Phase of harmonics after Quantization of sinusoids

On 24 Aug, 19:27, Jerry Avins <j...@ieee.org> wrote:
..
> Rune,
>
> Keep in mind that Sam is asking about quantization, not sampling, and
> that he generates both the signal and the quantization instants.

It doesn't matter. Even if he generates a discrete-time sinusoidal,
my arguments still hold if he introduces a random compunent oither
to the frequency or phase.

The only case when there exist an analytical expression for the
phase is in the idealized contitions where both the frequency
and phase of the sinusoidal are known, which is only of academic
interest.

Rune

Rune Allnor wrote:

   ...

> The only case when there exist an analytical expression for the
> phase is in the idealized contitions where both the frequency
> and phase of the sinusoidal are known, which is only of academic
> interest.

Or in the testing lab when characterizing codec distortion.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

>
>Consider the /difference/ between the original cosine and its quantized 
>version; the error signal. Each difference will be a small saw tooth. 
>The magnitude of each tooth will be greatest near the zero crossings of 
>the cosine and zero at the cosine's peaks. The result is the same as 
>suppressed-carrier double-sideband amplitude modulation of a sawtooth at

>the quantizing rate by the original cosine. It's spectrum is calculable.
>
>Jerry
>-- 
>Engineering is the art of making what you want from things you can get.
>�����������������������������������������������������������������������

The error signal is like a sawtooth with some bell shaped curves in
between. The bell shaped curve appears at the peak of the sinusoid. I have
seen analytical expressions for the amplitude of the quantized waveform
expressed as some Bessel functions, but I havent seen any analytical
expressions on the phase of the quantized waveform.
How can I know such an analytical expression for the phase?

Sammy

>
>Consider the /difference/ between the original cosine and its quantized 
>version; the error signal. Each difference will be a small saw tooth. 
>The magnitude of each tooth will be greatest near the zero crossings of 
>the cosine and zero at the cosine's peaks. The result is the same as 
>suppressed-carrier double-sideband amplitude modulation of a sawtooth at

>the quantizing rate by the original cosine. It's spectrum is calculable.
>
>Jerry
>-- 
>Engineering is the art of making what you want from things you can get.
>�����������������������������������������������������������������������

The error signal is like a sawtooth with some bell shaped curves in
between. The bell shaped curve appears at the peak of the sinusoid. I have
seen analytical expressions for the amplitude of the quantized waveform
expressed as some Bessel functions, but I havent seen any analytical
expressions on the phase of the quantized waveform.
How can I know such an analytical expression for the phase?

Sammy

SammySmith wrote:
>> Consider the /difference/ between the original cosine and its quantized 
>> version; the error signal. Each difference will be a small saw tooth. 
>> The magnitude of each tooth will be greatest near the zero crossings of 
>> the cosine and zero at the cosine's peaks. The result is the same as 
>> suppressed-carrier double-sideband amplitude modulation of a sawtooth at
> 
>> the quantizing rate by the original cosine. It's spectrum is calculable.
>>
>> Jerry
>> -- 
>> Engineering is the art of making what you want from things you can get.
> 
> The error signal is like a sawtooth with some bell shaped curves in
> between. The bell shaped curve appears at the peak of the sinusoid. I have
> seen analytical expressions for the amplitude of the quantized waveform
> expressed as some Bessel functions, but I havent seen any analytical
> expressions on the phase of the quantized waveform.
> How can I know such an analytical expression for the phase?
> 
> Sammy

What bell shape? One period of error is zero at its beginning and 
reaches the magnitude of the maximum difference at the other. The error 
grows nearly linearly within the interval if the number of sections per 
cycle of the cosine is large enough (32 segments are certainly enough). 
Assuming perfect straightness gives a result close enough to reality to 
provide a very good idea of the character. Second-order refinement -- 
assuming the segments are parabolic rather than straight -- gives 
excellent results with as few as 12 quantizing intervals per cycle and 
remains rather good with eight. Bessel functions are overkill.

Jerry
-- 
Engineering is the art of making what you want from things you can get.