The Effects of Time Weighting the Input to a Spectrum Analyzer

Abstract

The effects of time weighting the input to a spectrum analyzer are considered. The spectrum analyzer is assumed equivalent to a bank of similar filters whose overlapping passbands cover a region of spectral interest and whose outputs are measured at a single instant in time. The term transient selectivity is defined as the magnitude of the envelope of the response of a filter with center frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega + \Delta \omega</tex> , at the time instant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> , to an input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e^{j \omega t}</tex> over the interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 \leq t \leq T</tex> . The transient selectivity characteristics of an arbitrary bank of filters can be achieved using a bank of single-tuned filters and time varying the magnitude of the envelope of the input to the filter bank system. This property is called the time-weighting principle. Spectrum analyzer systems are employed to detect the presence and specify the frequency of fixed-duration constant-frequency signals immersed in noise. For a system whose objectives include distinguishing the presence of several simultaneous signals, two measures of system performance are the sharpness of the system's transient selectivity and the system's signal-to-noise ratio (SNR). Time weighting functions are presented which can realize arbitrarily sharp transient selectivity characteristics. An equation for the snr for arbitrary filters and arbitrary time weighting functions is also presented. The transient selectivity and snr performance for a bank of filters time weighted with a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sine^{N} \pi t/T</tex> weighting function is illustrated.