Sofware-Defined Radio Testbed for I/Q Imbalanced Single-Carrier Communication Systems

Abstract

An end-to-end testbed for In-phase and Quadrature (I/Q) Imbalance (IQI) communication systems based on Software-Defined Radio (SDR) is presented. The scenario under consideration is a Single-Input–Single-Output (SISO) single-carrier communication where the transmitter is heavily affected by IQI, whose effects are mitigated through digital signal processing at the receiver. The presented testbed is highly configurable, enabling the testing of different communication and IQI parameters. Crucial insights into the practical implementation of IQI mitigation techniques, specifically through the use of asymmetric signaling at the receiver, are provided. Initially, a detailed description of the mathematical framework is given. This framework serves as the foundation for the subsequent discussion on system implementation, effectively bridging the gap between research on IQI mitigation and its practical application in single-carrier architectures. Over-The-Air (OTA) Symbol Error Rate (SER) measurements for different constellations validate the receiver design and implementation. The source code of the presented testbed is publicly available.

1. Introduction

The evolution of modern wireless systems towards smart communication networks has sparked interest in developing wireless devices that are low-cost, compact, and highly energy-efficient [1]. Achieving these objectives is particularly challenging from a Radio Frequency (RF) design perspective, especially as the operating frequency increases. This complexity leads to hardware impairments in communication systems that, without appropriate countermeasures, can significantly degrade the quality of wireless communications [2,3].

In-phase and Quadrature (I/Q) Imbalance (IQI) is a hardware impairment caused by imperfections in the RF front-end components, especially in the mixer state. The impact of IQI on digital communications and the development of mitigation strategies have been extensively researched in recent years. The challenges and corresponding correction strategies differ based on the specific system, and whether the IQI affects the transmitter, the receiver, or both. A few, in-depth purely theoretical studies have been conducted on multi-carrier Orthogonal Frequency Division Multiplexing (OFDM) [4,5]; Multiple-Input–Multiple-Output (MIMO) [6,7], and Non-Orthogonal-Multiple-Access [8] systems. Recent studies have focused on single-carrier architectures, where the consideration of asymmetric signaling for symbol decision has proven beneficial [9,10].

IQI mitigation must be independently studied for single-carrier architectures, as the problem is inherently different from the one found in OFDM systems. In OFDM, IQI introduces crosstalk between mirror carrier frequencies, adding an additional layer of distortion. However, this characteristic also provides an opportunity to mitigate IQI in OFDM systems by incorporating redundancy in the transmission, albeit at the cost of a reduced data rate [11]. Furthermore, while many modern wireless protocols rely on OFDM, there is increasing interest in single-carrier systems, particularly for short-range communications and Internet of Things (IoT) applications [12]. The potential of this architecture for mm-Wave frequencies (ranging from 24 to 100 GHz) has also been explored in the literature [13,14,15]. This is particularly relevant as mm-Wave bands are expected to play a significant role in the future of wireless communications [16]. The advantages of single-carrier systems over OFDM include lower Peak-to-Average Power Ratio (PAPR), reduced sensitivity to phase noise, and lower latency.

The versatility of the Software-Defined Radio (SDR) architecture greatly aids in the prototyping and implementation of testbeds for wireless communications research, including measurements and tests related to IQI [17,18], IoT applications [19], and network positioning [20], among others. Nevertheless, an end-to-end testbed that allows for the adjustment of both IQI and communication parameters to compare theoretical and experimental results using Symbol Error Rate (SER) as a figure of merit remains to be seen in the literature. In this article, we present such a testbed for a specific scenario: a Single-Input–Single-Output (SISO) single-carrier communication system, in which the transmitter is significantly impacted by a configurable IQI, with its effects being mitigated at the receiver through digital signal processing. This setup replicates an uplink IoT scenario, where the transmitter is a low-cost device communicating with a more complex access point.

Our work diverges from previous IQI testbeds [17,21] by focusing on the single-carrier problem rather than on OFDM. Regarding [18], the purpose of that work is fundamentally different as it focuses on characterizing the IQI imbalance in USRP devices, but it does not address its correction. The main contributions of our work can be summarized as follows:

• We present an end-to-end hardware testbed for IQI communication systems. This testbed is designed with high adaptability; consequently, not only the IQI parameters are configurable, but also a wide range of system parameters, such as the constellation, frequency, and packet structure. Furthermore, the source code and files associated with this work are publicly available at https://github.com/alvpr/IQI-SDR-Testbed (accessed on 30 June 2024).
• We begin with a theoretical analysis of the proposed problem, summarizing key insights from previous works dedicated to the single-carrier architecture [9,10] and highlighting some considerations that have not been previously addressed.
• Throughout the receiver implementation, based on the consideration of asymmetric signaling, we encountered design aspects that are often overlooked in previous theoretical works, which typically focus on the final stage of symbol decision. These differences and their implications are discussed.
• We present Over-The-Air (OTA) measurements that closely align with theoretical simulations. Moreover, we compare the performance of an IQI-aware receiver with that of its unaware counterpart, demonstrating the effectiveness of the IQI-aware design. In this manner, this work narrows the gap between the theoretical analysis and practical implementation of IQI mitigation strategies in real-world systems, particularly for the study of asymmetric signaling. The extension of the proposed architecture to Multiple-Input–Multiple-Output (MIMO) is also discussed.

The remainder of this article is structured as follows. Section 2 presents the system model of the testbed. Section 3 describes the system implementation, focusing on crucial design considerations. Section 4 offers results from OTA measurements and compares them with simulations. The discussion is presented in Section 5. Finally, Section 6 summarizes the key conclusions.

Notation: Scalar values and scalar Random Variables (RV) are denoted as x and X, respectively. $E\left\{X\right\}$ denotes the expected value of X, $\left|x\right|$, the absolute value of x, $x^{\ast }$ its complex conjugate and $\mathrm{Re}\left\{x\right\}$, $\mathrm{Im}\left\{x\right\}$, its real and imaginary parts, respectively. A complex RV, N that follows a normal distribution with mean ${\mu }_N$ and variance $E\left\{{\left|N-{\mu }_N\right|}^2\right\}={\sigma }_N^2$ is denoted as $N\sim \mathcal{C}\mathcal{N}\left({\mu }_N,{\sigma }_N^2\right)$. The imaginary unit is denoted as j.

2. System Model

2.1. IQI Formulation

Let us consider a single-carrier, single-antenna system where the RF front-end of the transmitter is affected by IQI. We will note the signal constellation as X, a complex RV with zero mean and unit variance, i.e., $E\left\{X\right\}=0$, ${\sigma }_X^2=E\left\{{\left|X\right|}^2\right\}=1$, which takes values from an M-ary alphabet ${\mathcal{X}}_i=\{x_1,x_2,\cdots ,x_M\}$ with probability $1/M$ (equiprobability). Initially assuming a flat-fading channel and no IQI, the received signal would be given by

$$Y=H\sqrt{p}X+N,$$

(1)

where p denotes the transmitted signal power, H is a complex RV that models the channel effect, and N denotes the additive white Gaussian noise, $N\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_N^2\right)$.

As detailed in [9,10], IQI leads to a Widely Linear Transformation (WLT) of the transmitted constellation, which results in [9]

$$X_{\mathrm{IQI}}={\upsilon }_{1}X+{\upsilon }_2{X}^{\ast },$$

(2)

with ${\upsilon }_1$ and ${\upsilon }_2$ being given by

$${\upsilon }_1=\left(1+a_t{e}^{j{\theta }_t}\right)/2;{\upsilon }_2=1-{\upsilon }_1^{\ast }=\left(1-a_t{e}^{-j{\theta }_t}\right)/2,$$

(3)

accounting for the amplitude $\left(a_t\right)$ and phase $\left({\theta }_t\right)$ errors at the transmitter I/Q mixer stage. The Image Rejection Ratio (IIR) is defined as $\mathrm{IIR}={\displaystyle \frac{{\left|{\upsilon }_1\right|}^2}{{\left|{\upsilon }_2\right|}^2}}$. Note that ideal values are $a_t=1$, ${\theta }_t=0$, ${\upsilon }_1=1$, ${\upsilon }_2=0$, and $\mathrm{IIR}=\infty$. However, typical values of IIR reside between 20 dB and 40 dB [11,22]. Consequently, the received signal, after considering IQI, is expressed by

$$Y_{\mathrm{IQI}}=H\sqrt{p}({\upsilon }_{1}X+{\upsilon }_2{X}^{\ast })+N=H\sqrt{p}{X}_{\mathrm{IQI}}+N.$$

(4)

The fact that $X_{\mathrm{IQI}}$ is a WLT of X lets us draw a connection between two fields of study: the undesired effects of IQI and the deliberate study of discrete asymmetric and improper signaling for interference management techniques, [9,23,24,25,26]. $X_{\mathrm{IQI}}$ can be considered as a new constellation that takes values from the alphabet ${\mathcal{X}}_{\mathrm{IQI}}=\{x_{\mathrm{IQI}1},x_{\mathrm{IQI}2},\cdots ,x_{\mathrm{IQI}M}\}$, where $x_{\mathrm{IQI}m}={\upsilon }_1{x}_m+{\upsilon }_2{x}_m^{\ast }$.

A more detailed analysis of this WLT transformation provides further insights into this theoretical IQI model. Let us consider an arbitrary constellation point,

$$x_m=a_m{e}^{j{\theta }_m}=a_{m}cos{\varphi }_m+j{a}_{m}sin{\varphi }_m.$$

(5)

The resulting complex value after IQI, $x_{\mathrm{IQI}m}$, is

$$\begin{array}{cc}\hfill x_{\mathrm{IQI}m}& =v_1{x}_m+(1-v_1^{\ast })x_m^{\ast }=x_m^{\ast }+j2\mathrm{Im}\left\{v_1{x}_m\right\}\hfill \\ \hfill & =\mathrm{Re}\left\{x_m\right\}-j\mathrm{Im}\left\{x_m\right\}+j2\mathrm{Im}\left\{\left({\displaystyle \frac{1+a_t{e}^{{\theta }_t}}{2}}\right)x_m\right\}\hfill \\ \hfill & =a_{m}cos{\varphi }_m+j{a}_t{a}_{m}sin\left({\varphi }_m+{\theta }_t\right)\hfill \end{array},$$

(6)

where the comparison with Equation (5) illustrates how the real part of $x_m$ remains unchanged while the imaginary component is transformed.

From this point, it is of interest to evaluate how the energy of the constellation points is affected, i.e., the variance of the RV X. With that purpose, we define the complementary variance ${\tilde{\sigma }}_X^2=E\left\{X^2\right\}$ and the circularity coefficient, $\kappa$, of X as [25]

$$\kappa ={\displaystyle \frac{\left|E\left\{X^2\right\}\right|}{E\left\{{\left|X\right|}^2\right\}}}={\displaystyle \frac{\left|{\tilde{\sigma }}_X^2\right|}{{\sigma }_X^2}},$$

(7)

where $0\le \kappa \le 1$. The circularity coefficient measures the degree of impropriety since X is proper if $\kappa =0$, improper if $\kappa >0$, and maximally improper when $\kappa =1$ [27]. M-QAM constellations are referred to as proper since ${\tilde{\sigma }}_{X_{\mathrm{QAM}}}^2=0$ and ${\kappa }_{\mathrm{QAM}}=0$. On the other hand, M-PAM constellations are maximally improper since ${\sigma }_{X_{\mathrm{PAM}}}^2={\tilde{\sigma }}_{X_{\mathrm{PAM}}}^2=1$, and thus ${\kappa }_{\mathrm{PAM}}=1$. Applying the IQI WLT transformation to any generic constellation, X, and assuming ${\sigma }_X^2=1$, yields

$$\begin{array}{cc}\hfill E\left\{{\left|X_{\mathrm{IQI}}\right|}^2\right\}& =E\left\{\left(v_{1}X+v_2{X}^{\ast }\right)\left(v_1^{\ast }{X}^{\ast }+v_2^{\ast }X\right)\right\}=\left({\left|v_1\right|}^2+{\left|v_2\right|}^2\right){\sigma }_X^2+2\mathrm{Re}\left\{v_1{v}_2^{\ast }\right\}{\tilde{\sigma }}_X^2\hfill \\ \hfill & ={\displaystyle \frac{1+a_t^2}{2}}+\left({\displaystyle \frac{1-a_t^{2}cos2{\theta }_t}{2}}\right){\tilde{\sigma }}_X^2,\hfill \end{array}$$

(8)

from where the final results in terms of how the variance of each type of constellation is affected are

$$\left\{\begin{array}{c}\begin{array}{c}{\sigma }_{X_{\mathrm{IQI},\mathrm{QAM}}}^2={\displaystyle \frac{1+a_t^2}{2}}\hfill \\ {\sigma }_{X_{\mathrm{IQI},\mathrm{PAM}}}^2=1+{\displaystyle \frac{a_t^2}{2}}\left(1-cos2{\theta }_t\right)\hfill \end{array}.\hfill \end{array}\right.$$

(9)

Note that the final variance values are not dependent on the number of constellation points, M, but only on the IQI parameters. For example, considering an $\mathrm{IIR}=20$ dB with parameters $a_t=0.835$ and ${\theta }_t=5^{\circ }$, we get ${\sigma }_{X_{\mathrm{IQI},\mathrm{QAM}}}^2=0.8486$ and ${\sigma }_{X_{\mathrm{IQI},\mathrm{PAM}}}^2=1.0053$.

This analysis showcases an inaccuracy in the IQI model that has not been highlighted in previous works [9,10]. For M-QAM constellations, the variance is reduced by the effects of IQI, as expected. However, for M-PAM constellations, which use only the real part of the signal, one would expect the variance to remain constant. However, the variance slightly increases due to a rise in the imaginary part of the constellation points. Nevertheless, the provided example illustrates how this increase is marginal. Thus, it does not limit the applicability of the model, especially considering that the imaginary part of the signal is generally ignored in both theoretical models and hardware implementations for M-PAM constellations.

2.2. Receiver

On the receiver side, there is a fundamental distinction between what we will denote as IQI-aware and IQI-unaware receivers. An IQI-aware receiver is assumed to have knowledge of the IQI, characterized by $a_t$ and ${\theta }_t$, and therefore of ${\upsilon }_1$, ${\upsilon }_2$, and $X_{\mathrm{IQI}}$. Note that this might be subjected to detection [11,28] if the system does not know those parameters beforehand. On the contrary, an unaware receiver is designed without accounting for IQI at the transmitter and therefore expects the unaltered constellation X. The optimal Maximum Likelihood (ML) receiver accounting for IQI is

$${\widehat{x}}_{\mathrm{IQI}}=\underset{x_{\mathrm{IQI}m}}{arg\; min}\left\{\left|y_{\mathrm{D}}-x_{\mathrm{IQI}m}\right|\right\},$$

(10)

with $y_{\mathrm{D}}$ being the digital sample at the end of the receiver chain, after the product $h\sqrt{p}$ has been compensated for but before symbol decision (Section 3). Note that h denotes a realization of the channel coefficient H. This rule remarkably coincides with the Minimum Euclidean Distance (MED) receiver, considering $X_{\mathrm{IQI}}$ as an asymmetric and improper constellation, as the constellation in use. In contrast, an unaware receiver would implement an MED receiver based on X, yielding

$$\widehat{x}=\underset{x_m}{arg\; min}\left\{\left|y_{\mathrm{D}}-x_m\right|\right\},$$

(11)

which is sub-optimal as it does not account for IQI and thus does not consider the transformation in amplitude and phase produced by $a_t$ and ${\theta }_t$.

3. System Implementation

3.1. General Considerations

The proposed testbed, shown in Figure 1, consists of two USRP N210 devices connected to a single laptop via Ethernet cables. A wireless link is established between these devices, which act as transmitter and receiver. Baseband digital signal processing is carried out by the laptop in real-time using the open-source GNU Radio software [29], based on Python and C++. The USRP devices handle ADC/DAC conversion and include homodyne RF front-ends. The power gain of the transmitter and receiver, along with the carrier frequency and sampling rate, are all adjustable dynamically via software within the operational range supported by the hardware.

At the transmitter, this setup enables the intentional generation of IQI. Given a baseband digital sample, x, a digital WLT can generate its IQI version, $x_{\mathrm{IQI}}$, based on specific values of ${\upsilon }_1$ and ${\upsilon }_2$, as denoted in Equation (2). Therefore, any IQI characterized by the parameters described in Section 2 can be accurately replicated with the proposed scheme depicted in Figure 2. Furthermore, white noise can be added to the digital sample to simulate an AWGN channel, in addition to the noise that will be introduced by the wireless link. Despite the alterations being applied to the baseband digital signal, once DAC is performed, the resulting analog electromagnetic signal will mimic the added effects as if they were inherently caused by the hardware. Finally, it is important to note that the proposed testbed is also affected by the intrinsic IQI of the USRP devices. Notes on IQI values for the USRP N210 model equipped with the UBX40 daughterboard can be found in [30,31], showing an IRR of more than 30 dB in both transmission and reception. Since our study focuses on mimicking the IQI found in low-cost devices, whose IIR might be around 20 dB, this intrinsic IQI is considered negligible compared to the replicated values.

3.2. Packet Design and SER Measurement

The communication link is designed to transmit small packets of information, simulating an IoT uplink communication scenario where the transmitter is significantly affected by IQI. The structure of the packet is depicted in Figure 3. An access code of 8 bytes is employed to identify the start of a new packet. The access code is followed by a Cyclic Redundancy Check 32 (CRC32) code for error detection. The CRC32 sequence of 4 bytes is calculated using the payload symbols and provides an estimation of the symbol error probability, noted as $P_{\mathrm{E}}$. Although the length of the payload is adjustable, it is set by default to 100 bytes.

The probability of successfully decoding a packet payload without errors, $P_{\mathrm{SPD}}$, is $P_{\mathrm{SPD}}={\left(1-P_{\mathrm{E}}\right)}^L$, where L is the number of symbols in the packet, which is $L={\displaystyle \frac{N}{{log}_{2}M}}$ for an M-ary constellation, with N being the length of the payload in bits. By taking logarithms in the $P_{\mathrm{SPD}}$ expression, we obtain

$$P_{\mathrm{E}}=1-{10}^{{log}_{10}\left(P_{\mathrm{SPD}}\right)/L},$$

(12)

where $P_{\mathrm{SPD}}$ might be estimated by the Successful Packet Decoding Rate (SPDR). Using the measured value of SPDR, instead of the theoretical $P_{\mathrm{SPD}}$, in Equation (12), we obtain the measured SER that is, indeed, an estimation of $P_{\mathrm{E}}$.

3.3. Receiver Design

The receiver design is critical as it determines the awareness of IQI and, consequently, the ability to mitigate its effects. Figure 4 illustrates the digital baseband processing scheme at the receiver. Blocks that are influenced by IQI at the transmitter, leading to changes in the constellation (as detailed in Section 2), are marked as Constellation-Dependent (CD). In contrast, blocks that are unaffected by changes in the constellation are labeled as Constellation-Independent (CI). All blocks operate in real-time, processing data continuously on a sample-by-sample basis. A brief explanation of each block is as follows:

• Automatic Gain Control (AGC): This block corrects the fading effect caused by the power coefficient and channel realization, $\sqrt{p}\left|h\right|$. Note that this correction is not related to the power or analogue gain, and its only purpose is to scale the constellation symbols with two goals: first, to ensure the values of the digital samples fall within the expected ranges for the blocks downstream; second, to adjust the symbols to their proper decision regions.
  To perform its task, the block implements an adaptive filter that multiplies the incoming digital samples by a gain constant. This constant ensures that the variance of the output samples matches the desired value, ${\sigma }_{X_{\mathrm{IQI}}}^2$. The gain constant is updated every 100 $\mathsf{\mu }$s. Changes in the scale due to ADC/DAC conversion are also corrected using this method. Since this block must know ${\sigma }_{X_{\mathrm{IQI}}}^2$, which differs from ${\sigma }_X^2$; as shown in Equation (9), it is considered CD.
• Time synchronization: This block performs time synchronization and matched filtering of the incoming signal, removing, or at least reducing, Inter-Symbol Interference (ISI). Under the assumption of a flat-fading channel, it can be generally regarded as CI. However, note that symbol equalization for a channel that introduces ISI would be CD.
  For clock recovery, the Time synchronization block uses a polyphase filter bank with 32 sub-filters, each introducing equally spaced time offsets from 0 to $2\pi$, where $2\pi$ corresponds to the time offset of one symbol. The selection of the appropriate filter arm is dynamically controlled by a second-order control loop. This loop bases its decision on the first differential of the signal after it has been delayed by the chosen sub-filter. An output close to zero in the first differential indicates that the clock offset is corrected, which is why this output is used as the error signal for the control loop. After selection, the output from the appropriate filter arm passes through a Root-Raised Cosine (RRC) digital filter to remove ISI. Finally, the signal is downsampled to one sample per symbol, correcting for an oversampling added at the transmitter (Section 4) to aid the clock recovery algorithm. Employing 32 sub-filters ensures that the maximum ISI noise factor remains below the quantization noise of a 16-bit value, as further explained in [32], where the algorithm is described in more detail.
• Phase synchronization: This block is responsible for phase synchronization, essentially de-rotating the complex digital sample whose phase has been mainly shifted by h. Along with the AGC block, it plays a crucial role in ensuring correct symbol decision, and it is CD. The implementation of this block is based on a digital Costas Loop [33].
• Decision: This block performs ML decisions of the incoming symbol. It is clearly a CD block, as detailed in Section 2.

Some theoretical works [8,9] assume that IQI-unaware and IQI-aware receivers estimate and compensate for the channel coefficient h in the same manner. Under this assumption, the signal before symbol decision, $y_D$, as described in Equations (10) and (11), would be identical, the only difference being in the decision regions applied. However, in any practical implementation, an IQI-unaware receiver will attempt to indirectly compensate for the effects of IQI by adjusting the received points to align (by rotating and escalating) with the expected constellation X. To illustrate this point, consider $p=1$ and $h=1$ and IQI parameters of $a_t=0.835$ and ${\theta }_t=5^{\circ }$ for a 64-QAM constellation. Figure 5 illustrates the three receivers analyzed, in this case through simulation: (a) IQI-unaware, as modeled in [9], where it is assumed that only the product $\sqrt{p}h$ is corrected; (b) IQI-aware, where both the channel effect is corrected and the decision regions are adjusted to $X_{\mathrm{IQI}}$; and (c) IQI-unaware in a practical situation, where the receiver attempts to adjust the points to align with X.

Note that under scenario (a), some points fall outside their decision regions, making communication impossible, even in the ideal case without noise. However, the more realistic scenario (c) ensures all points fall within their respective decision regions. However, since it is unaware of the IQI effects, it cannot adapt its decision regions as effectively as an aware receiver does in scenario (b).

4. Methodology and Obtained Results

For the presented results, the testbed shown in Figure 1 is configured with a carrier frequency of 900 MHz and a sampling rate of 5 Msps. Using four samples per symbol and roll-off factor in the RRC filter of $\alpha =0.35$ results in a bandwidth of $1.6875$ MHz. The transmitted power is approximately $-27$ dBm. A low value of IIR (see Section 2) is selected, $\mathrm{IIR}=20$ dB, with parameters $a_t=0.835$ and ${\theta }_t=5^{\circ }$, which introduce a considerable IQI [9,11].

SER values are extracted using the CRC32 code of each packet (Section 3.2). To carry out measurements across different SNR values, a digital pre-distortion of additive white noise is applied as depicted in Figure 2. Consequently, we define the SNR at the transmitter, ${\mathrm{SNR}}_{\mathrm{TX}}$, as ${\mathrm{SNR}}_{\mathrm{TX}}={\displaystyle \frac{E\left\{{\left|X\right|}^2\right\}}{{\sigma }_{\mathrm{TX}}^2}}={\displaystyle \frac{1}{{\sigma }_{\mathrm{TX}}^2}}$. Note that, for the sake of choosing a common reference point, the variance of X is considered rather than that of $X_{\mathrm{IQI}}$. Figure 6 depicts several examples of captured constellation points by the tested before symbol decision, $y_D$ in Figure 4, for an IQI-aware receiver. Different constellations and ${\mathrm{SNR}}_{\mathrm{TX}}$ values are displayed. The graphs illustrate how the digital receiver chain successfully adjusts the received points to align with the noiseless constellation $X_{\mathrm{IQI}}$, marked with red crosses, as detailed in Section 3.3.

Figure 7 presents the results obtained for QPSK, 16-QAM, 64-QAM, and 16-PAM constellations. Five scenarios are presented: software simulations without IQI (Simulation no IQI); IQI-aware software simulations (Simulation IQI-aw.), and measurements (Meas. IQI-aw.), whose detection is illustrated in Figure 7b; IQI-unaware simulations (Simulation IQI-un.), whose detection is depicted in Figure 7a; and IQI-aware testbed measurements (Meas. IQI-aw.), corresponding to Figure 7c. For the measurements, the IQI-aware receiver is configured such that all CD blocks in the baseband scheme, Figure 4, have $X_{\mathrm{IQI}}$ as their reference constellation. On the other hand, the IQI-unaware receiver uses X as the reference constellation. Noise variance at the receiver can be modeled as ${\sigma }_{\mathrm{N}}^2={\sigma }_{\mathrm{TX}}^2+{\sigma }_{\mathrm{PH}}^2$, where ${\sigma }_{\mathrm{PH}}^2$ denotes the noise variance of the physical channel. For most of the analyzed cases, ${\sigma }_{\mathrm{TX}}^2\gg {\sigma }_{\mathrm{PH}}^2$ and ${\sigma }_N^2\approx {\sigma }_{\mathrm{TX}}^2$, so the SNR values are identified with ${\mathrm{SNR}}_{\mathrm{TX}}$, $\mathrm{SNR}={\displaystyle \frac{1}{{\sigma }_{\mathrm{N}}^2}}\approx {\displaystyle \frac{1}{{\sigma }_{\mathrm{TX}}^2}}$. However, this assumption starts to fail when ${\mathrm{SNR}}_{\mathrm{TX}}\ge 25$ dB, as the noise introduced by the physical channel is no longer negligible. For those cases, additional points are depicted (Meas. IQI-aw. Disp.) for the IQI-aware receiver, where the dispersion of received points, $y_D$, with respect to the expected noiseless points, $x_{\mathrm{IQI}}$, is measured. This is achieved by averaging ${\displaystyle \frac{1}{{\left|{\widehat{x}}_{\mathrm{IQI}}-y_{\mathrm{D}}\right|}^2}}$, where ${\widehat{x}}_{\mathrm{IQI}}$ is the closest point in the constellation to the digital sample $y_{\mathrm{D}}$, corresponding to the decision criteria expressed in Equation (10).

5. Discussion

From the results obtained, the following key considerations are highlighted:

• Measurements with an IQI-aware receiver are consistent with simulations. These matched results validate the receiver implementation. The slight differences observed at higher values of $\mathrm{SNR}$ (low ${\sigma }_{\mathrm{N}}^2$) are attributed to the fact that, at these levels, the perturbations introduced by the hardware channel, ${\sigma }_{\mathrm{PH}}^2$, become comparable to o higher than the pre-distortion noise, ${\sigma }_{\mathrm{TX}}^2$. This assumption is confirmed by measuring the dispersion of points at the receiver, which provides an accurate estimation of ${\sigma }_N^2$ for the whole range of the tested SNRs.
• The performance of an IQI-unaware receiver significantly degrades as the constellation number of points, M, increases. In denser constellations, the impact of an uncompensated IQI becomes more pronounced.
• An exception to the previous point are maximally improper or asymmetric constellations, such as 16-PAM. Results show the robustness of these constellations against IQI, even for unaware receivers. This experimental finding aligns with the theoretical conclusions presented in [9]. Consequently, PAM or, in general, maximally improper constellations, are a compelling option for IQI-unaware scenarios, albeit at the cost of lower power efficiency compared to QAM.
• IQI-unaware measurements exhibit better performance than IQI-unaware software simulations. As explained in Section 3.3, this is due to the receiver attempting to adjust the points to the constellation X before symbol decision, whereas software IQI-unaware simulations are assumed to only correct the product $\sqrt{p}h$.

It is important to note that the proposed design can be applied to MIMO systems by replicating the same strategy for each independent receiver chain. Nonetheless, it must be considered that multiple transmission chains may experience different IQI values [6,7]. Another promising direction for future research is the experimental validation of IQI mitigation techniques in mm-Wave frequencies. As noted in the introduction, single-carrier architectures exhibit significant potential in this domain [13,14,15]. Additionally, devices operating at these frequencies are more susceptible to IQI imbalance due to increased hardware impairments [16]. However, current SDR commercial devices do not support these operating frequencies, limiting the rapid prototyping of research testbeds to study the problem.

6. Conclusions

This article presents the design, implementation, and validation of an IQI communication system testbed based on SDR. As a representative example, a single-carrier scenario where the transmitter is highly affected by IQI has been considered. This scenario is akin to an IoT communication uplink. The SDR paradigm offers remarkable flexibility, enabling the accurate replication of IQI effects and facilitating the implementation of signal processing techniques for their mitigation. OTA measurements have been carried out for both IQI-aware and IQI-unaware receivers. Results show that denser constellations are more susceptible to IQI, but its effects are significantly mitigated by an IQI-aware receiver. Maximally improper constellations are an exception, demonstrating robustness against IQI, even with an unaware receiver. These findings align with previous theoretical works, though this paper also addresses inaccuracies in the theoretical modeling of IQI-unaware receivers. The source code and files of the proposed testbed are publicly available. Future research directions include expanding the proposed testbed to incorporate MIMO scenarios and mm-Wave frequencies.