SNR of quantisation noise

I have been trying in Matlab to model and study the SNR equation of
quantisation noise which states that for n bit representation:

(1) SNR(dB) = 6.02*n + 1.76 for a single tone signal.

(2) when measured by FFT, add 10 *log10(fft resolution/2).

(3) A further change occurs when relative bandwidth changes 
with respect to Fs e.g. upsampling: add 10*log10(Fs/(2*bandwidth))

I managed to model the first two sections but find it difficult 
to understand and quantify the effect of upsampling/downsampling. 
Surely it does have effect but this varies with applied method 
(or chosen filter attenuation) while section (3) above implies fixed
effect.

Can anybody please shed some light on explaining and quantifying the effect
of sampling rate change on SNR.

Regards

kadhiem Ayob

On Oct 4, 3:25&#4294967295;pm, "kaz" <kadhiem_ayob@n_o_s_p_a_m.yahoo.co.uk> wrote:
> I have been trying in Matlab to model and study the SNR equation of
> quantisation noise which states that for n bit representation:
>
> (1) SNR(dB) = 6.02*n + 1.76 for a single tone signal.
>
> (2) when measured by FFT, add 10 *log10(fft resolution/2).
>
> (3) A further change occurs when relative bandwidth changes
> with respect to Fs e.g. upsampling: add 10*log10(Fs/(2*bandwidth))
>
> I managed to model the first two sections but find it difficult
> to understand and quantify the effect of upsampling/downsampling.
> Surely it does have effect but this varies with applied method
> (or chosen filter attenuation) while section (3) above implies fixed
> effect.
>
> Can anybody please shed some light on explaining and quantifying the effect
> of sampling rate change on SNR.
>
> Regards
>
> kadhiem Ayob

SNR is always specified over a specific bandwidth, usually fs/2 Hz

If you reduce that bandwidth via signal processing, via FFT or
oversampling, the SNR reduces in direct proportion to the reduction in
bandwidth

On Oct 4, 3:40&#4294967295;pm, steve <bungalow_st...@yahoo.com> wrote:
> On Oct 4, 3:25&#4294967295;pm, "kaz" <kadhiem_ayob@n_o_s_p_a_m.yahoo.co.uk> wrote:
>
>
>
>
>
> > I have been trying in Matlab to model and study the SNR equation of
> > quantisation noise which states that for n bit representation:
>
> > (1) SNR(dB) = 6.02*n + 1.76 for a single tone signal.
>
> > (2) when measured by FFT, add 10 *log10(fft resolution/2).
>
> > (3) A further change occurs when relative bandwidth changes
> > with respect to Fs e.g. upsampling: add 10*log10(Fs/(2*bandwidth))
>
> > I managed to model the first two sections but find it difficult
> > to understand and quantify the effect of upsampling/downsampling.
> > Surely it does have effect but this varies with applied method
> > (or chosen filter attenuation) while section (3) above implies fixed
> > effect.
>
> > Can anybody please shed some light on explaining and quantifying the effect
> > of sampling rate change on SNR.
>
> > Regards
>
> > kadhiem Ayob
>
> SNR is always specified over a specific bandwidth, usually fs/2 Hz
>
> If you reduce that bandwidth via signal processing, via FFT or
> oversampling, the SNR reduces in direct proportion to the reduction in
> bandwidth- Hide quoted text -
>
> - Show quoted text -

opps, SNR increases...

On Oct 4, 3:40&#4294967295;pm, steve <bungalow_st...@yahoo.com> wrote:

>
> SNR is always specified over a specific bandwidth, usually fs/2 Hz
>
> If you reduce that bandwidth via signal processing, via FFT or
> oversampling, the SNR reduces in direct proportion to the reduction in
> bandwidth

Does signal processing resulting in bandwidth reduction
always reduce the SNR, or are there schemes (e.g. matched
filtering) that remove "out-of-band" noise etc (thereby effectively
decreasing the bandwidth) that might also increase the SNR?

Thanks Steve,

I am aware of the rule but wanted to model it and see by myself. In the
case of single tone I upsampled the signal using various filters but got
different results each time depending on much power residue was left in the
stop band. How can we explain these results with the theoretical rule.

Thanks

Kadhiem

On Tue, 04 Oct 2011 12:46:44 -0700, dvsarwate wrote:

> On Oct 4, 3:40&nbsp;pm, steve <bungalow_st...@yahoo.com> wrote:
> 
> 
>> SNR is always specified over a specific bandwidth, usually fs/2 Hz
>>
>> If you reduce that bandwidth via signal processing, via FFT or
>> oversampling, the SNR reduces in direct proportion to the reduction in
>> bandwidth
> 
> Does signal processing resulting in bandwidth reduction always reduce
> the SNR, or are there schemes (e.g. matched filtering) that remove
> "out-of-band" noise etc (thereby effectively decreasing the bandwidth)
> that might also increase the SNR?

Steve meant to say the SNR increases (see his correction).  Which is true 
to an extent: as long as the overall bandwidth is capturing the whole 
signal, reducing the bandwidth reduces noise without reducing signal, and 
the SNR goes up.  As soon as you start biting into the signal, then (a) 
your SNR isn't going to improve, and (b) your signal is getting mangled.

-- 
www.wescottdesign.com

kaz wrote:

> I have been trying in Matlab to model and study the SNR equation of
> quantisation noise which states that for n bit representation:

Quantization noise is not a noise, but a nonlinear distortion. Thereby 
common confusion.

Hi,

"assuming" that my quantization noise is white, the in-band part scales
proportionally with the bandwidth that I consider "in-band". Simple as
that.

On Tue, 04 Oct 2011 14:25:11 -0500, kaz wrote:

> I have been trying in Matlab to model and study the SNR equation of
> quantisation noise which states that for n bit representation:
> 
> (1) SNR(dB) = 6.02*n + 1.76 for a single tone signal.

Presumably this is for a single tone signal that takes up the whole range 
of the ADC.

> (2) when measured by FFT, add 10 *log10(fft resolution/2).

I'm not sure just where this comes from, but I suspect that the author is 
not being rigorous about his definitions -- the total signal power vs. 
the total noise power is going to be the same after a Fourier transform 
as before, so the SNR won't change.  So either his definitions are loose, 
or he's taking windowing into account?

> (3) A further change occurs when relative bandwidth changes with respect
> to Fs e.g. upsampling: add 10*log10(Fs/(2*bandwidth))

Again, you need to understand what the author is thinking.

> I managed to model the first two sections but find it difficult to
> understand and quantify the effect of upsampling/downsampling. Surely it
> does have effect but this varies with applied method (or chosen filter
> attenuation) while section (3) above implies fixed effect.
> 
> Can anybody please shed some light on explaining and quantifying the
> effect of sampling rate change on SNR.

Simply changing the sampling rate and looking at the output of the 
quantization step isn't going to change the SNR: you have the same amount 
of signal (dictated by the input amplitude) and the same amount of noise 
(dictated by the ADC step size).  There's nothing you can do about either 
of those.

If your quantization noise occurs as broadband noise (which is often, but 
not always*, the case) then what _does_ change is the power spectral 
density of the noise.  The noise _power_ remains the same, but when you 
sample faster that noise power is spread out over more spectrum -- so the 
power spectral density of the noise goes down proportionally to your 
increase in sampling rate.

So if your quantization noise is, indeed, broadband, and if you sample at 
some sampling rate and then filter the ADC output with a filter 
appropriate to your signal bandwidth and sampling rate, then with 
increasing sampling rate you will be filtering out more and more of the 
noise (but no more of the signal) and you will thus enjoy an improved SNR.

* It is nearly always the case that quantization noise will be 
broadband.  With most 16-bit or better ADCs it _is_ the case because the 
ADC has sufficient thermal noise to guarantee that the quantization noise 
is scrambled.  Only if you are using an ADC that has significantly less 
excess noise than quantization noise will this happen -- but it can 
happen.

-- 
www.wescottdesign.com

>
>
>kaz wrote:
>
>> I have been trying in Matlab to model and study the SNR equation of
>> quantisation noise which states that for n bit representation:
>
>Quantization noise is not a noise, but a nonlinear distortion. Thereby 
>common confusion.
>

But for a single sine tone I compute the ratio of mean signal power/mean of
out of band power from dc to Nyquist. If I upsample it by factor of 2 then
I  measure the same ratio over new bandwidth just like I do with fft
resolution. In the case of fft, the out of band power spreads evenly across
and results are predictable but this does not happen with upsampling. 

Regards

Kadhiem