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In the resulting series, f a 1 ¼ f a 2 ¼ f a 3 ¼ 25 Hz. Because of the aliasing phenomenon, the
75- and 125-Hz components would be completely indistinguishable from the 25-Hz component.
The discrete series would be described by
yðrdtÞ ¼ ðB 1 þ B 2 þ B 3 Þsin2pð25Þrdf r ¼ 0; 1; 2; . . .
where B 1 ¼ A 1 ; B 2 ¼ ÀA 2 ; and B 3 ¼ A 3 . Note from Figure 7.3 that when the original frequency
component is out of phase with the alias frequency, the corresponding amplitude is negative. Clearly,
this series misinterprets the original analog signal in both its frequency and amplitude content.
COMMENT Example 7.2 illustrates the potential for misrepresenting a signal with improper
sampling. With signals for which the frequency content is not known prior to their measurement,
sample frequencies should be increased according to the following scheme:
Sample the input signal at increasing sample rates using a fixed total sample time,
and examine the time plots for each signal. Look for changes in the shape of the waveform
(Fig. 7.2).
Compute the amplitude spectrum for each signal at increasing sample rates and compare the
resulting frequency content.
Always use (anti-aliasing) analog filters set at the Nyquist frequency.
Amplitude Ambiguity
For simple and complex periodic waveforms, the DFT of the sampled discrete time signal remains
unaltered by a change in the total sample period Ndt provided that (1) the total sample period
remains an integer multiple of the fundamental period T 1 of the measured continuous waveform, that
is mT 1 ¼ Ndt where m is an integer; and (2) the sample frequency meets the sampling theorem
criterion. If both criteria are met, the amplitudes associated with each frequency in the periodic
signal, the spectral amplitudes, are accurately represented by the DFT. This means that an original
periodic waveform can be completely reconstructed from a discrete time series regardless of the
sample time increment used. The total sample period defines the frequency resolution of the DFT:
df ¼
1
Ndt
¼
f s
N
ð7:10Þ
The frequency resolution plays a crucial role in the reconstruction of the signal amplitudes, as noted
below.
An important difficulty arises when Ndt is not coincident with an integer multiple of the
fundamental period of y(t): The resulting DFT cannot exactly represent the spectral amplitudes of
the sampled continuous waveform. This is exaggerated when Ndt represents only a relatively few
fundamental periods of the signal. The problem is brought on by the truncation of one complete
cycle of the signal (Fig. 7.4) and from spectral resolution, because the associated fundamental
frequency and its harmonics are not coincident with a center frequency of the DFT. However, this
error decreases either as the value of N dt more closely approximates an exact integer multiple of T 1
or as f s becomes very large relative to f m .
This situation is illustrated in Figure 7.4, which compares the amplitude spectrum resulting
from sampling the signal yðtÞ ¼ 10 cos 628t over different sample periods. Two sample periods of
266 Chapter 7 Sampling, Digital Devices, and Data Acquisition
14:43:49 Page 266
In the resulting series, f a 1 ¼ f a 2 ¼ f a 3 ¼ 25 Hz. Because of the aliasing phenomenon, the
75- and 125-Hz components would be completely indistinguishable from the 25-Hz component.
The discrete series would be described by
yðrdtÞ ¼ ðB 1 þ B 2 þ B 3 Þsin2pð25Þrdf r ¼ 0; 1; 2; . . .
where B 1 ¼ A 1 ; B 2 ¼ ÀA 2 ; and B 3 ¼ A 3 . Note from Figure 7.3 that when the original frequency
component is out of phase with the alias frequency, the corresponding amplitude is negative. Clearly,
this series misinterprets the original analog signal in both its frequency and amplitude content.
COMMENT Example 7.2 illustrates the potential for misrepresenting a signal with improper
sampling. With signals for which the frequency content is not known prior to their measurement,
sample frequencies should be increased according to the following scheme:
Sample the input signal at increasing sample rates using a fixed total sample time,
and examine the time plots for each signal. Look for changes in the shape of the waveform
(Fig. 7.2).
Compute the amplitude spectrum for each signal at increasing sample rates and compare the
resulting frequency content.
Always use (anti-aliasing) analog filters set at the Nyquist frequency.
Amplitude Ambiguity
For simple and complex periodic waveforms, the DFT of the sampled discrete time signal remains
unaltered by a change in the total sample period Ndt provided that (1) the total sample period
remains an integer multiple of the fundamental period T 1 of the measured continuous waveform, that
is mT 1 ¼ Ndt where m is an integer; and (2) the sample frequency meets the sampling theorem
criterion. If both criteria are met, the amplitudes associated with each frequency in the periodic
signal, the spectral amplitudes, are accurately represented by the DFT. This means that an original
periodic waveform can be completely reconstructed from a discrete time series regardless of the
sample time increment used. The total sample period defines the frequency resolution of the DFT:
df ¼
1
Ndt
¼
f s
N
ð7:10Þ
The frequency resolution plays a crucial role in the reconstruction of the signal amplitudes, as noted
below.
An important difficulty arises when Ndt is not coincident with an integer multiple of the
fundamental period of y(t): The resulting DFT cannot exactly represent the spectral amplitudes of
the sampled continuous waveform. This is exaggerated when Ndt represents only a relatively few
fundamental periods of the signal. The problem is brought on by the truncation of one complete
cycle of the signal (Fig. 7.4) and from spectral resolution, because the associated fundamental
frequency and its harmonics are not coincident with a center frequency of the DFT. However, this
error decreases either as the value of N dt more closely approximates an exact integer multiple of T 1
or as f s becomes very large relative to f m .
This situation is illustrated in Figure 7.4, which compares the amplitude spectrum resulting
from sampling the signal yðtÞ ¼ 10 cos 628t over different sample periods. Two sample periods of
266 Chapter 7 Sampling, Digital Devices, and Data Acquisition