Increasing SDR Receiver Dynamic Range by ADC Diversity

Abstract

Nowadays, most radio implementations are based on software-defined radio (SDR) technologies. The capabilities of digital signal processing enable new applications like low power wide area networks (LPWAN), which are expected to play a decisive role in the upcoming Internet of Things. Centralized gateways, usually realized in an SDR architecture, are used to connect many thousands of objects to the internet. Due to the high variance of the received signal level, a high dynamic range is required for the SDR receiver front-end. In current receiver architectures, the dynamic range is mainly limited by the analog-to-digital converter (ADC). Several techniques have been proposed to extend the dynamic range by stacking multiple ADCs and driving them with different gain factors. Correlation of quantization noise was identified as key parameter to determine the dynamic range enhancement. This paper compares the proposed techniques and extends existing analysis tools for the use of arbitrary gain factors. Additionally, the influence of further noise sources like thermal noise and jitter are taken into account. The theoretical considerations are supported by simulations and measurements using a real LPWAN SDR implementation.

Appendix

With the following derivation, based on [26], Eq. 33 shall be brought into a more descriptive form. First, we concentrate on the expectation value, which is the only component in Eq. 33 dependent on the input values. In general, the expectation value of an arbitrary function Ψ(X) of a continuous random variable X is

$$ \text{E}\left\{ {\Psi}(X)\right\} ={\int}_{-\infty}^{\infty}{\Psi}(\xi)\cdot f_{X}(\xi)\text{d}\xi. $$

(48)

Since 𝜖(x) is a periodic function, the mapping of the quantization errors ν _i of both quantizers to the input values x is only unique within the step size q of the quantizer. Therefore, the integration is carried out step by step

$$\begin{array}{@{}rcl@{}} \text{E}\left\{ e^{j2\pi\frac{k+g_{2}l}{q}x}\right\} & =&{\int}_{-\infty}^{\infty}e^{j2\pi\frac{k+g_{2}l}{q}\xi}\cdot f_{X}(\xi)\cdot(h*{\Delta})(\xi)\text{d}\xi\\ & =&\sum\limits_{n=-\infty}^{\infty}{\int}_{-\infty}^{\infty}f_{X}(\xi)\cdot\left[h(\xi)*\delta(\xi-nq)\right]\cdot e^{j2\pi\frac{k+g_{2}l}{q}\xi}\text{d}\xi, \end{array} $$

(49)

with the rectangular function

$$ h(x)=\left\{\begin{array}{ll} 1 & -\frac{q}{2}\leq x\leq\frac{q}{2}\\ 0 & \text{elsewhere}, \end{array}\right. $$

(50)

and the Dirac comb

$$ {\Delta}_{q}(x)=\sum\limits_{n-\infty}^{\infty}\delta\left(x-nq\right). $$

(51)

The integral in Eq. 49 is a Fourier transformation with $\omega =-2\pi \frac {k+g_{2}l}{q}$ , with this

$$\begin{array}{@{}rcl@{}} \text{E}\left\{ e^{j2\pi\frac{k+g_{2}l}{q}x}\right\} & =&\sum\limits_{n=-\infty}^{\infty}\left(F_{X}*H\right)\left(-\frac{k+g_{2}l}{q}\right)\cdot e^{j2\pi\frac{k+g_{2}l}{q}nq}\\ &=&\sum\limits_{n=-\infty}^{\infty}\left(F_{X}*H\right)\left(-\frac{k+g_{2}l}{q}\right)\cdot\frac{1}{q}\delta\\&&\times\left(-\frac{k+g_{2}l+n}{q}\right), \end{array} $$

(52)

where F(⋅) and H(⋅) are the Fourier transformations of f(⋅) and h(⋅), respectively.

Defining the characteristic function of the random variable X

$$ {\Phi}_{X}\left(j\omega\right)={\int}_{-\infty}^{\infty}e^{j\omega x}f_{X}(x)\text{d}x $$

(53)

and with the Fourier transformation of the rectangular function

$$ \mathcal{F}\left(h(ax)\right)=\frac{1}{\left|a\right|}\text{sinc}\left(\frac{\xi}{a}\right) $$

(54)

we get

$$\begin{array}{@{}rcl@{}} \text{E}\left\{ e^{j2\pi\frac{k+g_{2}l}{q}x}\right\} &=\!&\sum\limits_{n=-\infty}^{\infty}{\Phi}_{X}\!\left(\frac{k+g_{2}l}{q}\right)*\text{sinc}\left(k+g_{2}l\right)\cdot\delta\\&&\times\left(-\frac{k+g_{2}l+n}{q}\right). \end{array} $$

(55)

By substitution of Eqs. 55 in 33, we obtain the correlation coefficient of the quantization noise of two parallel ADCs quantizing differently scaled replicas of the same input signal depending on the the scaling factor g ₂

$$ \rho_{\nu_{1},\nu_{2}}=\frac{3}{\pi^{2}}\sum\limits_{_{k\neq0 }^{k=-\infty} }^{\infty}\sum\limits_{_{l\neq0}^{l=-\infty}}^{\infty}\frac{\left(-1\right)^{k+l+1}}{kl}{\Phi}_{X}\!\left(\frac{k+g_{2}l}{q}\right)*\text{sinc}\left(k+g_{2}l\right). $$

(56)