Talk:Quantization noise

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"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 = 2Q."

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

[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

SNRramp = m2 = 22N.

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)
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