View Full Version : Linear Combinations: sound sampling and mixers

04-04-2007, 12:12 AM
I have been looking at some past scholarship papers and had been scratching my head over this problem from quite a while. It was hinted that a similar question to this will be in the scholarship exam. Any help would be greatly appreciated as I am sitting for my papers in 2 days. The question is as follows:

High fidelity sound can recorded digitally by sampling a sound wave at the rate of 44,100 times a second. Thus a 10-second segment of sound can be represented by a vector in R441000. A sound technican at a jazz festival plans to record sound vectors with 2 microphones, 1 sound vector s from a microphone next to the saxophone player, and a second concurrent sound vector g from a microphone next to the guitar player. A linear combination of the two sound vectors will then be created by a "mixer" in a studio to produce the desired result.

Suppose that each microphone picks up all of the sound from its adjacent instrument, so the actual recorded vectors are u=s+0.06g for the saxophone, and v=g+0.12s for the guitar.

a) What linear combinations of u and v will recover the saxophone vector s?
b) What linear combinations of u and v will recover the guitar vector g?
c) What linear combination will produce an equal mix of s and g, that is 0.5(s+g)?


04-04-2007, 12:23 AM
For example, (a) asks to find scalars a, b such that au + av = s. That is, a*(s + 0.06g) + b*(g + 0.12s) = s. This must hold for all s, g so we may equate coefficients of s and g to solve for a and b.