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iir – is more efficient. Most of the time it does not have linear phase. Bessel filter is the type of IIR filter which has a lenear phase.

H(z) =

b0z – 2 +
b1z – 1 +
b2
a0z – 2 +
a1z – 1 +
a2

When sound travels through solids rather than air, the speed of propagation varies with the frequency (I believe acoustics folks talk about “bending” as the mechanism for propagation). Does it mean if 2 frequencies are mixed together and coupled into a solid, will the phase relationship between the two frequencies vary as the pickup point is moved along the surface of the solid.

Does this frequency dependent speed of prorogation imply non linearity thereby generating new frequencies when several frequencies are mixed together. Will the distortion also exist with single frequency excitation. Can this be modeled or characterized.

If the speed changes fairly slowly with frequency, just the
speed vs. frequency curve will be enough.

Consider that optical materials also change speed with frequency,
but away from resonances they can usually be considered linear.

Non-linear optics is done at high amplitude where things
do go non-linear. The displacement of the electrons is large
enough that non-linear effects become significant.

I think the effect of the sound speed as a function of frequency is
dispersion. I don’t think it implies nonlinearity at low amplitudes.
Here are a couple of documents that discuss measurements.

Improved Surface Wave Dispersion Models and Amplitude Measurements
ADA422916
http://stinet.dtic.mil/cgi-bin/GetTRDoc?AD=ADA422916&Location=U2&doc=…

Discrimination, Detection, Depth, Location, and Wave Propagation
Studies Using Intermediate Period Surface Waves in the Middles East,
Central Asia, and the Far East
ADA417753
http://stinet.dtic.mil/cgi-bin/GetTRDoc?AD=ADA417753&Location=U2&doc=…

Even if the topics aren’t an exact match to your interest, take a look
at the introductory sections.

Direct use of non-linearity at high drive levels is used in
“parametric” oscillators and amplifiers. Take a look at the discussion
of the non-linearity in this document as an example.

http://www.sea-acustica.es/Sevilla02/ult04021.pdf

Hi Joe,
I don’t think that this implies non-linearity or the creation of sum and
difference frequencies. Between any two locations the audio transmission
should be defined by an impulse response and a frequency response, just as
any linear system. As you mention, the frequency response will be a
constant amplitude and a changing phase, with respect to frequency. Off
hand, I don’t see any reason that this cannot be linear (someone smarter
might correct me!).

Propagation velocity being a function of wavelength is called
“dispersion” whether in optics or acoustics. Snell’s law applies in
either case. Acoustic lens structures exist and have been used to focus
and disperse sounds. Non-linearity, if involved at all, is a side effect
and not fundamental to the phenomenon.

My problem is that if I think of a linear mechanical system, I should
be able to define a spatial R-L-C lumped approximation, and once I
have done that I don’t see how propagation speed could depend on
wavelength (propagation speed in not the same as group delay at a
given spatial point; I am assuming no effects from rolloffs, the
system is flat at all frequencies that are injected). Try to think of
a R-L-C transmission line model where the propogation speed varies
with frequency; I can’t think of how this could happen in a linear
system.

Dispersion occurs on transmission lines also. They can be modeled (below
some critical frequency) with R-L-C components and no others. Instead of
my writing a treatise on transmission lines, try Google. Start with
http://www.ece.uci.edu/docs/hspice/hspice_2001_2-269.html

Waveguides are also dispersive. Unlike with transmission lines, it is
not even theoretically possible to avoid it. All optical fibers show
dispersion. The propagation velocity is the inverse of the refractive
index, and refractive index is a function of wavelength. No distortion
is involved.

I think this is different. Consider the following experiment. I drive
a narrow and thin strip of plastic at one end with a transducer of
some sort. I dial the frequency to 1KHz, and then I measure the
PHYSICAL wavelength somewhere near the end of the strip by moving 2
pickup microphones around until their outputs are in-phase. The
propogation speed is then calculated by dividing the physical
wavelength by 1ms. Let’s say the wavelength turns out to be 1 cm.
Now I double the input frequency to 2KHz, and repeat the experiment.
This time, though, the physical wavelength comes out to be .4cm
instead of the expected .5cm. indicating that the propogation speed
has changed.

I could run the exact same experiment with a transmission line, and I
would argue that no matter how much dispersion you get, the measured
wavelength will always be equal to the propogation velocity divided by
the frequency.

—-

> I have been told by knowledgeable acoustics people that when a sound
> wave travels through a solid rather than air, the speed of propagation
> varies with the frequency

Is it TRULY frequency dispersion, or is it amplitude dispersion? I
find brief mention of the difference here: “http://
imamat.oxfordjournals.org/cgi/content/abstract/1/3/269”. I seem to
recall from my SONAR work, for example, that water exhibits amplitude
dispersion, which causes sinusoidal waveforms to turn into sawtooth
waveforms. That does not, however, address frequency dispersion for
non-sinusoidal waveforms in that medium.

> My question is this; does the statement that the speed of propogation
> is frequency-dependant also imply non-linearity, and therefore have I
> generated new frequencies when I mix several frequencies together? Do
> I get distortion with only a single frequency excitation?

Again referring to the abstract mentioned above, media that are linear
but dispersive seem to be possible. If a medium is linear but
dispersive, I think that it can be treated like a linear filter with
nonlinear phase response — the kinds of filters that we encounter
routinely.

> If it is non-linear, assuming that I knew the relationship between the
> propagation speed and the frequency (for single frequencies), how
> would I characterize such a system mathematically?

Perhaps characterize the nonlinearities and the dispersion separately;
nonlinearities as distortion products, and dispersion as filters (as
above)?
Greg

————————–

>> Waveguides are also dispersive. Unlike with transmission lines, it is
>> not even theoretically possible to avoid it. All optical fibers show
>> dispersion. The propagation velocity is the inverse of the refractive
>> index, and refractive index is a function of wavelength. No distortion
>> is involved.

I should add that transmission lines need some dissipation in order to
exhibit dispersion, but waveguides don’t. They *always* exhibit
dispersion, which (to first order) is independent of loss.

> I think this is different. Consider the following experiment. I drive
> a narrow and thin strip of plastic at one end with a transducer of
> some sort. I dial the frequency to 1KHz, and then I measure the
> PHYSICAL wavelength somewhere near the end of the strip by moving 2
> pickup microphones around until their outputs are in-phase. The
> propogation speed is then calculated by dividing the physical
> wavelength by 1ms. Let’s say the wavelength turns out to be 1 cm.
> Now I double the input frequency to 2KHz, and repeat the experiment.
> This time, though, the physical wavelength comes out to be .4cm
> instead of the expected .5cm. indicating that the propogation speed
> has changed.

Propagation velocity is wavelength times frequency by definition.

> I could run the exact same experiment with a transmission line, and I
> would argue that no matter how much dispersion you get, the measured
> wavelength will always be equal to the propogation velocity divided by
> the frequency.

That’s true by definition. How do you conclude from it that the
propagation velocity is independent of frequency?

In the real world, dispersion is the expected state. Glass exhibits
chromatic aberration, water-wave velocities depend not only on the
water’s depth but also on the wavelength, waveguides have group
velocities that are asymptotically free-space at very high frequencies
and go to to zero at cut-off. (The phase velocity goes infinite at
cut-off.) Glass fibers are waveguides built from dispersive material.
Even their dispersion is dispersive!

Simple electrical transmissions are often thought of as dispersion free.
That’s a useful treatment because we try to keep losses small and it
turns out that such lines — including circuit-board traces — will be
free of dispersion if they’re lossless *AND* their component materials
have salient properties (e.g., dielectric constant) that are themselves
independent of frequency. Dielectric constant is the square of
refractive index. The dielectric constant of water measured at audio
frequencies is a little over 80, which should make its refractive index
about 9. That’s what it is at ELF, but it varies about 1.23 to 1.33 for
visible light. (Propagation velocity is c/n, where c is free-space
velocity and n is refractive index.)

We make simple models — models that allow us to ignore dispersion for
the most part. That makes it easy for us to get the mindset that
dispersion is rare and anomalous.* In fact, lack of dispersion is rare.

Jerry
____________________________________
* There is anomalous dispersion, but that’s a different subject! 🙂

(snip)

> Simple electrical transmissions are often thought of as dispersion free.
> That’s a useful treatment because we try to keep losses small and it
> turns out that such lines — including circuit-board traces — will be
> free of dispersion if they’re lossless *AND* their component materials
> have salient properties (e.g., dielectric constant) that are themselves
> independent of frequency. Dielectric constant is the square of
> refractive index. The dielectric constant of water measured at audio
> frequencies is a little over 80, which should make its refractive index
> about 9. That’s what it is at ELF, but it varies about 1.23 to 1.33 for
> visible light. (Propagation velocity is c/n, where c is free-space
> velocity and n is refractive index.)

In addition, note that both the dielectric constant and index of
refraction are complex. The imaginary part of the index of refraction
turns into absorption when it goes into exp(iwt). Ellipsometry:

http://en.wikipedia.org/wiki/Ellipsometry

is pretty much the science of complex index of refraction.

> We make simple models — models that allow us to ignore dispersion for
> the most part. That makes it easy for us to get the mindset that
> dispersion is rare and anomalous.* In fact, lack of dispersion is rare.

Far enough from resonance, (and for fields much less than those
inside atoms) dielectrics are pretty close to linear
Heaviside worked out the conditions needed to allow transmission lines
to be nondispersive. A lot of people would not believe him and the
early trans-Atlantic cables were horribly dispersive and in fact one
of them got destroyed when one engineer tried to overcome the
dispersion by increasing the source voltage to the point of arcing out
the cable. Imagine having to explain you just destroyed the cable just
after it had been run across the ocean! Do the cable’s owners dock
your paycheck in this case?

In addition, all real materials are non-linear, though often they
are used where linear is a very good approximation. (In the case
of both mechanical and electromagnetic waves.)

There is the field of non-linear optics, studying materials where
the light intensity (electric field) is high enough that materials
are non-linear.

http://en.wikipedia.org/wiki/Non-linear_optics

Now, can a (non-physical) material be linear in amplitude,
but non-linear in frequency? Assuming that the response to
electric or mechanical waves (P vs. E, for example) is an
analytic function then I believe the answer is no.
Can it be close enough that you can’t measure the difference?
Maybe so.

http://groups.google.com/group/comp.dsp/browse_thread/thread/68931bdcb7210c69

Google Research just published a paper detailing there predicted search triends and and deviations.

Here is the blog post and here is the paper  – On the Predictability of Search Trends – a worthy read.  Shows how google is all about innovation

For those you  aren’t a devoted listener of This American Life, seriously what do you listen to ? Just listened to the valentine’s day episode, “Somewhere Out There” ?

What caught my attention was that in the beginning of  of the program a Hayward physics grad explains how he and his fellow physics grad students used Drake Equation to determine the likelihood of finding the perfect mate – http://www.thislife.org/Radio_Episode.aspx?sched=1283. The numbers are pretty bleak “It’s amazing that out of the 6 billion people in the world we managed to find each other.”

Was trying to calculate if a house on sale for $200K with 1300 in rental income would make sense. Basic calculation showed it possibly will- The Buy to rent Ratio is 13 a little less than break even point.

The attached  excel sheet contains ratio for $200K and $500K house – http://spreadsheets.google.com/ccc?key=pXHHXgTvPZ5dqM-OuZXL1XA

Throughout the 1970s, ’80s and ’90s, the average rent ratio nationwide hovered between 10 and 14. In the last few years, though, it broke through that historical range and hit almost 19 by the time the housing market peaked, in 2006.

Courtesy : http://www.nytimes.com/2008/05/28/business/28leonhardt.html

Also the big picture gives some interesting food for thought http://bigpicture.typepad.com/comments/2008/05/housing-price-t.html

Now the interesting question is that in the current financial market does putting $200K make sense or will the housing prices continue to fall further …

Case Study : Hayward, CA

  • Rental Prices Jan 2008
  • rent.com Average Rental price $15000 (2bed/2bath)
  • craigslist.com Average Rental price $1250 to $1700  (2bed/2bath) mean ~1400

Other things to look into

  • proximity to Bart

An old carpenters’ adage to “measure twice, cut once.”  does not transpose into software. In software the time to market is more important – making measuring twice costlier than cutting (releasing) twice. The current software production model should be measure and cut – twice or thrice if needed.

Software design and production processes can and should be evolving—specifically, increasing the time spent refining products before they’re released as “finished.” Releasing (cutting) puts the product in the hands of testers and beta users making the earlier. Software specifications tend to evolve  over time  and involving customers in the earlier stage speeds up the evolution process. Cutting twice would make what gets shipped would be of higher quality while simultaneously prioritizes time-to-market.

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