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[[Digital signal processing
Modern [[General-purpose computing on graphics processing units|general purpose graphics processing units (GPGPUs)]] are considered having excellent throughput on vector operations and numeric manipulations by high degree of parallel computation. While processing digital signals, particularly multidimensional signals, often involves in a series of vector operations on massive amount of independent data samples, GPGPUs are now widely employed to accelerate multidimensional DSP, such as [[image processing]], [[Video processing|video codec]], [[Radar signal characteristics|radar signal analysis]], [[sonar signal processing]], and [[Ultrasound scan|ultrasound scanning]]. Conceptually, using GPGPU devices to perform multidimensional DSP is able to dramatically reduce the computation complexity compared with [[Cpu|central processing units (CPUs)]], [[Digital signal processor|digital signal processors (DSPs)]], or other [[Field-programmable gate array|FPGA]] accelerators.
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=== Digital Signal Processors (DSPs) ===
Digital signal processors are designed specifically to process vector operations. They have been widely used in DSP computations for decades. However, most digital signal processors are only capable of manipulating couple operations in parallel. This kind of designs is sufficient to accelerate audio processing (1-D signals) and image processing (2-D signals). However, with a large amount of data samples in multidimensional signals, this is still not powerful enough to retrieve computation results in real-time.
=== Supercomputer Assistance ===
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=== GPU Acceleration ===
[[Graphics processing unit|GPUs]] are originally devised to accelerate image processing and video stream rendering. Moreover, since modern GPUs' have good ability to perform numeric computations in parallel with a relatively low cost and better energy efficiency, GPUs are becoming a popular alternative to replace supercomputers performing multidimensional DSP.<ref>{{
== GPGPU Computations ==
[[File:SIMD GPGPU.jpg|alt= Figure illustrating a SIMD/vector computation unit in GPGPUs..|thumb|GPGPU/SIMD computation model.]]
Modern GPU designs are mainly based on [[SIMD]] computation paradigm<ref>{{
GPGPUs are able to perform an operation on multiple independent data concurrently with their vector or SIMD functional units. A modern GPGPU can spawn thousands of concurrent threads and process all threads in a batch manner. With this nature, GPGPUs can be employed as DSP accelerators easily while many DSP problems can be solved by [[Divide and conquer algorithms|divide-and-conquer]] algorithms. A large scale and complex DSP problem can be divided into bunch of small numeric problems and be processed altogether at one time so that the overall time complexity can be reduced significantly. For example, multiplying two {{math|''M'' × ''M''}} matrices can be processed by {{math|''M'' × ''M''}} concurrent threads on a GPGPU device without any output data dependency. Therefore, theoretically, by means of GPGPU acceleration, we can gain up to {{math|''M'' × ''M''}} speedup compared with a traditional CPU or digital signal processor.
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=== CUDA ===
[[CUDA]] is the standard interface to program [[Nvidia|NVIDIA]] GPUs. NVIDIA also provides many CUDA libraries to support DSP acceleration on NVIDIA GPU devices<ref>{{
=== OpenCL ===
[[OpenCL]] is an industrial standard which was originally proposed by [[Apple Inc.]] and is maintained and developed by [[Khronos Group]] now<ref>{{
[[File:OpenCL program execution flow.jpg|alt=Illustrating the execution flow of a OpenCL program/kernel|thumb|OpenCL program execution flow|285x285px]]
=== C++ Amp ===
[[C++ AMP|C++ Amp]] is a programming model proposed by [[Microsoft]]. C++ Amp is a [[C++]] based library designed for programming SIMD processors<ref>{{
=== OpenAcc ===
[[OpenACC|OpenAcc]] is a programming standard for parallel computing developed by [[Cray]], CAPS, [[Nvidia|NVIDIA]] and PGI<ref>{{
== Examples of GPU Programming for Multidimensional DSP ==
=== {{math|''m'' × ''m''}} Matrix Multiplication ===
Suppose {{math|'''A'''}} and {{math|'''B'''}} are two {{math|''m'' × ''m''}} matrices and we would like to compute {{math|1 = '''C''' = '''A''' × '''B'''}}.
<math>\mathbf{A}=\begin{pmatrix}
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\vdots & \vdots & \ddots & \vdots \\
B_{m1} & B_{m2} & \cdots & B_{mm} \\
\end{pmatrix}</math>
<math>\mathbf{C}=\mathbf{A}\times\mathbf{B}=\begin{pmatrix}
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\vdots & \vdots & \ddots & \vdots \\
C_{m1} & C_{m2} & \cdots & C_{mm} \\
\end{pmatrix},\quad C_{ij}=\sum_{k=1}^m A_{ik}B_{kj}</math>
To compute each element in {{math|'''C'''}} takes {{math|''m''}} multiplications and {{math|(''m''
// MxM matrix multiplication in C
void matrixMul(
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int id = row * size + col;
flot sum = 0.0;
for (int m = 0; m < size; m++) {
sum += (A[row * size + m] * B[m * size + col]);
}
C[id] = sum;
}
}
}
</
// MxM matrix multiplication in OpenCL
__kernel void matrixMul(
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size_t col = id % size;
float sum = 0.0;
// N iterations
for (int m = 0; m < size; m++) {
sum += (A[row * size + m] * B[m * size + col]);
}
C[id] = sum;
}
</syntaxhighlight>
=== Multidimensional Convolution (M-D Convolution) ===
Convolution is a frequently used operation in DSP. To compute 2-D convolution of two ''m'' × ''m'' signals, it requires {{math|''m''<sup>''2''</sup>}} multiplications and {{math|''m'' × (''m''
<math>y(n_1, n_2)=x(n_1,n_2)**h(n_1,n_2)=\sum_{k_1=0}^{m-1}\sum_{k_2=0}^{m-1}x(k_1, k_2)h(n_1-k_1, n_2-k_2)</math><
// 2-D convolution implementation in OpenCL
__kernel void convolution(
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size_t n2 = id % col;
float sum = 0.0;
// N x N iterations
for (int k1 = 0; k1 < size; k1++) {
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}
}
C[id] = sum;
}
</syntaxhighlight>
Note that, although the example demonstrated above is a 2-D convolution, a similar approach can be adopted for a higher dimension system. Overall, for a s-D convolution, a GPGPU implementation has time complexity ''Θ(n''<sup href="Category:GPGPU">''s''</sup>'')'', whereas a CPU implementation has time complexity ''Θ(n''<sup href="Category:GPGPU">''2s''</sup>'')''.
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<math>X(\Omega_1,\Omega_2,...,\Omega_s)=\sum_{n_1=0}^{m_1-1}\sum_{n_2=0}^{m_2-1}...\sum_{n_s=0}^{m_s-1}x(n_1, n_2,...,n_s)e^{-j(\Omega_1n_1+\Omega_1n_1+...+\Omega_sn_s)}</math>
Practically, to implement M-D DTFT, if assuming the system is separable, we can perform M times 1-D DFTF and matrix transpose with respect to each dimension. With a 1-D FFT operation, GPGPU can conceptually reduce the complexity from ''Θ(n''<sup href="Category:GPGPU">''2''</sup>'')'' to Θ(n'')'' by the following example of OpenCL implementation''.'' That is, to perform M-D DTFT, the complexity of GPGPU can be achieved by ''Θ(n''<sup href="Category:GPGPU">''2''</sup>'').'' While some GPGPUs' are also equipped hardware FFT accelerator internally, this implementation might be also optimized by invoking the FFT APIs or libraries provided by GPU manufactures<ref>{{
// DTFT in OpenCL
__kernel void convolution(
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X_re[id] = 0.0;
X_im[id] = 0.0;
for (int i = 0; i < size; i++) {
X_re += (x_re[id] * cos(2 * 3.1415 * id / size) - x_im[id] + sin(2 * 3.1415 * id / size));
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}
}
</syntaxhighlight>
== Real Applications ==
=== Digital Filter Design ===
Designing a digital filter in multidimensional area is a big challenge, especially [[Infinite impulse response|IIR]] filters. Typically it relies on computers to solve difference equations and obtain a set of approximated solutions. While GPGPU computation is becoming popular, several adaptive algorithms have been proposed to design multidimensional [[Finite impulse response|FIR]] and/or [[Infinite impulse response|IIR]] filters by means of GPGPUs<ref>{{
=== Radar Signal Reconstruction and Analysis ===
Radar systems usually require to reconstruct a numerous amount of 3-D or 4-D data samples in real-time. Traditionally, particularly in military, this needs supercomputers' support. Nowadays, GPGPUs are also employed to replace supercomputers to process radar signals. For example, to process [[Synthetic aperture radar|synthetic aperture radar (SAR)]] signals, it usually involves multidimensional [[Fast Fourier transform|FFT]] computations<ref>{{
=== Self-Driving Car ===
Many [[self-driving cars]] applies 3-D image recognition techniques to auto control the vehicles. Clearly, to accommodate the fast changing exterior environment, the recognition and decision processes must be done in real-time. GPGPUs are excellent devices to achieve the goal<ref>{{
=== Medical Image Processing ===
In order to have accurate diagnosis, 2-D or 3-D medical signals, such as [[ultrasound]], [[X-ray]], [[Magnetic resonance imaging|MRI]], and [[CT scan|CT]], often requires very high sampling rate and image resolutions to reconstruct images. By applying GPGPUs' superior computation power, it has been proved that we can acquire better quality on medical images<ref>{{
== References ==
{{Reflist}}
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