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0.0256 and 0.1024 second were used with dt ¼ 0.1 ms, and a third sample period of 0.08 second with
dt ¼ 0.3125 ms.
1 These sample periods provide for a DFT frequency resolution df of about 39, 9.8,
and 12.5 Hz, respectively.
Leakage
The two spectra shown in Figures 7.4a,b display a spike near the correct 100 Hz with surrounding
noise spikes, known as leakage, at adjacent frequencies. Note that the original signal y(t) cannot be
exactly reconstructed from either of these spectra. The spectrum in Figure 7.4c has been constructed
using a longer sample time increment (i.e., lower sample rate) than that of the spectra in Figures
7.4a,b and with fewer data points than the spectrum of Figure 7.4b. Yet the spectrum of Figure 7.4c
provides an exact representation of y(t). The N and dt combination used has reduced leakage to zero,
which maximizes the amplitude at 100 Hz. Its sample period corresponds to exactly eight periods of
y(t), and the frequency of 100 Hz corresponds to the center frequency of the eighth frequency
interval of the DFT. As seen in Figure 7.4, the loss of accuracy in a DFT representation occurs in the
form of amplitude ‘‘leakage’’ to adjacent frequencies. To the DFT, the truncated segment of the
sampled signal appears as an aperiodic signal. The DFT returns the correct spectral amplitudes for
both the periodic and aperiodic signal portions. But as a result, the spectral amplitudes for any
truncated segment are superimposed onto portions of the spectrum adjacent to the correct frequency.
Recall how the amplitude varied in Figure 7.2c. This is how leakage affects the time domain
reconstruction. By varying the sample period or its equivalent, the DFT resolution, one can
minimize leakage and control the accuracy of the spectral amplitudes.
If y(t) is an aperiodic or nondeterministic waveform, there may not be a fundamental period. In
such situations, one controls the accuracy of the spectral amplitudes by varying the DFT resolution
df to minimize leakage. As df tends toward zero, leakage decreases.
In summary, the reconstruction of a measured waveform from a discrete signal is controlled by
the sampling rate and the DFT resolution. By adherence to the sampling theorem, one controls the
frequency content of both the measured signal and the resulting spectrum. By variation of df, one can
control the accuracy of the spectral amplitude representation.
Programs Sampling.m, Aliasing.vi, and Aliasing frequency domain.vi explore the concept of
sample rate and amplitude ambiguity. Program Leakage.2.vi allows the user to vary sample rate and
number of points; it follows the discussion of Figure 7.4. Program DataSpec.m explores sampling
concepts on user-generated data series. Signal generation.vi explores sample rate effects on different
wave forms.
For an exact discrete representation in both frequency and amplitude of any periodic analog
waveform, both the number of data points and the sample rate should be chosen based on
the preceding discussion using the criteria of Equations 7.2 and 7.10. Equation 7.2 sets the
maximum value for dt, or equivalently, the minimum sample rate f s , and Equation 7.10 sets the
total sampling time N dt from which the data number N is estimated.
Waveform Fidelity
While choosing the sampling frequency in accordance with the sampling theorem ensures that there
will be no aliasing associated with frequency content, it does not address the ‘‘shape’’ of the
1 Many DFT (including fast Fourier transform [FFT]) algorithms require that N ¼ 2
M , where M is an integer. This affects the
selection of dt.
268 Chapter 7 Sampling, Digital Devices, and Data Acquisition
14:43:49 Page 268
0.0256 and 0.1024 second were used with dt ¼ 0.1 ms, and a third sample period of 0.08 second with
dt ¼ 0.3125 ms.
1 These sample periods provide for a DFT frequency resolution df of about 39, 9.8,
and 12.5 Hz, respectively.
Leakage
The two spectra shown in Figures 7.4a,b display a spike near the correct 100 Hz with surrounding
noise spikes, known as leakage, at adjacent frequencies. Note that the original signal y(t) cannot be
exactly reconstructed from either of these spectra. The spectrum in Figure 7.4c has been constructed
using a longer sample time increment (i.e., lower sample rate) than that of the spectra in Figures
7.4a,b and with fewer data points than the spectrum of Figure 7.4b. Yet the spectrum of Figure 7.4c
provides an exact representation of y(t). The N and dt combination used has reduced leakage to zero,
which maximizes the amplitude at 100 Hz. Its sample period corresponds to exactly eight periods of
y(t), and the frequency of 100 Hz corresponds to the center frequency of the eighth frequency
interval of the DFT. As seen in Figure 7.4, the loss of accuracy in a DFT representation occurs in the
form of amplitude ‘‘leakage’’ to adjacent frequencies. To the DFT, the truncated segment of the
sampled signal appears as an aperiodic signal. The DFT returns the correct spectral amplitudes for
both the periodic and aperiodic signal portions. But as a result, the spectral amplitudes for any
truncated segment are superimposed onto portions of the spectrum adjacent to the correct frequency.
Recall how the amplitude varied in Figure 7.2c. This is how leakage affects the time domain
reconstruction. By varying the sample period or its equivalent, the DFT resolution, one can
minimize leakage and control the accuracy of the spectral amplitudes.
If y(t) is an aperiodic or nondeterministic waveform, there may not be a fundamental period. In
such situations, one controls the accuracy of the spectral amplitudes by varying the DFT resolution
df to minimize leakage. As df tends toward zero, leakage decreases.
In summary, the reconstruction of a measured waveform from a discrete signal is controlled by
the sampling rate and the DFT resolution. By adherence to the sampling theorem, one controls the
frequency content of both the measured signal and the resulting spectrum. By variation of df, one can
control the accuracy of the spectral amplitude representation.
Programs Sampling.m, Aliasing.vi, and Aliasing frequency domain.vi explore the concept of
sample rate and amplitude ambiguity. Program Leakage.2.vi allows the user to vary sample rate and
number of points; it follows the discussion of Figure 7.4. Program DataSpec.m explores sampling
concepts on user-generated data series. Signal generation.vi explores sample rate effects on different
wave forms.
For an exact discrete representation in both frequency and amplitude of any periodic analog
waveform, both the number of data points and the sample rate should be chosen based on
the preceding discussion using the criteria of Equations 7.2 and 7.10. Equation 7.2 sets the
maximum value for dt, or equivalently, the minimum sample rate f s , and Equation 7.10 sets the
total sampling time N dt from which the data number N is estimated.
Waveform Fidelity
While choosing the sampling frequency in accordance with the sampling theorem ensures that there
will be no aliasing associated with frequency content, it does not address the ‘‘shape’’ of the
1 Many DFT (including fast Fourier transform [FFT]) algorithms require that N ¼ 2
M , where M is an integer. This affects the
selection of dt.
268 Chapter 7 Sampling, Digital Devices, and Data Acquisition