# Talk:Chirp

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## Linear chirp formula

Why was the linear chirp formula changed to:

${\displaystyle x(t)=\sin(2\pi \int _{0}^{t}f(t')t'dt')=\sin(2\pi (f_{0}+{\frac {k}{2}}t)t)}$

? — Omegatron 03:01, 31 October 2005 (UTC)

Because that is the correct formula for a chirped sine wave if the chirp function is given by

${\displaystyle f(t)=f_{0}+kt}$

For a function of the form

${\displaystyle x(t)=\exp(z(t))}$

the frequency is defined to be

${\displaystyle \nu (t)={\frac {1}{2\pi i}}{\frac {dz(t)}{dt}}}$

Lets pick an arbitrary chirp function:

${\displaystyle z(t)=2\pi if(t)t}$

Then:

${\displaystyle \nu (t)={\frac {1}{2\pi i}}{\frac {dz(t)}{dt}}=({\frac {df(t)}{dt}}t+f(t))}$

So for a linear chirp

${\displaystyle f(t)=f_{0}+kt}$

the frequency is

${\displaystyle \nu (t)=kt+kt+f_{0}=2kt+f_{0}}$

the chirp rate is

${\displaystyle {\frac {d\nu (t)}{dt}}=2k}$

In other words one of your equations is off by a factor of two. Either you need to define your linear chirp as

${\displaystyle \nu (t)=\nu _{0}+2kt}$

or your chirped sine wave as

${\displaystyle x(t)=\sin(2\pi \int _{0}^{t}\nu (t')t'dt')=\sin(2\pi (f_{0}+{\frac {k}{2}}t)t)}$

I haven't looked to see whether this means all the other chirp functions described are similarly wrong....

Korpela 05:58, 31 October 2005 (UTC)

Yes, the exponential chirp is similarly broken. In this case for a function of the type

${\displaystyle x(t)=\sin(2\pi f_{0}k^{t}t)}$

the instantaneous frequency is

${\displaystyle \nu (t)=f_{0}k^{t}(\ln(k)t+1)}$

If you define exponential chirp as one where the frequency has the form

${\displaystyle \nu (t)=f_{0}k^{t}}$

then the functional form of the waveform is

${\displaystyle x(t)=\sin(2\pi \int _{0}^{t}\nu (t')t'dt')=\sin({\frac {2\pi f_{0}}{\ln(k)^{2}}}\left[k^{t}(\ln(k)t-1)+1\right])=\sin(2\pi f_{1}\left[e^{k_{1}t}(k_{1}t-1)+1\right])}$

Neither is as simple as what's on the page already. You can make things look better by redefining k and t in one of the definitions. What is there now will confuse people who assume that that "k" and "f" in the frequency definition is the same as "k" and "f" in the waveform definition.

Korpela 17:22, 31 October 2005 (UTC) revised 18:48, 31 October 2005 (UTC)

Sorry, but I think there is a fundamental error in the previous reasoning: When you have an (angular) frequency ${\displaystyle \omega (t)}$ then the corresponding x(t) is

${\displaystyle x(t)=exp(i\int _{0}^{t}\omega (t')dt')=exp(2\pi i\int _{0}^{t}\nu (t')dt')}$

and not

${\displaystyle x(t)=exp(i\int _{0}^{t}\omega (t')t'dt')=exp(2\pi i\int _{0}^{t}\nu (t')t'dt')}$

Simple example to help against knots in the mind: Assume for a moment we wouldn't calculate a chirp and take ${\displaystyle \nu (t)=\nu =const.}$

${\displaystyle x(t)=exp(2\pi i\int _{0}^{t}\nu dt')=exp(2\pi i\nu t)}$

${\displaystyle x(t)=exp(2\pi i\nu \int _{0}^{t}t'dt')=exp(\pi it^{2})}$

Now let's see what we get for the chirps: Well, if we then start with the linear chirp and assume:

${\displaystyle f(t)=f_{0}+kt}$

then we get

${\displaystyle x(t)=\sin(2\pi \int _{0}^{t}f(t')dt')=\sin(2\pi \int _{0}^{t}(f_{0}+kt')dt')=\sin \left(2\pi (f_{0}+{\frac {k}{2}}t)t\right)}$

Checking the result:

${\displaystyle \nu (t)={\frac {d}{dt}}((f_{0}+{\frac {k}{2}}t)t)=f_{0}+kt}$

And for the exponential chirp with

${\displaystyle f(t)=f_{0}k^{t}}$

we get

${\displaystyle x(t)=\sin(2\pi f_{0}\int _{0}^{t}k^{t'}dt')=\sin(2\pi f_{0}\int _{0}^{t}exp(ln(k)t')dt')=\sin \left({\frac {2\pi f_{0}}{\ln(k)}}(k^{t}-1)\right)}$

I have tested these results, they produce correct chirps. SiriusGrey 00:31, 24 March 2006 (UTC)

## Graphs

I added the examples to the article, which I have now been told were wrong, and made the graphs with the same equations. I haven't put in the time to understand the differences regarding instantaneous frequency, but, in the meantime, can someone post some references that show these equations? (I don't see any references at all right now.) I want to know for certain that this is what people actually use to make Doppler-immune chirp radars and such. It's a little confusing. — Omegatron 14:37, 4 June 2006 (UTC)

## Nature of chirp signal

How can we say that a chirp signal is bandlimited as well as time limited? does the term time-limited make sense? please explain. —The preceding unsigned comment was added by Krishna2531985 (talkcontribs) 05:13, 23 January 2007 (UTC).

## Geometric same as exponential

Surely the geometric and exponential chirps are the same. If the frequency is k^t, say, then k^(t+const) = (k^const)*(k^t), ie the frequency at t2 = (t1 + c) is a fixed multiple of the frequency at t1. —Preceding unsigned comment added by 86.150.1.188 (talk) 18:01, 17 January 2008 (UTC)

I agree that geometric chirps and exponential chirps are the mathematically identical. (The "big O notation" article mentions another context where "geometric" and "exponential" are synonyms). --68.0.124.33 (talk) 16:14, 28 July 2008 (UTC)

## Exponential chirp vs hyperbolic chirp

Based on computer experimentation, I believe that the hyperbolic frequency modulated (aka linear period modulation, aka logarithmic phase modulation) chirp ${\displaystyle \phi _{i}(t)={\frac {ln(T_{0}+\beta t)}{\beta }}}$ has the property of ideal Doppler tolerance - is it possible for more than one type of waveform to have this property? That seems strange to me. If not, are these two different chirps somehow describing the same waveform? 146.6.204.185 (talk) 22:39, 22 March 2010 (UTC)

I think there is a mistake in the article - isn't it the case that the Doppler effect modifies the instantaneous frequency as ${\displaystyle f_{Doppler}(t)=cf_{Original}(ct)}$? In that case, ${\displaystyle f_{Doppler}(t)=cf_{Original}(t)}$ would be wrong, and that equation seems to be the basis of the claim that the exponential chirp is Doppler tolerant.

You are absolutely correct. The article is completely wrong about the doppler discussion. (Constant) Doppler simply scales the time axis, i.e., received signal is ${\displaystyle s_{Doppler}(t)=x(ct)}$ for some ${\displaystyle c}$. For doppler tolerance, we need the originally transmitted (non-time scaled) chirp to line up with the received stretched/compressed chirp. I.e., ${\displaystyle s_{Doppler}(t)=x(ct)=c_{1}x(t-shift)}$. This holds for the hyperbolic chirp, but not for the exponential chirp.

The entire claim of doppler tolerance hinges on the claim that ${\displaystyle f(t)_{\mathrm {Doppler} }=cf(t)_{\mathrm {original} }}$, which is simply not true. In fact, it is true that ${\displaystyle f(t)_{\mathrm {Doppler} }=c_{1}f(c_{2}t)_{\mathrm {original} }}$, as per the explanation above. Note that the instantaneous frequency is defined as ${\displaystyle f(t)=f_{0}k^{t}}$. Doppler results in ${\displaystyle c_{1}f(c_{2}t)=c_{1}f_{0}k^{c_{2}t}=c_{1}f_{0}(k^{c_{2}})^{t}}$, which cannot be written as ${\displaystyle f(ct)=c_{3}f(t)}$. — Preceding unsigned comment added by 24.15.198.199 (talk) 17:24, 2 December 2012 (UTC)

## Laser chirp

Noted that short-pulse (or high datarate) laser chirp interacts with optical fiber material disperion, increasing or decreasing pulse dispersion as the signal propagates. It does not change material dispersion, a fixed property of the fiber. Fabrice002 (talk) 12:49, 23 September 2011 (UTC)

## Ref. 2

Seems not existing anymore. — Preceding unsigned comment added by 93.71.78.42 (talk) 17:17, 25 May 2012 (UTC)

## Transferring New Content

I have written an overview on the derivation and properties of chirp spectra, which is currently at User:D1ofBerks/sandbox (and called "The Spectrum of a Chirp"). Originally it was submitted, incorrectly, as a new article but it is actually an addition, so doesn't need an intro of its own. If the article is considered to be O.K., how do I get it transferred into the main article? I think it would fit best after the subsection 'Uses and occurrences' and before 'See also'. Please advise D1ofBerks (talk) 10:18, 16 May 2014 (UTC)

## Potential confusion in Expressions for Instantaneous Frequency and Phase of Exponential Chirp

In general, if a chirp has frequency ${\displaystyle f_{0}}$ at time ${\displaystyle t_{0}}$ and the frequency changes exponentially to ${\displaystyle f_{1}}$ at time ${\displaystyle t_{1}}$, the correct expression for instantaneous frequency is

${\displaystyle f(t)=f_{0}\left[{\frac {f_{1}}{f_{0}}}\right]^{\frac {t-t_{0}}{t_{1}-t_{0}}}=f_{0}k^{\frac {t-t_{0}}{t_{1}-t_{0}}}}$

Note that this gives the correct instantaneous frequencies at ${\displaystyle t=t_{0}}$ and ${\displaystyle t=t_{1}}$.

Assuming ${\displaystyle t_{0}=0}$, this simplifies to

${\displaystyle f(t)=f_{0}k^{\frac {t}{t_{1}}}}$

But the currently published expression is

${\displaystyle f(t)=f_{0}k^{t}}$

I feel some change is needed to make the exponent dimensionless (and the expression correct). And the expression for Instantaneous Phase should have similar changes.

If on the other hand, if ${\displaystyle t}$ describes some normalized/dimensionless time vector, this wasn't clear. — Preceding unsigned comment added by 65.112.5.1 (talk) 19:29, 26 July 2014 (UTC)

## Re Deletion of Spectrum of a Chirp Pulse

I have had section in "Chirp", describing spectra and their ripples, for some months now. Someone has decided to delete it. (i) The various techniques I discuss are from articles and books on chirps, so can hardly be described as unnecessary. (ii) The various spectral techniques are relevant to the methods by which chirps can be generated (a section I will add later) (iii) The present description of the spectra of chirps and their properties is lacking in detail. (iv) The techniques for generating the spectra are very relevant to pulse compression systems and their properties (to be added later to that article). Please advise D1ofBerks (talk) 22:17, 12 November 2014 (UTC)

@D1ofBerks, I'm the person that removed the material you developed for the Chirp page. I don't have strong feelings about that page, but my reaction to the material that I deleted was that it was extremely detailed, not likely to be generally of interest to a Wiki reader, not well integrated with the other material in the same page, and too much about general techniques for discovering properties of a time series (not just a chirp time series) and, so, not necessarily relevant to the chirp page per se. You might disagree with me, but let me suggest that these points be considered, and, then, taken over to Talk:Chirp. Sincerely and Thank You, DoctorTerrella (talk) 22:32, 12 November 2014 (UTC)
My concerns were the same as those expressed by DoctorTerrella. The material wandered too far away from the chirp, expressing concepts that were at once fairly inaccessible and appropriate to general spectral analysis. Binksternet (talk) 01:02, 13 November 2014 (UTC)

I decided to contribute the article after having seen a request for information while doing a search on some topic or other. It began (I think) "I have built a chirp pulse generator and when I inspected its spectrum it had ripple on it. Why are there ripples and how do I remove them?" As the request for help had been up for over two years I didn't response to it specifically, but thought that I could provide information that could answer his questions, maybe to help others.

As Wiki is an encyclopedia I thought I would start the discussion with early work on analytical solutions, though approximation methods and on to modern digital methods. All the spectra are of chirps. None of the work is original (although the figures are) and I believe is adequately referenced. I understand that the article may be difficult for a non-technical reader, but hopefully the figures help. The serious reader can progress further through the references. Without my sections, there is no forward guidance to anyone and the existing article doesn't answer the questions raised by the gentleman above.

My main interest, when I did my original studies in the mid 1990s, was on chirp pulse design and on pulse compression of these chirps for maritime and airborne radars (and I was proposing to write that up - but maybe I won't now!). The main features of interest in such pulses are: compression gain, main pulse width, close-in sidelobe levels and far-out sidelobe levels. Also of interest are reciprocal-ripple correction and Doppler tolerance. All these parameters can be linked to the chirp spectra in the article I wrote (but they can't to the original article you both prefer!). Also the compression methods are similar to the techniques used in generating chirps. (Of particular interest to me were the techniques for digital pulse compression - hence my emphasis on examples of spectral analysis of chirps by digital means).

Now a question to you both. With my sections now deleted, where else on Wiki do you find the information I provided?

I, personally think the original article is a bit lightweight and provides no real guidance for onward study. Also the mathematics on strange chirp laws is interesting but is of no relevance to the work I did. In addition, I was intending to expand the section on chirp pulse generation, sometime soon, and link it to my (previously) existing sections.

Finally, I must confess that having spent a considerable amount of time writing the article (and then learning how to get it into Wiki), I have been most disappointed to find it all axed in so brutal a fashion. D1ofBerks (talk) 15:25, 13 November 2014 (UTC)

Again, @D1ofBerks, this discussion properly belongs at Talk:Chirp, not on a personal page. Please move your discussion there so that other interested parties (gentlemen and ladies!) will expect to find it. Sincerely, DoctorTerrella (talk) 15:34, 13 November 2014 (UTC)
I moved this conversation from my talk page to here. I think more people should be allowed to share their thoughts.
One possible way forward is for D1ofBerks to create a new article containing the information. Perhaps the new article could be called Chirp analysis, Spectral ripple or Ripple correction. Another idea is that the disputed material could be worked into the existing article about pulse compression. In any of these cases, a brief summary of the material ought to be made here, with a link for the reader interested in more detail. Otherwise, I could be completely off base, and all of this new chirp analysis should stay here in this article. Dicklyon, what do you think? Binksternet (talk) 17:49, 13 November 2014 (UTC)
One other thing I would add to my multiple concerns with the material by @D1ofBerks: it is (and he/she seems to acknowledge) original research. My first suggestion to him/her would be to get it published in a formal domain (not Wiki). My thoughts, but, again, discussion would be useful … DoctorTerrella (talk) 17:54, 13 November 2014 (UTC)

I've just one quick comment at this stage, I think you misunderstood a comment I made about originality, only the figures are mine - to get over any copyright issues. They were done using the established data. All topics in the article can be linked to existing sources. D1ofBerks (talk) 18:08, 13 November 2014 (UTC)

### Also, some technical questions

1. How is the deleted material specifically relevant to chirps, as opposed to showing (in detail) how to discover an assumed property about a non-stationary time series?

2. My impression is that much of the methods discussed in the deleted material were essentially about trying (with difficulty) to apply Fourier analysis (which works best for discovering stationary periodic time series) to a time series that is non-stationary. The relevance of such an approach can be debated, and this causes me concern about its appropriateness in Wiki. — Preceding unsigned comment added by DoctorTerrella (talkcontribs)

Can you explain what you mean by stationary? Binksternet (talk) 19:33, 13 November 2014 (UTC)
It is a somewhat qualitative term, but basically it means that the properties of representative durations of the time series do not change over time. So, a sinusoid with constant amplitude, frequency, and phase is "stationary" -- it just keeps oscillating the same way over all time, forever and ever. Transients signals, such as exponentially decaying functions, are not stationary. A chirp is certainly not stationary, since its instantaneous frequency is changing all the time. While scientists and technical types often decompose data time series into Fourier components (those being stationary sinusoids), such a decomposition is not always especially appropriate if the time series being decomposed is not actually stationary. Yes, it can be done, an algorithm can produce a "result", but whether or not it makes sense is something that needs to be justified, and it can depend on the details of the analysis being undertaken. Fourier transforms of differential equations can facilitate their solution, of course, but just pushing a nonstationary data time series into a Fourier transform should be considered with care. Often it is better to use a Laplace transform for transient signals, those that are basically impulse responses and which, ultimately, decay back to quiescence. As for chirps, I'm not sure, but I believe that a "chirplet transform" is what is needed, but this is now taking me outside of my comfort zone. DoctorTerrella (talk) 20:08, 13 November 2014 (UTC)
Thanks for taking the time to explain. It's not completely clear to me, but that's probably because I work with this stuff in the form of a finished tool rather than as a tool designer.
I use FFT analysis quite a bit in my work in audio, setting up loudspeakers for optimum reproduction of an audio signal in a performance space. The dual FFT section of Smaart software running on my laptop compares the desired audio signal (input) with the signal as heard in the space (output), with frequency response problems displayed so that the system engineer can make intelligent corrections. As such, the only "stationary" parts of that process I can imagine would be the architectural interactions resulting in standing waves, or if "stationary for all time" can be reduced to "stationary during the measurement period" then it can also mean interactions between the microphones on stage and whatever loudspeakers are aimed at the audience. I would not typically need a chirp test signal in my work, but I am aware of their historic and current value in many other fields. Binksternet (talk) 23:39, 13 November 2014 (UTC)
@Binksternet, of course, I am unfamiliar with what exactly you are doing. But, again, it is possible to describe any time series as the superposition of Fourier components. That is the "completeness" theorem of Fourier analysis. Such a decomposition is not, however, unique. One could, I'm sure, describe a time series in terms of lots of other different wiggly functions. For you, I imagine that you might be using an FFT for a finite duration of data, looking at its Fourier content, then, maybe, looking at another finite duration of data, comparing the Fourier content with the first duration, or, maybe you are using a sliding window, within which you apply an FFT, etc. All of that can be fine. Just depends on what you want to do. It is also useful to simply look at data in the time domain (without FFT). And, sometimes, the signal of interest is perfectly obvious when inspected in the time domain. So, for example, a straight line function looks pretty simple in the time domain, but take a FFT of a finite duration of that straight line and you end up something complicated; its spectrum will be red, but that might not be considered to be a very convenient description of a straight line! Similarly, a chirp function might look simple in the time domain, but, as we've been discussing, it can have a complicated description in terms of Fourier components. Consider all of this in one of the contexts in which Fourier analysis was first developed -- for the analysis of tides. There we have something that is very appropriately described in terms of "stationary" Fourier components, or, at least, it is a convenience for analyzing tidal data of a finite duration! Tidal data are often modeled as the sum of specific Fourier components with a discrete set of frequencies that can a priori be predicted from physics. Some random noise element is also superimposed to model irregular tide gauge data that are realized during storms. Enough for now, I think, DoctorTerrella (talk) 14:17, 14 November 2014 (UTC)

I'm afraid you've misunderstood the term 'stationary' with respect to chirps, so none of the above explanation is relevant. In attempting to derive the frequency spectrum of a chirp pulse by means of Fourier analysis, there is a point in time, at a given frequency, at which the phase rate slows to zero (hence it is called 'the stationary PHASE method'. It is the contribution to the integral around this point that provides most of the answer at that frequency. The result is only approximate but was better than nothing in the days before digital methods. It is still useful when deriving non-linear chirps. The method is particularly applicable to oscillatory waveforms and lots of articles on the stationary phase method can be found in the literature and some, with particular relevance to chirps, are given in my article.

I thought my article simply summarized several methods of deriving the spectrum of a chirp, namely analytical methods, stationary phase methods and digital methods, and displaying the results. I did not attempt to widen the discussion to other topics, so I don't understand why you wish to do so. The spectra are important because their characteristics play an important part in the final outcome, when chirps are compressed - which I originally intended to write up. As all the topics I include can be found in the literature on chirps, so I do not understand why they can be considered irrelevant or not about chirps. I suggest you have a quick look at 'Radar Signals' by Cook and Bernstein before commenting further. Also, please bring in someone into the discussion who has the relevant expertise D1ofBerks (talk) 11:46, 14 November 2014 (UTC)

### Notability

This most certainly needs a separate article, linked from and summarized (one paragraph max) in chirp. Also, although it's not WP:OR, it's bordering on WP:SYNTHESIS, so please be careful about the drawing conclusions not given in the sources cited. The tone should be more encyclopedic and less tutorial; e.g., it shouldn't explain what is a Fourier transform or why the FFT is preferable in practice. Also, figures in the DFT section seem to have wrong x-label: it should be in frequency units, not time units (i.e., hertz nor seconds). Finally, very little is said about the phase spectrum, which is crucial to discriminate between a chirp and an impulse function, both having similar power spectra. Fgnievinski (talk) 12:15, 14 November 2014 (UTC)

@D1ofBerks,I also suggest that you develop a new article, as both @Binksternet and @Fgnievinski have suggested, with context, explanatory motivation, etc. DoctorTerrella (talk) 12:33, 14 November 2014 (UTC)

Thank you for your comments. It is true that phase doesn't feature very highly in the article, which is a bit remiss of me. You are right to point out that phase is just as important as amplitude in what makes a chirp a chirp, rather than say an impulse (which has a similar amplitude response). However, I have several excuses to offer in that regard! Firstly, in the topics I cover, the literature doesn't discuss phase in much detail either. Secondly, the stationary phase approximation doesn't provide or require detailed phase information. Thirdly, the plots look a bit boring when compared to the 'pretty' amplitude plots! Fourthly, with digital processing, one is usually dealing with numerical lists of I/Q data, so although phase is 'in there' it doesn't feature prominently. Finally, when a chirp pulse is compressed by its matched filter, the final spectrum is very nearly phase-less. Consequently, designers concentrated on the consequences of the Fresnel ripples (and finding ways of reducing them), as purely an amplitude problem. However, I think a paragraph, verbally describing the phase characteristics of a chirp and referring to relevant literature, would improve the article - and needn't be too long.

Of course, if one is designing a dispersive filter in order to produce or compress a chirp pulse, then phase is what it is all about. However, this topic belongs in another section, the one on the methods of chirp generation, which I once intended to expand.

I do not believe that I have drawn any new conclusions about anything in the paper. All the topics are now quite old. The first paper on chirps in radar, in the public domain, was in the Bell Systems Journal in 1960 and even the methods of digital processing are decades old now. Lastly, please give more info regarding the incorrect figure, as I can't find it. (Can I get the article, in its state when it was deleted back into my Sandbox, as I think it has had a few alterations since I submitted it?) D1ofBerks (talk) 10:58, 15 November 2014 (UTC)

@D1ofBerks, in the spirit of constructive progress, yes, I'm cool with restoring the material I originally deleted, but please, then, move to a separate work in progress Sandbox. I'm glad to see that we all appreciate the utility of explanatory paragraphs for this material. Indeed, when I look at technical books, I usually prefer those with more words and less equations. Intuition is essential, maths can then follow from there. Thank you, DoctorTerrella (talk) 15:42, 15 November 2014 (UTC)
No need to sandbox it, it can go into an article of its own; improvements can be made live. Fgnievinski (talk) 16:35, 15 November 2014 (UTC)
O.K., whatever works. DoctorTerrella (talk) 16:39, 15 November 2014 (UTC)
The problematic figures have x axis labeled "number of samples" but y axis labeled power spectral density; the x-label should be in frequency units. Also, it's funny you say the stationary phase method doesn't require phase information; you're working on the real (or imaginary) components, so its amplitude times cosine of phase -- phase is present. Finally, if the phase spectra plots look unpleasant, try unwrapping phase. Fgnievinski (talk) 16:35, 15 November 2014 (UTC)

I am happy with the article being set up separately, I was most unhappy that it might disappear entirely! I do concede that in its original location did make that article unwieldy. It will need an introduction of course, but that shouldn't be too demanding a requirement. I'll also add something about phase, but I'm not sure I want more diagrams!

With regard to stationary phase methods and the problem of phase in detail. Initially, in the method, phase is a parameter occurring as Theta double dot in the equations. It can be used for developing linear or non-linear frequency characteristics (Theta dot), but these characteristics are only approximations - when the final chirps are developed the actual phase characteristics are similar to, but not the same as, those first chosen. In the examples I give, the frequency versus time characteristics are first developed, from which chirp pulses can be produced. These chirps will have actual magnitude and phase characteristics, as you rightly point out, but it is not necessary to plot them out to finalize a design. In any case, they aren't exactly what one started with and it does mean, as I mentioned in the article, that awkward issues like Fresnel ripples are not addressed.

Now back to the diagrams. The diagrams in question are for sampled data. The first diagram shows a base-band signal sweeping though zero frequency in the time domain. There are 128 amplitude samples of 'I' data and 128 samples of 'Q' data. The sample numbers are for a time sequence T0, T1, T2,. .., T127. The diagram shows the 128 'I' samples, only, plotted against sample number (i.e. sample number is for time). If the total duration of the samples is T seconds, then the sampling frequency is 128/T Hz. When the FFT is taken, 128 'I/Q' samples result. Now the sample number represents frequency, with zero frequency at N = 0 (and repeated at N =128). The point N = 64 corresponds to half the sampling frequency, i.e. the Nyquist limit. Sample numbers N = 0 to N = 64 are for increasing positive frequencies and sample numbers N = 128 to N = 64 are for negative frequencies, as shown in the second figure. The y axis shows I(squared) plus Q(squared). I hope that's clear - and that I've got it right! D1ofBerks (talk) 19:59, 15 November 2014 (UTC)

As you well described, "sample number" is ambiguous; I'd strongly recommend multiplying the sample number by the sample time spacing and frequency spacing, then labeling x axes as time or frequency and their respective units. Fgnievinski (talk) 00:00, 16 November 2014 (UTC)

On reflection, I think I can do a better job on explaining the phase issues you raised regarding the stationary phase method, so here goes: Consider a constant amplitude pulse (in the time domain), which is required to have the spectral profile of, say, the Hamming window. The stationary phase method will derive the frequency and phase characteristics, versus time, that will provide this profile. So we do have the phase characteristic in the time domain, as you pointed out. Armed with the amplitude and phase characteristics, we can plot plots of the real and imaginary parts of the chirp, against time, should we wish. We now have a non-linear chirp that apparently fulfills the initial requirement.

However, what the method doesn't provide us with are accurate details of amplitude and phase in the frequency domain. Hence my earlier excuse about the paucity of phase information with the method. In addition, the data that is available is only an approximation of the true situation, so even the magnitude versus frequency characteristic which aimed to match the Hamming profile, will not be an accurate representation of the actual chirp spectrum.

Returning now to the labeling of axes. My intention was to show that with digital processing one is dealing, basically, with lists of data. The computer doesn't know about time or frequency - it just accepts in and churns out sequences of numbers. The graphical displays are just an aid to the old fashioned analogue engineer (like me!). Following your concerns though, some further thought on the labeling is needed. D1ofBerks (talk) 15:49, 16 November 2014 (UTC)