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We have seen that all periodic functions may be represented in the form 
.
This is the time domain model for the signal as it expresses the amplitude as
a function of time. Using the identity sin(A+B) = sin(A)cos(B) + sin(B)cos(A),
we may express 
Thus we may express all periodic functions in the form
.
Using eiθ = cos(θ) + i sin(θ)
and e-iθ = cos(-θ) + i sin(-θ) = cos(θ)
- i sin(θ), we may derive cos(θ) = ½(eiθ
+ e-iθ) and sin(θ) = ½(eiθ
- e-iθ). By substitution: 
,
where c0=a0, for k>0 ck=½(ak-ibk),
and for k<0 ck=½(ak+ibk).
Positive frequencies correspond to anticlockwise rotating vectors
and negative frequencies to clockwise rotating vectors. The coefficients ck
can be obtained from the equation: 
,
where T is the fundamental period of the signal; T = 2π/ω.
General (non-periodic) signals may be represented in the form
where 
in angular frequency; or 
,
in frequency. H(ω) (or sometimes G(f) ) is called the
Fourier transform of h(t).
We note that this complex function model for audio signals is only valid as a vector space; the linear combinations of signals perserve the model but products do not.
The fourier transform H(ω) is the angular frequency domain model for the signal h(t) as the domain is the angular frequency and value is the energy present at that frequency. The function G(f) is the frequency domain model for the signal h(t).
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