Leland B. Jackson

Frequency estimation by linear prediction

The application of linear prediction to frequency estimation for sinusoidal signals in noise is investigated. It is shown that improved performance is obtained by processing a complex-valued version of the real-valued input signal, with the corresponsing sampling rate reduced by one-half. The case of a single sinusoid in white noise is studied in detail, including the eigenvalues of the covariance matrix, zeros of the inverse filter polynomial, frequency bias, and frequency variance as a function of input SNR and prediction order.

Frequency and bearing estimation by two-dimensional linear prediction

Simultaneous frequency and bearing estimation using 2-D spectral analysis of the space-time data array is investigated. The spectral estimates are generated using 2-D linear prediction. It is shown that single-quadrant prediction can lead to severe asymmetry and bias in the estimated spectra; while a certain combination of the results for two adjacent quadrants yields well-behaved spectral estimates.

An improved Martinet/Parks algorithm for IIR design with unequal numbers of poles and zeros

The Martinet/Parks (1978) algorithm for equiripple IIR filters with unequal numbers of poles and zeros has been modified to converge for a wider range of designs, including an important extraripple case. Several designs are compared with respect to complexity, coefficient sensitivity, and group delay. The MATLAB program implementing the algorithm is available by e-mail. >

Linear prediction in cascade form

The autocorrelation and covariance methods of linear prediction are formulated in terms of an inverse digital filter in cascade form, rather than the traditional direct form, to allow pole locations in the system model to be readily estimated and constrained. Iterative solution of the corresponding nonlinear normal equations is described. Applications to speech analysis and the compensation of biomedical signals are briefly discussed.

Discrete Fourier Transform

In chapter 6, we investigated the definition and properties of the discrete-time Fourier transform X(e jω ), with ω being a continuous frequency variable, and found it to be very useful for analyzing a wide variety of signals and systems of theoretical interest. However, much of the practice of digital signal processing is done in computers where we cannot evaluate a continuum of frequencies ω, nor can we input and store an infinite-duration sequence x(n). Hence, for actual data sequences, as opposed to theoretically defined signals, we cannot compute the Fourier transform, in general.

FIR Filter Design Techniques

As we saw in section 5.3, an FIR filter is easily constrained to have one of two particularly useful properties: namely, linear-phase or linear-plus-90°-phase response corresponding to even or odd symmetry, respectively, in its impulse response. Therefore, techniques for the design of FIR filters are of considerable interest. The transformation techniques of the preceding chapter are not applicable to FIR design because, in general, they produce filters with poles, as well as zeros, and thus with infinite-duration impulse responses. The four general techniques most commonly employed for FIR design are described in this chapter.

A correction to impulse invariance

The classic equations for discrete time filter design by impulse invariance are not correct, in general. In particular, if the impulse response of the causal continuous-time filter is discontinuous at t=0, then the initial sample h[0] of the discrete-time impulse response should equal one-half the value of the discontinuity, not the full value as classically given. Otherwise, the resulting frequency response does not equal the sum of shifted versions of the original response as expected. This effect will typically be seen only in low order filters such as first order lowpass filters or second order bandpass filters.

On the relationship between digital Hilbert transformers and certain low-pass filters

Designs of symmetric Hilbert transformers are shown to be easily derived from corresponding designs for symmetric half-band low-pass filters, and vice versa. The latter filter type is particularly useful for interpolation or smoothing in sample-rate alteration.

Iterative Method for Nonlinear FM Synthesis of Radar Signals

The problem of synthesizing a time-domain signal with a given energy spectral density (ESD) often arises in the field of signal processing. Many solutions have been proposed and successfully used over the years. However the problem of synthesizing a time-domain signal with constraints for a given ESD has not been investigated sufficiently. We propose a solution to one such constraint where the amplitude of the complex-valued time-domain signal is required to be unity. This is equivalent to phase modulating a unit amplitude signal, such that its ESD matches a desired energy spectral density. We provide an algorithm for this solution and apply it to a real problem encountered in radars.

Frequency-domain Steiglitz-McBride method for least-squares IIR filter design, ARMA modeling, and periodogram smoothing

The classic Steiglitz-McBride (mode-1) time-domain algorithm for least-squares approximation of desired impulse responses for IIR digital filters or ARMA signal models is reformulated in the frequency domain to allow the direct least-squares approximation of either complex-valued or magnitude-only frequency responses, as well as power-density spectra, including periodograms. The resulting (stable) designs in the complex-valued case with both magnitude- and phase-response specifications can be either causal or noncausal, as appropriate to the phase, while the magnitude-only designs can always be made causal and minimum-phase. The periodogram models provide effective spectral smoothing without the need for averaging of data blocks, although averaging can be used, if desired, to reduce the computation. The filter coefficients can be either real- or complex-valued, corresponding to conjugate-symmetric or asymmetric frequency responses, respectively.