[Martin Taylor 2007.12.17.14.39]
[From Erling Jorgensen (2007.12.17 1355 EST)]
Martin Taylor 2007.12.16.11.10
Bill Powers (2007.12.17.0110 MST)
I've been trying to follow this discussion of degrees of freedom,
though it gets muddled for me at times. Every so often, a term like
"bandwidth" comes along, & I have yet to fully wrap my brain around
that notion in a way that matters & that allows me to use it correctly
in a subsequent discussion. (Some of us do not have much mathematical
or engineering background, alas...)
As usual, you ask cogent questions. Thank you. I'll answer some of them by implication in responses to Bill, but some here.
Bandwidth, like most of the concepts in the discussion, is simple unless you go into it in detail. I've been trying to keep to the simple side of all of them, becasue unless the simple core of te concept is understood, there's not much point in working through the subtleties.
Any waveform can be represented in a lot of different ways. One way is to tell what its value is at a lot of different "samples". A sample is just the value of the waveform at a moment in time. If you have a waveform lasting T seconds, and you sample N times per second, you have NT samples.
Fourier discovered that another way to represent any waveform is as the sum of a set of sinusoids, each of which has a magnitude m. To specify a sinusoid is to give values to m, f and b in the expression m*sin(ft+b). Setting t to any value in that expression gives the sample value of the sinusoid at moment t. f represents frequency, and b represents phase -- where along its wiggly waveform was the sinusoid at te moment you choose to call zero.
What Fourier discovered was that you can describe a waveform as accurately as you want if you use a set of sinusoids with frequencies that are all the multiples of a "fundamental" frequency f0 that completes exactly one cycle in the T seconds of the waveform you are describing. You then have a set of sinusoids plus a constant that gives you the overall average value (a sinusoid that completes an integer number of ccycles has an average value of zero). Each sinusoid is described by two valuesm m and b in the previous paragraph.
If you have used sinusoids up to a frequency fn = n*f0, you have described a waveform exactly. That waveform has a bandwidth W = fn. It describes your original waveform exactly, except for wiggles or steps that are faster than the top frequency.
Bandwidth doesn't necessarily have to start at zero frequency. If you can describe your waveform without using f0 ... fl ("ell" for lower), and have a good representation using only sinusoids of frequencies fl ... fh, that representation has a bandwidth W = fh - fl.
Since each sinusoid of frequency fx is specified by its magnitude m and its phase b, the Fourier representation uses 2W samples per second, or 2WT samples in all. All these values can be independently set, so each represents a degree of freedom. There are no other degrees of freedom in the band-limited waveform, but those 2WT+1 degrees of freedom can be used in different ways. One of those ways is to specify the values of 2WT sample values along the time of the waveform.
In other words once every 2W'th of a second, you can have a new independent value of a variable with a bandwidth of W. To put it another way, a varioable with a bandwidth W has 2W degrees of freedom per second.
The definitions of terms definitely help. For instance --
1. A degree of freedom represents a variable that can take more than
one state.That just seems to define a variable: something that can exist in
more than one state. If something is a variable, it can take more
than one state. To me, a degree of freedom refers to a relationship
between different variables.A question arises for me, with Bill's continuing comment:
If it is possible for two variables to
be set to any pair of values at the same time, as we can place a>point (x,y) in Cartesian coordinates anywhere in the plane, the
relationship has two degrees of freedom. If x + y = constant, that
relationship has one degree of freedom since specifying the value of
one variable also specifies the value of the other.I am thinking of the situation where a color is exactly specified
by the weightings of three more primary colors. (I realize there is
some slack in the choice of the primary colors, an instance of the
concept of "rotation" of the coordintates, if I have this right.)So the weights of the three primary colors (whatever they may be)
constitute three degrees of freedom. But if a particular weighted
sum combination of the three is desired, for instance in specifying
a background color on a computer screen, then choosing the weights
of two of them no longer leaves any freedom to vary the weight of
the third.In this situation, it appears to me as though the third degree of
freedom from the weight of that third color has been "transferred"
to the higher level combination. There remain three degrees of
freedom, but their distribution has altered. Is this correct?
Yes. That's precisely the situation.
My other question has to do with temporal order as a degree of
freedom. ...
Consider the following example. I can choose different objects to
point at with my finger, & so in the language of this discussion, the
objects are different "values" of the pointing degree of freedom. If I
add a temporal degree of freedom, I can point to all of them (in turn).
The speed at which I can point (is that an output "bandwidth"?) would
seem to become the unit of the different "values" of that temporal
degree of freedom.Are we still just dealing with two degrees of freedom here, one for
pointing & one for time, or does time effectively multiply the
available degrees of freedom? Your words, Bill, seem to imply the
latter.
Time does multiply the available degrees of freedom.
The finger angle has one degree of freedom, but the waveform of the variable finger angle over time has 2WT degrees of freedom, where W represents how fast you can move the finger to new choices of pointing location.
One of the omitted subtleties is that the multiplication is not always direct. If you have an x variable and a y variable, you have two degrees of freedom for any sample (I've called that "instantaneous" degrees of freedom). Let's say that when looked at individually as functions of time, x and y each has a bandwidth W. You would think that over T seconds they would have between them 2*2WT degrees of freedom. However, if the movements of x and y are at all correlated, then there are fewer than 2*2WT degrees of freedom over T seconds. I don't want to be concerned with this kind of subtlety yet, but I thought I should mention it as the kind of thing we have to be wary about.
Thanks for the care with which you are each conducting this discussion.
I'm glad you are finding it to be useful.
Martin