Signals and Systems - Electrical Engineering

1.3 Continuous-Time Signals 73

FIGURE 1.4
(a) Continuous-time signal, and its
(b) delayed, (c) advanced, and
(d) reflected versions.

(a)

(c)

t

x(t)

−τ t

x(t+τ)

(b)

(d)

τ

t

x(t−τ)

t

x(−t)

(c) x(−t)isreflected.

(d) x(−t−τ)isreflected and shifted leftτseconds, whilex(−t+τ)isreflected and shifted rightτseconds.

RemarksWhenever we combine the delaying or advancing with reflection, delaying and advancing are
swapped. Thus, x(−t+ 1 )is x(t)reflected and delayed, or shifted to the right, by 1. Likewise, x(−t− 1 )is
x(t)reflected and advanced, or shifted to the left by 1. Again, the value x( 0 )of the original signal is found in
x(−t+ 1 )at t= 1 , and in x(−t− 1 )at t=− 1.

nExample 1.4

Consider an analog pulse

x(t)=

{

1 0≤t≤ 1

0 otherwise

Find mathematical expressions forx(t)delayed by 2, advanced by 2, and the reflected signalx(−t).

Solution

The delayed signalx(t− 2 )can be found mathematically by replacing the variabletbyt−2 so that

x(t− 2 )=

{

1 0≤t− 2 ≤1 or 2≤t≤ 3

0 otherwise

The valuex( 0 )(which inx(t)occurs att=0) inx(t− 2 )now occurs whent=2, so that the signal

x(t)has been shifted to the right two units of time, and since the values are occurring later, the

signalx(t− 2 )is said to be “delayed” by 2 with respect tox(t).