Signals and Systems - Electrical Engineering

4.8 Time and Frequency Shifting 271

we then have that

x(t−t 0 )=

∑

k

Xkejk^0 (t−t^0 )=

∑

k

[

Xke−jk^0 t^0

]

ejk^0 t

x(t+t 0 )=

∑

k

Xkejk^0 (t+t^0 )=

∑

k

[

Xkejk^0 t^0

]

ejk^0 t

so that the Fourier coefficients {Xk} corresponding to x(t) are changed to {Xke∓jk^0 t^0 } for
x(t∓t 0 ). In both cases, they have the same magnitude|Xk|but different phases.

In a dual way, if we multiply the above periodic signalx(t)by a complex exponential of frequency
 1 ,ej^1 t, we obtain a so-calledmodulated signal y(t)and its spectrum is shifted in frequency by 1
with respect to the spectrum of the periodic signalx(t). In fact,

y(t)=x(t)ej^1 t

=

∑

k

Xkej(^0 k+^1 )t

indicating that the harmonic frequencies are shifted by 1. The signaly(t)is not necessarily periodic.
SinceT 0 is the period ofx(t), then

y(t+T 0 )=x(t+T 0 )ej^1 (t+T^0 )

and for it to be equal toy(t), then 1 T 0 = 2 πM, for an integerM6=0 or

 1 =M 0 M>> 1

which goes along with the condition that the modulating frequency 1 is chosen much larger than
 0. The modulated signal is then given by

y(t)=

∑

k

Xkej(^0 k+^1 )t=

∑

k

Xkej^0 (k+M)t=

∑

`

X`−Mej^0 `t

so that the Fourier coefficients are shifted to new frequencies 0 (k+M).

To keep the modulated signal real-valued, one multiplies the periodic signalx(t)by a cosine of
frequency 1 =M 0 forM>>1 to obtain a modulated signal

y 1 (t)=x(t)cos( 1 t)

=

∑

k

0.5Xk[ej(k^0 +^1 )t+ej(k^0 −^1 )t]

so that the harmonic components are now centered around± 1.