INTRODUCTIONIMPLEMENTED ARITHMETIC UNITSREAL LIFE APPLICATIONOPTIMIZATION CRITERIAINTERFACESSOFTWARE TO GENERATE TEST VECTORSTEST PLANLANGUAGE, PLATFORM, AND TOOLSREFFERENCESMULTIPLIER FREE LAGRANGE INTERPOLATOR OPTIMIZED FOR SPECTRAL BEAMFORMING PROJECT II FINAL SPECIFICATION GMU ECE 645 Group Members: Troy CherasaroTable of Contents Introduction ...............................................................................3 Implemented arithmetic units .................................................3 Real life application ...................................................................4 Optimization Criteria................................................................5 Interfaces ....................................................................................6 Software to Generate Test Vectors ........................................7 Test Plan .....................................................................................8 Language, Platform, and tools.................................................8 refferences ..................................................................................8 2INTRODUCTION Beam-forming is a computation that is essential to sonar signal processing. There are different ways to approach the beam-forming problem. Two important types of beam-forming are time domain beam-forming and frequency domain or spectral beam-forming. Both types involve a number of intermediate pre or post processing stages that enhance the integrity of the final result. Probably one of the most common intermediate stages used in beam forming is an interpolating operation. This operation is commonly used in systems that utilize data obtained by sampling continuous signals at regular intervals to predict intermediate values between two sampling points. The LaGrange interpolator is a linear interpolator that has a maximally flat frequency response and has superior performance in the pass band and the stop band. This is desirable in sonar signal processing because it limits distortion and minimizes attenuation. The ideal interpolator for sonar signal processing is characterized as an “all pass” filter that has only delay properties. This project will focus on developing a Lagrange interpolator architecture optimized for an FPGA implementation of a spectral beam-former. The driving requirements for such an architecture will be derived from the fact that this design would need to be instantiated in parallel to handle as many channels of data as possible on a single FPGA device. Each interpolator will also need to run fast enough to accommodate the real time through-put which is a function of the sampling frequency of the array. Since actual sonar array sampling frequencies are classified information, the sampling frequency of one channel of CD quality music will be used in place of actual sampling frequencies to derive the requirements for the purposes of this project. IMPLEMENTED ARITHMETIC UNITS Two main arithmetic units will be required for this project: o A digital integrator which is simply composed of a fast adder and an accumulator register. o A tree adder that implements multi-operand addition Both arithmetic units will handle 2’s complement signed binary numbers. Both adders will be subject the optimization criteria. It will be shown that the optimization criteria of the two arithmetic units are interdependent. The digital integrator will require very basic control inputs for enabling and resetting its register and will be scalable to k bits. The tree adder will be combinatorial, but may include a pipelining option in the case that it is in the critical path. The number of operands may be scaleable, but 4 operands is desired for the target application. The output will have to be larger than the input in order to avoid overflow and overflow signals. No control signals for the tree multiplier are projected at this time. 3REAL LIFE APPLICATION The signal processing flow for a fairly generic spectral beam-former (with the exception of the LaGrange interpolator) is shown in Figure 1. 1D of PCornerTurnlanarArray DataSpatialFFTLagrangeInterpolatorSpectralFFTSpatial BeamData Figure 1: Spectral beam-former One could argue that a multiplier free architecture of a Lagrange interpolator is much better suited for this FPGA application. The proposed multiplier free architecture under consideration in Figure 2 was taken from [1]. Figure 2: Multiplier free Lagrange interpolator LaGrange interpolators are usually implemented using a standard FIR filter or Farrow structure approach that requires high frequency multipliers. Even though some FPGAs have embedded high frequency multipliers these days, such devices are much more expensive and the multiplier resources on more economically feasible devices are severely limited. These limited resources are easily consumed by other intermediate processing stages such as Fast-Fourier Transforms. A Lagrange interpolator that conserves multipliers seems optimal. 4Notice that this architecture is comprised of a low frequency section (1/T) and a high frequency section (1/T‘). The interpolation factor is L = T/T’ [1]. The coefficients and input sequence products update at a slower rate. The interpolator runs an integer number of times faster producing L-1 intermediate or interpolated results. Figure 3 provides a example of the multiplier-free Lagrange interpolator timing. The interpolated outputs are represented by y[mT + kT’]. TT’An[mT]y[mT+kT’] y[mT+T’]y[mT+2T’] y[mT+3T’]k123 0An[mT] An[mT+T] Figure 3: LaGrange interpolator timing OPTIMIZATION CRITERIA The optimization criteria will be prioritized as follows: o Throughput o Area The interpolator must be able to handle all the data for one dimension of the array in real time. The latency of the arithmetic units will be matched to this rate, and after that, the area will be minimized. The throughput that the interpolator will have to achieve is ultimately a function of the size of the array, sampling frequency, the size of the FFTs in the processing flow, and the interpolation factor. For purposes of this project, one dimension of the planar array will be 20 elements each sampled at 44.1KHz, the spectral FFT will have a block size of 2048, and the spatial FFT will have a block size of 256. Three points will be interpolated between samples using the