Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 103

Notice that time scaling also changes the frequency content of the signal. For instance, a signalx(t)=
ej^0 tis periodic of periodT 0 = 2 π/ 0 , whilex(αt)=ejα^0 thas a periodT 0 /αor a frequencyα 0 ,
which is larger than the original frequency of 0 whenα >1 and smaller than 0 when 0< α <1.

RemarksWe can thus summarize the above as follows:

n If x(t)is periodic of period T 0 then the time-scaled signal x(αt),α6= 0 , is also periodic of period T 0 /|α|.

n The support in time of a periodic or nonperiodic signal is inversely proportional to the support in frequency

for that signal.

n The frequencies present in a signal can be changed by modulation—that is, multiplying the signal by

a complex exponential or, equivalently, by sines and cosines. The frequency change is also possible by

expansion and compression of the signal.

n Reflection is a special case of time scaling withα=− 1.

nExample 1.22

Letx 1 (t), for 0≤t≤T 0 , be one period of a periodic signalx(t)of periodT 0. Representx(t)in terms

of advanced and delayed versions ofx 1 (t). What would bex( 2 t)?

Solution

The periodic signalx(t)can be written as

x(t)=···+x 1 (t+ 2 T 0 )+x 1 (t+T 0 )+x 1 (t)+x 1 (t−T 0 )+x 1 (t− 2 T 0 )+···

=

∑∞

k=−∞

x 1 (t−kT 0 )

and the contracted signalx( 2 t)is then

x( 2 t)=

∑∞

k=−∞

x 1 ( 2 t−kT 0 )

and periodic of periodT 0 /2. n

nExample 1.23

An acoustic signalx(t)has a duration of 3.3 minutes and a radio station would like to use the

signal for a three-minute segment. Indicate how to make it possible.

Solution

We need to contract the signal by a factor ofα=3.3/ 3 =1.1, so thatx(1.1t)can be used in the

three-minute piece. If the signal is recorded on tape, the tape player can be run 1.1 times faster

than the recording speed. This would change the voice or music on the tape, as the frequencies

x(1.1t)are increased with respect to the original frequencies inx(t). n