IRIS Metadata Workshop, Cairo 2009

IRIS Metadata Workshop, Foz do Iguacu 2010

International Training Course, Turkey 2010

E. Wielandt:

Analog to Digital Conversion

Title

So why don‘t 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?

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.

Why do we digitize analog seismic signals?

Let‘s have a look at the process of quantization. This is our analog input signal

First step: sampling at equidistant times n*t. The time increment t (here 160 milliseconds) is called the sampling interval.

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/(2*t). The power density is thus inverse to the sampling interval.

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 V. 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 per count.

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.

Quantization noise is only the theoretical minimum noise.

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 isn‘t always possible.

Step functions are especially difficult (actually, impossible) to represent by samples.

This problem is however solved by anti-alias (low-pass) filtration.

Signals whose bandwidth lies entirely within the Nyquist bandwidth 1/(2*t) can be exactly reconstructed from (noiseless) samples, by interpolation with the sin(t)/t function

There are two ways of limiting the bandwith:

1. With an external, analog anti-alias filter. Its corner frequency can normally not be higher than half of the Nyquist frequency.

2. 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.

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.

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“.

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.

Minimum-phase“ is normally maximum-phase

The term „minimum phase“ was coined with respect to a Fourier transformation with exp(-jt) in the inverse transformation (synthesis). Most seismologists use however exp(jt). 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“.

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:

One-bit D/A Decision Logic

IntegratorDifferencer

Correction of bitstream quantization errors

Analog in

Digital out

Schematic of an actual 24-bit digitizer (Quanterra)

„Calibration“ of digitizers

Digitizers normally don‘t need to be calibrated if the manufacturer’s 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.

Don’t 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!

Post-filtration from zero-phase to min-del or vice versa

zero-phase (recorded)

minimum-delay

low-pass filtered

Digital all-pass filters are available to change the phase of a seismic record between zero-phase and min-del. See Frank Scherbaum‘s 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.

Another example of

post-filtration(with two different methods)

(Dieter Stoll, Lennartz

Electronics)

The End