Signals and Systems - Electrical Engineering

422 CHAPTER 7: Sampling Theory

where the Fourier coefficients{Dk}are

Dk=

1

Ts

T∫s/ 2

−Ts/ 2

δTs(t)e−jkstdt=

1

Ts

T∫s/ 2

−Ts/ 2

δ(t)e−jkstdt

=

1

Ts

T∫s/ 2

−Ts/ 2

δ(t)e−j^0 dt=

1

Ts

The last equation is obtained using the sifting property of theδ(t)and that the area of the impulse

is unity. Thus, the Fourier series of the sampling signal is

δTs(t)=

∑∞

k=−∞

1

Ts

ejkst (7.5)

and the sampled signalxs(t)=x(t)δTs(t)is then expressed as

xs(t)=

1

Ts

∑∞

k=−∞

x(t)ejkst

with Fourier transform

Xs()=

1

Ts

∑∞

k=−∞

X(−ks) (7.6)

where we used the frequency-shift property of the Fourier transform, and letX()andXs()be

the Fourier transforms ofx(t)andxs(t), respectively.

n Discrete-time Fourier transform: The Fourier transform of the sum representation ofxs(t)in the

second equation in Equation (7.4) is

Xs()=

∑

n

x(nTs)e−jTsn (7.7)

where we used the Fourier transform of a shifted impulse. This equation is equivalent to Equa-

tion (7.6) and will be used later in deriving the Fourier transform of discrete-time signals.

Remarks

n The spectrum Xs()of the sampled signal, according to Equation (7.6), is a superposition of shifted

analog spectra{X(−ks)}multiplied by 1 /Ts(i.e., the modulation process involved in the sampling).

n Considering that the output of the sampler displays frequencies that are not present in the input, according

to the eigenfunction property the sampler is not LTI. It is a time-varying system. Indeed, if sampling

x(t)gives xs(t), sampling x(t−τ)whereτ6=kTsfor an integer k will not be xs(t−τ). The sampler is,

however, a linear system.