Quantization effects for a wideband beamformer (sonar arrays)

Detection of a sinusoid of unknown frequency in wideband noise is performed efficiently by the FFT (fast Fourier transform). The detector performs a hypothesis test on the magnitude of the FFT output. When the FFT is implemented, error due to arithmetic roundoff coefficient quantization limits the accuracy of the transform and degrades the detection performance. When the FFT is used as a detector of an unknown sinusoidal signal, the coefficient quantization error is significant and increases with the FFT length. The decimation is analyzed in time for a radix-2 FFT. The FFT output error is defined to be the maximum magnitude of the difference between the true FFT and FFT computed with the quantized coefficients. An upper bound on the error is derived by a deterministic analysis and is verified to be close to the actually measured error. Using the functional form of the bound and scaling it to fit the measured error, an empirical formula for the error is derived. The probability of detection of the quantized-coefficient FFT is computed using the empirical error formula. The probability of detection curves is presented as a function of the FFT length. Simulations indicate that when a sufficient number of bits is used to quantize the coefficients, the probability of detection does not significantly degrade.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fixed-point arithmetic is used to provide faster and smaller implementations in many digital signal processing applications, including medical imaging, at the expense of decreased accuracy. In particular, when a Fast Fourier Transform (FFT)-Inverse Fast Fourier Transform (IFFT) pair are required as part of the calculation, the error introduced into the calculations can be significant. For some applications, such as Fourier Domain Optical Coherence Tomography (FD-OCT), this degradation is unacceptable. Our study shows that using a conventional fixed-point FFT-IFFT pair, such as Xilinx's FFT core, can produce an average 6-bit error for a 1024-point FFT using 12-bit input data in a 32-bit arithmetic system. The majority of the error is caused by quantization effects, particularly on the phase information of input signal. For this reason, in phase sensitive applications such as FD-OCT, the error dominates the fixed-point calculation: 78% in 16-bit and 51% in 32-bit systems. This work presents a parameterized 32 to 4096-point integer FFT implementation for FPGAs that uses a Split-Radix algorithm to reduce the number of multiplies and improve latency. Integer FFTs are perfectly reconstructible, with zero reconstruction error. Here, we specifically analyze a 1024-point Integer Split-Radix FFT (Int-SRFFT) and IFFT pair that perfectly reconstructs the original 12-bit input data using 22-bit arithmetic, compared to the average 6-bit error for a 1024-point FFT using 32-bit fixed-point arithmetic. The pipelined architecture of this design has a latency of 29.06us for a 1024-point FFT, and a throughput of more than 34 thousand 1024-point FFTs/second for a 22-bit datapath at an operating frequency of 274MHz. Although our Int-SRFFT is perfectly reconstructible, compared to Xilinx's fixed-point FFT, it has ~6% more flipflops, ~63% more LUTs, 5.3x more BRAMs, and a ~44% increase in latency. However, compared to a previous fixed-point SRFFT design on an FPGA, our throughput is 15.5x greater.

Orthogonal Frequency Division Multiplexing (OFDM) is one of the promising contendents in the race of very high data rate delivery, for 4G wireless systems. The conventional OFDM systems, so far deployed make use of Fast Fourier Transform (FFT) as their basic building block. Besides the numerous advantages, the restriction over the data rate imposed by the FFT in the form of cyclic prefix may be intolerable for the next generation OFDM systems. So, researches are on the way to replace FFT with a more suitable transform which is redundant free. DWT OFDM is one such system in which FFT in the conventional OFDM is replaced with Discrete Wavelet Transform (DWT). Using simulations, already it has been established that in ideal conditions, the performance of DWT OFDM is better than that of FFT OFDM. As phase noise can be considered as the major RF impairment and one of the leading performance limiting factors of the wireless systems, this paper analyses the Bit Error Rate (BER) performance of the DWT OFDM in presence of phase noise. BER performances of both DWT OFDM and FFT OFDM are analysed and compared in presence of phase noise. For this, Matlab Simulink model have been developed for both types of OFDM, and simulation is carried over in presence of phase noise for different modulation schemes. In all cases the BER performances of DWT OFDM is found to be fairly ahead that of FFT OFDM.

Investigation of instantaneous and local transmembrane pressure in rotating and vibrating filtration (RVF) module: Comparison of three impellers

Re: Hoffer et al.: Update on intraocular lens power calculation study protocols: the better way to design and report clinical trials (Ophthalmology. 2020; Jul 9 [Epub ahead of print

The radix- $2^{k}$ algorithm plays a crucial role in the pipelined implementation of fast Fourier transform (FFT). This paper presents a fixed-point analysis and hardware evaluation of radix- $2^{k}$ FFT under the framework of the single-path delay feedback (SDF) and multi-path delay commutator (MDC) pipelined structure. The investigation is carried out with variable operating word-lengths to ensure the generality. Furthermore, the main streams to fulfill FFT coefficients weighting, namely, the approach using complex multipliers and the one adopting memoryless CORDIC units, are both considered in the analysis. Based on these derivations, a joint optimization of radix- $2^{k}$ algorithm and operating word-length is discussed to achieve a reasonable trade-off between computational accuracy and hardware expenditure. Simulations and experiments indicates that the derived SQNR is reliable to unfold the quantization effects of fixed-point radix- $2^{k}$ FFT. In addition, the proposed joint optimization strategy is capable of providing better solutions to implement the radix- $2^{k}$ FFT processor efficiently.

This paper describes a GPS (Global Positioning System) signal acquisition method using sample-based shifting and FFT (Fast Fourier Transform). Before the digital IF (Intermediate Frequency) signal is processed with the correlation algorithm, an averaging algorithm is applied to reduce the FFT operation. The principle of sample-based shifting is presented and its influence on the maximum correlation value is also simulated and analyzed. The result of simulation indicates that this method can increase the maximum correlation value and reduce the number of correlation operations.

To detect interharmonics whose amplitudes are often small, a detection method based on ANF and interpolated FFT is proposed. In the method, the parallel structure of the ANF (adaptive notch filter) is firstly used to detect the fundamental and harmonic components, in which the fundamental frequency signal with high amplitude is detected with the main ANF and then the measured base frequency is output to sub-parallel ANF structure so as to extract the harmonic components. Then a separated interpolated fast Fourier transformation (FFT) analysis of interharmonic components will be effectively completed without the influence of the fundamental and harmonic components. A detection platform based on the proposed method is implemented in LabVIEW. The results of simulation and experiments show the effectiveness of the method.

Without the response variation constraint, the main‐lobe responses of wideband beamforming usually vary with the frequencies of interest, and the beam pointing may deviate from the incident angle of the desired signal. If the interference‐plus‐noise covariance matrix (INCM) is directly replaced by the sample covariance matrix, the desired signal in the sample data may result in self‐null problem, which can lead to the degradation of the performance of wideband beamformers. Through frequency division, a novel wideband adaptive beamforming is proposed by reconstructing the INCM and solving the convex optimisation problem. Using fast Fourier transform (FFT), the time‐domain samples are transformed into the frequency field, and the covariance matrix of each subband can be obtained from the frequency‐domain data. In the spatial‐frequency domain, the INCM and signal‐plus‐noise covariance matrix can be respectively reconstructed by integrating the stacked steering vector and the Capon spatial spectrum over the corresponding angular region. Then, we establish the optimisation criteria, that is, minimising the output interference‐plus‐noise power, minimising the main‐lobe spatial response variation, and constraining the response deviation of the desired signal. A novel convex optimisation model can be established, and then the weight coefficients for time‐domain wideband beamforming can be obtained by solving the optimisation problem. The simulation and experimental results demonstrate that the proposed beamformer can provide better performance compared with other tested beamformers and maintain good robustness in various applications.

The analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms [1] with a common fast algorithm similar to the FFT algorithm. This class includes the BInary FOurier REpresentation (BIFORE) transform (BT) [2], the complex BT (CBT) [3], and the discrete Fourier transform (DFT). Expressions for the mean square error (MSE) in the two-dimensional BT, CBT, and FFT are derived. In the case of white input data, the mean square error-to-signal ratio is derived for the multidimensional generalized transforms. The error-to-signal ratio for the one-dimensional FFT derived by Kaneko and Liu is modified with improvement. Some comparisons among BIFORE, DFT, and Haar transforms are also included. The theoretical results for the two-dimensional FFT and BIFORE have been verified experimentally. The experimental results are in good agreement with the theoretical results for lower order sequences, but deviate as the order increases due to the actual manner of rounding in the digital computer.

The effects that anesthesia has on the surgical patient can be characterized with the electroencephalogram (EEG). These effects are typically quantified through frequency analysis of the EEG signal. The fast Fourier transform (FFT) is usually employed for this purpose. In recent years, the fast Walsh transform (FWT) has been proposed as an alternative to the FFT for signal analysis because it is computationally more efficient, requiring less time to complete on a digital computer. This paper statistically evaluates the quantitative and dynamic differences between the results of Walsh and Fourier analysis of the EEG done for the purpose of characterizing the effects of anesthesia. This paper shows that there are quantitative differences between the results of the two frequency analysis techniques but they are dynamically equivalent, both showing essentially the same changes due to the effects of anesthesia. The efficiency of the FWT has been reconfirmed. These results support the use of the FWT in place of the FFT for tracking the effects of anesthesia on surgical patients with the EEG.

The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in $${\cal O}(N^d \log N)$$ arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only $${\cal O}(N \log^d N)$$ arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.

The complex-logarithmic number system (CLNS), which represents each complex point in log/polar coordinates, may be practical to implement the Fast Fourier Transform (FFT). The roots of unity needed by the FFT have exact representations in CLNS and do not require a ROM. We present an error analysis and simulation results for a radix-two FFT that compares a rectangular fixed-point representation of complex numbers to CLNS. We observe that CLNS saves 9–12 bits in word-size for 256–1024 point FFTs compared to the fixed-point number system while producing comparable accuracy. The consequence of the word-size advantage is that the number of full adders required for CLNS is significantly smaller than for an equivalent fixed-point implementation. The major cost of CLNS is the memory, which unlike conventional LNS, is addressed by both real and imaginary parts. Table-reduction techniques can mitigate this. The simplicity of the CLNS approach requires significantly fewer full adders, which pays for some or all of the extra memory. In applications needing the magnitude of the complex parts, such as a power spectrum, the CLNS approach can actually require less memory than the conventional approach.

Optical interferometry plays an important role in the topographical surface measurement and characterization in precision/ultra-precision manufacturing. An appropriate surface reconstruction algorithm is essential in obtaining accurate topography information from the digitized interferograms. However, the performance of a surface reconstruction algorithm in interferometric measurements is influenced by environmental disturbances and system noise. This paper presents a comparative analysis of three algorithms commonly used for coherence envelope detection in vertical scanning interferometry, including the centroid method, fast Fourier transform (FFT), and Hilbert transform (HT). Numerical analysis and experimental studies were carried out to evaluate the performance of different envelope detection algorithms in terms of measurement accuracy, speed, and noise resistance. Step height standards were measured using a developed interferometer and the step profiles were reconstructed by different algorithms. The results show that the centroid method has a higher measurement speed than the FFT and HT methods, but it can only provide acceptable measurement accuracy at a low noise level. The FFT and HT methods outperform the centroid method in terms of noise immunity and measurement accuracy. Even if the FFT and HT methods provide similar measurement accuracy, the HT method has a superior measurement speed compared to the FFT method.

Analog implementations of decoders have been widely studied by considering circuit complexity, as well as power and speed, and their integration with other analog blocks is an extension of analog decoding research. In the front-end blocks of orthogonal frequency-division multiplexing (OFDM) systems, combination of an analog fast Fourier transform (FFT) with an analog decoder is suitable. In this article, the implementation of a 16-symbol FFT processor based on analog complementary metal-oxide-semiconductor current mirrors within circuit and system levels is presented, and the FFT is implemented using a butterfly diagram, where each node is implemented using analog circuits. Implementation details include consideration of effects of transistor mismatch and inherent noises and effects of circuit non-linearity in OFDM system performance. It is shown that not only can transistor inherent noises be measured but also transistor mismatch can be applied as an input-referred noise source that can be used in system- and circuit-level studies. Simulations of a radix-2, 16-symbol FFT show that proposed circuits consume very low power, and impacts of noise, mismatch and non-linearity for each node of this processor are very small.