E1C07 09/14/2010
14:43:48 Page 263
the frequency content of a measured signal accurately, the sample rate must be more than twice the
highest frequency contained in the measured signal.
Denoting the maximum frequency in the analog signal as f m , the sampling theorem requires
f s > 2f m
ð7:2Þ
or, equivalently, in terms of sample time increment,
dt <
1
2 f m
ð7:3Þ
When signal frequency content is important, Equations 7.2 and 7.3 provide a criterion for the
minimum sample rate or maximum sample time increment, respectively, to be used in converting data
from a continuous to a discrete form. The frequencies that are extracted from the DFT of the resulting
discrete series provide an accurate representation of the original signal frequencies regardless of the
sample rate used, provided that the requirements of the sampling theorem are satisfied.
Alias Frequencies
When a signal is sampled at a sample rate that is less than 2 f m , the higher frequency content of the
analog signal takes on the false identity of a lower frequency in the resulting discrete series. This is
seen to occur in Figure 7.2d, where, because f s < 2 f m , the 10-Hz analog signal is observed to take on
the false identity of a 2-Hz signal. As a result, we misinterpret the frequency content of the original
signal! Such a false frequency is called an alias frequency.
The alias phenomenon is an inherent consequence of a discrete sampling process. To illustrate
this, consider a simple periodic signal:
yðtÞ ¼ Csin ½2pf t þ fðf Þ
ð7:4Þ
Suppose y(t) is measured with a sample time increment of dt, so that its discrete time signal is given
by
fyðrdtÞg ¼ Csin ½2pf rdt þ fðf Þ r ¼ 0; 1; . . . ; N À 1
ð7:5Þ
Now using the identity sinx ¼ sin ðx þ 2pqÞ, where q is any integer, we rewrite {y(rdt)} as
Csin ½2pf rdt þ fðf Þ ¼ Csin 2pf rdt þ 2pq þ f f
ð Þ
½
¼ Csin 2p f þ
m
dt
rdt þ f f
ð Þ
h
i
ð7:6Þ
where m ¼ 0, 1, 2, . . . (and hence, mr ¼ q is an integer). This shows that for any value of dt, the
frequencies of f and f þ m/dt are indistinguishable in a sampled discrete series. Hence, all
frequencies given by f þ m/dt are the alias frequencies of f. However, by adherence to the sampling
theorem criterion of either Equation 7.2 or 7.3, all m ! 1 are eliminated from the sampled signal, and
thus this ambiguity between frequencies is avoided.
This same discussion applies equally to complex periodic, aperiodic, and nondeterministic
waveforms. This is shown by examining the general Fourier series used to represent such signals.
A discrete series such as
fy rdt
ð Þg ¼
X 1
n¼1
C n sin 2pnf rdt þ f n f
ð Þ
½
ð 7:7Þ
7.2 Sampling Concepts 263
14:43:48 Page 263
the frequency content of a measured signal accurately, the sample rate must be more than twice the
highest frequency contained in the measured signal.
Denoting the maximum frequency in the analog signal as f m , the sampling theorem requires
f s > 2f m
ð7:2Þ
or, equivalently, in terms of sample time increment,
dt <
1
2 f m
ð7:3Þ
When signal frequency content is important, Equations 7.2 and 7.3 provide a criterion for the
minimum sample rate or maximum sample time increment, respectively, to be used in converting data
from a continuous to a discrete form. The frequencies that are extracted from the DFT of the resulting
discrete series provide an accurate representation of the original signal frequencies regardless of the
sample rate used, provided that the requirements of the sampling theorem are satisfied.
Alias Frequencies
When a signal is sampled at a sample rate that is less than 2 f m , the higher frequency content of the
analog signal takes on the false identity of a lower frequency in the resulting discrete series. This is
seen to occur in Figure 7.2d, where, because f s < 2 f m , the 10-Hz analog signal is observed to take on
the false identity of a 2-Hz signal. As a result, we misinterpret the frequency content of the original
signal! Such a false frequency is called an alias frequency.
The alias phenomenon is an inherent consequence of a discrete sampling process. To illustrate
this, consider a simple periodic signal:
yðtÞ ¼ Csin ½2pf t þ fðf Þ
ð7:4Þ
Suppose y(t) is measured with a sample time increment of dt, so that its discrete time signal is given
by
fyðrdtÞg ¼ Csin ½2pf rdt þ fðf Þ r ¼ 0; 1; . . . ; N À 1
ð7:5Þ
Now using the identity sinx ¼ sin ðx þ 2pqÞ, where q is any integer, we rewrite {y(rdt)} as
Csin ½2pf rdt þ fðf Þ ¼ Csin 2pf rdt þ 2pq þ f f
ð Þ
½
¼ Csin 2p f þ
m
dt
rdt þ f f
ð Þ
h
i
ð7:6Þ
where m ¼ 0, 1, 2, . . . (and hence, mr ¼ q is an integer). This shows that for any value of dt, the
frequencies of f and f þ m/dt are indistinguishable in a sampled discrete series. Hence, all
frequencies given by f þ m/dt are the alias frequencies of f. However, by adherence to the sampling
theorem criterion of either Equation 7.2 or 7.3, all m ! 1 are eliminated from the sampled signal, and
thus this ambiguity between frequencies is avoided.
This same discussion applies equally to complex periodic, aperiodic, and nondeterministic
waveforms. This is shown by examining the general Fourier series used to represent such signals.
A discrete series such as
fy rdt
ð Þg ¼
X 1
n¼1
C n sin 2pnf rdt þ f n f
ð Þ
½
ð 7:7Þ
7.2 Sampling Concepts 263