Have you ever used a gong as a reverb unit, or listened to a flock of birds singing a violin arpeggio? Did you ever wonder what sound you’d get if you could pour water through a cymbal? What if Howlin’ Wolf had recorded inside a motorcycle engine instead of the Chess Records studio?
Don’t worry, EM has not been taken over by a gang of surrealist poets. But a poet might actually be helpful in evoking the strange and wonderful quality of sonic hybrids (like those I mentioned) that you can produce through convolution. When two signals are convolved, their spectra are multiplied. The output signal partakes of the timbral and temporal attributes of both sources, and convolution coils the signals together inextricably.
Engineers have known convolution as a fundamental operation of digital signal processing (DSP) for decades. However, information about convolution hasn’t yet reached many musicians outside the academic and research communities. We hope to change that a bit, because many of its applications—including reverberation and other spatial effects, filtering, and cross-synthesis—are of interest to electronic musicians.
Convolution and Spectrum Multiplication
Strictly speaking, the term convolution refers to a sample-by-sample operation on two signals; this is called direct convolution. I won’t discuss the details of direct convolution, because it is seldom, if ever, used in the real world and is terribly inefficient. Instead, convolution software usually implements an analysis/resynthesis process called spectrum multiplication.
Spectrum multiplication is mathematically equivalent to direct convolution. The process begins with a fast Fourier transform (FFT) analysis of the spectra of two input signals. The analyzed spectra are then multiplied. Finally, the output signal is resynthesized through a process called inverse FFT (IFFT). This may sound like a lot of computation, but a modern computer processor can scream through a lengthy spectrum multiplication in almost no time.
When the spectra of two signals are multiplied, like frequencies reinforce each other, while unlike frequencies weaken or disappear. This effect is called spectral intersection. In an effective convolution, the two input signals should have at least some energy in a common frequency range. If you convolve piano and clarinet samples, both at middle C, the spectra will have many common frequencies. Odd-numbered harmonics, which are abundant in the clarinet tone, will be strongly reinforced in the output spectrum.
But if you convolve the highest note on a piano with the lowest note of a bass clarinet, you’ll get a rather faint signal because only the clarinet’s weak upper harmonics will intersect with the piano note’s spectrum. Looking at it in another way, you could say that the piano spectrum had “filtered out” the bass clarinet’s fundamental and lower harmonics. Convolution is, in fact, intimately tied to filtering, as you’re about to see.
For the purposes of this article, spectrum multiplication and convolution can be considered as synonymous. From now on, I’ll stick to the simpler term convolution. This is an excerpt from the following article: Convolution Number Nine.
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