PowerPointPrsentation

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Title
International Training Course 2008 OVSICORI-UNA He
redia, Costa Rica E. Wielandt Analog to
Digital Conversion
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Why do we digitize analog seismic signals?
Of course, because we want to store and analyze
them with digital computers. Because, unlike
analog data, digital data can be stored, copied
and distributed without loss of quality. The
secret is quantization. The information is
represented by integer numbers. Even in the
presence of noise, integer numbers can be
restored by rounding. Since computers use a
binary representation, we need only to
distinguish between two digits, 0 and 1. So
analog noise in the physical record can be
tolerated up to half of the quantization
interval. Digital samples are also quantized in
time. They need not be available in real time.
So why dont we have analog computers? How is
that possible? Digital information is physically
still stored on analog media (e.g. in
semiconductors and magnetic or optical disks).
All these are fundamentally noisy. Digital
processing may produce unpredictable delays so
how can timing accuracy be ensured?
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Lets have a look at the process of quantization.
This is our analog input signal
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First step sampling at equidistant times nDt.
The time increment Dt (here 160 milliseconds) is
called the sampling interval.
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Second step rounding to integer numbers
(counts). Each count represents a small
increment q of the input signal, the
quantization interval.
Rounding is equivalent to adding noise. The power
of this noise is under certain assumptions equal
to q2 /12 in a bandwidth from zero to the Nyquist
frequency, 1/(2Dt). The power density is thus
inverse to the sampling interval.
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Quantization noise is only the theoretical
minimum noise.
In order to minimize quantization noise, we use
high-resolution (24 bit) digitizers. They
typically have a differential input, with a range
of 10 V per input (40 V p-p), and a quantization
interval (LSB-equivalent) of 40 V / 224 2.4 mV.
With a standard broadband seismometer that has a
generator constant of 1500 Vs/m, this translates
into a ground motion of 1.6 nm/s. Digitizers also
exhibit ordinary semiconductor noise, which
usually predominates at low frequencies, say
below 1 Hz. A 24-bit digitizer does not
necessarily resolve 24 bits the last few bits
may represent internal noise. On the other hand,
the best seismic digitizers can resolve more than
24 bit. For the rest of this lecture, we will
ignore quantization noise.
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The problem of aliasing do the digital samples
represent the input signal?
The blue and the green waveforms produce
identical samples at the given sampling rate.
Thus, a reconstruction of the analog signal isnt
always possible.
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Step functions are especially difficult
(actually, impossible) to represent by samples.
This problem is however solved by anti-alias
(low-pass) filtration.
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Signals whose bandwidth lies entirely within the
Nyquist bandwidth 1/(2Dt) can be exactly
reconstructed from (noiseless) samples, by
interpolation with the sin(t)/t function

• There are two ways of limiting the bandwith
• With an external, analog anti-alias filter. Its
  corner frequency can normally not be higher than
  half of the Nyquist frequency.
• With oversampling and subsequent decimation. The
  corner frequency of digital anti-alias filters
  can be up to 80 of the Nyquist frequency.
  However, their step response is ugly. We must
  choose between zero-phase (linear-phase) and
  minimum-delay filters. Zero-phase filters produce
  precursors to high-frequency onsets min-del
  filters delay the signal.

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The sin(t)/t function. This is the analog
equivalent of one sample (the red square).It
also represents the impulse response of an
ideally sharp, zero-phase low-pass filter at the
Nyquist frequency. Note that the digitizer
samples the zero crossings and produces only one
nonzero sample. However, if the input pulse were
time-shifted by one-half of the sampling
interval, the digitizer would sample the extrema.
The digital representation of a pulse can thus be
quite different, depending on the exact start
time.
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This is what a step at time zero looks like if
seen through the Nyquist bandwidth. The digitizer
now samples the extrema hence the problem of
spurious precursors. If the analog step is
however shifted in time by one-half sampling
interval, its digital representation looks much
nicer. The overshooting of the analog signal is
mathematically known as the Gibbs phenomenon.
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The concept of zero-phase and minimum-phase
(minimum-delay)
The lowermost two signals have the same amplitude
spectrum but different phase spectra. mdfilt2 has
zereo phase with respect to the time of sample
100. Mdfilt1 is minimum-delay w.r.t. that origin
time. Note the delayed onset.
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Minimum-phase is normally maximum-phase
The term minimum phase was coined with respect
to a Fourier transformation with exp(jwt) in the
inverse transformation (synthesis). Most
seismologists use however exp(-jwt). Then the
causal signal with the smallest possible delay
the minimum-delay signal - has the largest
possible (although negative) phase. Instead of
the ambiguous term minimum-phase, whose exact
meaning depends on the definition of the Fourier
transformation, it is preferable to use
minimum-delay.
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How do digitizers actually work?

• There are many ways to digitize analog electric
  signals. Seismologists need
• High resolution
• Low sampling rates (compared to other technical
  applications)
• Moderate absolute accuracy
• The basic circuit of most low-frequency
  digitizers is like this

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Schematic of an actual 24-bit digitizer
(Quanterra)
One-bit D/A
Decision Logic
Digital out
Correction of bitstream quantization errors
Integrator
Differencer
Analog in
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Calibration of digitizers

• Digitizers normally dont need to be calibrated
  if the manufacturers specifications are clear
  and complete.
• You may want to check the scale factor, normally
  in microvolts per count, by connecting a battery
  and a digital voltmeter to the input.
• Dont care about the filter coefficients, or
  poles and zeros! Just find out whether you have
  zero-phase or minimum-delay filters.
• For all frequencies lower than one-quarter of the
  sampling rate, (that is, one-half of the Nyquist
  frequency) you may assume that the response is
  flat and the phase
• of a zero-phase filter is zero
• of a minimum-delay filter represents a constant
  delay whose magnitude you can experimentally
  determine by recording a time signal (such as
  from the pps output of a GPS receiver)
• - To be certain, do the same test for zero-phase
  filters!

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Post-filtration from zero-phase to min-del or
vice versa
Digital all-pass filters are available to change
the phase of a seismic record between zero-phase
and min-del. See Frank Scherbaums Book On Poles
and Zeros. However, a normal low-pass
filtration with a recursive (IIR) filter is in
general sufficient or even preferable in order to
remove the undesired features from a zero-phase
record. Such filters are part of most
data-processing software packages. In order to
compare digital seismograms to analog ones, the
response of the analog system should be digitally
simulated.
zero-phase (recorded)
minimum-delay
low-pass filtered
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Another example of post-filtration(with two
different methods)(Dieter Stoll, Lennartz
Electronics)
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The End
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Quantization noise is only the theoretical
minimum noise.
In order to minimize quantization noise, we use
high-resolution (24 bit) digitizers. They
typically have a differential input, with a range
of 10 V per input (40 V p-p), and a quantization
interval (LSB-equivalent) of 40 V / 224 2.4 mV.
With a standard broadband seismometer that has a
generator constant of 1500 Vs/m, this translates
into a ground motion of 1.6 nm/s. Digitizers also
exhibit ordinary semiconductor noise, which
usually predominates at low frequencies, say
below 1 Hz. A 24-bit digitizer does not
necessarily resolve 24 bits the last few bits
may represent internal noise. On the other hand,
the best seismic digitizers can resolve more than
24 bit. For the rest of this lecture, we will
ignore quantization noise.
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Minimum-phase is normally maximum-phase
The term minimum phase was coined with respect
to a Fourier transformation with exp(jwt) in the
forward transformation. Most seismologists use
however exp(-jwt). Then the causal signal with
the smallest possible delay the minimum-delay
signal - has the largest possible (although
negative) phase. Instead of the ambiguous term
minimum-phase, whose exact meaning depends on
the definition of the Fourier transformation, it
is preferable to use minimum-delay.
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Calibration of digitizers

• Digitizers normally dont need to be calibrated
  if the manufact-urers specifications are clear
  and complete.
• You may want to check the scale factor, normally
  in microvolts per count, by connecting a battery
  and a digital voltmeter to the input.
• Dont care about the filter coefficients, or
  poles and zeros!
• For all frequencies lower than one-quarter of the
  sampling rate, (that is, one-half of the Nyquist
  bandwith) you may assume that the response is
  flat and the phase
• of a zero-phase filter is zero
• of a minimum-delay filter represents a constant
  delay whose magnitude you can experimentally
  determine by recording a time signal (such as
  from the pps output of a GPS receiver)
• - To be certain, do the same test for zero-phase
  filters!

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Post-filtration from zero-phase to min-del or
vice versa
Digital all-pass filters are available to change
the phase of a seismic record between zero-phase
and min-del. See Frank Scherbaums Book On Poles
and Zeros. However, a normal low-pass
filtration with a recursive (IIR) filter is in
general sufficient or even preferable in order to
remove the undesired features from a zero-phase
record. Such filters are part of most
data-processing software packages. In order to
compare digital seismograms to analog ones, the
response of the analog system should be digitally
simulated.
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