Introduction to Computer Music: Volume One

6. Principles of Audio-rate Frequency Modulation | page 4

Reflected sidebands

What happens to the negative values that we inevitably compute for lower sidebands past a certain integer multiple (i.e. is there such a thing as a 'negative' frequency)? Using our two methods of calculating sidebands, say we had a carrier ƒ of 200 Hz and a modulator ƒ of 400 Hz -- that would give us our 1:2 C:M ratio. If we calculate the n=1 pair, we get an upper sideband of 600, but a lower sideband of -200 using our first method, or a relative frequency of -1 to the carrier. These sidebands in the negative domain are called reflected sidebands, because they bounce back from zero at their absolute value 180 degrees out of phase with their sideband partners. So for both methods of calculating frequency, we simply remove the minus sign, expressed mathematically as absolute or |-200|. If these frequencies do not bounce back on top of other frequencies, then the phase reversal is inaudible. However, as is particularly true in harmonic spectra, when they do bounce back on top of other partials, phase cancellation or summation has a great impact on the timbre. For example, if you had a positive sideband at 400 Hz and a negative sideband at 400 Hz but half the strength of the positive one, only half the amplitude of the positive one will survive. If both were at equal strength, neither would be heard since they would completely cancel each other out. If they were both positive or both negative, they would be summed. In our example above of a Cƒ=200 Hz and Mƒ=400 Hz, the lower sideband of the n=1 pair would reflect back on the carrier frequency (n=0), or |C-M|=C (|-1|=1). Who will survive will be a mystery to be solved below when we can calculate the relative strength of each.

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