Efficient Frequency Estimation Algorithm Based on Chirp-Z Transform

Spectrum analysis is very important in geological hazards of infrasonic signal observation systems. The spectrum of infrasonic signal is a dense spectrum which can leads to potential erroneous spectrum analysis. Hereby we propose a dense spectrum analysis algorithm combining all phase Fast Fourier Transform (apFFT) and Chirp Z-transform (CZT) to analyse dense low frequency signal. This is called all phase Chirp Z transform (apCZT). The apFFT spectrum analysis can reduce spectrum leakage, but does not enhance resolution while the CZT vice versa. The novel algorithm apCZT can suppress spectral leakage and improve the resolution at the same time. Simulation results demonstrate that the apCZT algorithm can distinguish the frequencies whose intervals are less than the ordinary frequency resolving power of Discrete Fourier Transform (DFT) and apFFT. The apCZT it is not only suitable for infrasonic signals but also in other dense spectrum analysis applications, such as voice, vibration, noise, electrocardiography, radar signals, power system harmonics and other engineering practice.

Diffraction, Fourier Optics and Imaging

This paper presents a method to derive efficient frequency estimators from the Discrete Fourier Transform (DFT) of the signal. These estimators are very similar to the phase-based Discrete Fourier Spectrum (DFS) interpolators but have the advantage to allow any type of analysis window (and especially non-rectangular windows). As a consequence, it leads to better estimations in the case of a complex tone (cisoid) perturbed by other cisoids. Overall, our best estimator leads to results similar to those of phase vocoder and reassignment estimators but at a lower complexity, since it is based on a single Fast Fourier Transform (FFT) computation.

Abstract. Today Discrete Fourier Transforms (DFTs) are applied in various radio standards based on OFDM (Orthogonal Frequency Division Multiplex). It is important to gain a fast computational speed for the DFT, which is usually achieved by using specialized Fast Fourier Transform (FFT) engines. However, in face of the Software Defined Radio (SDR) development, more general (parallel) processor architectures are often desirable, which are not tailored to FFT computations. Therefore, alternative approaches are required to reduce the complexity of the DFT. Starting from a matrix-vector based description of the FFT idea, we will present different factorizations of the DFT matrix, which allow a reduction of the complexity that lies between the original DFT and the minimum FFT complexity. The computational complexities of these factorizations and their suitability for implementation on different processor architectures are investigated.

Chapter 10 - Discrete and Fast Fourier Transforms (DFT, FFT)

In this paper, a new synthetic frequency estimator of sinusoid signal based on discrete Fourier transform (DFT) amplitudes is proposed. Compared with the existing methods, the proposed method has the advantages of low computational complexity and high precision. The method has two main stages. Firstly, N-point sampled data in the time domain DFT is performed and the coarse estimation is completed by searching the location of the spectrum line with maximum amplitude. Secondly, the synthetic fine estimation is performed by utilizing both the two samples values of Discrete-Time Fourier Transform (DTFT) of the original signal and the maximum amplitude spectrum line and its right or left neighbor. To verify the proposed method, simulation is conducted. The result shows that the root mean square error (RMSE) of frequency estimation of proposed synthetic method is close to the Cramer-Rao Lower Bound, and it outperforms the Candan method. This method is also comparable to Fang method and RCTSL method with zero-padding when conducted under medium and high SNR condition. Moreover, without zeropadding, the proposed method has much lower computational complexity than other known estimators with zero-padding.

A narrowband active vibration control algorithm with frequency estimator based on lattice notch filter is proposed to suppress the primary signal with multi line spectra. A adaptive iir (infinite impulse response) lattice gradient notch filter was adapted to estimated the frequency of the multi-harmonics buried in the primary signal, then the reference signal is filtered by the adaptive band pass filter. With the reference signal filtered by the adaptive band pass filter, active control algorithm is implemented. Simulations of the frequency estimation of the multi-sinusoidal signal are conducted and simulation results confirm that the proposed algorithm could achieve effective performance when the frequency of the primary disturbance is unknown. Active control of vibration has finds a lot application such as fans, engines, and air conditioners in nowadays. The basic concept of active vibration control is to suppress the primary disturbance with the secondary force which is of same amplitude and opposite phase. Signals generated by the rotating machines such as fans, diesel engines, and air conditioners is characterized with discreet frequency which cause damage to operator in the working environment. Narrowband active vibration control (NAVC) is an effective method to suppress such vibration with multi-line spectra. The application of NAVC has been implemented taken into applications since 1970's. A lot of algorithms have been proposed for suppress the primary disturbance with multi-spectral. There exists two families of algorithm in AVC system, which composed of filtered-X least mean square (FXLMS) algorithm and filtered-X recursive least square algorithm (FXRLS). Due to its simplicity and implementation efficiency, FXLMS algorithm is the most popular adaptive algorithm to suppress the primary disturbance compared to FXRLS algorithms. In the application of NAVC systems, the performance of NAVC systems is directly correlated with the reference signal and the primary signal. NAVC algorithm could not achieve effective performance if the correlation between the primary signal and the reference signal is weak. Frequency estimation of the primary signal is an effective method to adapt to improve the relativity between the reference signal and the primary signal. To estimate the frequency of a sinusoidal signal, a number of algorithm has been proposed to exploit the predictable nature of sinusoidal primary disturbance and characterize the frequency of the sinusoidal primary disturbance, such the Kalman algorithm, music, FFT (fast Fourier transform), etc. such method could be capable of tracking nonstationary or slowly-varyin g primary signals. However, they have some shortcomings in the application of NAVC systems. a lot of Kalman-based algorithms have been proposed for the frequency estimation of the sinusoidal signal in many applications such as the electrical power system, however, it requires much more computation burdens which is much more complicated compared the other algorithms. FFT methods have been proposed in the past to estimated the frequency of the primary signal, but it is has been found that the estimated result could not be accessed effectively in real-time of AVC system. Adaptive notch filter has been proposed for the frequency estimation of the sinusoidal signal in a lot of applications. Adaptive lattice filter shows much more stability and performance compared to the bilinear adaptive notch filter. In this paper, the adaptive lattice notch filter is introduced into an adaptive vibration control system to estimate the frequency of synthesized desired signal. Then, the reference signal is generated to suppress the primary vibration in the active vibration control system.

Estimating the frequencies of multiple superimposed exponentials in noise is an important problem due to its various applications in engineering. In order to obtain good inhibition of spectral leakage and improve the estimation accuracy, an improved Chirp-Z transform (CZT) algorithm is proposed for high-precision frequency estimation. Firstly, the proposed algorithm analyzes the characteristics of the CZT spectrum and utilizes the CZT spectrum to construct a bias correction factor for frequency bias estimation. Then, an expression between the bias correction factor and the frequency estimation error is derived to obtain a more accurate estimate of the frequency bias. Finally, the frequency estimate of the CZT is corrected by the estimated frequency bias so as to obtain a higher frequency estimation accuracy. Compared with the conventional CZT algorithm, the proposed improved CZT algorithm achieves a higher frequency estimation accuracy by correcting the frequency estimate of the CZT method using the estimated frequency bias. The proposed improved CZT algorithm is verified using simulation studies and experimental results, and the results show that it has a higher accuracy and better robustness than the existing methods.

This paper revisits the characteristics of windowing techniques with various window functions involved, and successively investigates spectral leakage mitigation utilizing the Welch method. The discrete Fourier transform (DFT) is ubiquitous in digital signal processing (DSP) for the spectrum analysis and can be efficiently realized by the fast Fourier transform (FFT). The sampling signal will result in distortion and thus may cause unpredictable spectral leakage in discrete spectrum when the DFT is employed. Windowing is implemented by multiplying the input signal with a window function and windowing amplitude modulates the input signal so that the spectral leakage is evened out. Therefore, windowing processing reduces the amplitude of the samples at the beginning and end of the window. In addition to selecting appropriate window functions, a pretreatment method, such as the Welch method, is effective to mitigate the spectral leakage. Due to the noise caused by imperfect, finite data, the noise reduction from Welch’s method is a desired treatment. The nonparametric Welch method is an improvement on the periodogram spectrum estimation method where the signal-to-noise ratio (SNR) is high and mitigates noise in the estimated power spectra in exchange for frequency resolution reduction. The periodogram technique based on Welch method is capable of providing good resolution if data length samples are appropriately selected. The design of finite impulse response (FIR) digital filter using the window technique is firstly addressed. The influence of various window functions on the Fourier transform spectrum of the signals is discussed. Comparison on spectral resolution based on the traditional power spectrum estimation and various window-function-based Welch power spectrum estimations is presented.

This paper describes an implementation of the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). We show how the computation of the DFT and IDFT may be performed in Java and show why such operations are typically considered slow. This is a multi-part paper, in part 2, we discuss a speed up of the DFT and IDFT using a class of algorithms known as the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). Part 3 demonstrates the computation of the PSD (Power Spectral Density) and applications of the DFT and IDFT. The applications include filtering, windowing, pitch shifting and the spectral analysis of re-sampling.

The Fast Fourier Transform (FFT) is an efficient algorithm to calculate the Discrete Fourier Transform (DFT), which is often employed in Digital Signal Processing (DSP) and communication. In FFT, complex multiplication and addition units in the butterfly module consume most of the hardware resources. Compared to the addition operation, multiplication is more complicated. In this paper, the multiplier in the complex multiplication unit of the FFT is approximated. Four Radix-4 Booth multipliers with different approximation levels are proposed to reduce the hardware complexity. The pipeline FFT and the parallel FFT based on the proposed approximate multipliers are implemented and extensively evaluated. Compared with the state-of-the-art FFT designs, the LUTs amount is reduced up to 20.3% and 29.1% for pipeline and parallel FFTs, respectively. The power is reduced up to 69.9% for pipeline FFT, and the delay is reduced up to 45.7%. Moreover, the PSNR is reduced by less than 1dB in both pipeline FFT and parallel FFT. Proved by experiment results, the overall performance of the proposed designs is better than FFT designs using other approximate multipliers.

In many applications, when the representation of a discrete-time signal or a system in the frequency domain is of interest, the Discrete-Time Fourier Transform (DTFT) and the z-transform are often used. In the case of a discrete-time signal of finite length, the most widely used frequency-domain representation is the Discrete Fourier Transform (DFT), which is simply composed of samples of the DTFT of the sequence at equally spaced frequency points, or equivalently, samples of its z-transform at equally spaced points on the unit circle. A generalization of the DFT, introduced in this chapter, is the Nonuniform Discrete Fourier Transform (NDFT), which can be used to obtain frequency domain information of a finite-length signal at arbitrarily chosen frequency points. We provide an introduction to the NDFT and discuss its applications in the design of 1-D and 2-D FIR digital filters. We begin by introducing the problem of computing frequency samples of the z-transform of a finite-length sequence. We develop the basics of the NDFT, including its definition, properties and computational aspects. The NDFT is also extended to two dimensions. We propose NDFT-based nonuniform frequency sampling techniques for designing 1-D and 2-D FIR digital filters, and present design examples. The resulting filters are compared with those designed by other existing methods.

To assess the insulation condition of capacitive apparatus, on-line monitoring of dielectric loss angle is an effective measure. Harmonic analysis algorithm is used to calculate dielectric loss angle traditionally, this method can effectively suppress the DC (Direct Current) offset and harmonic interference. However, when the power system frequency fluctuates, the synchronous sampling condition is difficult to meet, so, traditional harmonic analysis method can produce leakage of spectrum and picket-fence effect, which greatly affects the accuracy of dielectric loss angle measurement. In order to reduce this adverse effect, this paper proposes a method called hamming windowed interpolated DFT (Discrete Fourier Transform) algorithm based on modified ideal sampling frequency. Time series analysis based on modified ideal sampling frequency can be used to modify the time series in non-synchronous sampling condition to meet synchronous sampling condition. Hamming windowed interpolated DFT algorithm can inhibit leakage of spectrum and picket-fence effect. Firstly this paper analyzes the principle of traditional DFT algorithm and the reason that causes leakage of spectrum and picket-fence effect in non-synchronous sampling condition. Secondly, the principle of hamming windowed interpolated DFT algorithm based on modified ideal sampling frequency is elaborated. Finally this paper presents the simulation result which shows that the improved approach has higher accuracy.

The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.

This work studies the spectrum of discrete-time Uniform-sampling pulse width modulation (UPWM) signals originating from stochastic input signals. It demonstrates that for ergodic input sequences of independent and identically distributed random variables, the Discrete Fourier Transform (DFT) of the UPWM signals can be directly estimated from the input signal’s statistics. Consequently, it is shown that if the input signal can be modeled as such a random sequence, only statistical information of the sequence is required for the accurate estimation of the DFT of the UPWM signal. This is achieved here by proving that the DFT estimators obtained by observation of the input sequence within a time window are consistent estimators of the DFT coefficients of the underlying random process. Moreover, for signals whose generalized probability density functions can be expressed as functions of a small number of parameters, the DFT coefficients can be estimated or even calculated via closed-form expressions with linear complexity. Examples are given for input signals derived from symmetric and asymmetric distributions. The results are validated by comparison with evaluations of the UPWM signal’s DFT via the Fast Fourier Transform (FFT). The proposed method provides a mathematical framework for the analysis and design of UPWM systems whose inputs have known statistical properties.