Talk:Quantization noise

From Wikipedia, the free encyclopedia

You have new messages (last change).

Physics Portal

This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.

???

This article has not yet received a rating on the assessment scale. [FAQ]

???

This article has not yet received an importance rating within physics.

Help with this template

Please rate this article, and then leave comments here to explain the ratings and/or to identify the strengths and weaknesses of the article.

"The noise is additive and independent of the signal when the number of bits $Q$ is greater than 4, that is, more than 16 digitizing levels, $L = 2 Q$ ."

Why?

Also, the letter L is used for both "load" and "number of levels" — Omegatron 14:07, August 30, 2005 (UTC)

1 Derivation
- 1.1 Thesis
2 Don't Merge
3 Error In Example?

[edit] Derivation

Trying to find a derivation for the quantization noise power equations and subsequent SNR per bit resolution values. I can't find the type of derivation I am thinking of online. Some scribbled, incoherent notes from class (I was probably asleep, and can't find the source I copied them from):

$\sigma^2_{e_t} =$

	Signed magnitude	Two's complement
Truncation	$5 \cdot 2^{-2b} \over 24$	$2^{-2b} \over 12$
Round-off	$2^{-2b} \over 12$	$2^{-2b} \over 12$

$\sigma^2_{e_t} = {5 \over 2} \sigma^2_{e_r}$

LSB-1: $\sigma^2_{e_l} = {2^{-2b} \over 3}$

$\sigma^2_{e_l} = 4 \sigma^2_{e_r}$

Mean squared error or quantization noise

${1 \over \Delta} \int_{-\Delta/2}^{\Delta/2} \epsilon^2\, d \epsilon = {1 \over \Delta} \left( {\epsilon^3 \over 3} \right) \Bigg|_{-\Delta/2}^{\Delta/2} = {1 \over \Delta} \left( {\Delta^3 \over 8 \cdot 3} + {\Delta^3 \over 8 \cdot 3} \right) = {\Delta^2 \over 12}$

with a note making sure I remember that it's not always Δ²/12. The external link I included shows a more thorough derivation of q²/12. — Omegatron 06:04, 17 October 2005 (UTC)

I keep forgetting that Google Print exists. [1]

[edit] Thesis

I have previously derived the SNR formulas in my thesis. Sorry for the unconventional symbols, i use Q for LSB (in volts) and N for the number of resolution bits and m=2^N for the number of levels. umax and umin are the max and min voltage corresponding to the quantized interval of Q*m

The noise power is assumed to be uniform:

$P_\mathrm{noise}=\frac{1}{T}\int_{0}^{T}\left[(\frac{t}{T}-\frac{1}{2})Q\right]^{2}dt=2\int_{0}^{1/2}(tQ)^{2}dt=\frac{Q^{2}}{12}$

where Q is the quantization step (often called LSB):

$Q=\frac{u_\mathrm{max}-u_\mathrm{min}}{m}$

In the case of a uniformly distributed signal (ramped, triangle etc) the signal power is

$P_\mathrm{signal}^\mathrm{ramp}=\frac{1}{T}\int_{0}^{T}\left[(\frac{t}{T}-\frac{1}{2})(u_\mathrm{max}-u_\mathrm{min})\right]^{2}dt=\frac{(mQ)^{2}}{12}$

where i use m=2^N for the number of levels (does not have to be a power of two!). And thus the SNR

$SNR ramp = m 2 = 2 2 N .$

In decibels this is

$\mathrm{SNR_{ramp}}[\textrm{dB}]=10\log_{10}(2^{2N})=2N\cdot10\log_{10}(2)=6.02N$

Next case: A sine wave test tone. This one has more power than the ramped, namely:

$P_\mathrm{signal}^\mathrm{sine}=\frac{1}{T}\int_{0}^{T}\left[\sin(2\pi\frac{t}{T})\frac{u_\mathrm{max}-u_\mathrm{min}}{2}\right]^{2}dt=\frac{(mQ)^{2}}{8}$

And the SNR becomes

$\mathrm{SNR_{sine}}=\frac{3}{2}m^{2}=\frac{3}{2}2^{2N}.$

in decibels:

$\mathrm{SNR_{sine}}[\textrm{dB}]=10\log_{10}(\frac{3}{2}2^{2N})=2N\cdot10\log_{10}(2)+10\log_{10}(3/2)=6.02N+1.76$

/ Johan Stigwall

Wonderful. Thank you. — Omegatron 14:44, 6 December 2005 (UTC)

[edit] Don't Merge

Quantisation error is not Quantisation noise - it's a complicated topic and worthy of several pages to make the distinctions needed, especially when weighting and dither are brought in. --Lindosland 16:18, 5 February 2006 (UTC)

Well they're not even close to a page right now, so I figured the subjects were close enough to merge into one topic. — Omegatron 01:57, 13 March 2006 (UTC)

[edit] Error In Example?

I think there is a discrepancy in the formula for sawtooth SNR and the example given. The formula is approximated as 6.02 · n, where n=4 for 16 bit audio. However, the example provided comes up with the answer 6.02 · 16 = 96.3 dB, which is not in agreement with the formula.

One stinky bum 01:35, 14 January 2007 (UTC)

Retrieved from "http://en.wikipedia.org../../../q/u/a/Talk%7EQuantization_noise_512d.html"

Category: Unassessed physics articles