Digital Signal Processing Reference
In-Depth Information
3.
Draw the transformed
t
-axis directly below the
t-axis.
4.
Plot
y(t)
on the
t
-axis.
Three transformations of time (the independent variable) have been de-
scribed. Three equivalent transformations of the amplitude of a signal (the depen-
dent variable) are now defined.
We now consider signal-amplitude transformations. One application of these
transformations is in the amplification of signals by physical amplifiers. Some
amplifiers not only amplify signals, but also add (or remove) a constant, or dc,
value. A second use of amplitude transformations is given in Chapter 4, in appli-
cations of Fourier series. Amplitude transformations follow the same rules as
time transformations.
The three transformations in amplitude are of the general form
y(t) = Ax(t) + B,
(2.8)
where
A
and
B
are constants. For example, consider The value
yields amplitude reversal (the minus sign) and amplitude scaling
and the value shifts the amplitude of the signal. Many physical
amplifiers invert the input signal in addition to amplifying the signal. (The gain is
then a negative number.) An example of amplitude scaling is now given.
y(t) =-3x(t) - 5.
A =-3
( ƒAƒ = 3),
B =-5
Amplitude transformation of a signal
EXAMPLE 2.2
Consider the signal of Example 2.1, which is shown again in Figure 2.5(a). Suppose that this
signal is applied to an amplifier that has a gain of 3 and introduces a bias (a dc value) of
as shown in Figure 2.5(b). We wish to plot the amplifier output signal
-1,
y(t) = 3x(t) - 1.
We first plot the transformed amplitude axis, as shown in Figure 2.5(a). For example, when
Figure 2.5(c)
x(t) = 1, y(t) = 2.
shows the desired plot of the transformed signal
■
y(t).
Time and amplitude transformation of a signal
EXAMPLE 2.3
Next we consider the signal
t
2
y(t) = 3x
¢
1 -
≤
- 1,
