bdb@becker.UUCP (Bruce Becker) (06/09/90)
In article <1990Jun6.170458.25618@athena.mit.edu> tldavis@athena.mit.edu (Timothy L. Davis) writes: |In article <48870@seismo.CSS.GOV> black@beno.CSS.GOV (Mike Black) writes: |>I always though combining two samples was not possible in the time domain. |>If you add two opposite phase sine waves you'll get a null response. It's |>necessary to do a Fourier Transform to the frequency domain, add the |>amplitudes, and invert the transform. This will give you twice the amplitude |>of the original two sine waves (this would seem to be the desired effect). |>Am I totally off-base in this rather long-held belief? |>Mike... | |I'm afraid you are, Mike. First of all, the Fourier transform is a linear |operator: F[a x(t) + b y(t)] = a F[x(t)] + b F[y(t)]. Thus adding in the |frequency domain has the same result as adding in the time domain. Linearity |holds for continuous, discrete, and mixed versions of the Fourier transform and |Fourier series. | |Second, two opposite-phase sinusoids SHOULD cancel each other. Have you |ever played a wind instrument? While tuning, the rhythmic beating of the |blended sound of two horns is the result of the sinusoids going in and out of |phase. When the beating stops, you are in tune (same frequency). If you are |more careful, you can play a long note to be both in tune and in phase with |another player (w.r.t. a particular point in space), so that the volume of |the summed sound waves of your two horns is near maximal. Of course, there |are harmonics generated in the horn and the phase difference of the fundamental |depends on the position of the instruments and the listener and the room |acoustics, but my basic tenent remains that the sound pressure levels |generated by each instrument can be algenraically summed to give the sound |which would be produced by the instruments playing together. In yet another instance of the interplay of art and technology, I recommend you investigate the work "Still and Moving Lines of Silence in Families of Hyperbolas", by Alvin Lucier. This piece, composed in 1983, is recorded on an album of the same name, on Lovely Records. A quote from the record jacket: "Still and Moving Lines of Silence in Families of Hyperbolas is based upon interference phenomena between two or moresound waves. When closely tuned musical tones are sounded, audible beats - bumps of loud sound produced as the sound waves coincide - occur at speeds determined by the difference between the pitches of the tones. The larger the difference, the faster the beating. At unison, bo beating occurs. Furthermore, if each tone originates from a separate source, the beats spin in elliptical patterns through space, from the higher source to the lower one." -- ,u, Bruce Becker Toronto, Ontario a /i/ Internet: bdb@becker.UUCP, bruce@gpu.utcs.toronto.edu `\o\-e UUCP: ...!uunet!mnetor!becker!bdb _< /_ "I still have my phil-os-o-phy" - Meredith Monk