Operand Folding Hardware Multipliers

This paper describes a new accumulate-and-add multiplication algorithm. The method partitions one of the operands and re-combines the results of computations done with each of the partitions. The resulting design turns-out to be both compact and fast. When the operands' bit-length $m$ is 1024, the new algorithm requires only $0.194m+56$ additions (on average), this is about half the number of additions required by the classical accumulate-and-add multiplication algorithm ($\frac{m}2$)

Data path analysis for dynamic circuit specialisation

Dynamic Circuit Specialisation (DCS) is a method that exploits the reconfigurability of modern FPGAs to allow the specialisation of FPGA circuits at run-time. Currently, it is only explored as part of Register-transfer level design. However, at the Register-transfer level (RTL), a large part of the design is already locked in. Therefore, maximally exploiting the opportunities of DCS could require a costly redesign. It would be interesting to already have insight in the opportunities for DCS from the higher abstraction level. Moreover, the general design trend in FPGA design is to work on higher abstraction levels and let tool(s) translate this higher level description to RTL. This paper presents the first profiler that, based on the high-level description of an application, estimates the benefits of an implementation using DCS. This allows a designer to determine much earlier in the design cycle whether or not DCS would be interesting. The high-level profiling methodology was implemented and tested on a set of PID designs

Truncated Binary Multipliers with minimum Mean Square Error: analytical characterization, circuit implementation and applications

In the wireless multimedia word, DSP systems are ubiquitous. DSP algorithms are computationally intensive and test the limits of battery life in portable device such as cell phones, hearing aids, MP3 players, digital video recorders and so on. Multiplication and squaring are the main operation in many signal processing algorithms (filtering, convolution, FFT, DCT, euclidean distance etc.), hence efficient parallel multipliers are desirable. A full-width digital nxn bits multiplier computes the 2n bits output as a weighted sum of partial products. A multiplier with the output represented on n bits output is useful, as example, in DSP datapaths which saves the output in the same n bits registers of the input. Note that the truncated multipliers are useful not only for DSP but also for digital, computational intensive, ASICs where the bit-widths at the output of the arithmetic blocks are chosen on the basis of system-related accuracy issues. Hence 2n bits of precision at the multiplier output are very often more than required. A truncated multiplier is an nxn multiplier with n bits output. Since in a truncated multiplier the n less-significant bits of the full-width product are discarded, some of the partial products are removed and replaced by a suitable compensation function, to trade-off accuracy with hardware cost. Several techniques have been proposed in the Literature following this basic idea. The difference between the various circuits is in the choice and the implementation of the compensation circuit. The correction techniques proposed in the Literature are obtained through exhaustive search. This means that the results are only available for small n values and that the proposed approach are not extendable to greater bit widths. Furthermore the analytical characterization of the error is not possible. In this dissertation an innovative solution for the design and characterization of truncated multipliers is presented. The proposed circuits are based on the analytical calculation of the error of the truncated multiplier. This approach allows to have the description of a multiplier characterized by a minimum mean square error which gives a fast and low power VLSI implementation. Furthermore the analytical approach yields to a closed form expression of the mean square error and maximum absolute error for the proposed truncated multipliers. In this way the a priori knowledge of the output error is available. The errors are known for every bit width of the multiplier and it is also possible to decide, for a given bit width, which correction circuit has to be used in order to obtain a certain error. This analytical relation between the error and the parameters of hardware implementation is extremely important for the digital designer, since now it is possible to select the suitable implementation as a function of the desired accuracy. Proposed truncated multipliers overcome the previously proposed truncated multipliers since provide lower error, lower power dissipation, lower area occupation and also provide higher working frequency. The circuits are also easily implemented and allow an automatic HDL description as a function of bit width and desired error. The complete description of the errors for the truncated multipliers allows the use of these circuits as building blocks for more complex systems. It will be shown how the proposed multiplier can be used to design low area occupation FIR filters and an efficient PI temperature controller

Adaptive and hybrid schemes for efficient parallel squaring and cubing units

Squaring (X2) and cubing (X3) units are special operations of multiplication used in many applications, such as image compression, equalization, decoding and demodulation, 3D graphics, scientific computing, artificial neural networks, logarithmic number system, and multimedia application. They can also be an efficient way to compute other basic functions. Therefore, improving their performances is a goal for many researchers. This dissertation will discuss modification to algorithms to compute parallel squaring and cubing units in both signed and unsigned representation. After that, truncated technique is applied to improve their performance. Each unit is modeled and estimated to obtain its area, delay by using linear evaluation model. A C program was written to generate Hardware Description Language files for each unit. These units are simulated and verified in simulation. Moreover, area, delay, and power consumption are calculated for each unit and compared with those ones in previous approaches for both Virtex 5 Xilinx FPGA and IBM 65nm ASIC technologies

Fast Modular Reduction for Large-Integer Multiplication

The work contained in this thesis is a representation of the successful attempt to speed-up the modular reduction as an independent step of modular multiplication, which is the central operation in public-key cryptosystems. Based on the properties of Mersenne and Quasi-Mersenne primes, four distinct sets of moduli have been described, which are responsible for converting the single-precision multiplication prevalent in many of today\u27s techniques into an addition operation and a few simple shift operations. A novel algorithm has been proposed for modular folding. With the backing of the special moduli sets, the proposed algorithm is shown to outperform (speed-wise) the Modified Barrett algorithm by 80% for operands of length 700 bits, the least speed-up being around 70% for smaller operands, in the range of around 100 bits

Low-Power Design of Digital VLSI Circuits around the Point of First Failure

As an increase of intelligent and self-powered devices is forecasted for our future everyday life, the implementation of energy-autonomous devices that can wirelessly communicate data from sensors is crucial. Even though techniques such as voltage scaling proved to effectively reduce the energy consumption of digital circuits, additional energy savings are still required for a longer battery life. One of the main limitations of essentially any low-energy technique is the potential degradation of the quality of service (QoS). Thus, a thorough understanding of how circuits behave when operated around the point of first failure (PoFF) is key for the effective application of conventional energy-efficient methods as well as for the development of future low-energy techniques. In this thesis, a variety of circuits, techniques, and tools is described to reduce the energy consumption in digital systems when operated either in the safe and conservative exact region, close to the PoFF, or even inside the inexact region. A straightforward approach to reduce the power consumed by clock distribution while safely operating in the exact region is dual-edge-triggered (DET) clocking. However, the DET approach is rarely taken, primarily due to the perceived complexity of its integration. In this thesis, a fully automated design flow is introduced for applying DET clocking to a conventional single-edge-triggered (SET) design. In addition, the first static true-single-phase-clock DET flip-flop (DET-FF) that completely avoids clock-overlap hazards of DET registers is proposed. Even though the correct timing of synchronous circuits is ensured in worst-case conditions, the critical path might not always be excited. Thus, dynamic clock adjustment (DCA) has been proposed to trim any available dynamic timing margin by changing the operating clock frequency at runtime. This thesis describes a dynamically-adjustable clock generator (DCG) capable of modifying the period of the produced clock signal on a cycle-by-cycle basis that enables the DCA technique. In addition, a timing-monitoring sequential (TMS) that detects input transitions on either one of the clock phases to enable the selection of the best timing-monitoring strategy at runtime is proposed. Energy-quality scaling techniques aimat trading lower energy consumption for a small degradation on the QoS whenever approximations can be tolerated. In this thesis, a low-power methodology for the perturbation of baseline coefficients in reconfigurable finite impulse response (FIR) filters is proposed. The baseline coefficients are optimized to reduce the switching activity of the multipliers in the FIR filter, enabling the possibility of scaling the power consumption of the filter at runtime. The area as well as the leakage power of many system-on-chips is often dominated by embedded memories. Gain-cell embedded DRAM (GC-eDRAM) is a compact, low-power and CMOS-compatible alternative to the conventional static random-access memory (SRAM) when a higher memory density is desired. However, due to GC-eDRAMs relying on many interdependent variables, the adaptation of existing memories and the design of future GCeDRAMs prove to be highly complex tasks. Thus, the first modeling tool that estimates timing, memory availability, bandwidth, and area of GC-eDRAMs for a fast exploration of their design space is proposed in this thesis

Profile-directed specialisation of custom floating-point hardware

We present a methodology for generating floating-point arithmetic hardware designs which are, for suitable applications, much reduced in size, while still retaining performance and IEEE-754 compliance. Our system uses three key parts: a profiling tool, a set of customisable floating-point units and a selection of system integration methods. We use a profiling tool for floating-point behaviour to identify arithmetic operations where fundamental elements of IEEE-754 floating-point may be compromised, without generating erroneous results in the common case. In the uncommon case, we use simple detection logic to determine when operands lie outside the range of capabilities of the optimised hardware. Out-of-range operations are handled by a separate, fully capable, floatingpoint implementation, either on-chip or by returning calculations to a host processor. We present methods of system integration to achieve this errorcorrection. Thus the system suffers no compromise in IEEE-754 compliance, even when the synthesised hardware would generate erroneous results. In particular, we identify from input operands the shift amounts required for input operand alignment and post-operation normalisation. For operations where these are small, we synthesise hardware with reduced-size barrel-shifters. We also propose optimisations to take advantage of other profile-exposed behaviours, including removing the hardware required to swap operands in a floating-point adder or subtractor, and reducing the exponent range to fit observed values. We present profiling results for a range of applications, including a selection of computational science programs, Spec FP 95 benchmarks and the FFMPEG media processing tool, indicating which would be amenable to our method. Selected applications which demonstrate potential for optimisation are then taken through to a hardware implementation. We show up to a 45% decrease in hardware size for a floating-point datapath, with a correctable error-rate of less then 3%, even with non-profiled datasets

Area-Optimized Fully-Flexible BCH Decoder for Multiple GF Dimensions

Recently, there are increasing demands for fully flexible Bose Chaudhuri Hocquenghem (BCH) decoders, which can support different dimensions of Galois fields (GF) operations. As the previous BCH decoders are mainly targeting the fixed GF operations, the conventional techniques are no longer suitable for multiple GF dimensions. For the area-optimized flexible BCH decoders, in this paper, we present several optimization schemes for reducing hardware costs of multi-dimensional GF operations. In the proposed optimizations, we first reformulate the matrix operations in syndrome calculation and Chien search for sharing more common sub-expressions between GF operations having different dimensions. The cell based multi-m GF multiplier is newly introduced for the area-efficient flexible key-equation solver. As case studies, we design several prototype flexible BCH decoders for digital video broadcasting systems and NAND flash memory controllers managing different page sizes. The implementation results show that the proposed fully-flexible BCH decoder architecture remarkably enhances the area-efficiency compared with the conventional solutions.112Ysciescopu