Otis M. Solomon

The use of DFT windows in signal-to-noise ratio and harmonic distortion computations

The discrete Fourier transform (DFT) can be used to compute the signal-to-noise ratio (SNR) and harmonic distortion of a waveform recorder. When the data record contains a non-integer number of cycles of the sine wave, energy leaks from the sine wave and its harmonics to adjacent frequencies. A.L. Benetazzo et al. (1992) describe a windowed DFT method for computing the RMS value of a sine wave from the magnitude of the main lobe of its DFT and recommend the use of minimum energy windows. We present criteria for choosing the DFT window. A constraint for the window coefficients is derived to insure that quantization error does not influence the estimate of the amplitude of a sine wave from the main lobe of its DFT. >

Bounds on least-squares four-parameter sine-fit errors due to harmonic distortion and noise

Least-squares sine-fit algorithms are used extensively in signal processing applications. The parameter estimates produced by such algorithms are subject to both random and systematic errors when the record of input samples consists of a fundamental sine wave corrupted by harmonic distortion or noise. The errors occur because, in general, such sine-fits will incorporate a portion of the harmonic distortion or noise into their estimate of the fundamental. Bounds are developed for these errors for least-squares four-parameter (amplitude, frequency, phase, and offset) sine-fit algorithms. The errors are functions of the number of periods in the record, the number of samples in the record, the harmonic order, and fundamental and harmonic amplitudes and phases. The bounds do not apply to cases in which harmonic components become aliased.<<ETX>>

The effects of windowing and quantization error on the amplitude of frequency-domain functions

The author describes how windows modify the magnitude of a discrete Fourier transform and the level of a power spectral density computed by Welchs method. For white noise, the magnitude of the discrete Fourier transform at a fixed frequency has a Rayleigh probability distribution. For sine waves with an integer number of cycles and quantization noise, the theoretical values of the amplitude of the discrete Fourier transform and power spectral density are calculated. The authors show the signal-to-noise ratio in a single discrete Fourier transform or power spectral density frequency bin is related to the normal time-domain definition of the signal-to-noise ratio. The answer depends on the discrete Fourier transform length, the window type, and the function averaged. >

Windows modify the amplitude of frequency domain functions

The discrete Fourier transform and power spectral density are often used in analyzing data from analog-to-digital converters. These analyses normally apply a window to the data to alleviate the effects of leakage. It is shown how windows modify the magnitude of a discrete Fourier transform and the level of a power spectral density computed by Welchs method. For white noise, the magnitude of the discrete Fourier transform at a fixed frequency has a Rayleigh probability distribution. For sine waves with an integral number of cycles and quantization noise, the theoretical values of the amplitude of the discrete Fourier transform and power spectral density are calculated. It is demonstrated that the signal-to-noise ratio in a single discrete Fourier transform or power spectral density frequency bin is related to the normal time-domain definition of the signal-to-noise ratio. The answer depends on the discrete Fourier transform length, the window type and the function averaged.<<ETX>>

Rational parametric coherence estimation via convolved correlations

In this paper, the magnitude squared (MS) coherence is computed by estimating the parameters of a rational model. The parameters are constrained so that the estimated MS coherence is real-valued on the unit circle. The method entails first estimating the auto- and cross-correlation lags from raw data sequences. These lag estimates are then used to define two auxiliary sequences, the convolution of the cross-correlation function with itself and the convolution of the two autocorrelation functions. The MS coherence parameters will nearly satisfy a homogeneous set of equations involving these auxiliary sequences. This system of linear equations is solved via an eigenspace decomposition. The algorithm is compared with two traditional periodogram based estimation methods.

Estimation of time-varying coherence functions

Two methods for computing an estimate of the time-varying magnitude squared (MS) coherence between two data sequences are described. The first method estimates the time-varying parameters of two autoregressive moving average (ARMA) transfer functions, whose product approximates the time-varying MS coherence function. The ARMA parameters are computed by an exponentially weighted recursive least squares algorithm. The second method estimates the time-varying MS coherence function by substituting time-varying estimates of the auto- and cross-spectra into usual the definition of the MS coherence function. The time-varying spectra are obtained by bandpass filtering, squaring and averaging operations.