*“Immediately following that you say:*

"This process is significantly more involved than simply ?connecting the dots? between sample points. Today it involves extremely sophisticated means of reconstructing the waveform, using filters that are highly complex mathematical systems utilizing ?oversampling,? ?upsampling,? ?linear phase, equiripple FIR? designs and much more.

This is the part that I am curious about. How is the waveform actually reconstructed from the sample data? How are over sampling, up-sampling, FIR filter(s) used to do this reconstruction?

Many thanks for your time!

Nathan”

Well Nathan, the wording “highly complex mathematical systems” is just a different way of saying “beyond the scope of this thread”. A lot of what we are talking about can fall into that category. But given that we are here, trying to communicate the best we can, I’ll take a stub at it.

Let’s think of a sloping straight line. You could take a pen and draw it on a paper. That is an “analog line”. After the AD conversion, you do not have an analog line. You have equally spaced dots. That is the “digital data”. How do you make it back into an analog line? You plot the dots on a different page, and you connect them.

Of course, life would be simple if we could connect the dots with straight lines for all shapes. But try it with a circle; say with only four equally spaced dots (90 degrees apart) the outcome is a square! With 8 dots (45 degrees apart)the shape (an octagon - eight sided daimond like) gets rounder looking but it is not a circle. But either way, with straight lines you end up with sudden change in direction (corners).

Suppose I tell you that you are not allowed to perform an immediate change in direction. No corners. You can not go for a sudden or sharp bend in the line. In a sense I am telling you to “round the corners”. Everything that moves slow is “left intact’ but any attempt to make a fast change is encountered with a lot of “rounding and smoothing”. With these “rules”, the square (or octagon) curve will be smoother, more circle like…

Rounding and smoothing is what a low pass filter does. “Low pass” means to allow slow changes to pass through intact. But low pass (passing low frequencies) is a “high reject”, thus not allowing for fast changes - sharp corners - quick changes, thus rejecting the high frequency content.

There are many mechanisms and forms for such filtering (smoothing) action. For analog low pass, think again of that big swimming pool – you can fill it and empty it given some time, but with some limited size pipe, it takes time. It can not happen at very high speed. The same is true for charging and discharging a capacitor with limited current. That description is good one for a very elemetry filter. A more complex network of capacitors and resistors and other parts makes a much better smoothing machine and a better filter.

Here is the amazing part: When you build a filter that is good enough to pass all the slow motion (low frequencies, everything below Nyquist) but reject all the fast motion (high frequencies above Nyquist), you end up with the perfect smoothing machine that will give you back the precise curve. It will connect the 4 dots of the circle with a perfect circle!!!

The closer you get to passing the energy below Nyquist and rejecting the energy above Nyquist, the closer the curve will be to the original circle “or audio waveform”. A perfect filter yields the perfect curve fitting. In fact, the perfect filter is the perfect curve fitter; they are one in the same!

Does it always work? Well, it does work if some takes care of some relationship in the conversion - The relationship between how fast the signal can change and how tightly spaced the samples are.

How fast the signal can change is set by the audio bandwidth. “40KHz audio” changes faster than "20KHz audio", so it needs twice as many samples… Once your audio bandwidth is less then 1/2 the sample rate, your filter is a perfect curve fitter. If the audio bandwidth is more than 1/2 the sample rate, the filter is not a perfect smoother and the reconstructed wave is not a perfect match to the original wave. That is Nyquist theorem on a gut level.

BR

Dan Lavry