Editing Additive synthesis (section)

==Discrete-time equations==
In digital implementations of additive synthesis, [[discrete signal|discrete-time]] equations are used in place of the continuous-time synthesis equations.  A notational convention for discrete-time signals uses brackets i.e. <math>y[n]\,</math> and the argument <math>n\,</math> can only be integer values.  If the continuous-time synthesis output <math>y(t)\,</math> is expected to be sufficiently [[bandlimited]]; below half the [[sampling rate]] or <math>f_\mathrm{s}/2\,</math>, it suffices to directly sample the continuous-time expression to get the discrete synthesis equation. The continuous synthesis output can later be [[Nyquist&ndash;Shannon sampling theorem|reconstructed]] from the samples using a [[digital-to-analog converter]]. The sampling period is <math>T=1/f_\mathrm{s}\,</math>.

Beginning with ({{EquationNote|3}}),

: <math>y(t) = \sum_{k=1}^{K} r_k(t) \cos\left(2 \pi \int_0^t f_k(u)\ du + \phi_k \right)</math>

and sampling at discrete times <math> t = nT = n/f_\mathrm{s} \,</math> results in

:<math> \begin{align}
 y[n] & = y(nT) = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \int_0^{nT} f_k(u)\ du + \phi_k \right) \\
      & = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} \int_{(i-1)T}^{iT} f_k(u)\ du + \phi_k \right) \\
      & = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} (T f_k[i]) + \phi_k \right) \\
      & = \sum_{k=1}^{K} r_k[n] \cos\left(\frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \right) \\
       \end{align} </math>
where

: <math>r_k[n] = r_k(nT) \,</math> is the discrete-time varying amplitude envelope
: <math>f_k[n] = \frac{1}{T} \int_{(n-1)T}^{nT} f_k(t)\ dt \,</math> is the discrete-time [[Finite difference|backward difference]] instantaneous frequency.

This is equivalent to

: <math> y[n] = \sum_{k=1}^{K} r_k[n] \cos\left( \theta_k[n] \right) </math>

where

:<math> \begin{align}
    \theta_k[n] &= \frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \\
                &= \theta_k[n-1] + \frac{2 \pi}{f_\mathrm{s}} f_k[n] \\
       \end{align} </math> for all <math>n>0\,</math><ref name="RodetDepalle_FFTm1"/>
and
: <math> \theta_k[0] = \phi_k. \,</math>