In this paper, we propose a novel approximate multiplier with error-improved logarithmic and antilogarithmic conversion. In logarithmic multiplication, the forward and backward conversions to the logarithm domain introduce errors in the product. To recover accuracy, we apply an offset when logarithm and antilogarithm are computed, and find suitable values for these offsets in order to make the approximation error with zero mean. The hardware structure of the proposed multiplier is described for signed arithmetic, and a novel hardware-efficient approach, able to compute the sign of the output, is also proposed. Error metrics and synthesis analyses in a 28nm CMOS technology show that our multiplier offers the best accuracy-hardware performance for MRED-2. Furthermore, remarkable results are achieved also in JPEG compression, exhibiting SSIM and PSNR metrics competitive with the state-of-the-art.

In this paper we propose a novel approximate floating-point divider based on bidimensional linear approximation. In our approach, the mantissa quotient is seen as a function of the two input mantissas of the divider. The domain of this two-variable function is partitioned into$nx \times ny$subregions, named tiles, where$nx, ny$are chosen as powers of two. In each tile the quotient is approximated with a linear combination of the input mantissas. To achieve fine accuracy, an optimization problem is formulated within each tile to determine the optimal coefficients for the linear combination, which minimize the Mean Relative Error Distance (MRED) of the divider. Furthermore, to make hardware implementation more effective, the minimization problem is appropriately modified to search for optimal quantized coefficients. The hardware structure of the divider only requires a small look-up table to store the linear approximation coefficients, and a carry save adder tree. The proposed architecture is highly tunable at design-time over a wide range of accuracy, depending on the number of tiles chosen for the approximation. The obtained results demonstrate error performance and hardware features superior to the state-of-the-art. The proposed dividers define the Pareto front, considering the trade-off between power-delay-product vs. MRED and area-delay-product vs. MRED, for MRED in the range of$4\times 10^{-3}-2\times 10^{-2}$. Application results for JPEG compression and tone mapping further highlight the strength of our proposal, which exhibits Structural Similarity Index (SSIM) very close to 1 in all cases and Peak Signal-to-Noise Ratio (PSNR) up to 45 dB.

Approximate arithmetic circuits sacrifice computing accuracy in exchange for improvements in power, area, and speed. Many approximate binary multipliers that use 4-2 compressors have been proposed but most of the proposals have error performances that depend on the order in which the inputs are connected to the compressors. This complicates the design and prevents a fair comparison among different approximate multipliers. The paper proposes the input order invariant approximate 4-2 compressors, whose behavior remains consistent regardless of the order of input signals. We derive the complete set of such compressors and utilize them to synthesize 8-bit multipliers. Our analysis reveals that only a limited subset of input order invariant approximate 4-2 compressors offers an optimal balance between error and power savings. Furthermore, we demonstrate that this optimal set of 4-2 compressors can be strategically distributed within the columns of the multiplier to further enhance the trade-off between error and power efficiency. The proposed circuits, once implemented in a 14 nm FinFET standard cell technology, favorably compare against the state of the art.

Floating-point division involves the computation of the ratio (1$+$Mx)/(1$+$My), whereMxandMyrepresents the mantissas of the input values. In this paper, we propose a new method for approximating this operation using a linear function ofMx, with coefficients that depend onMy. The coefficients are calculated to minimize the Mean Relative Error Distance (MRED) of the approximation. To this end, the range of My is partitioned in N sub-intervals where the minimization ofMREDis formulated as a linear programming problem, whose solution gives optimal coefficient values. The hardware implementation requires a small lookup table, two multipliers and an adder. An aggressive coefficients quantization is exploited to further optimize the design. ObtainedMREDimproves by increasing$N$, ranging from 1.4% to 0.33%. Implementation results in a 28nm CMOS technology show that the proposed design outperforms the state-of-the-art, offering the best trade-off between hardware complexity and accuracy. Results for two image processing applications, change detection and JPEG compression, demonstrate remarkable performance, with SSIM very close to 1 and PSNR values exceeding 50dB.

In this paper, a novel implementation is proposed for the Delayed LMS (DLMS) filter, able to reduce the power dissipation while preserving regime performances. The approach relies on the observation that the error signal is small in magnitude and oscillates around zero when the circuit is close to the convergence point. Therefore, the most significant bits of the error signal continuously toggle from positive to negative values causing high switching activity in the multipliers of the feedback section. This paper proposes to employ a sign-modulus representation of the error signal, to substantially reduce the switching activity of the feedback path of the filter. Additional approximation techniques are also devised to further reduce power dissipation. Comparisons with the state-of-the-art show that the proposed filter is the only one able to approach the MSE of the exact implementation with a remarkable reduction of power dissipation. A test-chip in TSMC 28nm CMOS technology has been realized to experimentally verify the validity of our technique. The experimental results show the possibility of saving up to 45.4% of power consumption with respect to the exact implementation of the filter.

Approximate multipliers are used in error-tolerant applications, sacrificing the accuracy of results to minimize power or delay. In this paper we investigate approximate multipliers using static segmentation. In these circuits a set of$m$contiguous bits (a segment of$m$bits) is extracted from each of the two$n$-bits operand, the two segments are in input to a small$m\times m$internal multiplier whose output is suitably shifted to obtain the result. We investigate both signed and unsigned multipliers, and for the latter we propose a new segmentation approach. We also present simple and effective correction techniques that can significantly reduce the approximation error with reduced hardware costs. We perform a detailed comparison with previously proposed approximate multipliers, considering a hardware implementation in 28 nm technology. The comparison shows that static segmented multipliers with the proposed correction technique have the desirable characteristic of being on (or close to) the Pareto-optimal frontier for both power vs normalized mean error distance and power vs mean relative error distance trade-off plots. These multipliers, therefore, are promising candidates for applications where their error performance is acceptable. This is confirmed by the results obtained for image processing and image classification applications.

In this paper we propose an energy-efficient approximate multiplier which uses a new approximate 4-2 compressor. The proposed compressor has a low error probability and its error conditions can be easily detected. This, as previously shown in the literature, makes it possible to implement error recovery, when the compressor is used in the partial product reduction phase of a multiplier. Simulation results show that proposed approximate multipliers exhibit a sensible reduction in Mean Error Distance and in maximum Error Distance, compared to previous art. Application to an image processing task shows an improvement of about 8dB in peak signal-to-noise ratio. Implementation results in 28nm CMOS show that the electrical performance of multipliers designed with the novel circuit are close to the one obtained with previously proposed approximate compressors.

Approximate computing relaxes algorithm precision constraints to improve digital circuit performance. Adaptive filters based on least-mean-square (LMS) algorithm constitute a standard in many DSP applications. The LMS algorithm, being an approximation of the Wiener filter, is inherently imprecise, and constitutes a fertile ground to employ approximate hardware techniques with the additional challenge related to the presence of a feedback path for coefficients update. In this paper, approximate LMS adaptive filters are explored for the first time, by employing approximate multipliers. A system identification scenario is adopted to assess the algorithm behavior. The analysis reveals that the choice of the approximate multiplier topology should be carefully examined, otherwise the stability and convergence performance of the algorithm can be compromised. We propose a novel approximate multiplier able to reduce the power dissipation in adaptive LMS filters up to 29% with tolerable convergence error degradation.