Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 18

plot. We notice that the 40-Hz signal is adequately sampled, since the sampled values clearly come from
the analog version of the 40-Hz sine wave. However, as shown in the second plot, the sine wave with
a frequency of 90 Hz is sampled at 100 Hz. Since the sampling rate of 100 Hz is relatively low compared
with the 90-Hz sine wave, the signal is undersampled due to 2 f max ¼ 180 > f s . Hence, the condition of
the sampling theorem is not satisfied. Based on the sample amplitudes labeled with the circles in the
second plot, we cannot tell whether the sampled signal comes from sampling a 90-Hz sine wave (plotted
using the solid line) or from sampling a 10-Hz sine wave (plotted using the dot-dash line). They are not
distinguishable. Thus they are aliases of each other. We call the 10-Hz sine wave the aliasing noise in
this case, since the sampled amplitudes actually come from sampling the 90-Hz sine wave.
Now let us develop the sampling theorem in frequency domain, that is, the minimum sampling rate
requirement for sampling an analog signal. As we shall see, in practice this can help us design the anti-
aliasing filter (a lowpass filter that will reject high frequencies that cause aliasing) that will be applied
before sampling, and the anti-image filter (a reconstruction lowpass filter that will smooth the
recovered sample-and-hold voltage levels to an analog signal) that will be applied after the digital-to-
analog conversion (DAC).
Figure 2.5 depicts the sampled signal x s ðtÞ obtained by sampling the continuous signal xðtÞ at
a sampling rate of f s samples per second.
Mathematically, this process can be written as the product of the continuous signal and the
sampling pulses (pulse train):

x s ðtÞ¼xðtÞpðtÞ (2.1)

where pðtÞ is the pulse train with a period T ¼ 1 =f s . From spectral analysis, the original spectrum
(frequency components) Xðf Þ and the sampled signal spectrum X s ðf Þ in terms of Hz are related as

N

1
T

X s ðfÞ¼ Xðf nf s Þ (2.2)

n¼ N

pt ()

1

t

T

xt ()

ADC
encoding

xt xtpt
s

() = () ()

x ()

x s (0

x T
s

()

x T
s

() 2

t t

T

FIGURE 2.5
The simplified sampling process.