Adaptive quantization based binary hypothesis detection method 1-bit MIMO receiver
Patent Information
- Application Number
- CA3110925
- Authority / Receiving Office
- CA · CA
- Patent Type
- Patents
- Current Assignee / Owner
- Filing Date
- 2021-03-02
- Publication Date
- 2026-09-15
- Estimated Expiration
- 2041-03-02
Abstract
Description
Adaptive Quantization Based Binary Hypothesis Detection Method in 1-Bit MIMO Receiver FIELD OF THE INVENTION
[0001] This present invention relates generally to binary hypothesis testing in a multiple-input, multiple-output (MIMO) receiver relying on 1-bit quantization, and in particular to binary measurements acquired by using binary window comparators to adaptively quantize received signals. BACKGROUND OF THE INVENTION
[0002] In a standard digital receiver, the most straightforward approach to process analog signals in a digital signal processing unit is to quantize received signals sufficiently finely for a faithful reproduction by using high-resolution analog-to-digital converters (ADCs). The problem is that as the number of antennas at a receiver increases, using high-resolution ADCs becomes increasingly inefficient.
[0003] Using high-resolution ADCs comes at the cost of energy consumption which increases exponentially with the number of bits. Hence, in a receiver having a large number of antennas where each antenna is attached to a pair of ADCs (each per signal dimension), energy consumption becomes inconceivable. In addition, the resultant hardware complexity may make practical implementation of the system challenging. Moreover, the amount of information that needs to be processed, especially at high 1 CA 3110925 Date reçue / Received date 2025-03-06 sampling rates, is potentially enormous, making information processing very expensive.
[0004] In order to address these challenges, high-resolution ADCs can be replaced with low-resolution ADCs (e.g., 1-3 bits) and information loss will have to be accepted. With the emergence of massive MIMO systems, 1-bit ADC has become a popular research topic. As well, 1-bit quantization is relevant in wireless sensor networks due to energy and bandwidth constraints in these networks comprising a large number of power-constrained cheap sensors. Several studies have shown that massive MIMO systems using 1-bit quantization can achieve adequate data rates, providing the loss of information resulting from 1-bit quantization is compensated for by using large antenna arrays.
[0005] A conventional zero-threshold 1-bit ADC (aka, comparator or 1-bit quantizer) is commonly used in the literature, which retains the sign of a signal based on whether the signal level exceeds zero voltage. A conventional 1-bit quantizer presents two major challenges. Due to the high nonlinear dependence of quantization noise on the input signal of the 1-bit quantizer, the mathematical analysis becomes intricate and intractable. There are certain conditions where it is difficult to extract useful information about an embedded signal (such as interference or information-carrying signal in noise ) from a received signal. In some cases, it is even impossible due to the significant information loss.
[0006] One application of massive binary measurements acquired at a communication receiver is binary hypothesis testing, which can be used for a variety of purposes, including detecting jamming attack (i.e., a presence or absence of an attack on a receiver during a legitimate communication) and in radar systems for detecting the presence or absence of a target from a reflected signal. This invention is primarily 2 CA 3110925 Date reçue / Received date 2025-03-06 intended for binary hypothesis testing based on the binary measurements (i.e., 1-bit quantized signals) acquired at a MIMO receiver employing 1-bit quantization.
[0007] The following are some motivations for the present invention. As far as jamming detection (i.e., viewed as a binary hypothesis testing problem) in a MIMO receiver is concerned, existing studies typically assume the receiver or the BS has access to high-precision observations (theoretically, unquantized signals) and have not examined jamming detection within a 1-bit quantized system.
[0008] A subspace-based method can detect and possibly suppress significant amounts of jamming energy when the signal and jamming lie in a smaller dimensional space relative to the number of antennas at the BS (i.e., the dimension of received signal space), where the signal space and jamming space can be identified. Despite this, the subspace-based method does not always work with 1-bit quantization at the MIMO receiver because the signal space and jamming space are highly intertwined in a nonlinear manner due to the high nonlinearity of 1-bit quantization. As a result, the two spaces cannot be easily identified or isolated, making jamming detection extremely difficult.
[0009] There are a number of energy-based detectors which are useful for detecting pilot attacks (i.e., an adversary attack during channel training to ruin the receiver’s ability to estimate the channel of a legitimate user by contaminating its pilot signal). The asymmetry between the received signal power levels between legitimate parties (i.e., the BS and its users) enables detection of jamming by using an energy-based detector. Since the energy of binary measurements (i.e., 1 or -1) at the MIMO receiver is constant as well as the fact that stronger jamming power can dominate the output of the conventional 1-bit quantizer, energy-based detectors are inapplicable under conventional 1-bit quantization. 3 CA 3110925 Date reçue / Received date 2025-03-06
[0010] Pilot-data based channel acquisition methods can theoretically mitigate pilot attack in the asymptotic sense of data length. However, with conventional 1-bit quantizers at a MIMO receiver, it is well-known that an optimal channel estimator (based on so-called Bussgang decomposition) saturates as signal power increases. As a consequence, the quality of channel estimation cannot be arbitrarily enhanced with increasing pilot signal power. Instead, it continues to deteriorate after a certain interference level. A smart attacker can adapt his power so it dominates the output of a conventional 1-bit quantizer and hence pilot-data-based channel acquisition methods become less useful under conventional 1-bit quantization.
[0011] It may be possible to estimate some statistical parameters of the attacker using unused pilots in a network which are made unknown to the attacker, although extension to a multi-antenna jammer is not yet possible. It is not applicable, however, to 1-bit quantized systems that utilize conventional 1-bit quantizers because a significant amount of information regarding the presence of an attacker is already lost. For practical implementation of the estimation process, some statistical parameters (relating not only to legitimate users but also to attackers) must be known beforehand, where such a priori information about an attacker may not be feasible.
[0012] Log-likelihood ratio testing (LRT) is a powerful tool for distinguishing between two hypotheses (i.e., the null and alternative hypotheses) in which, at least, the statistical model employed to design the sufficient statistic test threshold is assumed to be known. Binary hypothesis testing based on generalized likelihood ratio test (GLRT) provides satisfactory performance for detecting a multiple-antenna attacker in unquantized MIMO systems. However, when the conventional 1-bit quantizer is used, the unknown distribution of quantization noise makes binary hypothesis testing difficult. 4 CA 3110925 Date reçue / Received date 2025-03-06
[0013] Despite the difficulties associated with binary hypothesis testing in a MIMO receiver (for example, a massive MIMO system’s base station) using conventional 1- bit quantizers, which have been discussed primarily in terms of jamming detection, this is a more general challenge concerning binary hypothesis testing based on binary measurements acquired by a MIMO receiver using conventional 1-bit quantizers.The result is that binary hypothesis testing can be difficult or even impossible for binary measurements obtained through the use of conventional 1-bit quantizers at a MIMO receiver.
[0014] In order to facilitate binary hypothesis testing and overcome the challenges inherent in conventional 1-bit quantization, it is desirable to provide a low-complexity detection method in a MIMO receiver based on an adaptive 1-bit quantization method in lieu of the conventional 1-bit quantization. This will bring many advantages and a variety of new applications. SUMMARY OF THE INVENTION
[0015] The present invention discloses a method for an efficient binary hypothesis testing in a MIMO receiver employing binary quantization, which comprises an adapted binary window comparator (BWC) for quantizing a signal into binary digits and a very low-complexity bit density detector (BDD) for deciding which one of two hypotheses gave rise to the binary measurements. The adapted BWC makes it possible to recapture a significant amount of information about the presence of an embedded signal in a received waveform, which could be a challenge through using a conventional zero-threshold 1-bit quantizer due to the substantial information loss and its inherent high non-linearity. 5 CA 3110925 Date reçue / Received date 2025-03-06
[0016] The adapted BWC is purposely designed so that it generates Bernoulli random variables with a fixed success rate under a null hypothesis model, which is assumed known at the receiver. This is achieved by adjusting the lower and upper thresholds of BWC to be proportional to the variance of the model in null hypothesis and symmetrical around zero, yielding a window range which is symmetrical around zero. In addition, the proportionality constant of the upper and lower thresholds can be further pre-selected to maximize the detectability power of the receiver. Success rate of a generated binary digits being fixed is important, as this will work as a reference point for detecting a deviation resulting from a signal to be detected.
[0017] The BDD relies essentially on three parameters: the density of 1’s (number of 1’s) in the observed binary digits, test threshold, and a randomization parameter between 0 and 1. The test threshold and randomization parameter are computed offline, where the purpose of the latter is to meet a desired probability of false alarm. The detector uses one of two approaches to determine which one of two hypotheses gave rise to the observed binary digits, wherein the decision is based on a first approach if the sum of binary digits is equal to the pre-computed test threshold, but is based on a second approach if the sum of binary digits is not equal to the pre-computed test threshold. In the first approach, decision is merely based on a simple binomial test, wherein hypothesis H1 or H0 is accepted if the sum of binary digits is greater or less than the pre-computed test threshold, respectively. For the second approach, a uniformly distributed random number X ∈ [0, 1] is generated, wherein hypothesis H1 or H0 is accepted as true if X is less than or equal to the randomization parameter or greater than randomization parameter, respectively.
[0018] Compared to the traditional zero-threshold 1-bit quantizer whose quantization noise does not allow a tractable probabilistic model, the adapted BWC alongside 6 CA 3110925 Date reçue / Received date 2025-03-06 BDD brings many advantages and new applications, which are described as follows. 1. The binary quantization based on an adapted BWC can overcome the problem of significant information loss inherent in the conventional 1-bit quantizer, which is necessary for binary hypothesis testing. 2. The adapted BWC combined with BDD can achieve a significant fraction of performance that can be achieved under unquantized signal with log-likelihood based detection. Further, the performance of the proposed detectability method increases monotonously with the variance of the signal to be detected. 3. According to the invention, any deviation from the reference point (fixed success rate under null hypothesis) is translated into change in variance which is used for detection. This is not an easy task using conventional zero-threshold 1-bit quantizer, especially under symmetrical hypotheses models around zero, where 0’s and 1’s are equally likely in both models. 4. While binary hypothesis detection methods utilizing a conventional 1-bit quantizer are scarce due to its high non-linearity and intractability of associated probabilistic modeling, the proposed detection method enjoys a very low complexity which is linear with the total number of observations. 5. The present invention works with both scalar-valued and matrix-valued hypothesis since the proposed detection method depends only on the sum of all observed binary digits no matter how they are ordered. 6. The present invention can be beneficial for a variety of applications comprising detecting the presence of an unknown signal embedded in Gaussian noise, detecting jamming in MIMO system, especially massive MIMO system where 7 CA 3110925 Date reçue / Received date 2025-03-06 number of observations are large, and it can have application in wireless sensor networks where energy and hardware complexity is of important concerns. BRIEF DESCRIPTION OF THE DRAWINGS
[0019] FIG. 1A is a logic gate diagram of an adjusted binary window comparator used in the present invention;
[0020] FIG. 1B shows the operation of the binary window comparator embodied in FIG. 1A;
[0021] FIG. 2 is a block diagram of a mathematically equivalent model of the binary window comparator embodied in FIG. 1A;
[0022] FIG. 3 illustrates a probabilistic output analysis for the binary window comparator in FIG. 1A (or equivalently FIG. 2), in response to two probability models.
[0023] FIG. 4 is a block diagram of a MIMO receiver comprising a set of binary window comparators and binary density detector, which is a binary hypothesis detection unit;
[0024] FIG. 5 is a flowchart of detection method for binary density detector;
[0025] FIG. 6 illustrates a massive MIMO system utilizing the present invention to detect jamming.
[0026] FIGs. 7-11 are plots of numerical results for various embodiments of the present invention. 8 CA 3110925 Date reçue / Received date 2025-03-06 DETAILED DESCRIPTION
[0027] The present invention provides a method for an efficient binary hypothesis testing in a MIMO receiver employing binary quantization, which comprises an adapted binary window comparator (BWC) for quantizing a signal into binary digits and a very low-complexity bit density detector (BDD) for deciding which one of two hypotheses gave rise to the binary measurements.
[0028] Much research on 1-bit quantization is focused on a conventional zerothreshold 1-bit quantizer (aka a 1-bit ADC, comparator, or hard-limiter) that retains the sign of an input signal. For several years, great effort has been devoted to the study of performance of large-scale MIMO systems, particularly, so-called 5G massive MIMO systems employing conventional 1-bit quantization at the base station, because of the simplicity it brings to the physical layer and insignificant energy consumption. However, there are some limitations of using the conventional zerothreshold 1-bit quantizer including a significant information loss and the fact that its output can be rapidly dominated by a stronger signal.
[0029] In typical wireless communication, Rayleigh fading channels, randomness of data, interference and background noise give rise to a received signal admitting a symmetrical or slightly skewed density around zero. As a result, the average number (averaged over all randomness) of 1’s and 0’s after conventional quantization is equal asymptotically. When a receiver is equipped with a sufficient number of antennas (e.g., massive MIMO), a balance between the numbers of 1’s and 0’s is even expected over a single realization of channel, as a consequence of the law of large numbers. 9 CA 3110925 Date reçue / Received date 2025-03-06
[0030] One important scenario where acquiring useful information by the conventional zero-threshold 1-bit quantizer may be intricate is the situation where a receiver is under jamming attack which is intentionally launched by one malicious adversary or more to degrade system performance or / and eavesdrop on communication. According to the above discussion, extracting information about the presence of jamming from the quantized bits seems intricate. Another scenario is when one needs to detect the presence of a signal embedded in noise. Both the above two scenarios fall under the framework of binary hypothesis testing.
[0031] From an information-theoretic perspective, it is easy to think of n independent bits as if they were generated by one of two discrete memoryless sources S0 and S1 with Pr( 1|S0) = Pr(1|S1) = 0.5, then both sources have the same entropy, i.e., H(S0) = H(S1) = 1. For a sufficiently large n, the typical sets T n 0 and T n 1 of sequences generated by S0 and S1 admit the same cardinality in the asymptotic sense, i.e., |T n 0 | = |T n 1 | = 2n as n → ∞ and hence both typical sets are completely overlapping. Thus there will be no hope to divide the n-dimensional space into two disjoint regions from which one can distinguish if a binary sequence was likely generated by either S0 or S1. To increase the distance between the two above typical sets, an imbalance between the number of 1s and 0s under both hypotheses (H0 = S0,H1 = S1) is purposely induced by a binary window comparator (BWC) 100. Specifically, the density of 1s under the null hypothesis H0 is kept fixed at some small value p0 and hence any deviation from p0 will likely signify data being generated under the alternative hypothesis H1.
[0032] The present invention utilizes a BWC 100 and provides a very low complexity binary hypothesis detector referred to as bit density detector (BDD) 410 where the detection method is shown by the flowchart in FIG. 5. Note that the BWC is 10 CA 3110925 Date reçue / Received date 2025-03-06 purposely adapted to facilitate detection by BDD. The BDD comprises of two decision approaches, wherein in a first approach the decision on which hypothesis is true is simply based on a binomial test, wherein the number of 1’s in the observed binary sequence at the output of the BWC is computed and compared to a pre-defined threshold. In a second approach, the BDD utilizes a randomization process to make a decision. This randomization process is important to maintain a fixed probability of false alarm, PF .
[0033] FIG. 1A shows a BWC 100 represented by a logic gate diagram. It comprises two operational amplifiers (opamps) 103 working as comparators and a NAND gate 104. The voltages of non-inverting input 101 and inverting input 102 are set to vH = λ (upper threshold) and vL = −λ (lower threshold), respectively. The BWC 100 is a memoryless device which detects whether an input voltage is outside a window predetermined by the upper and lower voltage thresholds. FIG. 1B shows the operation of our BWC embodied in FIG. 1A, which can be mathematically defined by Qw(a) = 8>>>< >>>: 1 if |a| > λ, 0 −λ ≤ a ≤ λ. (1) where a ∈ R denotes the signal voltage level. For a complex-valued input, a pair of BWCs are used to quantize the real and imaginary parts separately and independently.
[0034] FIG. 2 shows an equivalent model for the BWC which essentially has the same operation, where it consists of an absolute value (or modulus) operator 210 followed by a simple λ-threshold comparator 220. A conventional zero-threshold 1- bit quantizer (aka comparator, 1-bit ADC or hard limiter) is defined by block 220 with λ = 0. 11 CA 3110925 Date reçue / Received date 2025-03-06
[0035] FIG. 3 illustrates the probabilistic response of a BWC to two discrete independent and identically distributed (IID) Gaussian processes each of zero-mean; a first process corresponds to null hypothesis H0 : Yj ∼ CN(0, σ2 0) whereas a second to alternative hypothesis H1 : Yj ∼ N(0, σ2 1), where j = 1, 2, · · · , n and σ2 0 and σ2 1 represent the corresponding variances of these processes and σ1 ≥ σ0. Each of the corresponding binary processes (under H0 or H1) at the output of the BWC is viewed as an IID Bernoulli random process Bj with different success rates, i.e., H0 : Bj ∼ Ber(p0) and H1 : Bj ∼ Ber(p1), where p0 and p1 denote the corresponding success rates. Evidently, p0 or p1 is simply the probability that Yj ∼ N(0, σ2 0) or Yj ∼ N(0, σ2 1) lies outside a window range of width 2λ of the BWC, respectively, which is equivalent to the probability the BWC being in on-state given H0 or H1. Mathematically, we write p0 ≜ Pr(ON|H0) and p1 ≜ Pr(ON|H1). For instance, p0 is indicated by the shaded area 302 in FIG. 3.
[0036] By letting the number of observations n become sufficiently large, p0 and p1 can be seen as the density of 1’s in the output binary sequence, thanks to the law of large numbers. Therefore, it is sensible to use the density of 1’s in the observed binary sequence as a metric to decide which hypotheses (H0 or H1) was active when measurements have been acquired. To realize the above decision metric, we fix p0 to a small value. In other words, the entropy of the process Bj under H0 is held fixed. To that end, we let p0 be independent of σ0 by setting the upper and lower window thresholds such that vH = cσ0 vL = −cσ0 (2) where c > 0 is a constant that can be preselected to maximize a probability of detection. 12 CA 3110925 Date reçue / Received date 2025-03-06
[0037] Referring to FIG. 3 and using Eq. (2), we have p0 = 2 Z ∞ cσ0 1 q 2πσ2 0 exp " − y2 2σ2 0 # dy = 2Q(c) (3) p1 = 2 Z ∞ cσ0 1 q 2πσ2 1 exp " − y2 2σ2 1 # dy = 2Q(cσ0 / σ1) = 2Q(αc) (4) where Q(·) is the Q-function of the standard normal distribution and α = σ0 / σ1 ∈ [0, 1]. Our choice of vH and vL in Eq. (2) renders p0 fixed and independent of σ0, which is very important as p0 becomes a reference point in the detection process. Note also that p0 is a function of single parameter c which is known a priori and hence is computed beforehand.
[0038] FIG. 4 shows a multiple-input receiver 400 with M inputs. All received signals (or waveforms) y1(t), · · · , yM(t) are complex continuous-time baseband signals which are mutually independent. The real and imaginary parts of each signal are separated by blocks 402 and 404, respectively, sampled by a pair of samplers 406. The generated sampled signals {yℜ m,l, yℑ m,l } (m = 1, 2, · · · ,M, l = 1, 2, · · · ,N) are then quantized separately by a set of 2M BWCs 100, yielding corresponding binary digits {bℜ m,l, bℑ m,l } (m = 1, 2, · · · ,M, l = 1, 2, · · · ,N), where m and l denote the input and time indices, respectively. All 2MN binary digits {bℜ m,l, bℑ m,l } are collected by a bit density detector (BDD) 410 to decide which of two hypotheses gave rise to these binary digits, where a method of detection is illustrated in the flowchart in FIG. 5.
[0039] With reference to FIGs. 4 and 5, the basic inputs to the BDD are binary digits {bℜ m,l, bℑ m,l }, the number of observations n = 2MN, the BWC’s parameter c and the desired probability of false alarm PF . The BDD comprises two basic units: a randomized binomial test unit 420 and test parameters computation unit 430. We 13 CA 3110925 Date reçue / Received date 2025-03-06 feed the randomized binomial test unit by two parameters: a randomization parameter ζ 432 and test threshold γ 434, which are computed offline by test parameters computation unit 430.
[0040] Without loss of generality, we assume each signal or waveform ym(t) is the sum of transmitted signal xm(t) and noise zm(t), i.e., ym(t) = xm(t) + zm(t). We assume zm(t) ∼ CN(0, 2σ2 0) and xm(t) ∼ CN(0, 2σ2x ) and hence the sampled real and imaginary parts of ym(t) is given by yℜ m,l = xℜ m,l + zℜ m,l and yℑ m,l = xℑ m,l + zℑ m,l, respectively. Thus both the real and imaginary signals yℜ m,l and yℑ m,l are distributed as yℜ m,l, yℑ m,l ∼ N(0, σ2 0 + σ2x ). The two cases σ2x = 0 ( absence of transmitted signal) and σ2x > 0 (presence of transmitted signal) correspond to H0 and H1, respectively.
[0041] For notational simplicity, we replace the sampled transmitted signals {xℜ m,l, xℑ m,l }, noise samples {zℜ m,l, zℑ m,l }, sampled received signals {yℜ m,l, yℑ m,l } and the corresponding binary digits {bℜ m,l, bℑ m,l }, respectively, by {xj}, {zj}, {yj} and {bj} (j = 1, 2, · · · , n = 2MN). Thus, our hypothesis testing problem is expressed by H0 :{bj = Qw(yj)}nj =1 subject to yj = zj ∼ N(0, σ2 0), H1 :{bj = Qw(yj)}nj =1 subject to yj = xj + zj ∼ N(0, σ2 1). (5) where σ2 1 = σ2 0 + σ2x . According to Eq. (5), BDD 410 operates on the binary digits bj(j = 1, · · · , n) to determine if they have been caused by the received signal samples yj = zj (H0) or yj = xj + zj (H1). According to the probabilistic response of the BWC discussed previously, Eq. (5) can be equivalently recast as H0 : b1, · · · , bn I∼ID Ber(p0) H1 : b1, · · · , bn I∼ID Ber(p1) (6) where p0 = 2Q(c) and p1 = 2Q(cσ0 / σ1) are derived in Eqs. (3) and (4), respectively. 14 CA 3110925 Date reçue / Received date 2025-03-06
[0042] The operation of the BDD 410 is summarized in the flowchart in FIG. 5, which is discussed as follows. First, given n, c and PF , the test threshold γ is calculated then followed by the calculation of the randomization parameter ζ (step 520). The calculation of these two parameters is done as follows.
[0043] Using Eq. (6) the joint probability mass functions under both hypotheses are given by H0 : p({bj}|p0) = Yn j=1 pbj 0 (1 − p0)1−bj H1 : p({bj}|p1) = Yn j=1 pbj 1 (1 − p1)1−bj , (7) where p(·|θ) denotes joint probability mass function parameterized by θ. Thus the log-likelihood ratio can be expressed by Λ({bj}) = log 2 4 Qnj =1 pbj 1 (1 − p1)1−bj Qnj =1 pbj 0 (1 − p0)1−bj 3 5 = Xn j=1 bj log " p1(1 − p0) p0(1 − p1) # | {z } >0 +n log "1 − p1 1 − p0 # H1 ⋛ H0 γ′ (8) where γ′ is the detection threshold. After some mathematical manipulations, (8) boils down to the following sufficient statistic test: Sn = Xn j=1 bj H1 ⋛ H0 γ (9) where γ is the modified detection threshold (test threshold) given by γ ≜ γ′ − n log "1 − p1 1 − p0 #!, log " p1(1 − p0) p0(1 − p1) # . (10)
[0044] The BDD 410 compares the number of 1’s in the observed sequence {bj} to the threshold γ and decides H1 if Sn exceeds γ or accepts H0 otherwise. The test (9) does not depend on the unknown parameter p1. According to the Neyman-Pearson 15 CA 3110925 Date reçue / Received date 2025-03-06 criterion, it is a uniformly most powerful (UMP) test in the sense that for any value of p1 (or equivalently σ1), the resulting probability of detection PD is maximized for a fixed probability of false alarm PF . The threshold γ is determined by the distribution of Sn.
[0045] Since the distribution of yj under H0 is assumed known (i.e., known σ0), the threshold γ can be selected for a fixed probability of false alarm PF , resulting from accepting H1 when H0 was the correct hypothesis. The sufficient statistic Sn is a sum of IID Bernoulli random variables, hence it follows the binomial distribution under both hypotheses: H0 : Sn ∼ Bin(n, p0), H1 : Sn ∼ Bin(n, p1). (11)
[0046] Based on Eq. (9), it is desired that the BDD makes an error with a maximum probability of false alarm PF as Sn exceeds γ when H0 was the true hypothesis. Mathematically, PF ≥ Pr(Sn > γ|H0) = Xn l=γ+1 n l ! (2Q(c))l(1 − 2Q(c))n−l (12) where we have used p0 = 2Q(c). The summation in Eq. (12) can be easily computed numerically to find γ which can be given in terms of the binomial inverse cumulative distribution function (CDF), i.e., γ = Bin−1(1 − PF , n, 2Q(c)) (13) which returns the smallest integer γ such that the binomial CDF evaluated at γ is equal to or exceeds 1 − PF .
[0047] Since the test threshold γ in Eq. (12) is an integer number, it may happen that there is no integer that satisfies Eq. (12) with equality, in contrast with a contin- 16 CA 3110925 Date reçue / Received date 2025-03-06 uous random variable where a test threshold can be chosen to meet the probability of false alarm exactly. Also, there is a nonzero probability for the event Sn = γ which is not included in the test in Eq. (9), in sharp contrast to a continuous random variable where the probability at a single point is zero. Therefore, the resulting probability of false alarm will change with n, which is undesirable. The gap between the desired PF and resulting probability of false alarm after having chosen γ can be large and hence influences the probability of detection significantly.
[0048] In order to maintain the probability of false alarm at the desired level PF (satisfying Eq. (12) with equality), a randomization parameter ζ 432 (step 520) is included in the decision process. Having calculated the test threshold γ and randomization parameter ζ and fed back to the randomized binomial test unit 420, the operation of randomized binomial test unit 420 proceeds according to block 530 (steps 532-537) of the flowchart in FIG. 5, which is expressed mathematically by Decide: 8>>>>>>>>>>>>>< >>>>>>>>>>>>>: H1 if Sn > γ H0 if Sn < γ H1 with probability ζ if Sn = γ H0 with probability 1 − ζ if Sn = γ (14)
[0049] The calculation of ζ (step 520) is done as follows. Based on (14), (12) is modified as PF = Pr(Sn > γ|H0) | {z } ≤PF +ζPr(Sn = γ|H0) | {z } ΔPF (15) where ΔPF > 0 is a small probability increment needed to compensate for the gap 17 CA 3110925 Date reçue / Received date 2025-03-06 resulting from the closest solution to PF (first term). From Eq. (15), we have ζ = PF − Pr(Sn > γ|H0) Pr(Sn = γ|H0) = PF − Pnl =γ+1 n l (2Q(c))l (1 − 2Q(c))n−l n γ (2Q(c))γ (1 − 2Q(c))n−γ (16) where by definition the summation in the numerator of Eq. (16) is zero when the lower limit of the sum is greater than n. Note that when PF → 0(γ = n), ζ → 0 and when PF → 1(γ = 0), ζ → 1 and hence ζ ∈ [0, 1].
[0050] Using (14)-(16), the probability of detection PD of BDD is thus given by PD = Pr(Sn > γ|H1) + ζPr(Sn = γ|H1) = Xn l=γ+1 n l ! (2Q(αc))l (1 − 2Q(αc))n−l + PF − Pnl =γ+1 n l (2Q(c))l (1 − 2Q(c))n−l Q(c) Q(αc) γ 1−2Q(c) 1−2Q(αc) n−γ . (17) where from the previous discussion we write α = σ0 / σ1 = vuut σ2 0 σ2 0 + σ2x ∈ [0, 1] (18) which is a measure of the relative strength of transmitted signal x(t).
[0051] We summarize the BDD as follows. Given γ (Eq. (13)) and ζ (Eq. (16)) fed back to the randomized binomial test unit 430, it proceeds as follows. In step 531, the sum Sn of 1’s in the binary sequence bj(j = 1, 2, · · · , n = 2MN) is computed. In step 532, if Sn ̸= γ then a binomial test (based on log-likelihood)) is run, otherwise a randomized test takes place if Sn = γ . For the binomial test, H1 is accepted if Sn > γ, otherwise it accepts H0 if Sn < γ . For the randomized test, a uniform random variable X ∼ Uni(0, 1) is first generated (step 533). If X ≤ ζ(step 534) then 18 CA 3110925 Date reçue / Received date 2025-03-06 H1 is accepted, otherwise H0 is accepted instead. Then the detection stops (step 536).
[0052] Eq. (17) shows that the probability of detection PD can be maximized with respect to c. Note that PF and n are fixed beforehand, whereas α is not under control of the designer. The smaller σx, the larger the ratio α and hence the harder the detectability will be. Thus it is sensible to optimize c that maximizes PD under larger values of α or equivalently when σ2x is small. Unfortunately, Eq. (17) does not lead to a closed-form solution for c. However, the solution can be easily found numerically. FIG. 7 shows PD plotted against different values of c for different α = 0.8, 0.85, 0.9 (small variance change), two fixed PF = 1%, 5% and n = 200. As shown in FIG. 7, PD begins at PF for c = 0, increases to a maximum value, and subsequently decreases toward PF as c increases. FIG. 7 suggests that an excellent choice of c is about 1.6 under both probabilities of false alarm. Thus the lower and upper thresholds of the BWC are −1.6σ0 and 1.6σ0, respectively.
[0053] FIG.s 8 and 9 show the performance curves of the receiver 400, where solid lines and markers correspond respectively to analytical and simulated results. We assume xj ∼ N(0, σ2x ) and noise zj ∼ N(0, 1). Thus a received sample (real or imaginary) yj ∼ N(0, 1 + σ2x ) is quantized by a BWC with vH = 1.6, vL = −1.6. The output binary digits bj is then operated on by the BDD 410 to decide whether the transmitted signal is present. In FIG. 8, we vary σ2x while fixing the number of observations n = 100. In FIG. 9, we fix σ2x = 0.2 and change n. Referring to FIGs. 8 and 9, the performance curves show that the tradeoff becomes less stringent as the variance of transmitted signal and the number of samples increase. For instance, FIG. 9 suggests that for a fixed change in variance between both hypotheses, the probability of false alarm PF can made arbitrarily small while the probability of detection PD 19 CA 3110925 Date reçue / Received date 2025-03-06 converges to 1, asymptotically in the number of observations. Overall, FIGs. 8 and 9 show that the present invention is very efficient in distinguishing between two hypotheses models even with a coarsely quantized signal but in a controlled manner. Note that this is not possible if the zero-threhsold conventional 1-bit quantizer 220 is employed. Note that performance of our designed BWC converges to the conventional zero-threshold 1-bit quantizer as c → 0. Substituting c = 0 in Eq. (17) results in probability of detection equal to probability of false alarm when the conventional zero-threshold 1-bit quantizer 220 is used.
[0054] One use case of the present invention
[0055] FIG. 6 shows a scenario of jamming detection in 1-bit quantized massive MIMO system 600 operating in the time-duplex division (TDD) mode. The system includes a base station 610, equipped with a set of M antennas 611, a set of K legitimate single-antenna user equipment (UEs) 620, a multiple-antenna jammer 620. Both the UEs and jammer operate in the transmit mode whereas the base station in receive mode. UEs transmit their signals from a set of antennas 621 via channel H 640 whereas the jammer transmits its signal from a set of antennas 631 via channel G 650. The jammer comprises NJ ≥ 1 antennas and can be viewed as either a single jammer (collocated antennas) or geographically separated jammers (distributed, cooperative or non-cooperative devices) with different number of antennas.
[0056] The base station antennas 611 are attached to a set of RF chains 612. The analog-to-digital conversion of the baseband signal (in-phase and quadrature components) is achieved by a set of BWCs 100 as discussed before. The upper and lower thresholds of each BWC is set to 1.6σ0 and −1.6σ0, where we use c = 1.6. The in-phase and quadrature components of the baseband signal are quantized separately and independently by a pair of BWCs. ym(t)(m = 1, · · · ,M) denotes the received 20 CA 3110925 Date reçue / Received date 2025-03-06 bandpass signal impinged on m-th antenna 611 added to noise zm(t). yℜ m,l and yℑ m,l represent the in-phase and quadrature components of the baseband signal after RF chain, all associated with the m-th antenna 611 and at the l-th time index, and bℜ m,l and bℑ m,l are the corresponding (to in-phase and quadrature components) binary digits after quantization. All observed binary digits bℜ m,l, bℑ m,l (l = 1, 2, · · · ,N, m = 1, 2, · · · ,N) are operated on by a bit density detector 410 for detecting jamming. N is the number of observation intervals in samples or symbols. Although all antennas are equipped with BWCs which is followed here to simplify the exposition, however, it is unnecessary and hence a sufficient number of BWCs can be used to meet a certain level of detection performance.
[0057] We consider jamming detection during the uplink training phase which has a very deleterious impact on performance. The length of the pilot sequence sent by each UE in the system is τ . Without loss of generality, we consider a single-antenna jammer who sends the same pilot sequence of user k (pilot spoofing) to contaminate its channel estimate at the base station. It is worth noting that the present invention is also applicable for the case of a multiple-antenna jammer transmitting arbitrary pilot signal. We use hk, g ∼ CN(0, IM) to denote channel vectors (Rayleigh smallscale fading) corresponding to user k and the jammer, respectively. hk, g are columns of channel matrices H and G, respectively. We consider a flat Rayleigh block-fading channel model where all channel vectors stay constant over Nc symbols (coherence time of channel). In our model, H,G are assumed unknown, the large-scale coefficients and transmit power of UEs are also known, however, nothing known about the jammer. Alternatively, the base station has a means to know the variance of the received signal in the absence of jamming, which is feasible in practice.
[0058] Under hypotheses H0 (no jamming) and H1(with jamming) the (unquan- 21 CA 3110925 Date reçue / Received date 2025-03-06 tized) complex discrete-time signal vector yl = [yℜ 1,l + jyℑ 1,l, · · · , yℜ M,l + jyℑ M,l]T ∈ CM after the BWCs at the base station for time index l is given by H0 : yl = KX i=1 q βipihisi,l + wl H1 : yl = KX i=1 q βipihisi,l + q βJpJgsk,l + wl (19) where si,l is the pilot symbol sent from the i-th user at time index l, pi and q denote the average transmit powers of user i and jammer, βi and βJ are the large-scale coefficients (e.g., path loss and shadowing) of user i and jammer, respectively, and wl ∼ CN(0, IM) is white Gaussian which is assumed uncorrelated across time and space. Without loss of generality, we assume |si,l| = 1, i.e., constant modulus pilot symbols.
[0059] In the following analysis, we study a case where jamming detection is performed using only (one) pilot signal received per coherence time and N coherence blocks (independent realizations of channels), i.e., l = 0,Nc, 2Nc, · · · , (N−1)Nc. This means that the base station collects a total of 2MN binary digits over the observation interval. Jamming detection using all symbols during one coherence time is possible as long as K = τ , which is necessary for the received samples presented to the set of BWCs being mutually independent under H0 which is compliance with the present invention.
[0060] According to our channel model, conditioned on each hypothesis, the unquantized signal yl is complex Gaussian with IID entries, i.e., H0 : yl ∼ CN 0, σ2 0IM , H1 : yl ∼ CN 0, σ2 1IM l = 0,Nc, 2Nc, · · · , (N − 1)Nc (20) where N ≥ 1 is the number of channel blocks (number of collected vector-valued 22 CA 3110925 Date reçue / Received date 2025-03-06 samples) and σ2 0, σ2 1 are given by σ2 0 = KX i=1 piβi + σ2w , σ2 1 = KX i=1 piβi + pJβJ + σ2w . (21)
[0061] Define Y = [y0, yNc , · · · , y(N−1)Nc ] ∈ CM×N and d = Vec ([ℜ{Y}, ℑ{Y}]) ∈ R2MN where Vec(·) denotes the vectorization operator, i.e., we stack all real and imaginary components of Y in a column vector. All entries of d are IID N(0, σ2 0 / 2) under H0 and IID N(0, σ2 1 / 2) under H1. Let Qw(d) ≜ b = [b1, b2, · · · , b2MN]T be length- 2MN binary sequence observed at the base station where Qw(·) is a component-wise operator. Therefore, the jamming detection can be formulated by the following binary hypotheses: H0 : b1, · · · , b2MN I∼ID Ber(2Q(c)) H1 : b1, · · · , b2MN I∼ID Ber(2Q(cσ0 / σ1)) (22) where σ0 and σ1 are defined in (21). From Eq. (22) jamming detection boils down to distinguishing between two IID Bernoulli processes where the model in the null hypothesis is known. The BDD 410 at the base station operates on the observed bits b1, b2, · · · , b2MN according to the method of detection described in the flowchart in FIG. 5 to detect the presence of jamming.
[0062] FIG. 10 shows the performance of jamming detection in a massive MIMO system with K = 5 users, τ = K (pilot length) and users use mutually orthogonal pilots. For simplicity, we assume βJ = β1 = · · · = βK = 1 and p1 = · · · = pK = 0dB. Only one channel block (Nc = 1) is considered, and all pilot symbols are exploited for detection. Conditioned on H0, the case K = τ renders all received (unquantized) pilot symbols are IID Gaussian with zero-mean and variance σ2 0 = Kp+1 and hence the corresponding quantized bits are IID Bernoulli random variables. For comparison, we also simulate the corresponding unquantized system utilizing X2 v test borrowed 23 CA 3110925 Date reçue / Received date 2025-03-06 Table 1 Simulation parameters of Fig. 10 for BDD (1-bit quantized system) and X2 2τM test (unquantized system) with PF = 5%. M γ ζ γunq.(×103) 32 44 0.026 1.088 128 159 0.599 4.093 256 307 0.702 8.037 from the literature, which is derived using the log-likelihood ratio while considering all channels unknown at the base station. The parameters of BDD and X2 v test are shown in TABLE 1 . It is clear that the simulation results of BDD which relies on our BWC match perfectly with the analytical ones. Interestingly, we observe that our detector can achieve a significant fraction of the performance achieved when the base station has direct access to unquantized samples. As seen for most cases of interest (say, a region below 0 dB), the gap between the 1-bit quantized and unquantized is about 10%, proving that BDD combined with our BWC is a very effective method for jamming detection.
[0063] In FIG. 11 we consider the same scenario as in FIG. 10 except the jammer uses NJ antennas and hence the channel G is M × NJ . We also assume that the jammer sends a random pilot signal from its antennas, which is not a strict condition. Nevertheless, this assumption has been proven to be a good strategy for an attacker to circumvent a situation where UEs send random pilots from a pool of pilot sequences using a random seed which is perfectly known at the base station, making it hard for a potential attacker to know the exact pilot of a specific UE. Referring to FIG. 11, as expected, PD improves gradually as NJ increases since the performance of our detector improves when change in variance between H0 and H1 increases. Note that this is possible because the larger the variance under H1, the larger the density of 1’s outside the 3.2σ0-width window of each BWC. 24 CA 3110925 Date reçue / Received date 2025-03-06
Claims
What is claimed:
1. A method for deciding which of two hypotheses, H0 and H1, is more likely to have produced a set of observed bits in a multiple-input, multiple-output (MIMO) receiver employing one-bit quantization, the method comprising: (a) receiving, via M receive antennas in the MIMO receiver during N symbol intervals, complex discrete-time signals ym,l, where M and N are positive integers, and where m = 1 to M denotes the input index and l = 1 to N denotes the time index; (b) mapping, respectively and independently, by a first binary window comparator and a second binary window comparator at each receive antenna, each in-phase component yℜ m,l and each quadrature component yℑ m,l of each complex discrete-time signal ym,l, into corresponding bits bℜ m,l and bℑ m,l, wherein the first and second binary window comparators each have an upper threshold vH and a lower threshold vL, with vH = cσ0 and vL = −cσ0, where c is a positive real constant and σ0 is a standard deviation of the in-phase and quadrature components under hypothesis H0, wherein the bits bℜ m,l and bℑ m,l, for m = 1 to M and l = 1 to N constitute the set of observed bits; (c) computing a test threshold γ using γ = Bin−1(1−PF , n, p0), where Bin−1 denotes the binomial inverse cumulative distribution function (CDF) which returns the smallest integer γ ∈ [0, n] such that the binomial CDF with parameters n and p0, when evaluated at γ, is greater than or equal to 1 − PF , wherein: (i) n is a size of the set of observed bits, the size being given by the relation n = 2MN, 25 CA 3110925 Date reçue / Received date 2025-03-06 (ii) Q(·) is the Q-function of a standard normal distribution, (iii) p0 = 2Q(c), where p0 ∈ [0, 1], is a success rate, representing the probability that each observed bit in the set of observed bits equals 1 under hypothesis H0, (iv) PF is a desired probability of false alarm, representing the probability of erroneously rejecting hypothesis H0, and computing a randomization parameter ζ within the interval of real numbers [0, 1] according to the relation: ζ = PF − Pnk =γ+1 n k pk0 1 − p0 n−k n γ pγ0 1 − p0 n−γ , wherein ζ guarantees that the desired probability of false alarm PF is exactly achieved, and wherein c is precomputed through numerical optimization to maximize a probability of detection PD, representing the probability of correctly accepting hypothesis H1, wherein PD is expressed as a function of c, n, PF and an unknown ratio α defined as α = σ0 / σ1, where σ1 ≥ σ0 such that α ∈ [0, 1], and σ1 defining a standard deviation of the in-phase and quadrature components under hypothesis H1, wherein the numerical optimization accounts for uncertainty in α; (d) deciding, using one of two approaches, which of the two hypotheses H0 and H1 is more likely to have produced the set of observed bits, wherein the decision is based on a first approach if the sum of bits in the set of observed bits is equal to the test threshold γ, but is based on a second approach if the sum of bits in the set of observed bits is not equal to the test threshold γ, 26 CA 3110925 Date reçue / Received date 2025-03-06 wherein the first approach randomly selects a random number X uniformly distributed between 0 and 1, wherein hypothesis H1 is accepted as true if the random number X is less than or equal to the randomization parameter ζ, or hypothesis H0 is accepted as true if the random number X is greater than the randomization parameter ζ, and wherein, in the second approach, hypothesis H1 is accepted as true if the sum of bits in the set of observed bits is greater than the test threshold γ, or hypothesis H0 is accepted as true if the sum of bits in the set of observed bits is less than the test threshold γ.
2. The method of claim 1, wherein the in-phase and quadrature components of the complex discrete-time signals are independent, identically distributed Gaussian random variables with a common variance σ2 0 under hypothesis H0 and a common variance σ2 1 under hypothesis H1, wherein σ1 ≥ σ0.
3. The method of claim 2, wherein the success rate p0 is fixed and independent of the standard deviation σ0 of the in-phase and quadrature components under hypothesis H0, due to the thresholds vH = cσ0 and vL = −cσ0 that scale proportionally with σ0.
4. The method of claim 2, wherein the probability of detection PD is given by PD = Xn k=γ+1 n k ! (2Q(αc))k (1 − 2Q(αc))n−k + PF − Pnk =γ+1 n k (2Q(c))k (1 − 2Q(c))n−k Q(c) Q(αc) γ 1−2Q(c) 1−2Q(αc) n−γ .
5. The method of claim 4, wherein the numerical optimization of c to maximize PD comprises evaluating PD over a plurality of larger α-values such that σ2 1 is close to σ2 0, corresponding to small variance changes between hypotheses H0 27 CA 3110925 Date reçue / Received date 2025-03-06 and H1, wherein the smaller the variance change, the more difficult it becomes to decide between H0 and H1.
6. The method of claim 2, wherein the common variance σ2 0 is known a priori at the MIMO receiver and the common variance σ2 1 is unknown at the MIMO receiver. 28 CA 3110925 Date reçue / Received date 2025-03-06