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David Hansen and James Michelussi

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Introduction Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) FFT Algorithm – Applying the Mathematics Implementations of DFT and FFT Hardware Benchmarks Conclusion

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DFT In 1807 introduced by Jean Baptiste Joseph Fourier. allows a sampled or discrete signal that is periodic to be transformed from the time domain to the frequency domain Correlation between the time domain signal and N cosine and N sine waves X(k) = DFT Frequency Signal N = Number of Sample Points X(n) = Time Domain Signal W N = Twiddle Factor

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DFT (Walking Speed) Why is this important? Where is this used? allows machines to calculate the frequency domain allows for the convolution of signals by just multiplying them together Used in digital spectral analysis for speech, imaging and pattern recognition as well as signal manipulation using filters But the DFT requires N 2 multiplications!

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FFT (Jet Speed) J. W. Cooley and J. W. Tukey are given credit for bringing the FFT to the world in the 1960s Simply an algorithm for more efficiently calculating the DFT Takes advantage of symmetry and periodicity in the twiddle factors as well as uses a divide and conquer method Symmetry: W N r +N/2 = -W N r Periodicity: W N r+N = W N r Requires only (N/2)log 2 (N) multiplications ! Faster computation times More precise results due to less round-off error

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FFT Algorithm Several different types of FFT Algorithms (Radix-2, Radix-4, DIT & DIF) Focus on Radix-2 using Decimation in Time (DIT) method Breaks down the DFT calculation into a number of 2-point DFTs Each 2-point DFT uses an operation called the Butterfly These groups are then re-combined with another group of two and so on for log 2 (N) stages Using the DIT method the input time domain points must be reordered using bit reversal

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Butterfly Operation

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Bit Reversal

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8-Point Radix-2 FFT Example

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David Hansen Implementations of DFT and FFT

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DFT Implementation Nested For Loop, (N/2)*N Iterations… O(N 2 ) Cycles / Sample (123 cycles per inner loop iteration) Obvious Inefficiencies, cos and sin math.h functions Efficient assembly coding could reduce the inner loop to 3 cycles per iteration (1,536 cycles / sample) for (r=0; r<=samples/2; r++) { float re = 0.0f, im = 0.0f; float part = (float)r * -2.0f * PI / (float)samples; for (k=0; k

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C++ FFT Implementation void fft_float (unsigned NumSamples, float *RealIn, float *ImagIn, float *RealOut, float *ImagOut ) { for ( i=0; i < NumSamples; i++ ) { // Iterate over the samples and perform the bit-reversal j = ReverseBits ( i, NumBits ); } BlockEnd = 1; // Following loop iterates Log 2 (NumSamples) for ( BlockSize = 2; BlockSize <= NumSamples; BlockSize <<= 1 ) { // Perform Angle Calculations (Using math.h sin/cos) // Following 2 loops iterate over NumSamples/2 for ( i=0; i < NumSamples; i += BlockSize ) { for ( j=i, n=0; n < BlockEnd; j++, n++ ) { // Perform butterfly calculations } BlockEnd = BlockSize; }

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C++ FFT Implementation Bit-Reverse For Loop – N iterations Nested For Loops First Outer Loop – Log 2 (N) iterations Made use of sin/cos math.h functions Second Outer Loop – N / BlockSize iterations Inner Loop – BlockSize/2 iterations O(N + Log 2 (N) * N/BlockSize * BlockSize/2) O(N+N*Log 2 (N)) Cycles / Sample

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Assembly FFT Implementation Bit-Reverse Address Generation Hide Bit-Reverse operation inside first and second FFT Stages Sin and Cos values stored in a Look-Up-Table 256 Kbyte LUT added to Data1 Needed to grow Data1 Memory Space using LDF file Interleaved Real and Imaginary Arrays Quad Reads Loads 2 Complex Points per Cycle Supports the Real FFT for input signals with no Imaginary component 40% Algorithm-based Savings

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Assembly FFT Implementation Special Butterfly Instruction Can perform addition/subtraction in parallel in one compute block Speeds up the inner-most loop VLIW and SIMD Operations Performs simultaneous operations in both compute blocks Loop unrolling and instruction scheduling keeps the entire processor busy with instructions. Cycles per Sample

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Assembly FFT Implementation _BflyLoop: q[j2+=4]=r27:26; k5=k5+k9; fr6=r30*r12; fr16=r6-r7;; yr3:0=q[j0+=4]; k3=k5 and k4; fr15=r23*r4; fr24=r8+r18, fr26=r8-r18;; xr3:0=q[j0+=4]; r5:4=l[k7+k3]; fr7=r31*r13; fr25=r9+r19, fr27=r9-r19;; q[j1+=4]=r25:24; fr14=r30*r13; fr17=r14+r15;; q[j2+=4]=r27:26; k5=k5+k9; fr6=r2*r4; fr18=r6-r7;; yr11:8=q[j0+=4]; k3=k5 and k4; fr15=r31*r12; fr24=r20+r16, fr26=r20-r16;; xr11:8=q[j0+=4]; r13:12=l[k7+k3]; fr7=r3*r5; fr25=r21+r17, fr27=r21-r17;; q[j1+=4]=r25:24; fr14=r2*r5; fr19=r14+r15;; q[j2+=4]=r27:26; k5=k5+k9; fr6=r10*r12; fr16=r6-r7;; yr23:20=q[j0+=4]; k3=k5 and k4; fr15=r3*r4; fr24=r28+r18, fr26=r28-r18;; xr23:20=q[j0+=4]; r5:4=l[k7+k3]; fr7=r11*r13; fr25=r29+r19, fr27=r29-r19;; q[j1+=4]=r25:24; fr14=r10*r13; fr17=r14+r15;; q[j2+=4]=r27:26; k5=k5+k9; fr6=r22*r4; fr18=r6-r7;; yr31:28=q[j0+=4]; k3=k5 and k4; fr15=r11*r12; fr24=r0+r16, fr26=r0-r16;; xr31:28=q[j0+=4]; r13:12=l[k7+k3]; fr7=r23*r5; fr25=r1+r17, fr27=r1-r17;;.align_code 4; if NLC0E, jump _BflyLoop;

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DC FFT Test FFT Source ArrayFFT Output Magnitude

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Audio FFT Test FFT Source ArrayFFT Output Magnitude

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1024 Point DFT / FFT Comparison ImplementationCycles Per Sample DFT Implemented in C63, cycles / sample DFT Implemented in Assembly1,536 cycles / sample FFT Implemented in C cycles / sample FFT Implemented in Assembly11.35 cycles / sample

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1024 Point Radix-2 FFT Hardware Comparison Processor ArchitectureCycles Per SampleProcessor FrequencyExecution Time ADSP (SHARC)8.98 cycles / sample400 MHz22.99 µSec TigerSHARC (website)9.16 cycles / sample600 MHz15.63 µSec TigerSHARC (our results)11.35 cycles / sample600 MHz19.37 µSec TMS320C6000™ cycles / sample350 MHz41.33 µSec TMS320DM644x™7.59 cycles / sample594 MHz13.08 µSec

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Conclusion The FFT algorithm is very useful when computing the frequency domain on a DSP. FFT is much faster than a regular DFT algorithm FFT is more precise by having less errors created due to round off. The timed coding examples further support this claim and demonstrate how to code the algorithm. The Radix-2 FFT isn’t the fastest but it uses a less complex addressing and twiddle factor routine In this case (unlike in school) F is better then D.

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