Additive Synthesis - Discrete-time Equations

Discrete-time Equations

In digital implementations of additive synthesis, discrete-time equations are used in place of the continuous-time synthesis equations. A notational convention for discrete-time signals uses brackets i.e. and the argument can only be integer values. If the continuous-time synthesis output is expected to be sufficiently bandlimited; below half the sampling rate or, it suffices to directly sample the continuous-time expression to get the discrete synthesis equation. The continuous synthesis output can later be reconstructed from the samples using a digital-to-analog converter. The sampling period is .

Beginning with (3),

and sampling at discrete times results in

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

where

is the discrete-time varying amplitude envelope

is the discrete-time backward difference instantaneous frequency.

This is equivalent to

where

$\begin{align} \theta_k &= \frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k + \phi_k \\ &= \theta_k + \frac{2 \pi}{f_\mathrm{s}} f_k \\ \end{align}$ for all