Iterative algorithm for designing asymptotically optimal uniform scalar quantisation of the one‐sided Rayleigh density

IET CommunicationsVolume 15, Issue 5 p. 723-729 ORIGINAL RESEARCH PAPEROpen Access Iterative algorithm for designing asymptotically optimal uniform scalar quantisation of the one-sided Rayleigh density Aleksandra Jovanović, Corresponding Author Aleksandra Jovanović aleksandra.jovanovic@elfak.ni.ac.rs Faculty of Electronic Engineering, University of Niš, Niš, Serbia Correspondence Aleksandra Jovanović Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia. Email: aleksandra.jovanovic@elfak.ni.ac.rsSearch for more papers by this authorZoran Perić, Zoran Perić Faculty of Electronic Engineering, University of Niš, Niš, SerbiaSearch for more papers by this authorJelena Nikolić, Jelena Nikolić Faculty of Electronic Engineering, University of Niš, Niš, SerbiaSearch for more papers by this author Aleksandra Jovanović, Corresponding Author Aleksandra Jovanović aleksandra.jovanovic@elfak.ni.ac.rs Faculty of Electronic Engineering, University of Niš, Niš, Serbia Correspondence Aleksandra Jovanović Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, Niš, Serbia. Email: aleksandra.jovanovic@elfak.ni.ac.rsSearch for more papers by this authorZoran Perić, Zoran Perić Faculty of Electronic Engineering, University of Niš, Niš, SerbiaSearch for more papers by this authorJelena Nikolić, Jelena Nikolić Faculty of Electronic Engineering, University of Niš, Niš, SerbiaSearch for more papers by this author First published: 01 February 2021 https://doi.org/10.1049/cmu2.12114Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Design of optimal and asymptotically optimal quantisation subject to the mean squared error (MSE) criterion is a complex issue, even in the case of uniform scalar quantisation (USQ). The reason is that the MSE distortion dependence on the key designing parameter of USQ for source densities with infinite supports are complex and limit analytical optimisation of USQs. This issue of USQ design has been addressed for some source densities derived from the generalised gamma density. However, to the best of our knowledge, USQ for the one-sided Rayleigh density has not been studied in detail. This has prompted our research so that this study provides a detailed analysis of USQ for the one-sided Rayleigh density and proposes an iterative algorithm for its asymptotically optimal design. To estimate signal to quantisation noise ratio, we derive an asymptotic formula having reasonable accuracy for rates higher than 3 bits/sample. Our analysis can be useful in digital-to-analogue and analogue-to-digital conversion in diversity systems, orthogonal frequency division multiplexing systems and medical image processing. 1 INTRODUCTION Uniform scalar quantisation (USQ) is the earliest, the simplest and the most researched type of quantisation commonly studied for the source densities derived from the generalised gamma density (Gaussian, Laplacian, the two-sided Rayleigh density) [1-6]. Detailed analyses were conducted in the field of USQ for some of these densities. However, to the best of our knowledge, USQ for the one-sided Rayleigh density has not been thoroughly studied although this density is widely used to model: fading in diversity systems [7, 8], the amplitude of orthogonal frequency division multiplexing (OFDM) signals [9-13] and the noise variance in magnetic resonance imaging [14-16]. This implies that studying USQ for the one-sided Rayleigh density with the goal to maximise its performance would be important for digital-to-analogue and analogue-to-digital conversion (DAC/ADC) in diversity systems, OFDM systems and medical image processing. Also, as shown in [9-13], due to Rayleigh distribution of OFDM signals, the peak power can be much larger than the average power resulting in the high value of peak-to-average power ratio (PAPR) that can adversely affect the OFDM system. The value of PAPR can be reduced by using clipping and filtering that are highly correlated with DAC/ADC performance, particularly with the signal-to-quantisation-noise ratio (SQNR) [9, 10]. Therefore, it is important to specify USQ for the one-sided Rayleigh density that provides the lowest mean squared error (MSE) distortion, or equivalently, the highest SQNR. USQ is the most straightforward quantisation model that for a given number of quantisation levels N is characterised with only one parameter–support limit (xmax). Despite its simplicity, finding the xmax that achieves the lowest MSE distortion is not simple for source densities with infinite support [2, 3]. Namely, expressions for MSE distortion dependence on xmax for USQ of source densities with infinite support are complex, so minimising distortion over xmax does not generally yield an analytical expression for the optimal xmax [2, 3]. Besides, as N increases, the expression for distortion contains more terms, and determining the optimal xmax becomes more complex. Because of that, previous research has usually focused on finding suitable approximations for the optimal xmax. The most commonly used simple approximation for the optimal xmax is xmax ≈ 4σx [1], where σx is the standard deviation of the source density with infinite support. This approximation holds for a very limited range of N values since xmax has to increase with N [17]. For that reason, it is common to perform asymptotic analysis to provide a suitable approximation for the optimal xmax [2, 3]. In asymptotic analysis, the expression for MSE distortion in the function of xmax is replaced with an approximate expression (asymptotic MSE distortion) whose difference with respect to the exact MSE distortion asymptotically approaches zero as N increases. Although asymptotic analysis simplifies the expression for the MSE distortion, minimising the asymptotic distortion over xmax still remains complex [2, 3]. Therefore, the derivation of analytical expressions for the optimal and asymptotically optimal xmax for various quantisation models is a significant issue considered in numerous studies [2, 3, 17-25]. This issue has greater importance in uniform quantisation than in non-uniform quantisation [2, 3]. The constrained degree of freedom in uniform step size determination causes the values of the optimal and asymptotically optimal xmax to be smaller in uniform quantisation than in non-uniform quantisation. Consequently, in uniform quantisation, the overload distortion (the MSE distortion for amplitudes having an absolute value greater than xmax) has a greater influence on the total distortion and can significantly increase distortion, so determining the optimal and asymptotically optimal xmax is of greater importance in uniform quantisation than in non-uniform quantisation. To determine the xmax of USQ for the one-sided Rayleigh density, we perform asymptotic analysis to express MSE distortion as a function of xmax. We also develop a novel iterative algorithm for solving the equation obtained by equating the first derivative of the MSE distortion with respect to xmax to zero. The iterative algorithm we develop provides the asymptotically optimal xmax, yielding the asymptotically optimal USQ for the one-sided Rayleigh density. We also derive a formula for the SQNR of the asymptotically optimal USQ and examine the accuracy of this formula in order to show its usability. 2 ITERATIVE ALGORITHM FOR USQ DESIGNING In USQ, the cells are of the same length and the output levels are the midpoints of the cells, except for possibly the first and the last cells [1]. When N-level USQ is applied to the source having a one-sided density p(x) with infinite support, which is the case in this study, the cells are defined as [xi, xi+1), i = 1,2,…,N, where xi = (i–1) × Δ, i = 1, 2,…, N and xN+1 = + ∞ are the decision thresholds, while the output levels are yi = (i–1/2) × Δ, i = 1,2,…,N (see Figure 1). The cell width or uniform step size Δ is specified with Δ = xmax/N, where xmax is the support limit that divides the range of input USQ amplitudes into the granular region [0, xmax), inside which the absolute quantisation error is not greater than Δ/2, and the overload region [xmax,+ ∞), inside which the absolute quantisation error is greater than Δ/2 and can be unbounded. For a given N, USQ is completely specified with xmax. Hence, it is important to provide an algorithm for determining xmax such that the MSE distortion of the quantised signal is minimised. After relating the MSE distortion and the support limit xmax, we apply some asymptotic approximations and eventually perform the optimisation procedure. FIGURE 1Open in figure viewerPowerPoint An illustration of uniform scalar quantisation for the number of levels N = 8 and the amplitude x having the one-sided density p(x) with infinite support The MSE distortion of N-level USQ for the one-sided source density with infinite support, which is a sum of granular and overload distortions, can be expressed as a function of xmax as follows [1-3]: D N ( x max ) = D N gran ( x max ) + D N over ( x max ) D N gran ( x max ) = ∑ i = 1 N − 1 ∫ x i x i + 1 ( x − y i ) 2 p ( x ) d x + ∫ x N x max ( x − y N ) 2 p ( x ) d x D N over ( x max ) = ∫ x max + ∞ ( x − y N ) 2 p ( x ) d x . (1) The output levels and decision thresholds are functions of Δ and, hence, of xmax. So, xmax appears in Equation (1) not only in the limits of integrals but also in integrands. That is why, despite the simplicity of the USQ model, it is difficult, even impossible, to analytically determine the xmax that minimises overall distortion. Therefore, if USQ is applied to source densities with infinite support, the determination of the optimal value for xmax is indeed an open issue. This issue has been commonly addressed and solved somewhat using various mathematical approximations. However, to the best of our knowledge, USQ for the one-sided Rayleigh density has not been studied in detail. We simplify the MSE distortion expression (1) by applying asymptotic approximations; then, instead of designing USQ by minimising Equation (1), we design USQ by minimising asymptotic approximation of Equation (1). This optimisation procedure results in asymptotically optimal USQ and xmax, here denoted as xmaxopt. The asymptotic analysis assumes large N, that is, small width of the granular cells, which allows p(x) to be considered as constant within the granular cell, p(x) ≈ p(yi), x ∈ [xi, xi+1) and also yN ≈ xmax. Accordingly, as in [2, 3], we derive: D N gran ( x max ) ≈ D ∼ N gran ( x max ) = x max 2 12 N 2 D N over ( x max ) ≈ D ∼ N over ( x max ) = ∫ x max + ∞ ( x − x max ) 2 p ( x ) d x , (2)that is, for the total distortion we have: D N ( x max ) ≈ D ∼ N ( x max ) = x max 2 12 N 2 + ∫ x max + ∞ ( x − x max ) 2 p ( x ) d x . (3) Equation (3) shows that N-level USQ defined for a given xmax results in different distortion equations for different densities p(x). This means that USQ asymptotically optimal for one density cannot be asymptotically optimal for another one. We are interested in determining xmaxopt for the case where p(x) is the one-sided Rayleigh density [1]: p ( x ) = x σ 2 exp − x 2 2 σ 2 , 0 ≤ x < + ∞ . (4) By substituting Equations (4) in (3), we derive the following asymptotic formula for the distortion: D ∼ N ( x max ) = x max 2 12 N 2 + 2 σ 4 x max 2 exp − x max 2 2 σ 2 , (5)where the two-term approximation of the following asymptotic series representation is used [19]: ∫ a + ∞ exp − x 2 2 d x = exp − a 2 2 a 1 − 1 a 2 + 3 a 4 − + ⋯ . (6) Before tackling minimisation [2, 3, 20-22], we simplify Equation (5) by substituting t = x max 2 2 σ 2 , (7)replacing distortion minimisation over xmax with distortion minimisation over t, which results in a much simpler optimisation procedure. By substituting Equations (7) in (5), the asymptotic expression for the total distortion becomes: D ∼ N ( t ) = σ 2 t 6 N 2 + σ 2 t exp − t . (8) To determine t, denoted with topt, which provides the minimum of Equation (8), we equate the first derivative of Equation (8) to zero with respect to t: 1 6 N 2 − 1 t opt 2 + 1 t opt exp − t opt = 0 . (9) The above equation can be easily reformulated in an iterative equation for determining the asymptotically optimal t: t i = ln ( 6 ) + 2 ln ( N ) + ln 1 t i − 1 + 1 t i − 1 2 . (10) By recalling that t is related with xmax, as specified in Equation (7), and by assuming that the asymptotic approximation xmax4 >> 1 holds, we further modify the iterative equation into x max i σ = 2 ln ( 2 3 ) + ln ( N ) − ln x max i − 1 σ . (11) Instead of iteratively determining xmaxopt for each value of σ, it turns out, from Equation (11), that it is more convenient to determine xmaxopt normalised by σ, where xmax/σ is replaced with η: η i = 2 ln ( 2 3 ) + ln ( N ) − ln η i − 1 . (12) After determining ηopt it should be multiplied by σ in order to determine the asymptotically optimal value for xmax: x max opt = σ η opt . (13) Note that for σ = 1, ηopt is equal to xmaxopt, which means that determining ηopt is equivalent to designing asymptotically optimal USQ for the one-sided Rayleigh density having σ = 1. We initialise our algorithm by neglecting η(i–1) in Equation (12): η 0 = 2 ln ( 2 3 N ) . (14) We stop iterating when the reduction in distortion becomes negligible, that is, if it becomes smaller than some predefined relative error ε: δ D i = D ∼ N i − 1 − D ∼ N i D ∼ N i ≤ ε . (15) D ∼ N ( i ) denotes the distortion of USQ obtained after ith iteration and, hence, it has the following form obtained by replacing xmax/σ with η(i) in Equation (5): D ∼ N i = σ 2 η i 2 12 N 2 + 2 σ 2 η i 2 exp − η i 2 2 . (16) Pseudocode given below summarises the proposed iterative algorithm for asymptotically optimal USQ design. Eventually, we derive the asymptotic formula for SQNR of asymptotically optimal USQ. In doing so, we recall the following definition of SQNR [1]: SQNR = 10 log 10 σ x 2 D dB , (17)where σx2 is the variance of the input signal amplitudes. For the amplitudes with the one-sided Rayleigh density in Equation (4) having a mean value x ¯ = ∫ 0 + ∞ x p x d x = π 2 σ , (18)the variance is Opt_USQ_Rayleigh ( N, σ, ε) −-The design of asymptotically optimal USQ for the one-sided Rayleigh density −xmaxopt determining Require: Total number of levels N, density parameter σ, distortion accuracy ε < < 1 1: i : = 0 2: compute η(i) and D ∼ N ( i ) using Equations (14) and (16) 3: δD(i) : = 1 4: while δD(i) > ε do 5: i : = i+1 6: η (i-1) : = η (i) and D ∼ N ( i − 1 ) : = D ∼ N ( i ) 7: compute η (i) using Equation (12) 8: compute D ∼ N ( i ) and δD(i) using Equations (16) and (15) 9: end while 10: xmaxopt : = σ × η (i) and D ∼ N min : = D ∼ N ( i ) 11: return xmaxopt, D ∼ N min σ x 2 = ∫ 0 + ∞ x − x ¯ 2 p x d x = 2 − π 2 σ 2 . (19) By replacing σx2 with Equations (19) in (17) and D with the expression for the asymptotic distortion minimum D ∼ N min , obtained by substituting η(i) with ηopt in Equations (16), we derive the asymptotic formula for SQNR of asymptotically optimal USQ designed for the one-sided Rayleigh density: SQNR max AF = − 3.67 − 10 log 10 η opt 2 12 N 2 + 2 η opt 2 exp − η opt 2 2 (20) Equation (20) points that maximum SQNR does not depend on the Rayleigh density parameter σ. That is, for a given N and a different σ, the appropriate maximums of asymptotic SQNR have equal values. For the asymptotically optimal USQ we observe here, it is important to examine the accuracy of the derived asymptotic formula for SQNR. Accordingly, we calculate SQNR accurately by substituting the exact expression for the MSE distortion, given by Equation (1), into the basic definition of SQNR in Equation (17): SQN R max = − 3.67 + 20 log 10 σ − 10 log 10 ∑ i = 1 N ∫ x i + 1 x i x − y i 2 p x dx , (21)where xi and yi are defined by x i = i − 1 x max opt N , i = 1 , 2 , … , N , x N + 1 = + ∞ y i = i − 1 2 x max opt N , i = 1 , 2 , … , N . (22) More precisely, similarly as in [19, 20, 22, 24, 25], we estimate the relative error in SQNR calculation as δ SQNR = SQNR max AF − SQN R max SQN R max × 100 % , (23)where SQNRmaxAF and SQNRmax are calculated using Equations (20) and (21), respectively. 3 RESULT ANALYSIS In Table 1, we present the performance of asymptotically optimal USQ designed for the one-sided Rayleigh density by utilising our novel iterative algorithm. For the common value of distortion accuracy ε = 5 × 10–3 [1], for different values of σ and for different values of resolution R given by R = log 2 N bits sample , (24)we tabulate values for the asymptotically optimal xmax and the asymptotically optimal SQNR. In particular, we determine the asymptotically optimal SQNR by using the asymptotic formula we propose in Equation (20) and by using the exact formula in Equation (21). Since the normalised support limit ηopt depends only on N (see Equation 12), the asymptotic formula for maximum SQNR in Equation (20) does not depend on the input signal parameter σ. In other words, it is not necessary to know the value of σ in order to calculate the asymptotic formula for SQNR. For a given R (i.e. for a given N) and for a different σ, the asymptotic SQNR maxima are equal. On the other hand, the parameter σ has to be included during the exact SQNR formula calculation (see Equations 21, 22 and 13). Nevertheless, the SQNR results obtained for a given R and for a different σ are identical (see the eighth column in Table 1), confirming that the SQNR maximum does not depend on σ. The results obtained for SQNR, that is, SQNRmaxAF and SQNRmax, are very well matched. Furthermore, based on Equation (23), we estimate the accuracy of the asymptotic formula for SQNR and show that the proposed asymptotic formula has reasonable accuracy (smaller than 1%) for rates higher than 3 bits/sample. Besides this, if we take into account that the calculation of asymptotic formula in Equation (20) is far simpler than the calculation of exact formula in Equation (21), where summation should be repeated N times, we conclude that the asymptotic SQNR formula for straightforward performance assessment of asymptotically optimal USQ is beneficial. TABLE 1. Performances of asymptotically optimal uniform scalar quantisation (USQ) designed for one-sided Rayleigh density R σ η(0) m ηopt xmaxopt SQNRmaxAF SQNRmax δSQNR 3 1 3.645 3 2.978 2.978 14.80 14.51 1.96 0.5 1.489 14.51 1.96 5 14.892 14.51 1.96 4 1 4.008 3 3.347 3.347 19.99 19.88 0.55 0.5 1.673 19.88 0.55 5 16.734 19.88 0.55 5 1 4.340 3 3.687 3.687 25.29 25.26 0.10 0.5 1.844 25.26 0.10 5 18.437 25.26 0.10 6 1 4.648 3 4.006 4.006 30.68 30.69 0.04 0.5 2.003 30.69 0.04 5 20.028 30.69 0.04 7 1 4.937 3 4.305 4.305 36.14 36.16 0.07 0.5 2.152 36.16 0.07 5 21.524 36.16 0.07 8 1 5.211 3 4.588 4.588 41.66 41.68 0.07 0.5 2.294 41.68 0.07 5 22.941 41.68 0.07 9 1 5.470 2 4.862 4.862 47.22 47.25 0.06 0.5 2.431 47.25 0.06 5 24.311 47.25 0.06 10 1 5.718 2 5.119 5.119 52.83 52.85 0.05 0.5 2.559 52.85 0.05 5 25.594 52.85 0.05 Note: SQNR is signal-to-quantisation-noise ratio. Table 1 shows that our iterative algorithm quickly converges to the asymptotically optimal solution. The number of necessary iterations m amounts to only 3 for low and moderate rates and 2 for higher rates. The convergence speed increases with R, as expected, since the accuracy of the performed asymptotic analysis increases with R. We have already mentioned that optimal or asymptotically optimal USQ of the one-sided Rayleigh density has not been studied so far. Therefore, the fair comparison of our USQ model with other USQ models is quite limited. However, in order to perceive our contribution, we compare the considered asymptotically optimal USQ with globally optimal nonuniform SQ (Lloyd–Max scalar quantisation (LM-SQ)) from [14] for the one-sided Rayleigh density of unit variance (σx2 = (2–π/2) × σ2 = 1; see Table 2). Due to global optimality and non-linearity of the model, LM-SQ provides a higher SQNR than any other SQ model including our USQ. However, LM-SQ is designed by utilising a complex two-stage iterative LM algorithm whose complexity rapidly grows with R since an iteration consists of 2R+1+1 parameter updates. This leads to cumbersome execution of the LM algorithm for high R. On the other hand, USQ is the simplest and most frequently used quantisation model. Besides, although iterative, the algorithm we propose for USQ designing is a simple one since only one parameter is updated within the iteration. Also, one can observe from Table 1 that the number of necessary iterations decreases with the increase of R. All these features of our designing algorithm indicate its simplicity and efficiency. In comparison with LM-SQ, the USQ proposed in this study also provides the lower maximal values of quantised signal (see the underlined values in Table 2) and thus the lower value of peak power. This can be meaningful in PAPR reduction based on clipping and filtering in OFDM systems, where it is preferable to clip the smallest possible amplitude values that provide the desired SQNR. More precisely, PAPR reduction based on clipping and filtering affects (decreases) SQNR of DAC/ADC [9]. To avoid this negative effect, the proposed iterative algorithm can be helpful. Namely, the values for xmaxopt obtained by the iterative algorithm can be taken into account during the clipping threshold estimation. This way, PAPR can be reduced, whereby the performance of DAC/ADC can be within the asymptotically optimal values for USQ. Moreover, it was highlighted in [13] that the characteristics of the DAC/ADC, such as quantisation rate and support limit, are significant parameters influencing the transmission capacity and the distance of an optical OFDM modem. It is also important to identify optimum DAC/ADC parameters, as we do here, for USQ of the one-sided Rayleigh density. TABLE 2. Asymptotically optimal USQ versus Lloyd–Max scalar quantisation, both designed for the one-sided Rayleigh density of unit variance (σx2 = 1, σ = (2–π/2)–1/2) USQ LM-SQ R xi yi SQNRmax xi yi SQNRmax 3 0.000 0.284 14.51 0.000 0.502 15.32 0.568 0.853 0.762 1.022 1.137 1.421 1.260 1.497 1.705 1.989 1.733 1.969 2.273 2.557 2.218 2.467 2.841 3.125 2.748 3.028 3.410 3.694 3.371 3.714 3.978 4.262 4.212 4.711 +∞ +∞ 4 0.000 0.160 19.88 0.000 0.306 21.05 0.319 0.479 0.461 0.616 0.639 0.799 0.751 0.886 0.958 1.118 1.013 1.140 1.277 1.437 1.262 1.385 1.596 1.756 1.506 1.630 1.916 2.075 1.750 1.872 2.235 2.395 1.997 2.122 2.554 2.714 2.252 2.381 2.873 3.033 2.518 2.655 3.193 3.352 2.802 2.949 3.512 3.672 3.111 3.273 3.831 3.991 3.457 3.640 4.151 4.310 3.859 4.077 4.470 4.630 4.358 4.638 4.789 4.949 5.065 5.491 +∞ +∞ We also verify the validity of our theory by performing a MATLAB simulation. For a given R and for a corresponding xmaxopt from Table 1, we generate M = 107 Rayleigh variables, xi, i = 1, 2,…, M. After that, we realise USQ as follows: x ̂ i = x i N x max opt − 1 2 x max opt N , i = 1 , 2 , … , M , (25)where ⋅ denotes rounding to the nearest integer greater than or equal to the function argument. Since the variance of the input signal amplitudes is the mean value of all squared deviations of input amplitudes from their mean value and since the MSE distortion represents the mean value of all squared errors caused by quantisation, we determine the SQNR based on the following formula: SQN R ∗ = 10 log ∑ i = 1 M x i − 1 M ∑ j = 1 M x j 2 ∑ i = 1 M x i − x ̂ i 2 . (26) Figure 2, presents the results for SQNRs obtained analytically and by simulation, which matches well. This verifies the accuracy of our asymptotic analysis. FIGURE 2Open in figure viewerPowerPoint Signal-to-quantisation-noise ratioversusR 4 CONCLUSION In this study, we solved the important issue in designing asymptotically optimal USQ for the one-sided Rayleigh density. Specifically, we proposed a simple and effective iterative algorithm for determining asymptotically optimal support limit. We also derived an asymptotic SQNR formula having a reasonable accuracy for rates higher than 3 bits/sample, which is useful in performance assessment of asymptotically optimal USQ. Results showed that for a given R, the asymptotically optimal support limit is linearly proportional to the variance of the one-sided Rayleigh density for which the asymptotically optimal USQ is designed and that the asymptotically optimal value of SQNR does not depend on the designed variance. This analysis can be extended to variance mismatch quantisation of the one-sided Rayleigh probability density where the designed-for and applied-to sources have different variances. 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