A Real-Time Logarithmic-Frequency Phase Vocoder

Abstract

The phase vocoder, introduced over 20 years ago (Flanagan and Golden 1966), has been used musically for over 10 years (Moorer 1978); a review has been recently presented (Dolson 1986). A vocoder decomposes, in time, a complicated signal into simpler constituent elements. By this process of decomposition, it is possible to study the original signal, or to modify it, in ways that are not possible with the original, undecomposed signal. Vocoders traditionally use different frequency components as the means to separate the signal. This is based on the knowledge that the human ear has a set of filters in the cochlea that perform a frequency filtering function in real-time. Also, frequency filtering is readily accomplished with a filterbank. A filterbank used for the decomposition of a signal consists of a set of N bandpass filters which produce N output signals from one input signal. Figure 1 is a block diagram of this process. A similar representation in common use is called the Short-Time Fourier Analyzer (Portnoff 1980). This may be equivalent to a set of filters forming a filterbank in which the center frequencies are chosen linearly. A phase vocoder has an additional function. If one builds a bandpass filter from inductors, capacitors, and resistors, then the input and the output signals both are real. However, the output signal from a bandpass filter is a narrow-band bandpass signal, and could be considered the real part of a complex signal with both real and imaginary parts, or a carrier modulated in amplitude and phase. Knowledge of the frequency of the carrier, and the amplitude and phase modulation, allows recovery of the original real signal. Since the carrier frequency is the center frequency of the bandpass filter, the time-varying quantities are the amplitude and phase modulation. The time-varying signal with the amplitude and phase modulation is a complex modulation. A phase vocoder may be represented in at least two different ways. The output could be either the product of the complex modulation and the complex carrier, or a complex signal consisting of the complex modulation alone. The first method includes a complex transfer function creating a complex bandpass signal from a real input signal. The output from this filter has the carrier, and the amplitude and phase modulation, combined. This complex filter has a bandpass filtering function and suppresses negative input frequencies as well. The second approach results only in the amplitude and phase modulation, and the carrier is suppressed. This consists of a complex filter followed by a modulator, or a complex modulator followed by a complex lowpass filter. The modulation process is called heterodyning (Dolson 1986). These two approaches are shown in Fig. 2. In the phase vocoder implemented for this article, the first approach of retaining the carrier, and the amplitude and phase modulation is used, simply on the grounds that it is slightly easier to achieve. Signal filters can be characterized in many ways (Oppenheim and Schaefer 1975). However, the frequency response is useful to understand the functioning of the filter. A complex sinusoid is applied and the carrier frequency is swept across the frequencies of interest. The amplitude and phase of Computer Music Journal, Vol. 15, No. 1, Spring 1991, C 1991 Massachusetts Institute of Technology.