accelerating fast fourier transform for wideband channelization

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Accelerating Fast Fourier Transform for Wideband ChannelizationCarlo del Mundo*, Vignesh Adhinarayanan§, Wu-chun Feng*§

* Department of Electrical and Computer Engineering, § Department of Computer Science, Virginia Tech

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Accelerating Fast Fourier Transform for Wideband Channelization

Forecast

• Goal: Accelerate the Fast Fourier Transform (FFT) using graphics processing units (GPUs) – Replace fixed hardware ASICs with programmable

GPUs

Carlo del Mundo, [email protected], carlodelmundo.com

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Forecast

• Goal: Accelerate the Fast Fourier Transform (FFT) using graphics processing units (GPUs) – Replace fixed hardware ASICs with programmable

GPUs


http://upload.wikimedia.org/wikipedia/commons/7/79/SSDTR-ASIC_technology.jpgahttp://images.bit-tech.net/content_images/2011/12/amd-radeon-hd-7970-3gb-review/amd-radeon-hd7970-e.jpg

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Motivation

• FFT is a critical building blockacross many disciplines


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Motivation



http://www.ajnr.org/content/27/6/1230/F1.large.jpg

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Motivation




http://www.elektrodaily.com/wp-content/uploads/2013/02/shazam-app.png

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Motivation




http://www.elektrodaily.com/wp-content/uploads/2013/02/shazam-app.png

http://www.wireless.vt.edu/symposium/2012/tutorials/sessionA2.html

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Introduction• Wideband Channelization

– Purpose: To isolate channels within a wideband signal


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Figure: Stages in a PFB Channelizer http://www.wireless.vt.edu/symposium/2012/tutorials/sessionA2.html

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Introduction (Channelization)

• Algorithm: Polyphase filter bank (PFB) channelizer


Figure: Stages in a PFB Channelizer

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Introduction (Channelization)

• Algorithm: Polyphase filter bank (PFB) channelizer– Problem: FFT stage grows fastest in channelization


Figure: Stages in a PFB Channelizer

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Choosing the Right Processor

• Criteria: Programmability & Performance


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Choosing the Right Processor

• Criteria: Programmability & Performance



http://upload.wikimedia.org/wikipedia/commons/7/79/SSDTR-ASIC_technology.jpga

http://www.maximumpc.com/files/u154082/intel_cpu_socket3.jpg

http://fr.academic.ru/pictures/frwiki/70/Fpga_xilinx_spartan.jpg

http://images.bit-tech.net/content_images/2011/12/amd-radeon-hd-7970-3gb-review/amd-radeon-hd7970-e.jpg

http://upload.wikimedia.org/wikipedia/commons/7/79/SSDTR-ASIC_technology.jpga

http://www.maximumpc.com/files/u154082/intel_cpu_socket3.jpg

http://fr.academic.ru/pictures/frwiki/70/Fpga_xilinx_spartan.jpg


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Outline

• Motivation• Introduction• Background• Approach

– System-level optimizations– Algorithm-level optimizations

• Results– Optimizations in isolation– Optimizations in concert

• Conclusion


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Background (GPUs)

• GPU Memory Hierarchy


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Background (GPUs)

• GPU Memory Hierarchy– Global Memory


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Background (GPUs)

• GPU Memory Hierarchy– Global Memory


Memory Unit

Read Bandwidth (TB/s)

Global 0.17

Table: Memory Read Bandwidth for Radeon HD 6970

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Background (GPUs)

• GPU Memory Hierarchy– Global Memory– Image Memory


Memory Unit


L1/L2 Cache 1.35 / 0.45

Global 0.17


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Background (GPUs)

• GPU Memory Hierarchy– Global Memory– Image Memory– Constant Memory


Memory Unit


Constant 5.4

L1/L2 Cache 1.35 / 0.45

Global 0.17


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Background (GPUs)

• GPU Memory Hierarchy– Global Memory– Image Memory– Constant Memory– Local Memory


Memory Unit


Constant 5.4

Local 2.7

L1/L2 Cache 1.35 / 0.45

Global 0.17


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Background (GPUs)

• GPU Memory Hierarchy– Global Memory– Image Memory– Constant Memory– Local Memory– Registers


Memory Unit


Registers 16.2

Constant 5.4

Local 2.7

L1/L2 Cache 1.35 / 0.45

Global 0.17


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Outline




• Conclusion


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Approach• Act as the “human compiler”


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1. Derive a candidate set of optimizations for FFT on GPUs


Candidate Optimizations

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2. Apply optimizations in isolation



Optimizations in Isolation

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2. Apply optimizations in isolation3. Apply optimizations in concert



Optimizations in Concert

Optimizations in Isolation

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Approach

• System-level Optimizations (applicable to any application) 1. Register Preloading2. Vector Access/{Vector,Scalar} Arithmetic3. Constant Memory Usage 4. Dynamic Instruction Reduction5. Memory Coalescing6. Image Memory

• Algorithm-level Optimizations1. Transpose via LM2. Compute/Transpose via LM3. Compute/No Transpose via LM


C. del Mundo et al., “Accelerating Fast Fourier Transform for Wideband Channelization,” IEEE ICC, Budapest, Hungary, June 2013.

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Approach




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Approach


• Algorithm-level Optimizations1. Naïve Transpose (LM-CM)2. Compute/Transpose via LM (LM-CC)3. Compute/No Transpose via LM (LM-CT)



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System-level Optimizations


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1. Register Preloading (RP)– Load to registers first


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Without Register Preloading

79 __kernel void unoptimized(__global float2 *buffer) 80 { 81 int index = …; 82 buffer += index; 83 84 FFT4_in_order_output(&buffer[0], &buffer[4], &buffer[8], &buffer[12]);


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With Register Preloading

79 __kernel void optimized(__global float2 *buffer) 80 { 81 int index = …; 82 buffer += index; 83 84 __private float2 r0, r1, r2, r3; // Register Declaration 85 // Explicit Loads 86 r0 = buffer[0]; r1 = buffer[1]; r2 = buffer[2]; r3 = buffer[3]; 87 FFT4_in_order_output(&r0, &r1, &r2, &r3);

Without Register Preloading

79 __kernel void unoptimized(__global float2 *buffer) 80 { 81 int index = …; 82 buffer += index; 83 84 FFT4_in_order_output(&buffer[0], &buffer[4], &buffer[8], &buffer[12]);


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2. Vector Access (float{2, 4, 8, 16})


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a[0]


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a[0] a[1]


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a[0] a[1] a[2] a[3]


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– Scalar Math (VASM)


a[0] a[1] a[2] a[3]

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a[0] a[1] a[2] a[3]

+ =


– Scalar Math (VASM)• float + float

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– Vector Math (VAVM)• float4 + float4


a[0] a[1] a[2] a[3]

+ =

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a[0] a[1] a[2] a[3]

+ =

+ =



– Vector Math (VAVM)• float4 + float4

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Approach


• Algorithm-level Optimizations


1C. del Mundo, W. Feng. “Accelerating Fast Fourier Transform for Wideband Channelization,” IEEE ICC, Budapest, Hungary, June 2013.

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Approach


• Algorithm-level Optimizations1. Naïve Transpose (LM-CM)2. Compute/Transpose via LM (LM-CC)3. Compute/No Transpose via LM (LM-CT)


1C. del Mundo, W. Feng. “Accelerating Fast Fourier Transform for Wideband Channelization,” IEEE ICC, Budapest, Hungary, June 2013.

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Algorithm-level optimizations


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• Transpose – elements across the diagonal are exchanged



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• Transpose – elements across the diagonal are exchanged



4x4 matrix

Transposed matrix

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Original Transposed

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1. Naïve Transpose (LM-CM)



Local Memory

t0 t1 t2 t3

Original Transposed

Register File

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3. The pseudo transpose (LM-CT)



Original Transposed

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3. The pseudo transpose (LM-CT)– Idea:

• Load data to local memory



Original Transposed

Local Memory

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• Load data to local memory• Perform computation on

columns,



Original Transposed

Local Memory

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columns, then rows.



Original Transposed

Local Memory

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columns, then rows.

– Advantage: • Skips the transpose step



Original Transposed

Local Memory

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columns, then rows.

– Advantage: • Skips the transpose step

– Disadvantage:• Local memory has lower

throughput than registers.



Original Transposed

Local Memory

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Outline




• Conclusion


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Results (Experimental Testbed)


GPU Testbed

Device (AMD Radeon)

CoresPeak

Performance

(GFLOPS)

PeakBandwidth

(GB/s)

HD 7970 2048 3788 264

HD 6970 (VLIW) 1536 2703 176

HD 5870 (VLIW) 1600 2720 154

• Algorithm:– 1D FFT (batched), N = 16 pts– Cooley-Tukey Decomposition

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Results (in isolation)


IM: Image memory; RP: Register Preloading; LM-{CM, CT, CC}: Local Memory-{Communication Only; Compute, No Transpose; Computation and Communication}; VASM{n}: Vectorized Access & Scalar Math{floatn}; VAVM{n}: Vectorized Access & Vector Math{floatn}; CM: Constant Memory Usage; CGAP: Coalesced Access Pattern; LU: Loop unrolling; CSE: Common subexpression elimination; IL: Function inlining; Baseline: VASM2.

*Baseline refers to a functionally correct GPU implementation with VASM2 and no optimizations.

AMD Radeon HD 7970 (Scalar, non-VLIW)AMD Radeon HD 5870/6970 (VLIW)

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Improvements to Baseline (Max. % Increase)1. 160% - Minimize bus traffic via on-

chip optimizations (RP, LM-CC, LM-CT)


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100%




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100%

160%




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2. 40% - Coalesce memory accesses (CGAP)


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40%





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40%

0% (No Change)





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3. 20% - Use scalar math (VASM2/VASM4)


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20%






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20%

10%

41%






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20%






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20%

0% (No Change)






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Neutral/Detrimental to Baseline (Min. % Decrease)1. 20% - Naïve transpose (LM-CM),

40% - Constant Memory (CM-K, CM-L)






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20%








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20%

40%








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0% (No Change)








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40% - Constant Memory (CM-K, CM-L)2. 0% - Dynamic instruction reduction (LU,

CSE, IL)






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0% (No Change)



CSE, IL)






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CSE, IL)3. 18% - Avoid large vectors & vector math

(VASM16, VAVM8/16)


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(VASM16, VAVM8/16)

61%

39%

50%


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(VASM16, VAVM8/16)

53%

18%

34%


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Results (in concert)• Improvements (Max.

Increase)



*Baseline refers to a functionally correct GPU implementation with VASM2 and no optimizations.2 All implementations are coalesced (CGAP) and use VASM2.3 The speedups listed in the graph only applies to the Radeon HD 7970 (for brevity).

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Increase)




2.9x 2.4

x

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Increase)




2.9x 2.4

x

2.4x

1.8x

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Increase)– {RP + LM-CM} best on-

chip optimization




2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

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Results (in concert)• Improvements (Max. %


chip optimization– Use Constant Memory

(CM) for twiddle calculations




2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

6.5x

5.6x

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– Use global memory (instead of image memory)




2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

6.5x

5.6x

5.6x 5.6

x

5.6x

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2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

6.5x

5.6x

5.6x 5.6

x

5.6x

6.5x

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2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

6.5x

5.6x

5.6x 5.6

x

5.6x

6.5x

6.3x

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2.1x1.5

x

2.9x 2.4

x

2.4x

1.8x

6.5x

5.6x

5.6x 5.6

x

5.6x

6.5x

6.3x

2.4x

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2.9x 2.4

x

2.4x

2.1x

6.5x

2.4x

2.4x

6.5x

6.3x

1.8x1.5

x

5.6x

5.6x 5.6

x

5.6x

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– Optimal set for AMD GPUs

• RP – Register Preloading

• LM-CM – Transpose vialocal memory

• CM – Constant memoryusage

• CGAP – Coalesced Global Access Pattern

• VASM2 – Vector Access, Scalar Math (float2)Carlo del Mundo, [email protected],

carlodelmundo.com



2.9x 2.4

x

2.4x

2.1x

6.5x

2.4x

2.4x

6.5x

6.3x

1.8x1.5

x

5.6x

5.6x 5.6

x

5.6x

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Results (1D FFT 16-pts, GPU versions)


• Optimized GPU faster by factors of 14.5 over baseline GPU

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Conclusions

• Contributions:– A portable building block for FFT towards GPU-based radios– Architecture-aware insights for mapping and optimizing FFT across

three generations of AMD GPUs• Contact:

– Carlo del Mundo– [email protected]

• Optimal set for AMD GPUs– RP – Register Preloading– LM-CM – Transpose via

local memory– CM – Constant memory

usage– CGAP – Coalesced Global

Access Pattern– VASM2 – Vector Access,

Scalar Math (float2)




mailto:[email protected]

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Appendix Slides


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Introduction (FFT)

• Fast Fourier Transform (FFT)– A spectral method

• Key computational idiom for present and future applications (dwarf)§

List of Dwarfs1. Finite State Machine2. Circuits3. Graph Algorithms4. Structured Grid5. Dense Matrix6. Sparse Matrix7. Spectral Methods

8. Dynamic Prog.9. Particle Methods10. Backtrack/B&B11. Graphical Models12. Unstructured

Grids13. Map Reduce


§ Asanovic et al. A View of the Parallel Computing Landscape. CACM, 2009.

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Background (Optimizing on GPUs)1. RP (Register Preloading) - All data elements are first preloaded onto the register file of the

respective GPU. Computation is facilitated solely on registers.2. CGAP (Coalesced Global Access Pattern) - Threads access memory contiguously (the kth

thread accesses memory element k) 3. VASM2/4 (Vector Access, Scalar Math, float{2/4}) - Data elements are loaded as the

listed vector type. Arithmetic operations are scalar (float x float).4. LM-CM (Local Memory, Communication Only) - Data elements are loaded into local

memory only for communication. Threads swap data elements solely in local memory.5. LM-CT (Local Memory, Computation, No Transpose) - Data elements are loaded into

local memory for computation. The communication step is avoided by algorithm reorganization.

6. LM-CC (Local Memory, Computation and Communication) - All data elements are preloaded into local memory. Computation is performed in local memory, while registers are used for scratchpad communication.

7. CM-K (Constant Memory - Kernel Argument) - The twiddle multiplication stage of FFT is precomputed on the CPU and stored in the GPU constant memory for fast look up.

8. CSE (Common Subexpression Elimination) - A traditional optimization that collapses identical expressions in order to save computation. This optimization may increase register live time, therefore, increasing register pressure.

9. IL (Function Inlining) - A function's code body is inserted in place of a function call. It is used primarily for functions that are frequently called.

10. IM (Image Memory) – The use of a texture image replaces the use of global memory.


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Motivation (GPU FFT vs. CPU FFT)


* Device-Host Data Transfer Not Included

• GPU FFT outperforms CPU FFT by factors as high as 6.5*– 1D batched FFT, N = 16 pts

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Introduction (Channelizer Architecture)• Channelizer Architecture

– FIR Filtering, FFT, and Channel Mapping.


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S3: Constant Memory

• Fast cached lookup for frequently used data


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S3: Constant Memory

• Fast cached lookup for frequently used data


16 __constant float2 twiddles[16] = { (float2)(1.0f,0.0f), (float2) (1.0f,0.0f), (float2)(1.0f,0.0f), (float2)(1.0f,0.0f), ... more sin/cos values};

Without Constant Memory 61 for (int j = 1; j < 4; ++j) 62 { 63 double theta = -2.0 * M_PI * tid * j / 16; 64 float2 twid = make_float2(cos(theta), sin(theta)); 65 result[j] = buffer[j*4] * twid; 66 }

With Constant Memory61 for (int j = 1; j < 4; ++j)62 result[j] = buffer[j*4] *

twiddles[4*j+tid];

accelerating fast fourier transform for wideband channelization

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