1.3 Continuous-Time Signals 71
n Complex, composed of two sinusoids of frequency=π/2 rad/sec, phaseπ/4 in rad, and
amplitude
√
2 in 0≤t≤10, and it is zero outside that time interval. n
nExample 1.3
Consider the pulse signal
p(t)= 1 0 ≤t≤ 10
and zero elsewhere. Characterize this signal, and use it along withx(t)in Example 1.1, to represent
y(t)in the above example.
Solution
The analog signalp(t)is of finite support and real-valued. We have that
Re[y(t)]=x(t)p(t)
Im[y(t)]=x(t− 1 )p(t)
so that
y(t)=[x(t)+jx(t− 1 )]p(t)
The multiplication byp(t)makesx(t)p(t)andx(t− 1 )p(t)finite-support signals. This operation is
called time windowing as the signalp(t)only allows us to see the values ofx(t)whereverp(t)=1,
while ignoring the values ofx(t)whereverp(t)=0. It acts like a window. n
Examples 1.1–1.3 not only illustrate how different types of signal can be related to each other, but
also how signals can be be defined in shorter or more precise forms. Although the representations for
y(t)in Example 1.2 and in this example are equivalent, the one here is shorter and easier to visualize
by the use of the pulsep(t).
1.3.1 Basic Signal Operations—Time Shifting and Reversal
The following are basic signal operations used in the representation and processing of signals (for
some of these operations we indicate the system that is used to realize the operation):
n Signal addition—Two signalsx(t)andy(t)are added to obtain their sumz(t). Anadderis used.
n Constant multiplication—A signalx(t)is multiplied by a constantα. Aconstant multiplieris used.
n Time and frequency shifting—The signalx(t)is delayedτseconds to getx(t−τ), and advanced by
τto getx(t+τ). A signal can be shifted in frequency or frequency modulated by multiplying it
by a complex exponential or a sinusoid. Adelayshifts right a time signal, while amodulatorshifts
the signal in frequency.
n Time scaling—The time variable of a signalx(t)is scaled by a constantαto givex(αt). Ifα=−1,
the signal is reversed in time (i.e.,x(−t)), or reflected. Only the delay can be implemented in
practice.
n Time windowing—A signalx(t)is multiplied by a window signal w(t)so thatx(t)is available in the
support of w(t).
