


ARTICLE 

Year : 2011  Volume
: 1
 Issue : 2  Page : 8387 

Performances of Texas Instruments DSP and Xilinx FPGAs for CooleyTukey and Grigoryan FFT Algorithms
Narayanam Ranganadh, Parimal Patel, Artyom M Grigoryan
Department of Electrical Engineering, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 782490669, USA
Date of Web Publication  24Oct2011 
Correspondence Address: Narayanam Ranganadh Department of Electrical Engineering, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 782490669 USA
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.86639
Abstract   
Frequency analysis plays a vital role in the applications like cryptanalysis, steganalysis, system identification, controller tuning, speech recognition, noise filters, etc. Discrete Fourier transform (DFT) is a principal mathematical method for the frequency analysis. The way of splitting the DFT gives out various fast algorithms. In this paper, we present the implementation of two fast algorithms for the DFT for evaluating their performance. One of them is the popular radix2 CooleyTukey fast Fourier transform (FFT) algorithm and the other one is the Grigoryan FFT based on the splitting by the paired transform. We evaluate the performance of these algorithms by implementing them on the TMS320C5416 DSP and also on the VirtexII FPGAs. Finally, we show that the pairedtransformbased algorithm of the FFT is faster than the radix2 FFT; consequently, it is useful for higher sampling rates. We also discuss the performances of TMS DSP and Xilinx FPGAs and tradeoffs. Keywords: Discrete Fourier transform, fast algorithms, fast Fourier transform, frequency analysis, paired transforms
How to cite this article: Ranganadh N, Patel P, Grigoryan AM. Performances of Texas Instruments DSP and Xilinx FPGAs for CooleyTukey and Grigoryan FFT Algorithms. J Eng Technol 2011;1:837 
How to cite this URL: Ranganadh N, Patel P, Grigoryan AM. Performances of Texas Instruments DSP and Xilinx FPGAs for CooleyTukey and Grigoryan FFT Algorithms. J Eng Technol [serial online] 2011 [cited 2019 Nov 14];1:837. Available from: http://www.onlinejet.net/text.asp?2011/1/2/83/86639 
1. Introduction   
In the past decade, fast orthogonal transforms have been widely used in areas of data compression, pattern recognition and image reconstruction, interpolation, linear filtering, and spectral analysis. The suitability of unitary transforms in each of the abovementioned applications depends on the properties of their basis functions as well as on the existence of fast algorithms, including parallel ones. Since the introduction of the fast Fourier transform (FFT), Fourier analysis has become one of the most frequently used tools in signal/image processing and communication systems. The main problem when calculating the transform relates to the construction of the decomposition, namely, the transition to the short DFTs with minimal computational complexity. The computation of unitary transforms is a complicated and timeconsuming process. Since the decomposition of the DFT is not unique, it is natural to ask how to manage splittings and how to obtain the fastest algorithm of the DFT. The difference between the lower bound of arithmetical operations and the complexity of fast transform algorithms shows that it is possible to obtain FFT algorithms of various speeds ^{[1]} . One approach is to design efficient manageable split algorithms. Indeed, many algorithms make different assumptions about the transform length. The signal/image processing related to engineering research becomes increasingly dependent on the development and implementation of the algorithms of orthogonal or nonorthogonal transforms and convolution operations in modern computer systems. The increasing importance of processing large vectors and parallel computing in many scientific and engineering applications require new ideas for designing superefficient algorithms of the transforms and their implementations ^{[2]} .
In this paper, we present the implementation techniques and their results for two different fast DFT algorithms. The difference between the algorithm developments lies in the way the two algorithms use the splitting of the DFT. The two fast algorithms considered are radix2 and paired transform ^{[2]} algorithms. The implementation of the algorithms is done both on the TMS320C5416 digital signal processor and on the Xilinx ViretxII FPGAs. The performance of the two algorithms is compared in terms of their sampling rates and also in terms of their hardware resource utilization.
Section 2 presents the paired transform decomposition used in paired transform in the development of Grigoryan FFT. Section 3 presents the implementation techniques for the radix2 and paired transform algorithms on FPGAs. Section 4 presents the results. Finally, with Section 5, we conclude the work and put forward some suggestions for further sampling rate improvements.
2. Decomposition Algorithm of the Fast DFT Using the Paired Transform   
In this algorithm, the decomposition of the DFT is done by using the paired transform ^{[2]} . Let {x(n)}, n = 0:(N−1) be an input signal, N>1. Then the DFT of the input sequence x(n) is
which is in matrix form
where X(k) and x(n) are column vectors, and matrix is a permutation of X.
which shows that the applying transform is decomposed into short transforms F_{Ni} , i = 1:k. Let S _{F} be the domain of transform F, the set of sequences f over which F is defined. Let (D; σ) be a class of unitary transforms revealed by a partition σ. For any transform F ∈ (D; σ) the computation is performed by using paired transform in this particular algorithm. To denote this type of transform, we introduce "paired functions" ^{[2]} .
Let p, t ∈ period N, and let
Let L be a nontrivial factor of number N, and W _{L} = e^{2π/L} then the complex function
t = 0: (N/L1), p ∈ ot the period 0: N  1
is called the Lpaired function ^{[2]} . Based on these paired functions, the complete system of paired functions can be constructed. The totality of the paired functions in the case of N = 2^{r} is
Now considering the case of N = 2^{r} N_{1} , where N_{1} is odd, r is ≥1 for the application of the paired transform:
(a) The totality of the partitions is
(b) The splitting of F _{N} by the partition σ' is
(c) The matrix of the transform can be represented as
where is the diagonal matrix of the twiddle factors. The steps involved in finding the DFT using the paired transform are given below:
 Perform the Lpaired transform g = χ' _{N}(x) over the input x
 Compose r vectors by dividing the set of outputs so that the first L ^{r}^{−1} elements g _{1} ,g _{2} ,….,g _{L}^{r−1} compose the first vector X _{1} ; using the next L^{r−2} elements g _{Lr−1} + 1, …., g _{Lr−1+Lr−2} compose the second vector X _{2} , and so on.
 Calculate the new vectors Y _{k} , k = 1:(r−1) by multiplying elementwise the vectors X 1) by multiplying elementwise the vectors X _{k} by the corresponding turned factors .
 Perform the L _{r−k} point DFTs over Y_{k} , k = 1:r
 Make the permutation of outputs, if needed.
3. Implementation Techniques   
We have implemented various architectures for radix2 and paired transform processors on VirtexII FPGAs ^{[3]} . As there are embedded multipliers ^{[3]} and embedded block RAMs ^{[3]} available, we can use them without using the distributed logic, which economize some of the CLBs ^{[3]} . As most of the transforms are applied on complex data, the arithmetic unit always needs two data points at a time for each operand (real part and complex part); dualport RAMs are very useful in all these implementation techniques.
In the FFT process, the butterfly operation is the main unit on which the speed of the whole process of the FFT depends. So the faster the butterfly operation, the faster the FFT process. The adders and subtractors are implemented using the LUTs (distributed arithmetic). The inputs and outputs of all the arithmetic units can be registered or nonregistered.
Various possible implementations of multipliers one can consider are as follows.
3.1 Embedded Multiplier
 with nonregistered inputs and outputs,
 with registered inputs or outputs, and
 with registered inputs and outputs.
3.2 Distributed Multiplier
Distributed multipliers are implemented using the LUTs in the CLBs. These can also be implemented with the above three possible ways. Various considerations are made to implement the butterfly operation for its speed improvement and resource requirements. The results of these techniques are tabulated in [Table 1].
By observing the results, we can say that the butterfly operation with pipelined multipliers (pipelined to maximum extent possible ^{[4]} ) distributed arithmetic operations, and registering all the inputs and outputs of the all arithmetic units provides the fastest butterfly operation. The various architectures proposed for implementing radix2 and paired transform processors are single memory (pair) architecture, dual memory (pair) architecture, and multiple memory (pair) architectures. We applied the following two best butterfly techniques for the implementation of the processors on the VirtexII FPGAs ^{[3]} :
 One with distributed multipliers, with fully pipelined stages (best in the case of performance)
 One with embedded multipliers and one level pipelining (best in the case of resource utilization)
Single memory (pair) architecture [Figure 1] is suitable for single snapshot applications, where samples are acquired and processed thereafter. The processing time is typically greater than the acquisition time. The main disadvantage in this architecture is that while doing the transform process we cannot load the next coming data. We have to wait until the current data is processed. So we proposed dual memory (pair) architecture for faster sampling rate applications [Figure 2]. In this architecture, there are three main processes for the transformation of the sampled data: Loading the sampled data into the memories, processing the loaded data, and reading out the processed data. As there are two pairs of dual port memories available, one pair can be used for loading the incoming sampled data, while at the same time the other pair can be used for processing the previously loaded sampled data. For further sampling rate improvements, we proposed multiple memory (pair) architecture [Figure 3]. This is the best of all architectures in the case of very high sampling rate applications, but in the case of hardware utilization, it uses a lot more resources than any other architecture. In this model, there is a memory set, one arithmetic unit for each iteration. The advantage of this model over the previous models is that we do not need to wait until the end of all iterations (i.e., whole FFT process), to take the next set of samples to get the FFT process to be started again. We just need to wait until the end of the first iteration and then load the memory with the next set of samples and start the process again. After the first iteration, the processed data is transferred to the next set of RAMs, so the previous set of RAMs can be loaded with the next coming new data samples. This leads to the increased sampling rate.
Coming to the implementation of the pairedtransformbased DFT algorithm, there is no complete butterfly operation, as that in the case of radix2 algorithm. According to the mathematical description given in Section 2, the arithmetic unit is divided into two parts, addition part and multiplication part. This makes the main difference between the two algorithms, which causes the process of the DFT complete earlier than the radix2 algorithm. The addition part of the algorithm for the eightpoint transform is shown in [Figure 4]. The architectures are implemented for the 8point and 64point transforms. The radix2 FFT algorithm is efficient in the case of resource utilization and the paired transform algorithm is very efficient in the case of higher sampling rate applications.  Figure 4: The addition part of the eightpoint pairedtransformbased DFT
Click here to view 
4. The Implementation Results   
4.1 Results Obtained on the TMS320C5416 Digital Signal Processor
The software used for implemention on the DSP is the Texas Instruments Code Composer Studio. We used C programming language for implementation. [Table 2] and [Table 3] show the implementation results and the comparison between the two algorithms on the DSP processor.  Table 2: Performance comparison of the two algorithms on the DSP processor
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 Table 3: The sampling rate of both the algorithms (starting from N = 8 to N = 1024)
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[Table 2] shows that the paired algorithm of DFT is much faster than the radix2 algorithm. Going to higher transform lengths, the paired algorithm gives the higher percentage improvement over the radix2 algorithm. [Table 3] shows the time required and the sampling rates that the two algorithms can be operated at.
4.2 Results Obtained on VirtexII FPGAs
The hardware modeling of the algorithms is done by using Xilinx's system generator plugin software tool running under SIMULINK environment provided under the Mathworks's MATLAB software. The functionality of the model is verified using the SIMULINK
[Table 4] shows the implementation results of the two algorithms on the VirtexII FPGAs. From [Table 2], [Table 3] and [Table 4], we can see that the paired transform is always faster than the radix2 algorithm. Thus pairedtransformbased algorithm can be used for higher sampling rate applications. In military applications, while doing the process, only some of the DFT coefficients are needed at a time. For this type of applications, the paired transform can be used as it generates some of the coefficients earlier, and also it is very fast.  Table 4: The sampling rates and the resource utilization summaries for both the algorithms, implemented on the VirtexII FPGAs
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5. Conclusion   
In this paper, we have shown that both on DSPs and also on FPGAs the pairedtransformbased algorithm is faster and can be used at higher sampling rates than the radix2 FFT at an expence of high resource utilization.
In all implementations on FPGAs, the number of bits used for the data is 16. So, all the multipliers here are used as 16bit multipliers. The multipliers used were 18bit multipliers. For instance, if there are some applications using only 8bit data, then one can use the 40 dedicated multipliers as 80 multipliers, as two multiplications can be implemented by using a single embedded multiplier as long as the sum of the two product bits is less than 36 bits.
In the implementations on the DSP if the MAC engines are used explicitly, then there may be a possibility of a better comparison between the algorithms. Also, one can see some more speed improvement in the DFT processes.
References   
1.  J. W. Cooley, and J. W. Tukey, "An algorithm for the machine calculation of complex Fourier Series", Math. Comput., Vol. 19, pp. 297301, 1965. 
2.  A. Grigoryan, and S. S. Agaian, "Split Manageable Efficient Algorithm for Fourier and Hadamard transforms", Signal Processing, IEEE Transactions on, Vol. 48, no. 1, pp. 172183, 2000. 
3.  VirtexII platform FPGAs. Available from: http://direct.xilinx.com/bvdocs/publications/ds0312.pdf. [Last accessed on 2011 Jul 6]. 
4.  CC Studio getting started guide, TMS320C54x. 
Authors   
Mr. Narayanam Ranganadh did his Master's in Electrical and Computer Engineering at the University of Texas at San Antonio. He has undergone guidance for his Master's thesis under Dr. Parimal Patel and Dr. Artyom Grigoryan. He has undergone Master's level training and research at University of Ottawa, Canada, on the "brain stem speech evoked potentials" under the guidance of Dr. Hilmi Dajani.
[Figure 1], [Figure 2], [Figure 3], [Figure 4]
[Table 1], [Table 2], [Table 3], [Table 4]
