Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 405

Table 9.1. Signals reconstructed from samples of a sinusoidal tone x ( t ) = cos(2π f 0 t ) for different values of the
fundamental frequency f 0 ; the sampling frequency f s is kept constant at 6000 samples/s

Funadmental Original Reconstructed

frequency ( f 0 ) signal ( f 0 − mf s ) < f s / 2 signal Comments

cos(1000 π t ) f s > 2 f 0 cos(1000 π t )

(1) 500 Hz no aliasing

(2) 2.5 kHz cos(5000 π t ) f s > 2 f 0 cos(5000 π t ) no aliasing

(3) 2.8 kHz cos(5600 π t ) f s > 2 f 0 cos(5600 π t ) no aliasing

(4) 3.2 kHz cos(6400 π t ) 3200 − 1 6000 cos(5600 π t ) aliasing

(5) 3.5 kHz cos(7000 π t ) 3500 − 1 6000 cos(5000 π t ) aliasing

(6) 7 kHz cos(14000 π t ) 7000 − 1 6000 cos(2000 π t ) aliasing

(7) 10 kHz cos(20000 π t ) 10000 − 2 6000 cos(4000 π t ) aliasing

(8) 20 kHz cos(40000 π t ) 20000 − 3 6000 cos(4000 π t ) aliasing

(9) 1000 kHz cos(2 10 6 π t ) 10 6 − 167 6000 cos(4000 π t ) aliasing

of Table 9.1, are identical. Similarly, the reconstructed signals for the sinu-
soidal waveforms x ( t ) = cos(5000 π t ) and x ( t ) = cos(7000 π t ), listed in entries
(2) and (5) of Table 9.1, are also identical. Finally, the reconstructed signals
for the sinusoidal waveforms x ( t ) = cos(20 000 π t ), x ( t ) = cos(40 000 π t ), and
x ( t ) = cos(2 10 6 π t ), listed in entries (7)-(9) of Table 9.1, are the same. The
identical waveforms are the consequences of aliasing.

9.2 Practical appro aches to sampling

Section 9.1 introduced the impulse-train sampling used to derive the DT version
of a band-limited CT signal. In practice, impulses are difficult to generate and
are often approximated by narrow rectangular pulses. The resulting approach
is referred to as pulse-train sampling, which is discussed in Section 9.2.1. A
second practical implementation, referred to as the zero-order hold, is discussed
in Section 9.2.2.

9.2.1 Pulse-train sampling

In pulse-train sampling, the impulse train s ( t ) is approximated by a rectangular
pulse train of the form

∞ ∞

r ( t ) = p 1 ( t − kT s ) = p 1 ( t ) ∗ δ ( t − kT s ) , (9.17)

k =−∞ k =−∞

where p 1 ( t ) represents a rectangular pulse of duration τ ≪ T s , which is given
by

t
τ

p 1 ( t ) = rect .