Signals and Systems - Electrical Engineering

432 CHAPTER 7: Sampling Theory

FIGURE 7.5

Sampling of two sinusoids of

frequencies 0 = 1 and

 0 +s= 8 withTs= 2 π/s.

The higher-frequency signal is

undersampled, causing aliasing,

which makes the two sampled

signals coincide.

0 1 2 3 4 5 6

− 1

−0.5

0

0.5

1

t

x^1

(t

),

x^2

(t

),

x^1

(nT

)s

x 1 (t)

x 2 (t)

x 1 (nTs)

FIGURE 7.6

(a) Spectra of sinusoidsx 1 (t)andx 2 (t).

(b) The spectra of the sampled signalsx 1 s(t)

andx 2 s(t)look exactly the same due to the

undersampling ofx 2 (t). (a) (b)

1

8 6

1 6

1

− 1

8

8

X 1 (Ω) X 1 s(Ω)

X 2 (Ω) X 2 s(Ω)

Ω

Ω

Ω

Ω

− 8 − 8 − 6 − 1

− 8 − 6

· · ·

· · ·

· · ·

· · ·

− 1

7.2.4 Signal Reconstruction from Sinc Interpolation..............................

The analog signal reconstruction from the samples can be shown to be an interpolation using sinc

signals. First, the ideal low-pass filterHlp(s)in Equation (7.14) has as impulse response

hlp(t)=

Ts

2 π

∫s/ 2

−s/ 2

ejtd=

sin(πt/Ts)

πt/Ts

(7.15)

which is a sinc function that has an infinite time support and decays symmetrically with respect to the

origint=0. The reconstructed signalxr(t)is the convolution of the sampled signalxs(t)andhlp(t),

which is found to be

xr(t)=[xs∗hlp](t)=

∫∞

−∞

xs(τ)hlp(t−τ)dτ