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RE: All FIR filters are fatally flawed.

"When I said the test signal was at least seven octaves below the Nyquist frequency I was being absolutely precise and I assumed the superstars of digital audio, like you and Todd Krieger, would know exactly what I meant. I assumed too much."

When a DAC converts discrete samples into a continuous waveform a filter is needed to eliminate the images that appear above the Nyquist rate of the input sample rate. Whether this filter is done in the analog domain or by digitally upsampling to a much higher frequency, the same filter properties are required for reconstruction of the output waveform. The stop band of the filter must start at the Nyquist frequency corresponding to the input sample rate. Ringing in this filter will be determined by the filter response in the transition band located between the pass band and the stop band. In other words, ringing will be at or near the input Nyquist rate. The output sample rate is irrelevant to this discussion, although it does affect the filter design.

Even if a filter "rings" it will not produce ringing in the output unless it is provoked with a signal near the ringing frequency. So when you report that the two cycle waveform results in ringing at a non-harmonic frequency, this is not correct. What is happening is that the input waveform has energy at all frequencies up to the Nyquist frequency of the input sampling rate. Two cycles of a sine wave can be modeled as the product of a continuous sine wave and a pulse waveform (which is the sum of a positive and negative step function). The pulse waveform has energy at all frequencies and hence the product has energy at all frequencies. It makes no sense to say that this ringing is non-harmonic. The filter is not adding any frequency components that were not in the original signal.

"Here is the same signal processed by DAC W. It’s not pretty."

Without seeing the input, it is impossible to tell if DAC is doing a proper job of creating output. By input, I mean the actual samples, that is a bunch of dots on a graph, not a bunch of lines someone or something drew that connect the dots. Also, it is hard to tell from low resolution images how much of the ugliness has to do with image processing rather than audio processing.

Tony Lauck

"Diversity is the law of nature; no two entities in this universe are uniform." - P.R. Sarkar

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