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This report presents improved architecture designs and implementations for a fused floating-point add-subtract unit. The fused floating-point add-subtract unit is useful for DSP applications such as FFT and DCT butterfly operations. To improve the performance of the fused floating-point add-subtract unit, the dual path algorithm and pipelining technique are applied. The proposed designs are implemented for both single and double precision and synthesized with a 45nm standard-cell library. The fused floating-point add-subtract unit saves 40% of the area and power consumption and the dual path fused floating-point add-subtract unit reduces the latency by 30% compared to the traditional discrete floating-point add-subtract unit. By combining fused operation and the dual path design, the proposed floating-point add-subtract unit achieves low area, low power consumption and high speed. Based on the data flow analysis, the proposed fused floating-point add-subtract unit is split into two pipeline stages. Since the latencies of two pipeline stages are fairly well balanced the throughput of the entire logic is increased by 80% compared to the non-pipelined implementation. / text

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International Telemetering Conference Proceedings / October 25-28, 1999 / Riviera Hotel and Convention Center, Las Vegas, Nevada / Experimentation with various routines that create floating point values in MIL-STD-1750A 32 and 48 bit formats has uncovered several flaws that result in loss of precision in approximation and/or incorrect results. This paper will discuss approximation and key computational conditions in the creation of values in these formats, and will describe algorithms that create values correctly and to the closest possible approximation. Test cases for determining behavior of routines of this type will also be supplied.

The objective of this research is to design an efficient and flexible implementation of parallel matrix multiplication for FPGA devices by analyzing the computation and studying its design space. In order to adapt to the FPGA platform, the design employs blocking and parallelization. Blocked matrix multiplication enables processing arbitrarily large matrices using limited memory capacity, and reduces the bandwidth requirements across the device boundaries by reusing available elements. Exploiting the inherent parallelism in the matrix multiplication computation improves the performance and utilizes the available reconfigurable FPGA resources. The design is constructed by identifying the main design decisions and evaluating the alternatives for each one. The considered design decisions include the scheduling of block transfers, the scheduling of arithmetic operations in a block multiplication, the extent to which the parallelism is exploited, determining the block sizes and shapes, and the use of double buffers for storing matrix blocks. The choices offered by each decision are evaluated analytically in terms of their performance and utilization of FPGA resources. Based on this analysis, a detailed, flexible design that accommodates various alternative design choices is described. The design is optimized for matrices of floating-point elements, and for the FPGA target platform. Prior work is analyzed based on the considered design choices in order to identify the similarities and the differences. The proposed design is implemented using the VHDL hardware description language. The implementation is used to verify the correctness of the design and to confirm the analysis of the design decisions. Correctness is verified both by simulation using the ModelSim logic simulator, and in hardware through compiling the implementation using the Altera Quartus II CAD software and testing it on the Altera DE4 board, featuring a Stratix IV EP4SGX530C2 FPGA device. The implementation supports a range of parameters to facilitate the experimental evaluation of design choices. Experimental results show that the design scales linearly with respect to the consumed resources. Although increasing the system size reduces the maximum operating frequency, it also increases the parallelism, resulting in a higher performance. For instance, with 8 floating-point arithmetic units, the system runs at 320 MHz, which corresponds to a performance of 4 GFLOPS, whereas with 64 arithmetic units, it runs at 160 MHz, which corresponds to a performance of 16 GFLOPS. It is also shown that using a transfer schedule based on inner products reduces the transfer time by up to 50% compared to other schedules. Although using square blocks minimizes the number of required block multiplications, other non-square blocks minimize the transfer time, resulting in better total times. / Thesis (Ph.D, Electrical & Computer Engineering) -- Queen's University, 2013-08-03 12:46:13.484

The objective of this research is to design an efficient and flexible implementation of parallel matrix multiplication for FPGA devices by analyzing the computation and studying its design space. In order to adapt to the FPGA platform, the design employs blocking and parallelization. Blocked matrix multiplication enables processing arbitrarily large matrices using limited memory capacity, and reduces the bandwidth requirements across the device boundaries by reusing available elements. Exploiting the inherent parallelism in the matrix multiplication computation improves the performance and utilizes the available reconfigurable FPGA resources. The design is constructed by identifying the main design decisions and evaluating the alternatives for each one. The considered design decisions include the scheduling of block transfers, the scheduling of arithmetic operations in a block multiplication, the extent to which the parallelism is exploited, determining the block sizes and shapes, and the use of double buffers for storing matrix blocks. The choices offered by each decision are evaluated analytically in terms of their performance and utilization of FPGA resources. Based on this analysis, a detailed, flexible design that accommodates various alternative design choices is described. The design is optimized for matrices of floating-point elements, and for the FPGA target platform. Prior work is analyzed based on the considered design choices in order to identify the similarities and the differences. The proposed design is implemented using the VHDL hardware description language. The implementation is used to verify the correctness of the design and to confirm the analysis of the design decisions. Correctness is verified both by simulation using the ModelSim logic simulator, and in hardware through compiling the implementation using the Altera Quartus II CAD software and testing it on the Altera DE4 board, featuring a Stratix IV EP4SGX530C2 FPGA device. The implementation supports a range of parameters to facilitate the experimental evaluation of design choices. Experimental results show that the design scales linearly with respect to the consumed resources. Although increasing the system size reduces the maximum operating frequency, it also increases the parallelism, resulting in a higher performance. For instance, with 8 floating-point arithmetic units, the system runs at 320 MHz, which corresponds to a performance of 4 GFLOPS, whereas with 64 arithmetic units, it runs at 160 MHz, which corresponds to a performance of 16 GFLOPS. It is also shown that using a transfer schedule based on inner products reduces the transfer time by up to 50% compared to other schedules. Although using square blocks minimizes the number of required block multiplications, other non-square blocks minimize the transfer time, resulting in better total times. / Thesis (Ph.D, Electrical & Computer Engineering) -- Queen's University, 2013-08-03 12:46:13.484

<p>The effects of using limited precision floating point for intermediate storage in an embedded MP3 decoder are investigated in this thesis. The advantages of using limited precision is that the values need shorter word lengths and thus a smaller memory for storage. </p><p>The official reference decoder was modified so that the effects of different word lengths and algorithms could be examined. Finally, a software and hardware prototype was implemented that uses 16-bit wide memory for intermediate storage. The prototype is classified as a limited accuracy MP3 decoder. Only layer III is supported. The decoder could easily be extended to a full precision MP3 decoder if a corresponding increase in memory usage was accepted.</p>

<p>This master thesis consists of implementation and evaluation of an AEC, Acoustic Echo Canceller, algorithm in a floating-point architecture. The most important question this thesis will try to answer is to determine benefits or drawbacks of using a floating-point architecture, relative a fixed-point architecture, to do AEC. In a telephony system there is two common forms of echo, line echo and acoustic echo. Acoustic echo is introduced by sound emanating from a loudspeaker, e.g. in a handsfree or speakerphone, being picked up by a microphone and then sent back to the source. The problem with this feedback is that the far-end speaker will hear one, or multiple, time-delayed version(s) of her own speech. This time-delayed version of speech is usually perceived as both confusing and annoying unless removed by the use of AEC. In this master thesis the performance of a floating-point version of a normalized least-mean-square AEC algorithm was evaluated in an environment designed and implemented to approximate live telephony calls. An instruction-set simulator and assembler available at the initiation of this master thesis were extended to enable; zero-overhead loops, modular addressing, post-increment of registers and register-write forwarding. With these improvements a bit-true assembly version was implemented capable of real-time AEC requiring 15 million instructions per second. A solution using as few as eight mantissa bits, in an external format used when storing data in memory, was found to have an insignificant effect on the selected AEC implementation’s performance. Due to the relatively low memory requirement of the selected AEC algorithm, the use of a small external format has a minor effect on the required memory size. In total this indicates that the possible reduction of the memory requirement and related energy consumption, does not justify the added complexity and energy consumption of using a floating-point architecture for the selected algorithm. Use of a floating-point format can still be advantageous in speech-related signal processing when the introduced time delay by a subband, or a similar frequency domain, solution is unacceptable. Speech algorithms that have high memory use and small introduced delay requirements are a good candidate for a floating-point digital signal processor architecture.</p>

Nowadays, there are many commercial demands for decimal floating-point (DFP) arithmetic operations such as financial analysis, tax calculation, currency conversion, Internet based applications, and e-commerce. This trend gives rise to further development on DFP arithmetic units which can perform accurate computations with exact decimal operands. Due to the significance of DFP arithmetic, the IEEE 754-2008 standard for floating-point arithmetic includes it in its specifications. The basic decimal arithmetic unit, such as decimal adder, subtracter, multiplier, divider or square-root unit, as a main part of a decimal microprocessor, is attracting more and more researchers' attentions. Recently, the decimal-encoded formats and DFP arithmetic units have been implemented in IBM's system z900, POWER6, and z10 microprocessors.<p> Increasing chip densities and transistor count provide more room for designers to add more essential functions on application domains into upcoming microprocessors. Decimal transcendental functions, such as DFP logarithm, antilogarithm, exponential, reciprocal and trigonometric, etc, as useful arithmetic operations in many areas of science and engineering, has been specified as the recommended arithmetic in the IEEE 754-2008 standard. Thus, virtually all the computing systems that are compliant with the IEEE 754-2008 standard could include a DFP mathematical library providing transcendental function computation. Based on the development of basic decimal arithmetic units, more complex DFP transcendental arithmetic will be the next building blocks in microprocessors.<p> In this dissertation, we researched and developed several new decimal algorithms and architectures for the DFP transcendental function computation. These designs are composed of several different methods: 1) the decimal transcendental function computation based on the table-based first-order polynomial approximation method; 2) DFP logarithmic and antilogarithmic converters based on the decimal digit-recurrence algorithm with selection by rounding; 3) a decimal reciprocal unit using the efficient table look-up based on Newton-Raphson iterations; and 4) a first radix-100 division unit based on the non-restoring algorithm with pre-scaling method. Most decimal algorithms and architectures for the DFP transcendental function computation developed in this dissertation have been the first attempt to analyze and implement the DFP transcendental arithmetic in order to achieve faithful results of DFP operands, specified in IEEE 754-2008.<p> To help researchers evaluate the hardware performance of DFP transcendental arithmetic units, the proposed architectures based on the different methods are modeled, verified and synthesized using FPGAs or with CMOS standard cells libraries in ASIC. Some of implementation results are compared with those of the binary radix-16 logarithmic and exponential converters; recent developed high performance decimal CORDIC based architecture; and Intel's DFP transcendental function computation software library. The comparison results show that the proposed architectures have significant speed-up in contrast to the above designs in terms of the latency. The algorithms and architectures developed in this dissertation provide a useful starting point for future hardware-oriented DFP transcendental function computation researches.

Nowadays, there are many commercial demands for decimal floating-point (DFP) arithmetic operations such as financial analysis, tax calculation, currency conversion, Internet based applications, and e-commerce. This trend gives rise to further development on DFP arithmetic units which can perform accurate computations with exact decimal operands. Due to the significance of DFP arithmetic, the IEEE 754-2008 standard for floating-point arithmetic includes it in its specifications. The basic decimal arithmetic unit, such as decimal adder, subtracter, multiplier, divider or square-root unit, as a main part of a decimal microprocessor, is attracting more and more researchers' attentions. Recently, the decimal-encoded formats and DFP arithmetic units have been implemented in IBM's system z900, POWER6, and z10 microprocessors.<p> Increasing chip densities and transistor count provide more room for designers to add more essential functions on application domains into upcoming microprocessors. Decimal transcendental functions, such as DFP logarithm, antilogarithm, exponential, reciprocal and trigonometric, etc, as useful arithmetic operations in many areas of science and engineering, has been specified as the recommended arithmetic in the IEEE 754-2008 standard. Thus, virtually all the computing systems that are compliant with the IEEE 754-2008 standard could include a DFP mathematical library providing transcendental function computation. Based on the development of basic decimal arithmetic units, more complex DFP transcendental arithmetic will be the next building blocks in microprocessors.<p> In this dissertation, we researched and developed several new decimal algorithms and architectures for the DFP transcendental function computation. These designs are composed of several different methods: 1) the decimal transcendental function computation based on the table-based first-order polynomial approximation method; 2) DFP logarithmic and antilogarithmic converters based on the decimal digit-recurrence algorithm with selection by rounding; 3) a decimal reciprocal unit using the efficient table look-up based on Newton-Raphson iterations; and 4) a first radix-100 division unit based on the non-restoring algorithm with pre-scaling method. Most decimal algorithms and architectures for the DFP transcendental function computation developed in this dissertation have been the first attempt to analyze and implement the DFP transcendental arithmetic in order to achieve faithful results of DFP operands, specified in IEEE 754-2008.<p> To help researchers evaluate the hardware performance of DFP transcendental arithmetic units, the proposed architectures based on the different methods are modeled, verified and synthesized using FPGAs or with CMOS standard cells libraries in ASIC. Some of implementation results are compared with those of the binary radix-16 logarithmic and exponential converters; recent developed high performance decimal CORDIC based architecture; and Intel's DFP transcendental function computation software library. The comparison results show that the proposed architectures have significant speed-up in contrast to the above designs in terms of the latency. The algorithms and architectures developed in this dissertation provide a useful starting point for future hardware-oriented DFP transcendental function computation researches.