The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

Vocal Behavior and Vocal Loading Factors for Preschool Teachers at Work Studied with Binaural DAT Recordings

Preschool teachers are at risk for developing voice problems such as vocal fatigue and vocal nodules. The purpose of this report was to study preschool teachers' voice use during work. Ten healthy female preschool teachers working at daycare centers (DCC) served as subjects. A binaural recording technique was used. Two microphones were placed on both sides of the subject's head, at equal distance from the mouth, and a portable DAT recorder was attached to the subject's waist. Recordings were made of a standard reading passage before work (baseline) and of spontaneous speech during work. The recording technique allowed separate analyses of the level of the background noise, and of the subjects' voice sound pressure level, mean fundamental frequency, and total phonation time. Among the results, mean background noise level for the ten DCCs was 76.1 dBA (range 73.0-78.2), which is more than 20 dB higher than what is recommended where speech communication is important (50-55 dBA). The subjects spoke on an average of 9.1 dB louder (p < 0.0001), and with higher mean fundamental frequency (247 Hz) during work as compared to the baseline (202 Hz) (p < 0.0001). Mean phonation time for the group was 17%, which was considered high. It was concluded that preschool teachers do have a highly vocally demanding profession. Important steps to reduce the vocal loading for this occupation would be to decrease the background noise levels and include pauses so that preschool teachers can rest their voices.