BSA Flow Software Installation and User's Guide - CSI

S(f)
Spectrum and correlation estimates
f1
f1’
fc
True spectrum
Aliased high frequency
Distorted spectrum
f2
f1 = 60Hz
f1’ ≈ 70Hz
fc = 120Hz
f2 = 150Hz
Figure 7-87: Effect of aliasing on the calculated spectrum.
As clearly seen in Figure 7-87, aliasing will distort the true spectrum and
move the peak while increasing its height. Since the user was not originally
aware of the 150Hz component, and expected a peak at “about” 60Hz, he
may be inclined to accept the calculated peak-frequency f1’ although it is
false.
In general there are two ways to avoid aliasing:
• Use a sampling rate sufficiently high to satisfy the Nyquist sampling
criterion.
• Low-pass filter the signal to remove high frequency components.
As stated earlier, [Adrian & Yao (1987)] have shown that the sample and
hold process itself acts like a first-order low pass filter attenuating the
spectrum at frequencies above &n 2π . A first order lowpass filter will
however only attenuate 20 dB/decade above the cut-off frequency, so
considerable amounts of high frequency energy will be allowed to pass.
To minimize aliasing you should thus always aim at the highest possible
datarate.
With resampled data, the integrals in (7-67) can now be estimated as sums:
U
V
N
T
kn
= ( u − u) −i
Ufk T)
N
N
⎛
−1
⎞
∑ exp ⎜ 2π
⎟ ≅ ( ,
⎝ ⎠
k n
n=
0
N
T
kn
= ( v − v) −i
Vfk T)
N
N
⎛
−1
⎞
∑ exp ⎜ 2π
⎟ ≅ ( ,
⎝ ⎠
k n
n=
0
f
(7-74)
-where T/N=∆t is the resampling interval, and each value of k represent a
different frequency: fk=k/T, k=0, 1, 2, … , N/2.
If N, the number of samples, is a power of two, the above expressions can be
calculated very fast using FFT-algorithms, and Uk and Vk are multiplied like
in (7-68) to produce a spectrum estimate:
7-130 BSA Flow Software: Reference guide