just a code story: DMA noise estimation

Let's talk about "Delay-and-multiply approach" algorithm. It proposed in [Lin] and in the book [Tsui]. But I can't find evaluation of an impact of this algorithm to SNR. In the book wrote that this impact is significant and need further investiogation. Let's try to make some noise about it =)

Our incoming signal we can describe as:

$s(t)=C(t)\exp^{j\omega t}$ [1]

where C(t) - C/A code, exp() - carrier. The main thing of the DMA algorithm we should multiply s(t) and delayed conjugate copy of the s(t). Delayed signal can be expressed as:

$s(t-\tau)=C(t-\tau)\exp^{j\omega{(t-\tau)}}$ [2]

Multiply them:

$s_{new}(t)=s(t)s(t-\tau)^*=C(t)\exp^{j\omega{t}}C(t-\tau)\exp^{-j\omega{(t-\tau)}}=C(t)C(t-\tau)\exp^{j\omega{\tau}}$ [3]

yup, now we need to find only new Gold code [Gold] (C/A - code ) $C(t)C(t-\tau)$ , because $exp^{j\omega{\tau}}$ - is a constant.

But in real world we also have noise in the channel, so we must take into account it:

$s(t) = C(t)\exp^{j\omega{\tau}} + n(t)$ [4]

Let's make [2] operation with this:

$s_{new}=s(t)s(t=\tau)^{*}=(C(t)\exp^{j\omega{t}} + n_1(t))(C(t-\tau)\exp^{-j\omega(t-\tau)} + n_2(t))=$ [5]

$=C(t)C(t-\tau)exp^{j\omega\tau} + C(t)exp^{j\omega{t}n_2(t)} + C(t-\tau)exp^{-j\omega{(t-\tau)}}n_1(t) + n_1(t)n_2(t)$

We want to know how many noise we have after DMA operation, get variance from [5]:

$D[C(t)C(t-tau)\exp^{j\omega{t}}] = 1$ [6] $D[C(t-\tau)exp^{-j\omega{(t-\tau)}}n_1(t)] = D[C(t)exp^{j\omega{t}}n_2(t)]=D[n(t)]$ [7] $D[n_1(t)n_2(t)]=D[n^2(t)]$ [8]

So we can write variance of the [5]:

$D[s_{new}(t)]=n^2(t) + 2n(t) + 1$ [9]

Based on [9] we can calculate degrade SNR after DMA algorithm:

Our estimation exactly fit to our model. This model is really simple, we don't have interference and energy from every satellite are the same.

But again in real world we have 7,8 or 9 satellites online (picture 2) [GPS Standart]

I'll try to estimate it in near future.

[Lin] James B. Y. Tsui, David Lin. Acquisition scheme for software gps receiver. Proceeding of

the 11th Intrernational Technical Meeting ION-98, pages 317-325, September, 1998.

[Tsui] James Bao Yen Tsui. Fundamentals of global position system receivers: a software approach.

John Wiley and Sons, 2005.

[Gold]

[GPS Standart] GLOBAL POSITIONING SYSTEM STANDARD POSITIONING SERVICE SIGNAL SPECIFICATION, ANNEX A, STANDARD POSITIONING SERVICE PERFORMANCE SPECIFICATION

just a code story

Monday, April 16, 2012

DMA noise estimation

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