The chirp z-transform algorithm-a lesson in serendipity

t was the fall of 1968. I had finished my Ph.D. degree on speech synthesis at MIT Iin May of 1967 and started working as a member of the technical staff in Jim Flanagan’s group at AT&T Bell Laboratories in May of 1967. My initial assign­ ment was working on problems of speech analysis and synthesis using a serial formant speech synthesizer. Ron Schafer joined Jim Flanagan’s group at Bell Laboratories in the spring of 1968, having completed his Ph.D. at MIT in the area of homomorphic signal processing and its applications to speech process­ ing. Ron and I started working together almost immediately on the problem of formant estimation using homomorphic methods. The goal was to compute a smooth speech spectrum using homomor­ phic filtering methods and then to estimate the relevant formant fre­ quencies by peak picking the homo­ morphically smoothed spectrum. Most of the time this algorithm worked well; however once in awhile the formants were either too close to each other to resolve reli­ ably or had bandwidths that were just too high to find from simple peak picking methods. Ron and I investigated a number of speech enhancement methods with the goal of making formant estimation be more reliable and robust. None of the proposed methods worked as well as we would have liked. Next came the role of serendipity. I was attending an IEEE conference in New York City in the fall of 1968 when I ran into Charlie Rader from MIT Lincoln Laboratory. Charlie had become a good friend through our joint association with Ben Gold of Lincoln Laboratory and through our joint membership in the IEEE Signal Processing Group, a subgroup of IEEE’s group on audio and elec­ troacoustics. Charlie asked me what problems I was working on, and I brought up the issue of formant esti­ mation from homomorphically smoothed spectra and the problems that Ron and I faced as we tried to enhance the speech signal by appro­ priate signal weighting to reduce the formant bandwidth. Charlie asked me to define what I would consider to be the ideal solution to this prob­ lem, and after a bit of thought I told him that what we needed was a sig­ nal enhancement method that changed the unit circle in the z-plane (the usual curve for evaluating the spectrum of a real signal) into a spiral curve that increased gradually from the origin of the unit circle to a circle that curved gradually into the inside of the unit circle. Charlie told me that he had just heard a lecture at Sylvania Labs where Leo Bluestein had been working with chirp signals to convert a discrete Fourier trans­ form (DFT) for nonpower of two signals into an efficient convolution that could be computed using power of two DFTs. Charlie realized imme-