Oversampling Oscillator: Stuff or Calc?

[mystran]

Zero padding doesn't actually change anything except the sample rate. The aliased images are always there in digital audio, and by increasing the sample rate you raised the bar so that they now sit in the passband below half the sample rate. That's why you need to get rid of the aliases now revealed by the widened passband.

Well, yeah. Whether the images are there or not depends on how you interpret the data representing the digital signal. If you consider it representing a series of Dirac deltas, then your interpretation (images are always there) is correct. If you consider it as a representation for a series of critically band-limited cardinal sines (ie. the ideal reconstruction), then zero-stuffing adds new images.

Personally I like to think of signals in terms of the ideal reconstruction, but either approach leads to the same results, so it's really just a matter of which mental model you find more helpful.

Oh and .. in the z-transform plane, the "images" are simply what you get when you continuous going around the unit-circle (ie. DFT) in loops.

[xoxos]

http://www.analog.com/media/en/training ... MT-017.pdf

MT-017
TUTORIAL
Oversampling Interpolating DACs
by Walt Kester
INTRODUCTION
Oversampling and digital filtering eases the requirements on the antialiasing filter which
precedes an ADC. The concept of oversampling and interpolation can be used in a similar manner with a reconstruction DAC. For instance,
oversampling is common in digital audio CD
players, where the basic update rate of the data
from the CD is 44.1 kSPS. Early CD players used
traditional binary DACs and inserted "zeros" in
to the parallel data, thereby increasing the
effective update rate to 4-times, 8-times, or
16-times the fundamental throughput rate. The 4×,
8×, or 16× data stream is passed through a digital interpolation filter which generates the extra
data points. The high oversampling rate moves th
e image frequencies higher, thereby allowing a
less complex lower cost filter with a wider transiti
on band. In addition, there is an increase in the
SNR within the signal bandwidth because of the
process gain. The sigma-delta DAC architecture
uses a much higher oversampling rate and represen
ts the ultimate extension of this concept and
has become popular in modern CD players.

[earlevel]

Well, yeah. Whether the images are there or not depends on how you interpret the data representing the digital signal. If you consider it representing a series of Dirac deltas, then your interpretation (images are always there) is correct. If you consider it as a representation for a series of critically band-limited cardinal sines (ie. the ideal reconstruction), then zero-stuffing adds new images.

Personally I like to think of signals in terms of the ideal reconstruction, but either approach leads to the same results, so it's really just a matter of which mental model you find more helpful.

Oh and .. in the z-transform plane, the "images" are simply what you get when you continuous going around the unit-circle (ie. DFT) in loops.

Sure. Nothing wrong with viewing it as a model of cardinal sines—basically, how it will be when you send it out the DAC—but the reality is that the samples represent impulses (PCM—Pulse Code Modulation), and that reality works especially well in this case (you don't have to explain how new frequencies were created by adding zeros—it's obvious that the frequencies were there all along).

I'm not saying this to argue, I enjoy discussing the topic and hearing how other people view things. There is not just one useful way to look at things. I'm trying like heck to finish up a video on sampling theory, in my limited time, so it's nice to discuss and think.

[Richard_Synapse]

Repeating the samples will droop the frequency response, -3dB at the top of the passband, -1dB at 0.6 way.

Interesting thanks, wasn't aware of that (fwiw we use allpass networks so far, they should be perfectly flat and seem to be the most efficient solution possible, for the x2 case anyway).

Richard

[earlevel]

Repeating the samples will droop the frequency response, -3dB at the top of the passband, -1dB at 0.6 way.

Interesting thanks...

Edited my original post for better accuracy—it's actually 2/pi at the Nyquist frequency, -3.9 dB...

Edit for more Info: I thought I already said this, but was thinking of another thread on linear interpolation...the frequency response of zero-order hold (repeating samples) is the sinc function (sin(pi*x)/(pi*x)). (For first-order hold, linear interpolation, it's sinc^2—you'll see that's obvious if you note that it's convolution with a triangular distribution, which is the same as convolving two rectangular distributions, hence the "squared".)