I just had to stop reading about half way through. The writing style and organization is structurally incoherent and comes across as desultory gibberish with intolerably sloppy terminology and perspective. On the upside, it should be somewhat beneficial to the low caliber types who are confused by the subject matter.
Tim,
I like the article and totally agree that there's a lot of smoke and mirrors and misuse of sampling theorem ideas. I would definitely point newcomers to your description.
Please don't take the following as flaming. But I just can't resist throwing in a couple of my favourite pet peeves :-(
I echo the sentiments of an earlier comment about
fs >= 2fmax
versus
fs > 2fmax
If you look at e.g. Oppenheim & Wilsky, Signals & Systems pp 519 they clearly state in terms of proper greater than. However, if you look at Shannon&Weaver "The Mathematical Theory of Communication" P 86 "Band limited ensembles of functions" it's less clear (verbage rather than math): they say (more or less) "let f(t) contain no frequencies over W. Then you can sample at 2W". This would imply the geater-than or equal, which I suspect is incorrect.
In fact, in your article you point out that you cannot sample a sine wave at exactly 2*f unless you know the phase, which clearly contradicts the ">=" interpretation. (Actually you need the phase and the amplitude).
I'm not entirely sure of the history but I suspect that the band-limiting can actually be traced much father back to Fourier analysis in mid 1800's.
In the same vein, I've thought that one could do a neat write-up of how to reconstruct from the sampling of sin(2*pi*f*s) at a frequency approaching (but not equal to) 2f. If you describe how the almost-aliased-to-DC low frequency samples (regardless of phase) get turned back into a full amplitude signal through the magic of sinx/x reconstruction, it would be very cool. The samples would (where phase approached one side) be almost perfect (+1,-1,+1,-1) but as the phase drifted the other way, the envelope would attenuate (+0.0001, -0.0001, etc). Through the magic of summing an infinite series, you get your full-size plain old sine wave back. But that might be more math than the scope of this article.
I also would propose a change to sec 3.3 about repetitive signals. There is really no difference between this example and 3.4 - band limited signals! The only reason that the powerline sampling in sec 3.3 works is that you're talking about a bandlimited "modulation" signal that's modulating a 60Hz carrier! So it's not really the case that "sometimes you can sample slower than the Nyquist Thm would indicate"... this is a little misleading. I'd introduce it more like "some classes of signals, which appear to be fast, are actually low bandwidth, they're just a slowly modulated carrier, so you get a low Fs". (I'm not being precise but I hope you get my drift).
Cheers,
- Kenn
Hello Mike,
It seems you have overlooked that LTspice assumes a certain rise and fall time if you specify t_rise=0 or t_fall=0.
V1 Vin 0 DC 0 PULSE (0 1 0 0 0 2.5e-006 5e-006)
LTspice assumes in this case trise = fall = 10%*Twidth = 250ns
This is the reason why you see no overshoot.
Now we use fast edges.
V1 Vin 0 DC 0 PULSE (0 1 0 1n 1n 2.5e-006 5e-006)
You will see about 1.6x% overshoot. By the way, I am not sure how precise the choosen component values are to get the ideal Bessel filter response. Bessel filters have indeed overshoot in the step response.
Best regards, Helmut
Hi Helmut,
Thanks for taking a look at this, and for pointing out that LTspice changes the rise and fall times if you do not specify the values. I am still learning how LTspice works, and these unexpected variations from other SPICE programs can be major pitfalls.
In this case, the amount of overshoot is very small, and I'm not sure if the 1.6% is not caused by the way the component values are rounded.
I use this filter program for general filter design. The component values were calculated for the theoretical values and the inductors were rounded to 2 significant digits. This reflects the difficulty in obtaining precision inductors, particularly at VHF and UHF.
The capacitors were rounded to tenths of a pf. This is because the program is used mostly for work at VHF and UHF, where the caps are much smaller. It may well be that using the correct theoretical values for the components may reduce or eliminate the small amount of overshoot you measured.
However, there is still the practical matter of obtaining inductors and capacitors with the needed precision, and any practical filter will have imperfect components. I have found the Bessel and Equiripple to be much more tolerant of errors in component values than other filter types, and still give good performance.
If we are concerned about such small values of overshoot, we need to repeat this using the correct theoretical values for the components. Even so, an overshoot of 1.6% is not in the same category as the overshoot from Butterworth or other non-linear phase filters, which may have ten percent or more, depending on the sharpness of the cutoff. In practise, the Bessel and Equiripple are considered linear phase filters, and have negligible overshoot.
For example, a Butterworth or other nonlinear group delay filter may cause excessive timing errors when used to filter digital data. This can be minimized by changing to a Bessel or Equiripple filter.
So practically speaking, the Bessel and Equiripple distinguish themselves from other filters by the constant group delay through the filter bandpass, and the low or non-existant overshoot.
Regards,
Mike Monett
Antiviral, Antibacterial Silver Solution:
Tim, great article and I'm liking your book.
It is just me or did anyone else get a strong sense of deja-vu?
-Dave
-- David Ashley http://www.xdr.com/dash Embedded linux, device drivers, system architecture
Finally! Someone who sees my writing the way I do! I think we're in the minority, though.
I don't think of the intended audience as "low caliber", though -- just people who've had a different educational background, and whose back-brains don't light up with equations when they see a fountain, or a near miss on the freeway.
I have to say that this paper rather turned into a monster while I was writing it. I thought it was going to be between 2000 and 3000 words, with almost no math and very little real work. Instead it's about 5500 words, and I've got about a man-week into all those pretty charts and graphs (I should publish an appendix with "the making of..." along with all the math underneath).
As a reaction to this I haven't done my usual stage of letting it rest and getting back to it -- I was afraid I'd never do the "getting back to it" step. Instead I've put it outside without giving it time to get it's coat and boots on. If I can figure out how to tighten it up I certainly will -- assuming that I don't run away screaming at the thought of doing even _more_ work on it.
-- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Yes, I did write essentially that at the head of a posting soliciting suggestions for the article -- and you all helped.
(note to self -- add an acknowledgments section)
-- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Bessel filters do have a small overshoot, which decreases at higher orders.
2nd .43% 4th .84% 6th .64% 8th .34% 10th .06% note the *very low* value for higher orders. In fact the frequency response gets hardly better then. The Gauss filter has indeed zero ringing, but a much larger transition band.-- ciao Ban Apricale, Italy
You really need to look at theory versus what spice predicts. The input is an ideal step. As you do the convolution, the negative region of the impulse response subtracts from the result, making the signal drop in value, then the postive regions make the result increase in value, hence ringing.. The Gaussian impulse response is always positive, hence the convolution output can't decrease as time increases.
Linear phase alone is not enough to stop ringing.
The discrete time situation is easier to understand, especially if you consider a finite impulse response filter. The response of a FIR filter to a step input is the the sum of the tap coefficients from one to N. That is, the first output is the first tap. The second output is the sum of the first two taps, etc. If no tap is negative, then the output always rises, hence no ringing.
No circuits were simulated in writing this post. ;-)
Bwahahahaha! ROTFLMAO!
...Jim Thompson
-- | James E.Thompson, P.E. | mens | | Analog Innovations, Inc. | et | | Analog/Mixed-Signal ASIC's and Discrete Systems | manus | | Phoenix, Arizona Voice:(480)460-2350 | | | E-mail Address at Website Fax:(480)460-2142 | Brass Rat | |
Ban,
Thanks for the clarification. In most references, the Bessel is considered to have low, negligible, or no overshoot, especially when compared to Butterworth, Chebyshev and other types of filters. Your numbers confirm this.
In practise, it is difficult to obtain the exact component values needed for the theoretical performance. Not only are the values non-standard, but it may be difficult to get the inductor "Q" values used in most calculations. So we can assume there will be some deviation from the theoretical performance, and the overshoot will probably increase slightly. However, it is still low enough to be difficult to measure, and the terms "low", "neglible" or "no overshoot" are quite descriptive.
Regards,
Mike Monett
Antiviral, Antibacterial Silver Solution:
Please see Ban's post and my reply. Most references describe the Bessel as having low, negligible, or no overshoot. Ban's numbers show a theoretical overshoot of less than 1%, decreasing with higher order. In any practical LCR filter, the Bessel will have such low overshoot as to be difficult or impossible to measure. So it doesn't make sense to split hairs between
Well a few neurons were tortured trying to recall all that theory. No wait, make that extraordinarily rendered.
There are differences between analog and digital filters. Digital means an approximation of the desired characteristic in the passband, but the poles and zeros are modified to compensate for the modulation effects. aliasing is always happening, but it can be reduced below the noise floor with the analog input filter. Linear phase has a very undesirable side effect, it rings *before and after* the step, supposed to be more audible.
And Audio is a very forgiving métier, because the ears themselves function as reconstruction filters, suppressing all those high frequency artifacts.. If you want to display the signal with a digital scope or ECG, you better start with 5 to 10 times the upper frequency rolloff, look at the specs there, nobody even considers Nyquist adequate, exept programmers having no idea of reality.
-- ciao Ban Apricale, Italy
Nyquist has a lot to say about how far you *can* go and little to say about how far you *should* go. Those of us who are forced to push the Nyquist limits may have a better appreciation of this.
For example, compare the frequency response of a 5-pole IIR Butterworth low-pass filter designed to roll off at 1 KHz. with one designed to roll off at 10 KHz., both filters having been designed around a sampling frequency of 50 KHz. Things start to get nasty above Fs/10 and the harder you push Nyquist, the worse it gets.
Nyquist says you can put a sharp +/- 500 KHz analog filter around a 21.4 MHz IF and sample it at 2.14 MHz. What Nyquist doesn't say is what that does to your signal-to-noise ratio. TANSTAAFL!
Nyquist is often used by slick, hand-waving charlatans to over-sell their capabilities without having a clue about the real trade-offs involved. (And, I'll bet Nyquist didn't say
*that*, either.)
ction=20
ts..=20
And audio is a very unforgiving matter, because the distortions of the=20 audio amp are quickly growing towards the high frequencies. You won't=20 hear the high frq artifacts as they are, but you will very well hear the =
result of the nonlinear distortion of those.
Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
But only with the smelly markers. The water-based stuff doesn't do it for me.
robert
I see it that way as well. A pure sine signal of exactly f_0, sampled at exactly 2f_0, might come out as all-zero, or as a pulse train with alternating polarity and an amplitude of anything between 0 and V_p. There is not enough information to reconstruct the signal in this case.
However, if you sample the same signal at 2f_0+f_e (f_e being very small, think epsilon), it will come out as a pulse train with alternating polarity and the amplitude modulated by a beat frequency f_e. This would seem to be not enough information to reconstruct the signal, but in fact that's not true: A f_0 sine is the only possible input signal because the modulated pulse train contains frequency components greater than f_0 which couldn't have been in the input signal (which, as a prerequsitite to Nyquist, is brick-walled at f_0).
So, the way I see it is that the sampling frequency must be strictly greater than the highest frequency in the input.
Thanks, Tim, for a great write-up on the subject!
robert
Let me tell you that this article is so high-class from an educational point of view that it /deserves/ coat and boots. Next time anybody comes up to me and wants something explained about Nyquist I can just point them at your page, and I'm sure many others will do likewise.
The only thing I'd want is the whole thing as a pretty, nicely printable PDF document. But don't listen to me.
robert
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