Transformations; The Bilinear Transform - VTI Instruments EX1629 User Manual

The settling time is double this time (2τ
IIR filters
The group delay through the Bessel or Butterworth IIR filters is a function of cutoff frequency,
sampling frequency, and the number of poles. The group delay for the dc component is calculated
and reported to the digital board (τ
The total delay through the IIR = τ
The group delay will be provided to the end user. The samples will be group delay compensated.
Although it is possible to set each channel's IIR filter individually, it is highly recommended that
the channels on each digital board (channels 0 through 15, 16 through 31, and 32 through 47) be
configured to use the same filter. Currently, each EX1629 digital board calculates group delay
corrections based on the setting of the first channel (channels 0, 16, and 32), regardless of whether
these channels are included in a scan list. As a result, any channel that uses a different filter will
have data that has been incorrectly delay compensated.
It is possible to post-process the data from the EX1629 and compensate for group delay on a per
channel (per filter) basis. The group delay of all channels can be queried using the
vtex1629_get_IIR_filter_configuration() function. Group delay values for channels 0, 16, and 32
can then be used along with the group delays for the other channels to compute the additional
group delay (in samples) that data from each channel should be delayed or advanced.The data shift
can be calculated by using the following formula:
Data Shift = (D
channel in the group (channel 0, 16, or 32).
T
RANSFORMATIONS
The EX1629 utilizes two types of transformations: bilinear and matched Z. Both transformations
are discussed below.

The Bilinear Transform

The bilinear transformation (BLT) used in the EX1629 maps the entire jw axis in the s plane
exactly once to the unit circle in the z plane:
dc (s = 0) maps to dc (z = 1). As for the finite difference approximation (FDA), infinite frequency
(s = ∞) maps to half the sampling rate (z = -1) instead of z = 0 for the FDA. As a result:
damping characteristics are better preserved
no aliasing (mapping is one-to-one)
frequency axis remains warped away from dc
The real constant c > 0 allows one nonzero frequency (at s = jw
digital frequency (at
fashion:
EX1629 Filtering
g IIR
g IIR
_- D ), where D is the delay for the channel and D
x 0 x
1
1 z
s c
1
1 z
jw T
z e ). All other frequencies are warped, or distorted in a non-linear
d
jw T
1 e d
jw c jc tan
jw T
1 e d
).
g FIR
).
*cic_dec*fir_dec*20 µs
w T
d
2
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is the group delay for the first
0
) to map exactly to any desired
α
319