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GPU-Based FFT Computation for Multi-Gigabit WirelessHD Baseband Processing

Abstract

The next generation Graphics Processing Units (GPUs) are being considered for non-graphics applications. Millimeter wave (60 Ghz) wireless networks that are capable of multi-gigabit per second (Gbps) transfer rates require a significant baseband throughput. In this work, we consider the baseband of WirelessHD, a 60 GHz communications system, which can provide a data rate of up to 3.8 Gbps over a short range wireless link. Thus, we explore the feasibility of achieving gigabit baseband throughput using the GPUs. One of the most computationally intensive functions commonly used in baseband communications, the Fast Fourier Transform (FFT) algorithm, is implemented on an NVIDIA GPU using their general-purpose computing platform called the Compute Unified Device Architecture (CUDA). The paper, first, investigates the implementation of an FFT algorithm using the GPU hardware and exploiting the computational capability available. It then outlines the limitations discovered and the methods used to overcome these challenges. Finally a new algorithm to compute FFT is proposed, which reduces interprocessor communication. It is further optimized by improving memory access, enabling the processing rate to exceed 4 Gbps, achieving a processing time of a 512-point FFT in less than 200 ns using a two-GPU solution.

1. Introduction

As the data rates required for rich content applications rise, the throughput of wireless networks must also continue to increase in order to support them. Therefore, very high throughput wireless communications systems are now being considered [14]. As the throughput increases, the implementation of the highly compute intensive baseband functionality becomes challenging. Baseband (physical layer) processing occurs in between the radio frequency (RF) front-end and the medium access control (MAC) layer, and involves signal processing and coding on a data stream. The two most time and power consuming parts are the fast fourier transform (FFT) and the channel decoder. Therefore, any performance gains in these blocks could potentially improve the throughput of the whole system significantly. The acceleration of algorithms such as these is of critical importance for high throughput wireless communications systems.

The 3.8 Gbps throughput required by the WirelessHD "high-rate PHY" places the FFT and decoding blocks under the most computational strain relative to the other system components. The FFT computation must be completed in about 200 ns. For a WirelessHD modem with 512 subcarriers, this means that 2304 complex multiplications, 4608 complex additions and the ancillary operations such as loading the data into the input registers of the FFT processor must be completed in that time. This is a demanding deadline. In [5], a review of FFT execution times was carried out. The fastest quoted FFT speed was 5.5 s, with a "DSPlogic" FFT instantiated on a Virtex-II Pro 50 FPGA. This is about 28 times too slow for the FFT demanded by the WirelessHD standard. Hence, the current solutions are not capable of fulfilling the required specification.

In this paper, we focus on the implementation of the FFT and explore the feasibility of using the computational capability of a graphics processor (GPU) to achieve gigabit baseband throughput using the WirelessHD specification [6]. GPU is a massively parallel device, with a significant number of processing cores, whose processing ability can be exploited for general purpose use in high arithmetic intensity algorithms, where the parallel nature of its architecture can be exploited to maximum benefit. The NVIDIA compute unified device architecture (CUDA) platform [7] is used in order to access the computational capability provided by the GPU, which enables a programmer to utilize the parallel processing functionality of NVIDIA GPUs. The details of the GPU used in this paper are discussed in the following section.

There is a growing number of recently reported works on implementations of various applications on general-purpose GPU (GPGPU) technology [8]. To cite a few, in [9], the authors discuss the implementation of an image processing application using GPU. A MapReduce, a distributed programming framework originally proposed by Google for the ease of development of web search applications on a large number of commodity CPUs, framework is developed on a GPU in [10]. GPUs are used to accelerate the computational intensive operations in cryptographic systems [11]. The authors in [12] also use GPUs to accelerate radiative heat transfer simulation for a large form factor matrix. Acceleration of molecular modeling applications with graphics processors are presented in [13]. Authors in [14] discuss GPU implementation of sequence alignment algorithms used in molecular biology. Several other applications of general-purpose processing using GPUs are covered in a recent special issue [15].

There are also some reported FFT implementations on GPUs. In [16], the authors proposed an implementation to exploit the memory reference locality to optimize the parallel data cache. The same authors also reported their experiences with mapping nonlinear access patterns to memory in CUDA programming environment [17]. The researchers in [18, 19] addressed computation-intensive tasks such as matrix multiplication in implementing FFT on GPUs. In [20], the authors presented algorithms for FFT computation based on a Stockham formulation. Their algorithm attempts to optimize the radix with respect to the threads and the registers available to them. Their work, like ours, tries to use the memory and registers efficiently to increase the performance. In another paper by Govindaraju and Manocha [21], the authors also use a Stockham-based FFT algorithm for cache-efficient implementation.

Note that our aim is this work is not to come up with the fastest FFT algorithm but rather come up with a design that will accommodate FFT computation for the WirelessHD standard. The algorithms in the literature aim for the fastest implementation on the GPU and they do not take some features, for instance, memory transfer, into account. For example, in [20], it was specifically mentioned in Section .D, where the authors discuss the limitations of their work, their algorithm works only on data which resides in GPU memory. They added when the data must be transferred between GPU and system memory, the performance will be dramatically lowered. Hence, a comparison between the results of this paper (similarly others) and ours will not be an apple-to-apple comparison. In order to emphasize our contributions, though, we compare the original CuFFT algorithm with five difference proposed enhancements as well as our proposed FFT algorithm and also give GFLOPs performance for these. Reference [21] is another paper in the literature, which may seem similar to this work in the first instance; however, there are many differences: theirs use Stockham FFT algorithm, ours is based on Cooley-Tukey radix-2 FFT algorithm. They literally focus on the cache-efficiency, our work attempts to use the memory access efficiently but it does not go into cache structures. They exploit the nested loops in numerical algorithms, our algorithm exploits the fact that the butterfly selections follow a pattern when computing the multipliers in the FFT.

1.1. Contributions of This Paper

This paper makes several contributions while exploring the feasibility of achieving multigigabit baseband throughput.

  1. (1)

    A new algorithm for FFT computation is proposed. By grouping the butterfly operations, the proposed algorithm reduces the number of interprocessor communications. This leads into reducing the number of shared-memory accesses and hence eventually reducing the overall FFT computation time.

  2. (2)

    New load and save structures are proposed to improve the shared memory access. The order of accesses to the shared memory is modified to increase the amount of read and write coalescing. This has a significant impact on the algorithm performance.

  3. (3)

    New data types and conversions are developed. These 8-bit and 16-bit data types were not available in the existing CUDA environment and are used to overcome bandwidth limitations.

  4. (4)

    Last but not least, it is shown that multigigabit FFT computation can be achieved by software implementation on a graphics processor.

The rest of the paper is organized as follows. Section 2 gives an overview of the GPU used in this work. Section 3 describes the WirelessHD standard. CUDA FFT algorithm and enhancements are covered in Section 4. The new FFT algorithm is introduced in Section 5. Improvements for the shared memory access are also presented in this section. Finally, the paper concludes in Section 6.

2. The Graphics Processor

Most central processing units (CPUs) now have between 2 and 4 cores, serviced by varying amounts of on-chip cache as shown in Figure 1. The GPUs used in this work have 24 streaming multithreaded processors (SM) each with 8 cores, giving 192 cores in total. Each SM has a limited amount of cache known as shared memory which is accessible by all eight cores. Whilst the GPU cores are much less versatile than the CPU cores, and are clocked at lower frequencies, the combined processing capability for well designed parallel algorithms can easily exceed the capability of a CPU. This explains the emergence of the GPGPU technology for high performance applications.

Figure 1
figure1

An illustration of the differences between CPUs and GPUs.

2.1. Hardware: A Closer Look

The NVIDIA GPU architecture is constructed around SM processors. These consist of 8 scalar processors (SP), two special function units (SFU), a multithreaded instruction unit and a block of shared memory, as seen in Figure 2. Each GTX260 GPU used in this work has 24 SMs, each of which can run 1024 threads concurrently, so each SP handles 128 threads, giving a total of 24,576 threads per GPU. This vast amount of processing power is supported by a large amount of global memory. This is off-chip memory located on the GPU main board, primarily used to store data prior to and after calculation. A CUDA kernel call specifies the number of blocks, and threads per block, that must be processed. All the threads of a given block must execute on the same SM and as a block completes another will be scheduled if there are remaining blocks to be computed. Only one kernel may execute at any one time.

Figure 2
figure2

Streaming multiprocessors in detail.

Thread execution is implemented using a single instruction multiple thread (SIMT) architecture, where each SP executes threads independently. Threads are executed in parallel, in groups of 32 consecutive threads, called warps. Every instruction time, a warp that is ready to execute is selected and the same instruction is then issued to all threads of that warp, so maximum speed is obtained if there are no divergent branches in the code. For example, in an "IF" statement, if 17 threads follow one branch, and 15 the other, the 17 are suspended and 15 execute, and then 17 execute with the 15 suspended, so effectively both branches of the "IF" statement are processed.

2.2. Compute Unified Device Architecture

The CUDA platform is an extension of the C language which enables the programmer to access GPU functionality for parallel processing. It includes a set of additional function qualifiers, variable type qualifiers and built in variables. These are used to define and execute a kernel, a function called from the Host (PC) and executed on the Device (GPU). These functions are not written as explicitly parallel code, but the Device hardware automatically manages the threads that run in parallel.

The NVIDIA CUDA system uses an application programming interface (API) [22] which hides the complex architecture of the GPU. This hardware abstraction simplifies the task of coding for the GPU, and also has the advantage that the underlying architecture can change significantly in future products and the code designed for an older device will still work. The CUDA programming model [23] is based on a system where individual groups of threads use their own shared memory and local registers, with intermittent synchronization across all groups at defined points within the code. Therefore, maximum performance can be achieved if an algorithm is broken into sections that are each independent, where the individual sections can then be split into smaller divisions where data can be shared between those divisions.

2.3. Memory Access

Every CUDA thread may access several types of memory, each of which has a different access speed. The fastest is the thread level register space, which at full occupancy is limited to 16 registers per thread, although at maximum 32 are available. Occupancy is a measure of the number of active threads relative to the maximum number of threads that can be active for the given GPU and it is affected by the memory usage of threads and blocks. The next fastest is the shared memory, which is accessible by all the threads of a single block, of which there is a limit of 8 Kbytes per block at full occupancy, or a maximum of 16 Kbytes. The slowest access speed is for the global memory, of which for the GTX260s used in this project there is 896 Mbytes, which is accessible by all threads. In addition to this, there is another memory space which is read-only for the Device, known as the constant space, which the Host can copy data to, for applications such as look up tables.

The global memory access speed is due to the DRAM used being off chip, so although it is still on the GPU mainboard, the latency can be up to a few hundred clock cycles. This latency is hidden by the action of the thread schedulers, such that if a thread is waiting for data it will be de-scheduled and another is rescheduled. Once the data transaction is complete the thread will be reinstated on the list of threads to be processed. In this way, the vast number of threads hides the effect seen by such memory latency.

The interconnect between the GPU and CPU is provided by the PCI Express Bus, shown in Figure 3, which links the Host memory with the Device global memory. Data required for a kernel's execution must be loaded on to Device memory by the Host prior to the kernel being launched as the GPU cannot access Host memory, nor instigate transfers to or from it.

Figure 3
figure3

An overview of the GPU architecture.

3. WirelessHD

WirelessHD aims to enable consumer devices to create a wireless video area network (WVAN), for streaming High Definition (HD) video with resolutions of up to 1080P, 24 bit colour at 60 Hz, and also provide 5.1 surround sound audio. This papre has focussed only on the high rate physical (HRP) video link, as this is by far the most computationally intensive section of the specification [6]. The HRP link uses OFDM to achieve the 3.8 Gbps required for the streaming of the HD video in an uncompressed format.

WirelessHD utilizes the unlicensed bandwidth in the 60 GHz region, with a typical range of 10 m. This is achieved using smart antenna technology that adapts to environmental changes by focusing the receiver antenna in the direction of the incoming power from the transmitter using beam forming and steering. This enables both improvements in the quality of the line of sight link and also enables use of reflected paths if line of sight is not available.

There are many data, coding rates, and subcarrier modulation schemes used in WirelessHD, depending on the throughput required. The OFDM system outlined in the WirelessHD specification [6] uses 512 subcarriers of which 336 carry data, each modulated using 16 Quadrature Amplitude Modulation (QAM).

Figure 7 illustrates the data transmission process. To convert the N frequency domain symbols of each channel into the time domain signal transmitted, an N-point inverse FFT must be applied to the baseband signals of the N subcarrier OFDM system. In the receiver, an N-point FFT is used to switch from the time domain signal back to frequency domain data which can then be quantised and decoded. For WirelessHD, there are 2.538 giga samples per second using 512 sub carriers. Therefore, in the baseband of the receiver, depicted in Figure 4, a 512-point FFT must be computed every 226.95 ns in order to achieve a raw throughput of 5.9 Gbps. The key specifications for WirelessHD can be found in Table 1.

Table 1 WirelessHD specification parameters [6].
Figure 4
figure4

WirelessHD baseband receiver block diagram.

4. CUDA FFT Algorithm and Enhancements

In this section, we first utilize CuFFT [24], an FFT library included in the CUDA software development kit (SDK), to implement the FFT functionality required for the WirelessHD standard. CuFFT provides a set of standard FFT algorithms designed for the GPU. A benchmarking program obtained on the NVIDIA CUDA forum was used to assess the performance of this library. Minor modifications were made to the code, to enable it to run on the test system. An example CUDA FFT code is given in Figure 13. The execution of the algorithm is also discussed there. Tests showed that a 512-point FFT would be calculated in 7 s with the original CuFFT code using a single GPU. Since this time is not anywhere close to the required duration for the FFT calculation, we explored several methods to enhance the performance achieved by the CuFFT implementation. Below, we will describe these methods.

4.1. Method #1: Using Large Batch Size

The batch size is defined as the number of FFTs performed by the kernel in a single execution. Analysis showed that as the batch size increased, the processing time per FFT reduced towards the minimum of 4.15 s, shown in Figure 5. This shows that the overheads in initializing the kernel become insignificant to the execution time when the batch size rises above 8192 FFTs. All future code executions used a batch size greater than 16384 FFTs to eliminate the kernel overhead.

Figure 5
figure5

Graph showing the calculation time per FFT against batch size.

4.2. Method #2: Using Page-Locked Memory and Radix-2

Host memory that is page-locked, also known as pinned memory, ensures the assigned program memory is held in physical memory and is guaranteed not to be written to the paging file and removed from physical memory. This means the page-locked memory can be accessed immediately, rather than having to be restored from the paging file prior to access.

Using CUDA functionality, page-locked memory can be allocated as well as the regular pageable memory. The advantage of page-locked memory is that it enables a higher bandwidth to be achieved between the Device and Host. However, page-locked memory is a limited resource, and so the programmer must limit its allocation. The page-locked memory is also used by the computer operating system, so if large amounts are allocated for use with CUDA, then overall system performance may be affected, as it may force critical memory to be written to the paging file. The FFT program using the page-locked memory technique considered what gains may be achieved by utilizing this additional bandwidth.

Given the available radix sources, the radix-2 algorithm was the most appropriate for calculating a 512-point FFT. The CUDA FFT radix-2 code is based on the Cooley-Tukey algorithm, but the algorithm is parallelized such that there are 256 threads per FFT, each executing a single butterfly for each of the 9 stages of the 512-point FFT.

The performance gain of using page-locked memory and explicit execution of the radix-2 CUDA FFT source was analysed over a range of batch sizes. It was found that these modifications offered a significant improvement in the FFT calculation time, reducing it to 2.83 s per FFT when using 16384 FFTs per batch.

4.3. Method #3: Asynchronous Concurrency

In order to facilitate concurrent execution between the Host and the Device, some Device functions can be called asynchronously from the Host such that control is returned to the Host prior to the Device completing the task assigned. Asynchronous memory copies can be called to initiate Device-to-Host or Host-to-Device copies whilst a kernel is executing.

Kernels manage concurrency through streams. A stream is a sequence of operations which are processed in order; however, different streams can execute out of order with respect to one another. Since only one kernel is able to run on the Device at any one time, a queue of streams can be formed such that the memory copies of one stream can overlap with the kernel execution of another stream as shown in Figure 6. It should be noted that the memory copies themselves cannot overlap if maximum bandwidth is to be provided in a single copy, as all transfers utilize the same PCI Express Bus.

Figure 6
figure6

Illustration of streaming execution. This diagram assumes zero streaming overheads which would also affect performance.

Figure 7
figure7

An illustration of the data transmission process.

Asynchronous concurrency was implemented such that a batch of FFTs was divided equally between the number of available streams. Tests showed that in fact two streams proved to be the most efficient, reducing the total computation time to 2.54 s per FFT, and that batch size did not affect the 2-stream performance. Whilst streams decreased the calculation time by 290 ns, the overall performance remained significantly above that required to provide a gigabit throughput. This was due to the fact that the memory transfer time was a limiting factor in the performance of the algorithm, which streams cannot overcome.

4.4. Method #4: Reduced Accuracy for Input/Output Limitations

The previous subsection highlighted a significant challenge in the memory transfer limitations between the Device and Host. In order to target a solution to this, it was necessary to first consider why this was a performance bottleneck.

In the WirelessHD HRP link specification each FFT corresponds to 336 16-QAM data signals, equivalent to 1344 raw received bits. Using an outer code of rate 0.96 and inner code of rate 2/3 this represents 860 decoded data bits. The computation for this is achieved in 226.95 ns so fulfilling the 3.8 Gbps decoded data rate of the HRP link.

(1)

In a single FFT, 512 complex values must be copied to the Device, computations take place and the results be copied off again within the 226.95 ns deadline. When the memory copies alone are considered, in full 32-bit accuracy this requires an astounding bandwidth of 36 GBps

(2)

This represents a 76.2 : 1 ratio, relative to the decoded data rate. The reason for this bandwidth requirement is that the FFT is calculated prior to quantization, using magnitude and phase data from the receiver provided at a given accuracy. The accuracy of the complex signals must be sufficient to be able to mathematically separate the incoming signals into 16-QAM data channels and nulls precisely, so that quantization can occur.

The maximum benchmarked bandwidth for the PCI Express 1.0 bus on the motherboard used was 3.15 GBps. Given this, even with a 2-GPU solution, full 32-bit accuracy at gigabit data throughputs with the current hardware was not achievable.

There are three ways to overcome the bandwidth issues.

Hardware Upgrade

A PCI Express 2.0 compliant motherboard could be used.

Compression

Lossless compression could be employed to reduce the data transfer size. However, the compression/uncompression will add to the already tight schedule for the computation of the baseband algorithms in WirelessHD.

Reduced Accuracy Transfer

The majority of wireless communications systems use less than 16-bit accuracy and many less than 8-bit accuracy and therefore the possibility of reducing the accuracy of the data transferred between the Host and Device was explored. Since the GPU generally performs calculations in 32-bit floating point accuracy, additional processing time was required to convert the data prior to and post calculation.

4.4.1. Performance for the 16-Bit Reduced Accuracy Transfers

The LittleFloat, a 16-bit floating-point data type was created and integrated into the FFT code. It uses one sign bit, 5 exponent bits, and 10 mantissa bits. Tests showed the reduction in Host-to-Device and Device-to-Host transfer times had a significant impact on the calculation rate of the algorithm. The fastest processing time of 1.34 s per FFT was achieved using 4 streams. Analysis was carried out to provide a complete breakdown of the execution time and to find the performance of the conversion algorithm in the FFT calculation.

The tests undertaken to obtain this information are shown below.

(i)Instigate the memory transfers, but exclude the FFT kernel call.

(ii)Initiate memory transfers and also the kernel call but the function to run the FFT calculation was omitted so only the floating point conversions were processed.

(iii)Executed an empty kernel, to find the kernel execution overhead.

These tests were performed using a single stream to ensure the effects of streaming would not hide any processing time. The results are given in Table 2.

Table 2 Break down of calculation time.

The conversion time was relatively significant, taking 300 ns per conversion. Whilst the single stream calculation and conversion time took significantly longer than the 507 ns calculated earlier, streams almost completely hid this additional time, such that the overall calculation time was a few hundred nanoseconds above the memory transfer time.

4.4.2. Performance for the 8-Bit Reduced Accuracy Transfers

Consideration was given to whether 8-bit accuracy was sufficient for the FFT data. On a constellation diagram, 16-QAM uses 16 points to identify the values of 4 data bits. These are spaced equally about the origin in a grid, such that each point is equidistant from its neighbours. Figure 8 shows a 16-QAM constellation diagram with markers dividing the distance between points into 8. This illustrates the number of unique positions if 5-bit accuracy were used, giving 32 individual points on each row and column or 1024 individual points in total.

Figure 8
figure8

Constellation diagram of 16-QAM showing 5-bit accuracy divisions.

If 8-bit accuracy were used there would be 65536 unique points. Therefore it was determined that 8-bit accuracy would be sufficient to represent the requested data.

If an 8-bit data accuracy system (TinyFixed) was implemented it could reduce the transfer time such that it would be comparable with the calculation time. However, this in itself was not sufficient, the time taken for the floating point conversion had to be reduced, as a single conversion to or from the LittleFloat 16-bit data type exceeded the total processing time required for the entire WirelessHD FFT calculation.

Using a single stream, an FFT was calculated in 860 ns including memory transfer time and float conversion, and using 4 streams this was achieved in 534 ns. As the calculation time now approached low hundreds of nanoseconds, it became apparent that the use of streams added approximately 100 ns to the total computation time. This was verified by running the same program without any streaming code. In this case, the total execution time was approximately 100 ns less than the single stream execution time. Given that the total memory transfer took 320 ns, allowing for the 100 ns streaming overhead, the single stream time indicates calculation was achieved in 440 ns using the CUDA FFT algorithm, including conversion time. The conversion time of TinyFixed was significantly better than LittleFloat achieving 40 ns per conversion, so computation alone took approximately 360 ns. The calculation time breakdown is tabulated in Table 3.

Table 3 Break down of calculation time.

The overall computation performance significantly improved to 534ns per FFT on a single GPU. The balance of calculation to memory transfer time was significantly closer taking greater advantage of the functionality provided by streams.

4.5. Performance Summary of CuFFT and Proposed Enhancements

We provide a summary of the performances achieved by the original CuFFT and the proposed methods to improve it in Table 4. Though, there is a reduction in FFT time with the addition of each enhancement, we still cannot fulfil the WirelessHD specification. Therefore, it is necessary to consider the development of a new algorithm.

Table 4 Performance summary of CuFFT and proposed methods to improve it.

5. A New FFT Algorithm

In order to get a better FFT performance, the new algorithm needs to exploit the architecture of the GPU so as to maximise the processing throughput. Consideration was given to radix-8, and split-radix algorithms; however, the fine grained nature of the radix-2 algorithm offered a greater degree of flexibility than the other radices, as it was unknown how many butterflies would best fit per thread given the limited number of registers available. Also, the additional complexity of these algorithms was thought inappropriate given the scale of the parallelism required in the implementation of a new algorithm on the GPU. One of the limiting factors for performance is the interprocessor communication. References [25, 26] present implementations of the FFT without interprocessor communications, showing how the performance of the FFT could be enhanced significantly in avoiding communication with other processors by loading the entire input data to the processor. However, their direct application to the CUDA platform was not possible due to the limited register space available per thread. In this way, the CUDA architecture is limited in that since the register space is very small, only 16 per thread, if full occupancy was to be achieved then such a method could not be used. Nevertheless, the idea of limiting interprocessor communications could be applied.

5.1. Algorithm Overview

Basically, in order to reduce the interprocessor communications, our algorithm exploits the fact that the butterfly selections follow a pattern when computing the multipliers in the FFT. Limiting the interprocessor communication is possible by grouping 4 butterfly calculations into a single thread. If the correct butterflies are chosen, within each thread 3 stages of calculation can be implemented without interprocessor communication. Given that for a radix-2, 512-point FFT there are 9 stages, using this strategy, shared memory need only be accessed at two points in the entire calculation (see Figure 10). All other accesses within the calculation are to the internal thread registers, which have inherent speed enhancements as these are the fastest accesses. The details of the algorithms are given in the following subsection.

5.2. Algorithm Development

An 8-point FFT can be represented by two 4-point FFTs and a set of butterflies, and similarly the 4-point DFT can be seen as two 2-point FFTs and a set of butterflies. In Figure 9, first (a), a 2-point FFT is shown, then (b), a Decimation-in-Time decomposition of the 8-point FFT is shown. For the 512-point FFT the same process of decomposition can be used to form 9 stages of butterflies, where there are 256 butterflies per stage.

Figure 9
figure9

(a) 2-point FFT and (b) The decomposition of a DIT 8-point radix-2 FFT.

Figure 10
figure10

The workings of the new algorithm.

Implementing an 8-point FFT in a single thread, as using real and complex values would require all 16 available registers. Whilst implementing 4 stages, totalling 32 registers, would be possible, it would lower occupancy significantly and therefore impact performance.

Just as the first 3 stages could be grouped to form a single 8-point FFT, the next group of 3 butterfly stages could be grouped, if the correct data was selected. Using this method, shared memory only needed to be accessed after the 3rd and 6th set of butterfly stages. A Decimation-In-Frequency implementation was used as it enabled the inputs to be ordered at the beginning of the calculation.

For clarity of explanation, a group of 3 butterfly stages will be defined as a Set as shown in Figure 10. After Set 1 and 2, the 8 outputs of each thread are locally 8-bit, bit reversed like that of an 8-point FFT, but the outputs of Set 3 are globally, 512-bit, bit reversed. The simplest method of storing the computed data from Sets 1 and 2 in shared memory was to use similarly bit reversed location pointers so as to store data back in ordered form. In order to achieve this, a point of reference for the data was required. Throughout computation all data access was referenced relative to its original location in the input data.

Each Set used a similar pattern, whose exact design was tailored to the stages in the given set. To illustrate how the pattern was used, Figures 10 and 11 show a section of the 4th butterfly stage located in Set 2. The data used in Thread 0 is the same as that used in stages 5 and 6 however; the butterflies are paired differently and different multipliers are used. Thread 1 accesses data displaced by 64 locations, a pattern which is repeated for Threads 0–7. Each of these use the same multiplier due to their relative position within the butterfly block. Overall, by arranging a pattern so that Threads 8–15 access data offset by 1 place relative to that of Threads 0–7 and so on for a total of 64 threads, the required multipliers per thread could be assigned as shown in Table 5.

Table 5 Stage 4 multipliers.
Figure 11
figure11

A sample of a load structure used in the new algorithm.

Figure 11 shows the loading of data in groups of 8. Similarly for the 5th stage the threads are grouped into 16s and subsequently the 6th stage groups in 32s.

5.2.1. Performance

The single stream performance of the new algorithm improved upon the single stream CUDA FFT performance by 91 ns, taking 769 ns per 512-point FFT. Using four streams the total processing time dropped to 459 ns, an improvement of 75 ns or 14%, which would enable a throughput of 5.86 Gbps raw, or 3.75 Gbps decoded data rate using a 2-GPU solution.

Considering the algorithm performance, when allowing for the streaming overheads, conversion time and memory transfer time, the computation alone took no more than 269 ns per FFT, a 25% improvement on the CUDA FFT algorithm.

(3)

This was just under the WirelessHD requirement, and so it was necessary to optimize the algorithm in order to surpass this requirement.

5.3. Improved Memory Access

The shared memory accesses in the original algorithm were not optimal, limiting the level of coalescing achievable. Both memory save/load structures were therefore modified to improve their access performance and so improve the processing time.

The first shared memory access was more difficult to modify as the ordering constraints highlighted earlier meant any modification to the saving parameters needed to be counteracted by the load parameters to achieve the same overall load pattern during processing.

The easier of the two memory save/load structures to modify was the second transfer since the multipliers in the final Set are contained completely in each individual thread. This meant that the pattern outlined earlier to define the necessary multipliers were not necessary in this particular Set, which made significant modification to the save and load structure possible. However, this would have an effect on the save pattern used to store the data to global memory prior to transfer back to the Host, and so care had to be taken to organise data access to take this into account.

In Figure 12, a small sample of two load and save structures are shown for the first shared memory access. Each table only shows the upper 11 rows of each column of a total of 64.

(i)Each row of the "Data in Threads" table shows the computed elements labelled according to the equivalent input data locations.

(ii)Each "Data Structure" is arranged in columns where each column represents 64 locations, that is, columns begin with locations (0, 64, 128, 192, 256, 320, 384, and 448).

  1. (iii)

    The "Save" or "Load" patterns are arranged one row per thread, and map the thread data to a saved data location.

Figure 12
figure12

Sample of original load and save structures.

Figure 13
figure13

The CUDA FFT code [ 24 ] with explanation of its execution.

The original save structure reorders data back into ordered format such that location 0 holds data element 0, and so forth. For example the 4th element of the 1st thread is mapped via the 4th element of row 1 of the Save Structure to location 384 in shared memory. Whilst the original save structure used unit stride memory access down each column, the accesses across a row are not ordered, so performance was not maximized. The load structure was highly unordered giving slow performance.

A number of save and load structures were implemented, and their performance tested against the original algorithm. The best performance of 392 ns was obtained when using 4 streams. This was because the best usage of streams required a balance between kernel processing time and the memory transfer time.

5.4. Two-GPU Solution

A two-GPU solution is explored as well. A new motherboard with PCI-Express 2.0 is installed with two GTX260-based cards to perform at full bandwidth so achieving almost exactly the same computation times, the difference being no more than 4 ns. The average time per FFT per board is 394 ns, or 197 ns overall, giving 6.82 Gbps raw throughput, which corresponds to a decoded data rate of 4.36 Gbps.

5.5. Performance Summary

We summarize the performance of the original CuFFT and the proposed enhancements as well as the new FFT computation algorithm with its improvements in Table 6. The performance is given in terms of FFT computation time in addition to GFLOPs, which is calculated as follows, where N is 512 in this work.

(4)
Table 6 Performance summary of CuFFT, proposed methods to improve it and the new FFT algorithm

6. Conclusions

In order to achieve high throughput for the next generation wireless networks, it is essential to increase the throughput of wireless baseband processing. This requires acceleration of the most intensive algorithms, found in the baseband, such as the FFT and Viterbi algorithms which are critical to overall performance. This paper has introduced the architecture of the graphics card. It has also outlined the process of utilising the CUDA platform to expose the computational capability of the GPU and has shown that if applied to highly parallel algorithms, the processing power is impressive. The main objective of this work was to achieve gigabit baseband throughput using the WirelessHD specification. For the FFT algorithm this was achieved and subsequently surpassed, reaching a computation rate that was more than sufficient to fulfil the full WirelessHD specification, processing a 512-point FFT in less than 200 ns. This was equivalent to a raw throughput of 6.82 Gbps and a decoded data rate of 4.36 Gbps. This was achieved by overcoming a number of challenges, the major two of which were I/O limitations and the development of a new algorithm. This paper has presented the limitations of the PCI Express Bus linking the Device and Host, which was unable to transfer data sufficiently fast for full 32-bit accuracy. This was overcome by recognising it was not necessary to compute data to more than 8-bit accuracy as this provided 65536 unique points on a constellation diagram, of which 16-QAM uses 16 ideal locations. Since the GPU computes data in 32-bit accuracy, it was necessary to write an efficient conversion between 8-bit and 32-bit accuracy on the Device, which lead to a computation rate of 534 ns per FFT using the CUDA SDK FFT Algorithm. At this point, the CUDA SDK algorithm was a limiting factor and subsequently in order to achieve the highest computation rate, a new algorithm was developed. This minimized the interprocessor communication, so reducing the number of shared memory accesses. The new algorithm is further improved by modifying the order of accesses to the shared memory. Finally, a two GPU boards are installed to run this new algorithm, which achieved more than 35 times improvement in the FFT performance in terms of GFLOPs compared to that of the CUDA algorithm.

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Correspondence to Taskin Kocak.

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Hinitt, N., Kocak, T. GPU-Based FFT Computation for Multi-Gigabit WirelessHD Baseband Processing. J Wireless Com Network 2010, 359081 (2010). https://doi.org/10.1155/2010/359081

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Keywords

  • Fast Fourier Transform
  • Shared Memory
  • Global Memory
  • Fast Fourier Transform Algorithm
  • Compute Unify Device Architecture