Thanksgiving is behind us…it was a great day here in SoCal in spite of an hour delay on Pacific Coast Highway heading up to Ventura due to a traffic accident and a closed PCH near Trancas due to a water main failure on the way back. I guess I should stick to the Freeways.
I started talking about audio filters earlier this week. There are all sorts of audio filters and they’re useful at various stages in the audio production and reproduction path. I started the discussion by talking about the use of High Pass Filters during a live session to remove unwanted low frequencies. There are other analog filter types used during recording and mixdown sessions but I thought I would jump to the conversion side of the equation and talk about the need for LPF (Low Pass Filters) when converting analog to digital.
The Nyquist-Shannon Theorem deals with the conversion of a continuous signal into a numeric sequence. The following is Shannon’s version of the theorem:
If a function x(t) contains no frequencies higher than B cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
The essence of this statement of the theorem says that a frequency can be “completely determined” by a sampling frequency that is two times the original frequency value. The key here is the “completely determined” part. Yes, there are benefits to using sample rates that are higher…even much higher…than 2 times the highest frequency that you want to capture…but they aren’t absolutely necessary. But it does mean that you absolutely cannot present the conversion system with frequencies that are higher than the Nyquist Frequency (the sampling frequency divided by 2). This results in aliasing…a bad thing in digitizing audio.
I had a push pull with a person over at Gibson’s forum about this very issue. Craig Anderton wrote a piece on the world of high-resolution audio. His focus was on the benefits of making new recordings using higher sample rates. The comment is quoted below:
“There’s a theoretical reason 96Khz works so well. You need to sample any given waveform at minimum five times for truly accurate reproduction: Start (at the initial attack or zero amplitude); at peak positive amplitude; at the zero-cross point, at maximum negative amplitude, and end (final zero-cross). If the highest we can hear is 20,000 hertz, five samples of each wave (“hertz”) = 100,000 samples or 100KHz. Not many can hear 20,000 hertz, so 96K is usually considered close enough. The actual figure of 96Khz is used due to the mathematical nature of binary expansion.”
This completely misses the key point of the theorem. I used to believe similarly but revised my position after reading more carefully the Nyquist information. After I pointed this out to the writer, he responded:
“What’s known as the Nyquist-Shannon Theorem describes the MINIMUM number of samples to provide an INTERPOLATION of the analog signal (and a simple sinusoidal signal at that). For accurate REPRODUCTION, a greater sample rate is necessary. Some purists would argue that five samples would be grossly inadequate for a complex modulated envelope.”
His capitalizing of MINUMUM is ignores the “completely determined” statement in the theorem. The fact is that a sampling rate of 2x the highest frequency can absolutely reconstruct a complex musical sound. I didn’t want to get into a debate with him so I left the issue hanging.
But there’s an important issue that is required as we move from the theoretical to the real world. We need to have ideal LPFs with very steep slopes that maintain correct phase. Is this possible? Yes, it is.
To be continued…