# Phase of harmonics after Quantization of sinusoids

Started by August 20, 2008
```If I look at a spectrum of a uniformly quantized sinusoid, there are
harmonics at multiples of the input frequency.
Does someone know how can I predict the phase of those harmonics for the
case of single as well as two tones inputs?
As quantization is not a linear process, therefore the phase of the
harmonics should not be just multiple of the fundamental's phase. The
literature mostly talks about amplitude of the harmonics produced due to
quantization and not the phase...

Hope someone can help or point to some relevant text...

Regards,

Sammy

```
```SammySmith wrote:
> If I look at a spectrum of a uniformly quantized sinusoid, there are
> harmonics at multiples of the input frequency.

The difference between the quantized (but continuous) waveform and the
original is an amplitude-modulated approximate triangle wave at the
sampling frequency whose fundamental peaks when the original crosses
zero. (The modulating frequency is the original sinusoid. Why do you
think harmonics of the sampled wave are involved?) The largest triangle
is about one LSB peak. Because of the generally triangular nature of the
waveform, higher harmonics fall off with the square of the order.

...

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```On 21 Aug, 03:12, "SammySmith" <eigenvect...@yahoo.com> wrote:
> If I look at a spectrum of a uniformly quantized sinusoid, there are
> harmonics at multiples of the input frequency.
> Does someone know how can I predict the phase of those harmonics for the
> case of single as well as two tones inputs?

You can't, as that would require knowledge about the
exact time of the zero-crossing of the sinusoidal.

> As quantization is not a linear process, therefore the phase of the
> harmonics should not be just multiple of the fundamental's phase. The
> literature mostly talks about amplitude of the harmonics produced due to
> quantization and not the phase...

That's because finding the exact phase would require
knowledge about the exact time of the zero-crossing
of the sinusoidal.

Rune
```
```>On 21 Aug, 03:12, "SammySmith" <eigenvect...@yahoo.com> wrote:
>> If I look at a spectrum of a uniformly quantized sinusoid, there are
>> harmonics at multiples of the input frequency.
>> Does someone know how can I predict the phase of those harmonics for
the
>> case of single as well as two tones inputs?
>
>You can't, as that would require knowledge about the
>exact time of the zero-crossing of the sinusoidal.
>
>> As quantization is not a linear process, therefore the phase of the
>> harmonics should not be just multiple of the fundamental's phase. The
>> literature mostly talks about amplitude of the harmonics produced due
to
>> quantization and not the phase...
>
>That's because finding the exact phase would require
>knowledge about the exact time of the zero-crossing
>of the sinusoidal.
>
>Rune

Thanks for replying, but if I have a determinitic singal such as
cos(w1)[singe tone] or cos(w1)+cos(w2) [2 tone], then I would know the zero
crossings, then how can I know the phase of the harmonics produced after
quantization.

Sammy
```
```SammySmith wrote:
>> On 21 Aug, 03:12, "SammySmith" <eigenvect...@yahoo.com> wrote:
>>> If I look at a spectrum of a uniformly quantized sinusoid, there are
>>> harmonics at multiples of the input frequency.
>>> Does someone know how can I predict the phase of those harmonics for
> the
>>> case of single as well as two tones inputs?
>> You can't, as that would require knowledge about the
>> exact time of the zero-crossing of the sinusoidal.
>>
>>> As quantization is not a linear process, therefore the phase of the
>>> harmonics should not be just multiple of the fundamental's phase. The
>>> literature mostly talks about amplitude of the harmonics produced due
> to
>>> quantization and not the phase...
>> That's because finding the exact phase would require
>> knowledge about the exact time of the zero-crossing
>> of the sinusoidal.
>>
>> Rune
>
> Thanks for replying, but if I have a determinitic singal such as
> cos(w1)[singe tone] or cos(w1)+cos(w2) [2 tone], then I would know the zero
> crossings, then how can I know the phase of the harmonics produced after
> quantization.

Practical quantizing is done by a sampler, not by truncating a
continuous variable. Do you know the ticks of the quantizing clock
relative to the zero crossings of your signals? Is it constant? What are

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```>SammySmith wrote:
>>> On 21 Aug, 03:12, "SammySmith" <eigenvect...@yahoo.com> wrote:
>>>> If I look at a spectrum of a uniformly quantized sinusoid, there are
>>>> harmonics at multiples of the input frequency.
>>>> Does someone know how can I predict the phase of those harmonics for
>> the
>>>> case of single as well as two tones inputs?
>>> You can't, as that would require knowledge about the
>>> exact time of the zero-crossing of the sinusoidal.
>>>
>>>> As quantization is not a linear process, therefore the phase of the
>>>> harmonics should not be just multiple of the fundamental's phase.
The
>>>> literature mostly talks about amplitude of the harmonics produced
due
>> to
>>>> quantization and not the phase...
>>> That's because finding the exact phase would require
>>> knowledge about the exact time of the zero-crossing
>>> of the sinusoidal.
>>>
>>> Rune
>>
>> Thanks for replying, but if I have a determinitic singal such as
>> cos(w1)[singe tone] or cos(w1)+cos(w2) [2 tone], then I would know the
zero
>> crossings, then how can I know the phase of the harmonics produced
after
>> quantization.
>
>Practical quantizing is done by a sampler, not by truncating a
>continuous variable. Do you know the ticks of the quantizing clock
>relative to the zero crossings of your signals? Is it constant? What are

>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;

I assume that the number of samples per cycle donot change throughout the
quantization process...at the moment I am just assuming ideal conditions.
So the output after quantization is like a ZOH output. Can phase of the
harmonics be predicted for such an output?

Sammy
```
```On 23 Aug, 21:32, "SammySmith" <eigenvect...@yahoo.com> wrote:

> I assume that the number of samples per cycle donot change throughout the
> quantization process...at the moment I am just assuming ideal conditions.
> So the output after quantization is like a ZOH output. Can phase of the
> harmonics be predicted for such an output?

No.

Assume normalized sampling, T = 1, normalized quantization, dx = 1,
and a sinusoidal with amplitude A >> 1 so that the sine can be
approximated as a straight line in the vicinity of the zero
crossing.

At t= -1 you see the sample value -1, at t= 0 you see the
sample 0, and at time t = 1 you see the sample value 1:

(d,x) = {(-1,-1),(0,0),(1,1)}

Where was the zero crossing?

It turns out that the given samples can fit any sinusoidal
(or approximation to a sinusoidal) which is bounded by the
lines

y1 = x-1
y2 = x+1

on the interval t =[-1,1].

Basically, the zero crossing can be anywhere on the interval
t = [-1,1]. This uncertainty of the exact location of the
zero crossing is equivalent to a random phase in the
quantization noise.

Rune
```
```SammySmith wrote:
>> SammySmith wrote:
>>>> On 21 Aug, 03:12, "SammySmith" <eigenvect...@yahoo.com> wrote:
>>>>> If I look at a spectrum of a uniformly quantized sinusoid, there are
>>>>> harmonics at multiples of the input frequency.
>>>>> Does someone know how can I predict the phase of those harmonics for
>>> the
>>>>> case of single as well as two tones inputs?
>>>> You can't, as that would require knowledge about the
>>>> exact time of the zero-crossing of the sinusoidal.
>>>>
>>>>> As quantization is not a linear process, therefore the phase of the
>>>>> harmonics should not be just multiple of the fundamental's phase.
> The
>>>>> literature mostly talks about amplitude of the harmonics produced
> due
>>> to
>>>>> quantization and not the phase...
>>>> That's because finding the exact phase would require
>>>> knowledge about the exact time of the zero-crossing
>>>> of the sinusoidal.
>>>>
>>>> Rune
>>> Thanks for replying, but if I have a determinitic singal such as
>>> cos(w1)[singe tone] or cos(w1)+cos(w2) [2 tone], then I would know the
> zero
>>> crossings, then how can I know the phase of the harmonics produced
> after
>>> quantization.
>> Practical quantizing is done by a sampler, not by truncating a
>> continuous variable. Do you know the ticks of the quantizing clock
>> relative to the zero crossings of your signals? Is it constant? What are
>
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>
> I assume that the number of samples per cycle donot change throughout the
> quantization process...at the moment I am just assuming ideal conditions.
> So the output after quantization is like a ZOH output. Can phase of the
> harmonics be predicted for such an output?

Only if the sampler is phase locked to the sampled waveform. This is
forced in laboratory measurements of parameters that are difficult to
get any other way and occurs naturally in digitally generated waveforms.
With most signals, the number of samples per waveform cycle is rarely a
rational number, let alone an integer. What is your application?

In any case, the harmonics are of the sampling frequency. Of course, if
the sample clock is phase locked to the sampled waveform, harmonics of
the sample clock are also harmonics of the sampled waveform.

Jerry
--
Engineering is the art of making what you want from things you can get.
```
```Rune Allnor wrote:
> On 23 Aug, 21:32, "SammySmith" <eigenvect...@yahoo.com> wrote:
>
>> I assume that the number of samples per cycle donot change throughout the
>> quantization process...at the moment I am just assuming ideal conditions.
>> So the output after quantization is like a ZOH output. Can phase of the
>> harmonics be predicted for such an output?
>
> No.
>
> Assume normalized sampling, T = 1, normalized quantization, dx = 1,
> and a sinusoidal with amplitude A >> 1 so that the sine can be
> approximated as a straight line in the vicinity of the zero
> crossing.
>
> At t= -1 you see the sample value -1, at t= 0 you see the
> sample 0, and at time t = 1 you see the sample value 1:
>
> (d,x) = {(-1,-1),(0,0),(1,1)}
>
> Where was the zero crossing?
>
> It turns out that the given samples can fit any sinusoidal
> (or approximation to a sinusoidal) which is bounded by the
> lines
>
> y1 = x-1
> y2 = x+1
>
> on the interval t =[-1,1].
>
> Basically, the zero crossing can be anywhere on the interval
> t = [-1,1]. This uncertainty of the exact location of the
> zero crossing is equivalent to a random phase in the
> quantization noise.

Rune,

Keep in mind that Sam is asking about quantization, not sampling, and
that he generates both the signal and the quantization instants. I
imagine that his model is a simplified case, and that he doesn't realize
that it's too simplified to answer what he really wants to know.
Nevertheless, the question has an answer as posed, with the caveat that
the harmonics are best understood as those of the quantizer frequency.

A simple model suffices to caprure the essence of what happens under
Sam's ideal conditions. For definiteness, let the wave to be quantized
be cos(2pi*f*t), and be quantized n times per cycle. (The quantizer runs
at n*f*t.) Let the quantizing interval be small enough (n be large
enough) so that the curvature of the cosine wave is negligible in that
interval. Let one of the quantizing instants occur at t=0.

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.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Jerry Avins wrote:
(snip)

> Practical quantizing is done by a sampler, not by truncating a
> continuous variable. Do you know the ticks of the quantizing clock
> relative to the zero crossings of your signals? Is it constant? What are