

Rethinking the FFTBy J. Greg Nash, Centar LLC, Los Angeles, California Introduction It is well known that the discrete Fourier transform (DFT) is of central importance to many signal processing applications, in particular high datarate multicarrier applications such as wireless communications, which is the second largest market for semiconductor chips. Here a DFT is required in a large variety of wireless transmission protocols that are based on some form of orthogonal frequency division multiplexing (OFDM). What is unique about the wireless signal processing subdomain is that the FFT circuitry used to calculate a DFT is being simultaneously stressed in a number of ways as the transmission protocols attempt keep pushing data rates up and latencies down in a mobile, crowded environment and at the same time improving quality of service while lowering cost and power consumption. Examples of the variety of requirements are:
Finally, wireless protocols have been and always will be evolving [1] with different protocols used in different parts of the world. While there have been a multitude of different FFT designs proposed to meet this large present and future spectrum of requirements, they have historically been based on two algorithm/architecture models and unique instantiations of these models are typically applied to each separate application. In contrast Centar has created a completely new algorithm/architecture model that can be programmed for each separate application, yet at the same time can also provide better throughputs and lower power. The remainder of this note is a quick summary of this new model and its FFT circuit characteristics. Background Most high performance 1D FFT designs fall into one of two categories: either “pipelined” designs (Fig. 1), or serial “memorybased” designs (Fig. 2). The pipelined designs are typically obtained from different permutations of the “signal flow graph” by projecting them onto a linear pipelined array of memory/register and arithmetic hardware. The pipelined designs can be very computationally efficient, but lack programmability and must deal with stagetostage irregularities and the complex permutation networks, commutators and butterflies, which can ultimately limit clock speeds and add hardware. Fig. 1. Methodology for designing an 8point pipelined FFT. Memorybased designs consist of a plurality of large arithmetic units, where butterfly computations are performed, connected to a similar plurality of memories. The goal in such designs is to sequence data to/from the memories in such a way that available bandwidth is maximized. Fig. 2. Traditional memorybased FFT architecture. This design is more flexible because typically only a small number of butterfly units are used, so that by proper sequencing of data to/from the memories different functionality is possible. However, considerable complexity is usually required to achieve high speed either because many butterfly/memory units are required or because clock speeds must be very high. Additionally, the sequencing of data to from memories and butterfly units becomes very complex. Centar FFT Model Centar's design is also memorybased, but is architecturally very different than traditional memorybased approaches in that it is based on the fine grained array structure shown Fig. 3. It is seen to consist of many very small processing elements (PEs), each containing a multiplier/adder and a few registers with control logic. Each PE reads and writes to a typically small memory. Aggregate bandwidth is limited only by the number of PEs, e.g., no read/write conflicts can exist as in the memorybased model in Fig. 2.
Fig. 3. Centar array memorybased FFT architecture Although not shown in Fig. 3, each PE is locally connected to its neighbors and data moves only between adjacent PEs. Input and output data flow into and out of the array from the top (not shown), using the nearest neighbor interconnection network. Programmability Here the concept of “programmability” is used loosely to mean that the same array circuit can be used to perform any set of transform sizes at the appropriate throughput rates. Additionally, other special functionality, as for example cyclic prefix insertion, which normally isn’t available in FFT IP, can easily be added. The benefit of a programmable circuit is that the computational hardware can be highly optimized and then reused with different control circuitry to do different classes of computations. As long as a DFT of size N can be factored into powers of preferably small bases, a,b,c,…, e.g., N=anbmcp… , where n,m, and p are integers, programmability is achieved by performing appropriate sets of small DFTs. For example to do a 1024point transform a factorization could be N=424142, resulting in 16*4 16point DFTs followed by, 16*16 4point transforms and finally 16*4 16point transforms again. Since there are often many ways to factor a transform of size N, there are many potential orderings of small DFTs. The sequences of small DFTs are performed sequentially on the array in a way that all PEs are “active” almost all the time and thus the overall computation is very efficient. Scalability Scalability as used here is a measure of the throughput improvement (for a given FFT transform size) that can be obtained with a given amount additional circuit hardware. Ideally there should be a roughly linear relationship between the two. What’s unique about the memory model in Fig. 3 is that scalability is accomplished simply by duplicating the array or making it larger [2] or both, so that the computational structure remains the same and few design changes are needed. Alternatively, for the traditional pipelined FFT (Fig. 1) scaling is typically done by projecting the signal flow graph onto hardware that accepts more than the two samples per clock cycle as shown in Fig. 1; however, due to the irregularity of the signal flow graph there is no simple methodology for doing this and hardware typically grows faster than the throughput. For the memory based model in Fig. 2 there are inherent limits to scaling because of the difficulty in sequencing data to/from the memories in such a way that available bandwidth is maximized. Example Design As a basis for further discussion, two very common FFT (streaming) circuits were compiled using Altera’s software tools (Quartus II) with exactly the same settings (all synthesis options turned off) and the same target hardware (Stratix III EP3SE50F484C2 FPGA) [3]. The signaltoquantizationnoise (SQNR) results for the Altera circuits (IP v13.0) were obtained from a bitaccurate Matlab model created along with the circuit by the Altera FFT generators. Altera’s Timequest static timing analyzer was used to determine "worst case" maximum clock frequencies (Fmax) at 1.1V and 85C. Table 1. Example 256 and 1024 point streaming FFTs
Circuit comparisons shown in Table I are with Altera’s radix4 delayfeedback 20bit pipelined FFTs that use BFP (single exponent per FFT block) to achieve a similar SQNR as the Centar FFT circuits. Here the adaptive logic module (ALM) is the basic unit of a Stratix III FPGA (an 8input "adaptive" LUT, two registers plus other logic). Fmax or clock rate is the same as the complex data sample rate. Clock Speed (Fmax) High FPGA clock speeds are obtained by minimizing logic levels and keeping interconnect distances short. Because the logic in the simple PEs in Fig.3 is primarily multiplyadditions, which are performed in the embedded circuity, these are inherently fast. And because data movement is local, the interconnect delays for the critical path are less than the logic delays. Consequently, data rates >500MHz for the Stratix III 65nm technology are possible, whereas for most such FPGAs they are less than 400Mhz as seen in Table 1. This locality of data movement is particularly important as process technologies migrate to 22nm sizes and routing delays can become an increasingly dominant part of total the critical path. Throughput Throughputs are also enhanced because the algorithms used require fewer than the typical N cycles/DFT found in most pipelined streaming architectures. For example a 256 point transform can be performed in 165 clock cycles. Power Power dissipation is minimized by reducing LUT/register usage in the FPGA fabric and the nearest neighbor interconnection lengths are short, reducing capacitances. Here the Altera power per FFT (µ joules) is higher by 15/48% for the 256/1024 point transform sizes. It should be noted that most of the FFT design literature focuses on minimizing the number of multipliers and memory blocks, which can be useful in an ASIC context; however, modern FPGAs have large numbers of embedded multipliers (as many as 868 on a single FPGA in the Altera Stratix III series) and embedded memories, which result in very different design tradeoffs compared to ASIC designs. And because multipliers and memories are embedded, their contribution to power dissipation is typically much less than the LUT/register fabric. For example in the 256/1024point examples in Table 1, all the embedded components used only 14.4/14.7% of total power (minus I/O power). SQNR The dynamic range of an FFT circuit, which is usually characterized by the SQNR, is an important metric because of the tradeoff that exists between dynamic range and word lengths, e.g., longer word lengths are required for higher dynamic range. Centar uses both block floating point and floating point circuitry that’s an integral part of the design to provide higher SQNR values for a given fixedpoint word length. For example in Table 1, the Altera streaming block floating point design needs at least 4 more bits to yield an equivalent SQNR. Other nonblock floating point FFT designs would comparatively need more than 4 bits to maintain an SQNR equivalency. Non poweroftwo designs The underlying DFT algorithms Centar uses also allow nonpoweroftwo transforms. A good example is the LTE single carrier uplink “LTE SCFDMA” protocol that requires computation of 35 different nonpoweroftwo DFT transform sizes (12 to 1296 points) [3]. Centar’s circuit uses a single ROM memory to hold parameters that determine the specific factorizations and execution orderings used for loop index ranges in the verilog HDL coded control modules. (Any transform size or number of transforms can be performed by appropriate entries in this memory.) As can be seen in the normalized performance comparisons with other commercial circuits in Table 2 (average over 35 DFT sizes), this circuit is 2 to 3 times faster and the number of LUT/registers is relatively small (Table 3). Table 2. Relative times to compute an LTE resource block for single carrier uplink DFTs (lower is better).
Table 3. FPGA resource fabric usage for uplink DFT circuitry
Conclusion Centar’s goal is a single FFT algorithm/architecture with the programmability necessary to meet the variety of functional FFT demands of future wireless and other signal processing applications. Whereas traditionally such programmability must be traded off for performance and/or power dissipation, Centar’s array memorybased approach actually excels in providing both very high throughputs and low power. References [1] “5G Wireless Communication”, IEEE Communications Magazine, February 2014, Vol. 52, No.2. [2] J. Greg Nash, Computationally efficient systolic architecture for computing the discrete Fourier transform, IEEE Trans Sig. Process., V. 53, Dec. 2005, pp. 46404651. [3] J. Greg Nash, Highthroughput programmable systolic array FFT architecture and FPGA Implementations, presented at the 2014 International Conference on Computing, Networking and Communications (ICNC), CNC Workshop (ICNC'14  Workshops  CNC),Honolulu, HI, Feb 2014. [4] Xilinx Discrete Fourier Transform v3.1, DS615 Mar. 1, 2011. [5] Altera DFT/IDFT Reference Design, Application Note 464, May 2007.

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