Controlling Amplitude

Perhaps the most frequently used operation on electronic sounds is to change their amplitudes. For example, a simple strategy for synthesizing sounds is by combining sinusoids, which can be generated by evaluating the formula on Page , sample by sample. But the sinusoid has a constant nominal amplitude $a$, and we would like to be able to vary that in time.

Figure 1.4: The relationship between ``MIDI" pitch and frequency in cycles per second (Hertz). The span of 24 MIDI values on the horizontal axis represents two octaves, over which the frequency increases by a factor of four.

In general, to multiply the amplitude of a signal $x[n]$ by a factor $y \ge 0$, 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.