Topic: Do CD's reproduce lower frequencies much better than higher freque (Read 12724 times)
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## Do CD's reproduce lower frequencies much better than higher freque

##### 2008-12-30 07:14:43
The sample rate for compact disc digital audio is 44.1 kHz. This means that, due to the Shannon Sampling Theorem, CD's can effectively reproduce frequencies up to about 22 kHz, where each sample defines the sound wave's point twice per Hz. But is the sample rate for CD's the same for all frequencies, or is 44.1 kHz just its maximum - meaning does the sample rate for CD's vary depending on the frequency being sampled, or is it the same regardless?

I ask this because if it is constant regardless of the frequency being sampled, than CD's would be able to reproduce lower frequencies much better than higher frequencies. For example, a 100 Hz sound wave would have 5 times as many samples per Hz as a 500 Hz sound wave, thereby providing more data to reproduce it more accurately. But is this what actually happens?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #1 – 2008-12-30 07:56:15
I am not sure if it works that way. Sound will be sampled 44100 times per second, whether it is 20 Hz or 20 kHz, or just white noise, with 16bit precision. Maybe there is something I am missing in this.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #2 – 2008-12-30 08:24:32
No hlloyge, you're not missing anything.

But is the sample rate for CD's the same for all frequencies
The sample rate is fixed.  Now you might want to re-think what you said about each sample defining a wave twice per period since it is only true for one frequency: half the sample rate (AKA the Nyquist frequency).  Well, at least I think you mean to say twice per period (twice per Hertz makes no sense).

I ask this because if it is constant regardless of the frequency being sampled, than CD's would be able to reproduce lower frequencies much better than higher frequencies.
No, this conclusion isn't correct, at least not from a mathematical standpoint.  Thanks to oversampling, it's really not correct from a practical standpoint either.

surfasap, do you know what a sinc pulse is?

PS: Sorry about my edits.  I have a feeling someone more intimate with the subject will be writing something much more useful, so I'm doing my best to not look like an idiot.  Hopefully my premature response will not be quoted.  This one's bad enough.
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #3 – 2008-12-30 11:00:59
Shannon was a very clever man.

No frequency from 20hz to 22khz will be better reproduced than any other because they ALL will be reproduced perfectly.  Since perfect is perfect, there can be no debate about any being better reproduced than any other.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #4 – 2008-12-30 12:35:52
Putting it another way, a perfect 20 Hz sine wave sampled at just over 40 Hz can be reconstructed perfectly, just as a perfect 20 Hz sine wave sampled at 44100 Hz can be reconstructed perfectly.

If, however, the 20Hz waveform is not a perfect sine wave then it contains higher frequency components and will not be perfectly reconstructed when sampled at just over 40 Hz. What will be reconstructed will instead be a perfect 20 Hz sine wave that most closely matches the waveform.

If a 20 Hz non-sinusoidal waveform that contains no frequency components above 20 kHz is sampled at 44100 Hz then it will be reconstructed perfectly. So in effect what you gain from the higher sampling rate is the ability to reconstruct nuances in the waveform of lower frequencies that are the result of higher frequency components.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #5 – 2008-12-30 17:05:17
If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.

Why do people give Shannon all the credit and never speak of Nyquist?
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #6 – 2008-12-30 17:41:58
Why do people give Shannon all the credit and never speak of Nyquist?

Never even heard of Shannon...  only Nyquist.  Anyways, the issue is that audio is not this perfect world where there is frequency content up to exactly 22.05 kHz and nothing above that.  The result is that aliasing occurs because of the frequency content over half of the sample rate.  Certainly, analog (or digital with higher sample rate) filters are applied before capturing the audio at 44.1kHz, but these only attenuate the higher frequency content rather than actually eliminating it.  At the same time, frequencies under 22.05kHz are also attenuated somewhat.  So, we are stuck with two issues.  Antialiasing filters result in attenuated signals and they do not perfectly eliminate aliasing.  Both of these distortions show up most strongly at the higher frequencies.

So, given an ideal input (no frequency content over 22.05kHz at all), all frequencies within a CD's ability can be recreated with equal quality.  But, that rarely happens so yes, lower frequencies will, in general, be more accurate than higher frequencies.  How much so is dependent on the original signal, the filters (any analog processing), the sampling equipment (with it's own filtering effects), digital processing, digital resampling and even your CD player (or DAC, more specifically).

-kyle

http://en.wikipedia.org/wiki/Aliasing

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #7 – 2008-12-30 18:16:47
Content placed on a CD does not necessarily have to have originated from an analog signal originally sampled at 44.1kHz.

...but you're right, content above Nyquist has to be filtered out and this is a practical issue when dealing with signals that have energy in this region.  Out of fear that you're giving too much credence to the OP's logic, I think it should be stated that for frequency content not approaching Nyquist, lower frequency signals are not better reproduced than higher ones.

Since we're dropping wikipedia links, here's one concerning Shannon:
http://en.wikipedia.org/wiki/Information_theory
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #8 – 2008-12-30 18:42:48
If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.

Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #9 – 2008-12-30 18:51:36
Out of fear that you're giving too much credence to the OP's logic, I think it should be stated that for frequency content not approaching Nyquist, lower frequency signals are not better reproduced than higher ones.

Agreed.  I was providing reasons (brought about by the practical world) that the OP's conclusion (not logic) would have some validity.  Not that antialiasing filter attenuation or the remaining aliasing are a practical concern for me when considering the 'quality' of a recording or it's reproduction.

If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.

Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

Hence the 'less than half' requirement as opposed to 'less than or equal to half'.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #10 – 2008-12-30 18:57:12
No kidding.  One of the finer details that I clearly did not remember correctly.

Thanks for the well-articulated explanation, pdq.
Is 24-bit/192kHz good enough for your lo-fi vinyl, or do you need 32/384?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #11 – 2009-01-08 07:45:38
I ask this because if it is constant regardless of the frequency being sampled, than CD's would be able to reproduce lower frequencies much better than higher frequencies.

Wrong. Simply, plainly, flat-out-wrong.
Quote
For example, a 100 Hz sound wave would have 5 times as many samples per Hz as a 500 Hz sound wave, thereby providing more data to reproduce it more accurately. But is this what actually happens?

So what if it has "more samples"?  The quantization noise does not change. Read Shannon's paper for a mathematical proof.

If you're going for the "digital gaps," myth go read the adc tutorial I wrote at www.aes.org/sections/pnw/ppt.html

If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.

Why do people give Shannon all the credit and never speak of Nyquist?

The Nyquist conjecture was proven by Shannon.

And you can't sample accurately at exactly 1/2 the sampling frequency unless you have infinite time. The filter to ensure you obey the sampling theorem forces that.

So, given an ideal input (no frequency content over 22.05kHz at all), all frequencies within a CD's ability can be recreated with equal quality.  But, that rarely happens so yes, lower frequencies will, in general, be more accurate than higher frequencies.

Except that in the DC to 20kHz range, it's not hard to make a filter "good enough" so your point does not hold.
-----
J. D. (jj) Johnston

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #12 – 2009-01-08 13:20:56
I ask this because if it is constant regardless of the frequency being sampled, than CD's would be able to reproduce lower frequencies much better than higher frequencies. For example, a 100 Hz sound wave would have 5 times as many samples per Hz as a 500 Hz sound wave, thereby providing more data to reproduce it more accurately. But is this what actually happens?

If you replace the term "100 Hz sound wave" with "100 Hz sine wave" then the waveform can be exactly reproduced with just over 2 samples per period, and no amount of oversampling will improve on this.

If OTOH the sound wave is not sinusoidal then it contains frequencies higher than 100 Hz, and oversampling will retain more of these higher frequencies and therefore reproduce the original waveform more accurately.

The whole idea of only sampling at 44.1 kHz is that the human ear cannot distinguish a perfectly sinusoidal 20 kHz sound wave from a distorted one because the human ear cannot hear the overtones at 40 kHz etc., so the higher frequencies resulting from the distortion are superfluous and are not reproduced. So in this sense high frequencies are reproduced less accurately than low frequencies, but the difference is not audible.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #14 – 2009-01-08 14:10:56
If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.
Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

But if we can choose the moment when sampling begins, 40Hz is enough for representing all freqs <= 20Hz.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #15 – 2009-01-08 16:56:03
So, given an ideal input (no frequency content over 22.05kHz at all), all frequencies within a CD's ability can be recreated with equal quality.  But, that rarely happens so yes, lower frequencies will, in general, be more accurate than higher frequencies.

Except that in the DC to 20kHz range, it's not hard to make a filter "good enough" so your point does not hold.

I have no practical knowledge of what filters are used in this application, but I would be curious if you could tell me.  Filter type, order, cutoff frequency, phase effect...  It seems that we would want somewhere around a 96dB (round to 80dB for optimism and simplicity) cut by the Nyquist frequency (22.05kHz) while still retaining a pass-band to about 20kHz.  That leaves a ten percent frequency change to achieve the cut that a fourth order filter creates over 2 decades.

Without challenging the apparent fact that this can be achieved reasonably well, do you know how it's done?  Or, what trade offs are accepted to make it happen?

Hmm..  Is the signal attenuation simply accepted as stored on the CD, but undone by the recovery filter ? (I think that's the term for the DAC's filter that corresponds to the antialiasing filter)

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #16 – 2009-01-08 17:02:14
If we're just looking at the math, a 20 Hz sine wave can be reconstructed perfectly using a sample frequency of 40 Hz.  The "just over" part is not requirement.
Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

But if we can choose the moment when sampling begins, 40Hz is enough for representing all freqs <= 20Hz.

Yes, but then that additional information (how the sampling synchronized with the 20 Hz sine wave) would have to be applied during reconstruction as well.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #17 – 2009-01-08 17:17:32
I have no practical knowledge of what filters are used in this application, but I would be curious if you could tell me.  Filter type, order, cutoff frequency, phase effect...  It seems that we would want somewhere around a 96dB (round to 80dB for optimism and simplicity) cut by the Nyquist frequency (22.05kHz) while still retaining a pass-band to about 20kHz.  That leaves a ten percent frequency change to achieve the cut that a fourth order filter creates over 2 decades.

Without challenging the apparent fact that this can be achieved reasonably well, do you know how it's done?  Or, what trade offs are accepted to make it happen?

One way that Philips started using pretty early on is to take the 44 khz sample stream and upsample 4 or 8x that (interpolating the samples).  Then they could use a lot lower pole low pass filter to pass 20 khz and reject the ~200 khz.

TO the question of hitting samples that all fall on zero for a signal at 1/2 the sampling frequency, that's one reason  the sampling has to be greater than 2x.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #18 – 2009-01-08 17:36:29
Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

Not if sample-and-hold is used in the ADC, instead of theoretical instant sampling...

Regards,
Goran Tomas

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #19 – 2009-01-08 19:10:04
Without challenging the apparent fact that this can be achieved reasonably well, do you know how it's done?  Or, what trade offs are accepted to make it happen?
Gentle analogue filter > oversampled digital representation (e.g. 8x44.1kHz), steep digital filter ~ 20-22kHz, decimation to desired sample rate (e.g. 44.1kHz).

If you're asking about the design of the digital filter, there are plenty of books about this, but there's also plenty of software packages that will do the design for you.

As for compromises, it depends if you believe those who claim to hear audible differences between filters, or not. If human hearing typically stops at 20kHz, then it's easy to pass everything up to 20kHz perfectly (with a defined noise floor, and no other problems), add a bit of ringing above this, and have nothing above the noise floor anywhere above 22-24kHz (i.e. 96dB down for 16-bit audio, 144dB down for 24-bit audio).

Cheers,
David.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #20 – 2009-01-08 20:14:00
Gentle analogue filter > oversampled digital representation (e.g. 8x44.1kHz), steep digital filter ~ 20-22kHz, decimation to desired sample rate (e.g. 44.1kHz).

*sigh*  Sometimes I really let myself down...  Maybe next time I won't force some arbitrary connection between original A/D sample rate and the sample rate of the data in the file.

Thanks...

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #21 – 2009-01-08 20:53:29
Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

Not if sample-and-hold is used in the ADC, instead of theoretical instant sampling...

Regards,
Goran Tomas

I'm not sure what you mean by that?

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #22 – 2009-01-09 09:50:26

Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

Not if sample-and-hold is used in the ADC, instead of theoretical instant sampling...

I'm not sure what you mean by that?

You assume that the sample is taken in an instant and if in that instant the audio is in zero crossing, you would get 0 as a sample value, for a signal frequency that is exactly half of the sampling frequency. But, if you (analogue) sample-and-hold during the sample period, you would always acquire maximum value. Sample-and-hold is usually performed by a capacitor and a way of discharging it (ie. via a FET). At the beginning of a sample period, the capacitor is at zero. It's being fed by the input signal. As the amplitude of an input signal rises, so does the voltage across the capacitor. However, when the signal starts falling down, the capacitor remains at the maximum signal level. At the end of the sampling period, the capacitor would have the maximum value of the signal during the whole sampling period. That value is read and stored as a sample value, and the capacitor is discharged to zero. Then the process repeats for the next sampling period.

Regards,
Goran Tomas

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #23 – 2009-01-09 11:25:24

Depending on how the 40 Hz sampling synchronizes with the 20 Hz sine wave, you could get anything from full amplitude (sampled at positive and negative maxima) to zero amplitude (sampled at every zero crossing).

Not if sample-and-hold is used in the ADC, instead of theoretical instant sampling...

I'm not sure what you mean by that?

You assume that the sample is taken in an instant and if in that instant the audio is in zero crossing, you would get 0 as a sample value, for a signal frequency that is exactly half of the sampling frequency. But, if you (analogue) sample-and-hold during the sample period, you would always acquire maximum value. Sample-and-hold is usually performed by a capacitor and a way of discharging it (ie. via a FET). At the beginning of a sample period, the capacitor is at zero. It's being fed by the input signal. As the amplitude of an input signal rises, so does the voltage across the capacitor. However, when the signal starts falling down, the capacitor remains at the maximum signal level. At the end of the sampling period, the capacitor would have the maximum value of the signal during the whole sampling period. That value is read and stored as a sample value, and the capacitor is discharged to zero. Then the process repeats for the next sampling period.

Regards,
Goran Tomas

OK, now you know the maximum amplitude, but have no info about phase.  That shoots the criterion of perfect representation.

How does the cap hold the maximum value instead of following the voltage, or accumulating the average of the sample period?
Remember that the rise and fall of this 1/2 sampling freq. signal is superimposed on all the lower frequencies and may be 90 dB lower in magnitude.

## Do CD's reproduce lower frequencies much better than higher freque

##### Reply #24 – 2009-01-09 13:57:26
You assume that the sample is taken in an instant and if in that instant the audio is in zero crossing, you would get 0 as a sample value, for a signal frequency that is exactly half of the sampling frequency. But, if you (analogue) sample-and-hold during the sample period, you would always acquire maximum value. Sample-and-hold is usually performed by a capacitor and a way of discharging it (ie. via a FET). At the beginning of a sample period, the capacitor is at zero. It's being fed by the input signal. As the amplitude of an input signal rises, so does the voltage across the capacitor. However, when the signal starts falling down, the capacitor remains at the maximum signal level. At the end of the sampling period, the capacitor would have the maximum value of the signal during the whole sampling period. That value is read and stored as a sample value, and the capacitor is discharged to zero. Then the process repeats for the next sampling period.

Regards,
Goran Tomas

That certainly doesn't describe what I think of as sample and hold.

Nonetheless, are you saying that with such a circuit that it is not possible to get the exact same voltage reading every single time when sampling a 20 Hz sine wave at 40 Hz? All you have to do is sample from the negative maximum to the positive maximum, then from the positive maximum to the negative maximum, and both readings are equal to the positive maximum, giving no information about amplitude.