Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 17

After obtaining the sampled signal whose amplitude values are taken at the sampling instants, the
processor is able to process the sample points. Next, we have to ensure that samples are collected at
a rate high enough that the original analog signal can be reconstructed or recovered later. In other
words, we are looking for a minimum sampling rate to acquire a complete reconstruction of the analog
signal from its sampled version. If an analog signal is not appropriately sampled, aliasing will occur,
which causes unwanted signals in the desired frequency band.
The sampling theorem guarantees that an analog signal can be in theory perfectly recovered as long
as the sampling rate is at least twice as large as the highest-frequency component of the analog signal
to be sampled. The condition is described as

f s 2 f max
where f max is the maximum-frequency component of the analog signal to be sampled. For example, to
sample a speech signal containing frequencies up to 4 kHz, the minimum sampling rate is chosen to be
at least 8 kHz, or 8,000 samples per second; to sample an audio signal possessing frequencies up to
20 kHz, at least 40,000 samples per second, or 40 kHz, of the audio signal are required.
Figure 2.4 illustrates sampling of two sinusoids, where the sampling interval between sample points
is T ¼ 0 : 01 second, and the sampling rate is thus f s ¼ 100 Hz. The first plot in the figure displays
a sine wave with a frequency of 40 Hz and its sampled amplitudes. The sampling theorem condition is
satisfied since 2 f max ¼ 80 < f s . The sampled amplitudes are labeled using the circles shown in the first

Sampling condition is satisfied

1 40 Hz

0

-1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s ec . )

Sampling c onditi o n is no t satisfi e d

10 Hz

90 Hz

1

0

-1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s ec . )

FIGURE 2.4

Plots of the appropriately sampled signals and nonappropriately sampled (aliased) signals.