Controlling Amplitude

Conceptually at least, the simplest strategy for synthesizing sounds is by combining SINUSOIDS, which can be generated by evaluating the formula from section 1.1, sample by sample. The real sinusoid has a constant nominal amplitude $a$, and we would like to be able to vary that in time.

In general, to multiply the amplitude of a signal $x[n]$ by a constant $y \ge 0$, you can just multiply each sample by $y$, giving a new signal $y \cdot x[n]$. Any measurement of the RMS or peak amplitude of $x[n]$ will be greater or less by the factor $y$. More generally, you can change the amplitude by an amount $y[n]$ which varies sample by sample. If $y[n]$ is nonnegative and if it varies slowly enough, the amplitude of the product $y[n] \cdot x[n]$ (in a fixed window from $M$ to $M+N-1$) will be related to that of $x[n]$ by the value of $y[n]$ in the window (which we assume doesn't change much over the $N$ samples in the window).

In the more general case where both $x[n]$ and $y[n]$ are allowed to take negative and positive values and/or to change quickly, the effect of multiplying them can't be described as simply changing the amplitude of one of them; this is considered later in chapter 5.