Chapter 7 describes a simple extensible method of generating partially correlated narrow bands of noise by introducing a time offset between the bands. An alternative mathematical method is described here. It is possible to interpolate between the phases and amplitude of each component of the band of noise. This can be denoted in the following way:
i = Index of sinusoidal component (0£i£200).
x = Number denoting a certain parent band (0£x).
p = Factor of correlation (0£p£1)
p = 0 Þ Completely uncorrelated
p = 1 Þ Fully correlated
A(S)i = Amplitude of ith component of signal band.
f(S)i = Starting phase of ith component of signal band (0£f<2p).
A(M)i = Amplitude of ith component of masking band.
f(M)i = Starting phase of ith component of masking band (0£f<2p).
A(x)i = Amplitude of ith component of xth band (Rayleigh distribution).
f(x)i = Starting phase of ith component of xth band (0£f<2p).
So, for a correlated maskers:
A(S)i = A(0)i f(S)i = f(0)i
A(M)i = A(0)i f(M)i = f(0)i
i.e. A(M)i = A(S)i f(M)i = f(S)i
For entirely uncorrelated maskers:
A(S)i = A(0)i f(S)i = f(0)i
A(M)i = A(1)i f(M)i = f(1)i
i.e. A(M)i ¹ A(S)i f(M)i ¹ f(S)i
This allows the possibility of partial correlation:
A(S)i = A(0)i f(S)i = f(0)i
A(M)i = p.A(0)i + (1-p).A(1)i f(M)i = p.f(0)i + (1-p).f(1)i (1)
As a phase of 0 is equivalent to 2p, the phase calculation is an over-simplification and would not produce the correct results. A better way is firstly to calculate the value of | f(0)i - f(1)i | and then recalculate it when 2p is added to the smaller of f(0)i and f(1)i . If the value is smaller when 2p has been added, then 2p should be permanently added for the purposes of the
original calculation (1).
For example:
f(0)i = f/4 f(1)i = 6p/4
| f(0)i - f(1)i | = 5f/4 (2)
Add 2p to smaller of f(0)i and f(1)i:
f(0)i = 9p/4 f(1)i = 6p/4
| f(0)i - f(1)i | = 3p/4 (3)
(3) is smaller than (2), therefore f(0)i should be assumed to be 9p/4 for the purposes of calculation (1).