Multipliers for Floating-Point Double Precision and Beyond on FPGAs

International audienceThe implementation of high-precision floating-point applications on reconfigurable hardware requires a variety of large multipliers: Standard multipliers are the core of floating-point multipliers; Truncated multipliers, trading resources for a well-controlled accuracy degradation, are useful building blocks in situations where a full multiplier is not needed. This work studies the automated generation of such multipliers using the embedded multipliers and adders present in DSP blocks of current FPGAs. The optimization of such multipliers is expressed as a tiling problem where a tile represents a hardware multiplier and super-tiles are the wiring of several hardware multipliers making efficient use of the DSP internal resources. This tiling technique is shown to adapt to full or truncated multipliers. It addresses arbitrary precisions including single, double but also in the quadruple precision introduced by the IEEE-754-2008 standard and currently unsupported by processor hardware. An open-source implementation is provided in the FloPoCo project

High-Performance Accurate and Approximate Multipliers for FPGA-Based Hardware Accelerators

Multiplication is one of the widely used arithmetic operations in a variety of applications, such as image/video processing and machine learning. FPGA vendors provide high-performance multipliers in the form of DSP blocks. These multipliers are not only limited in number and have fixed locations on FPGAs but can also create additional routing delays and may prove inefficient for smaller bit-width multiplications. Therefore, FPGA vendors additionally provide optimized soft IP cores for multiplication. However, in this work, we advocate that these soft multiplier IP cores for FPGAs still need better designs to provide high-performance and resource efficiency. Toward this, we present generic area-optimized, low-latency accurate, and approximate softcore multiplier architectures, which exploit the underlying architectural features of FPGAs, i.e., lookup table (LUT) structures and fast-carry chains to reduce the overall critical path delay (CPD) and resource utilization of multipliers. Compared to Xilinx multiplier LogiCORE IP, our proposed unsigned and signed accurate architecture provides up to 25% and 53% reduction in LUT utilization, respectively, for different sizes of multipliers. Moreover, with our unsigned approximate multiplier architectures, a reduction of up to 51% in the CPD can be achieved with an insignificant loss in output accuracy when compared with the LogiCORE IP. For illustration, we have deployed the proposed multiplier architecture in accelerators used in image and video applications, and evaluated them for area and performance gains. Our library of accurate and approximate multipliers is opensource and available online at https://cfaed.tu-dresden.de/pd-downloads to fuel further research and development in this area, facilitate reproducible research, and thereby enabling a new research direction for the FPGA community

Comparison of reconfigurable structures for flexible word-length multiplication

Binary multiplication continues to be one of the essential arithmetic operations in digital circuits. Even though field-programmable gate arrays (FPGAs) are becoming more and more powerful these days, the vendors cannot avoid implementing multiplications with high word-lengths using embedded blocks instead of configurable logic. But on the other hand, the circuit's efficiency decreases if the provided word-length of the hard-wired multipliers exceeds the precision requirements of the algorithm mapped into the FPGA. Thus it is beneficial to use multiplier blocks with configurable word-length, optimized for area, speed and power dissipation, e.g. regarding digital signal processing (DSP) applications. <br><br> In this contribution, we present different approaches and structures for the realization of a multiplication with variable precision and perform an objective comparison. This includes one approach based on a modified Baugh and Wooley algorithm and three structures using Booth's arithmetic operand recoding with different array structures. All modules have the option to compute signed two's complement fix-point numbers either as an individual computing unit or interconnected to a superior array. Therefore, a high throughput at low precision through parallelism, or a high precision through concatenation can be achieved

Karatsuba with Rectangular Multipliers for FPGAs

Best paper awardInternational audienceThis work presents an extension of Karatsuba's method to efficiently use rectangular multipliers as a base for larger multipliers. The rectangular multipliers that motivate this work are the embedded 18×25-bit signed multipliers found in the DSP blocks of recent Xilinx FPGAs: The traditional Karatsuba approach must under-use them as square 18×18 ones. This work shows that rectangular multipliers can be efficiently exploited in a modified Karatsuba method if their input word sizes have a large greatest common divider. In the Xilinx FPGA case, this can be obtained by using the embedded multipliers as 16×24 unsigned and as 17×25 signed ones. The obtained architectures are implemented with due detail to architectural features such as the pre-adders and post-adders available in Xilinx DSP blocks. They are synthesized and compared with traditional Karatsuba, but also with (non-Karatsuba) state-of-the-art tiling techniques that make use of the full rectangular multipliers. The proposed technique improves resource consumption and performance for multipliers of numbers larger than 64 bits

Implementing Homomorphic Encryption Based Secure Feedback Control for Physical Systems

This paper is about an encryption based approach to the secure implementation of feedback controllers for physical systems. Specifically, Paillier's homomorphic encryption is used to digitally implement a class of linear dynamic controllers, which includes the commonplace static gain and PID type feedback control laws as special cases. The developed implementation is amenable to Field Programmable Gate Array (FPGA) realization. Experimental results, including timing analysis and resource usage characteristics for different encryption key lengths, are presented for the realization of an inverted pendulum controller; as this is an unstable plant, the control is necessarily fast

Design of approximate overclocked datapath

Embedded applications can often demand stringent latency requirements. While high degrees of parallelism within custom FPGA-based accelerators may help to some extent, it may also be necessary to limit the precision used in the datapath to boost the operating frequency of the implementation. However, by reducing the precision, the engineer introduces quantisation error into the design. In this thesis, we describe an alternative circuit design methodology when considering trade-offs between accuracy, performance and silicon area. We compare two different approaches that could trade accuracy for performance. One is the traditional approach where the precision used in the datapath is limited to meet a target latency. The other is a proposed new approach which simply allows the datapath to operate without timing closure. We demonstrate analytically and experimentally that for many applications it would be preferable to simply overclock the design and accept that timing violations may arise. Since the errors introduced by timing violations occur rarely, they will cause less noise than quantisation errors. Furthermore, we show that conventional forms of computer arithmetic do not fail gracefully when pushed beyond the deterministic clocking region. In this thesis we take a fresh look at Online Arithmetic, originally proposed for digit serial operation, and synthesize unrolled digit parallel online arithmetic operators to allow for graceful degradation. We quantify the impact of timing violations on key arithmetic primitives, and show that substantial performance benefits can be obtained in comparison to binary arithmetic. Since timing errors are caused by long carry chains, these result in errors in least significant digits with online arithmetic, causing less impact than conventional implementations.Open Acces

ARITHMETIC LOGIC UNIT ARCHITECTURES WITH DYNAMICALLY DEFINED PRECISION

Modern central processing units (CPUs) employ arithmetic logic units (ALUs) that support statically defined precisions, often adhering to industry standards. Although CPU manufacturers highly optimize their ALUs, industry standard precisions embody accuracy and performance compromises for general purpose deployment. Hence, optimizing ALU precision holds great potential for improving speed and energy efficiency. Previous research on multiple precision ALUs focused on predefined, static precisions. Little previous work addressed ALU architectures with customized, dynamically defined precision. This dissertation presents approaches for developing dynamic precision ALU architectures for both fixed-point and floating-point to enable better performance, energy efficiency, and numeric accuracy. These new architectures enable dynamically defined precision, including support for vectorization. The new architectures also prevent performance and energy loss due to applying unnecessarily high precision on computations, which often happens with statically defined standard precisions. The new ALU architectures support different precisions through the use of configurable sub-blocks, with this dissertation including demonstration implementations for floating point adder, multiply, and fused multiply-add (FMA) circuits with 4-bit sub-blocks. For these circuits, the dynamic precision ALU speed is nearly the same as traditional ALU approaches, although the dynamic precision ALU is nearly twice as large

Variation-aware high-level DSP circuit design optimisation framework for FPGAs

The constant technology shrinking and the increasing demand for systems that operate under different power profiles with the maximum performance, have motivated the work in this thesis. Modern design tools that target FPGA devices take a conservative approach in the estimation of the maximum performance that can be achieved by a design when it is placed on a device, accounting for any variability in the fabrication process of the device. The work presented here takes a new view on the performance improvement of DSP designs by pushing them into the error-prone regime, as defined by the synthesis tools, and by investigating methodologies that reduce the impact of timing errors at the output of the system. In this work two novel error reduction techniques are proposed to address this problem. One is based on reduced-precision redundancy and the other on an error optimisation framework that uses information from a prior characterisation of the device. The first one is a generic architecture that is appended to existing arithmetic operators. The second defines the high-level parameters of the algorithm without using extra resources. Both of these methods allow to achieve graceful degradation whilst variation increases. A comparison of the new methods is laid against the existing methodologies, and conclusions drawn on the tradeoffs between their cost, in terms of resources and errors, and their benefits in terms of throughput. In some cases it is possible to double the performance of the design while still producing valid results.Open Acces

Intrinsically Evolvable Artificial Neural Networks

Dedicated hardware implementations of neural networks promise to provide faster, lower power operation when compared to software implementations executing on processors. Unfortunately, most custom hardware implementations do not support intrinsic training of these networks on-chip. The training is typically done using offline software simulations and the obtained network is synthesized and targeted to the hardware offline. The FPGA design presented here facilitates on-chip intrinsic training of artificial neural networks. Block-based neural networks (BbNN), the type of artificial neural networks implemented here, are grid-based networks neuron blocks. These networks are trained using genetic algorithms to simultaneously optimize the network structure and the internal synaptic parameters. The design supports online structure and parameter updates, and is an intrinsically evolvable BbNN platform supporting functional-level hardware evolution. Functional-level evolvable hardware (EHW) uses evolutionary algorithms to evolve interconnections and internal parameters of functional modules in reconfigurable computing systems such as FPGAs. Functional modules can be any hardware modules such as multipliers, adders, and trigonometric functions. In the implementation presented, the functional module is a neuron block. The designed platform is suitable for applications in dynamic environments, and can be adapted and retrained online. The online training capability has been demonstrated using a case study. A performance characterization model for RC implementations of BbNNs has also been presented