation techniques for producing SSB, for example. Those frequencies that are lower than the carrier are negative with respect to the carrier.

Sorry, OM, but you seem to have equivocated yourself in a circle right back to where you jumped in with both feet in your desperation to insult me.

I suggest that you give up while you are still behind.

Why not put me back in your kill-file, and then I and the NG can resume the technical discussion for the benefit of Mr.Blair and other newcomers?

I don't think you got a reply to your point, Brian.

However, it should be borne in mind that on Mr Bean's planet the day is 23H20 long, so 'negative time' may well be possible. The time standard there (UBT, perhaps?) might, for example, have 2400 negative leap-seconds a day in order to stay aligned with, eg, UTC here.

I don't doubt that as a result of this some mathematical formulae might not have the same meaning as here.

With the proviso that Mr Reay was using a different example, this sounds remarkably like what he was saying.

So far I'd say it's Reay....1 Bean....0

And you haven't responded to his factual statement about the contents of Key Stage 3 Mathematics: was he right or not?

Be careful, a wrong answer will take you to Reay....2 Bean....0

Well, I have to admit to a slight error, it is actually Key Stage 4 Maths. Basic indice notation is in Key Stage 3 maths - normally up to year 9 in most schools- bu that is +postive integers only in the indice. I got confused as I was thinking of a school that is pioneering bringing forward K3 to finish in Y8

Doesn't change the maths, of course, a -ve sign in the indice indicates "1 over", at least in the world outside Chippenham.

It also explains why meter per second etc is often writen as ms^-1

Total rubbish.

a) The transforms you list are widely used in DSP and the explanation Brian gave covers these. (I can see why you call him Mr Reay- he is clearly far superior to you. )

b) I made a phasing SSB exciter without DSP.

BobD

Don't forget Mr Bean's other nonsense that Brian has seen off.

More like: Mr Reay 10 Bean the muppet 0

BobD.

Don't you understand it?

My maths is rusty but it is obvious.

How come you failed to acknowledge your other error, Boy Bean?

BobD