A&A 378, 700-709 (2001)
DOI: 10.1051/0004-6361:20011226
M. L. A. Kouwenhoven^{1} - J. L. L. Voûte^{2}
1 -
Astronomical Institute, Utrecht University, PO Box 80000, 3508 TA Utrecht, The Netherlands
2 -
Astronomical Institute "Anton Pannekoek", Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
Received 16 July 2001 / Accepted 29 August 2001
Abstract
We discuss the effect of digitisation on the signal-to-noise ratio of pulsed radio signals. We describe a general n-bit digitiser and show that a symmetric and equidistant digitiser has two free parameters: the threshold and the output value. We derive the best choice of these values for a 1, 1.5, 2, 4 and 8-bit digitiser and calculate the signal-to-noise ratio after digitisation of an undetected signal and of a detected signal with a Gaussian or a -distribution. Measurements made using PuMa, the new digital pulsar machine at the Westerbork Synthesis Radio Telescope, are presented and are shown to agree with the theoretical response of the digitiser.
Key words: instrumentation: miscellaneous - methods: data analysis - pulsars: general
A pulsar signal that is received by a radio telescope, can be described as amplitude modulated Gaussian noise (Rickett 1975). The periodic noise from the pulsar is weak and is added to steady noise from the sky background and the receiver. The signal-to-noise ratio (SNR) of the individual pulses of many pulsars is much less than one. In that case, the pulsation can be found by squaring (i.e. detecting) the signal and adding the pulses together (i.e. folding).
The group velocity of electromagnetic waves propagating in the interstellar medium is modified in a frequency-dependent way by the interaction between the waves and free electrons. This effect is called dispersion and causes the pulses from a pulsar to be smeared over significant bandwidths. There are two ways to remove this dispersion from a radio signal: coherently and incoherently.
In the coherent case the amplified voltages from the antenna are digitised and sampled. These recorded data are Fourier transformed off-line. Each Fourier component is rotated by a frequency dependent phase angle. After the inverse transformation a dispersion-free voltage time series is obtained (Hankins 1971).
For incoherent dedispersion the incoming bandwidth is split into a number of sub-bands (channels) by analogue filters or by a digital Fourier transform. The signal in each channel is then squared. The resulting power time series are delayed by the proper amount of time before the channels are added together. This delay can be applied on-line in hardware (Taylor & Huguenin 1971) or in off-line data processing after digitisation, sampling and recording. The resulting time series has a lower sampling rate than the original one: time resolution has been converted into frequency resolution.
Most radio pulsar observations include at least one digitisation and sampling step. The signal that comes from a radio telescope is always analogue, which means that it is continuous both in time and in its value. Sampling is the process which makes a signal discrete in time. No information is lost in this process, provided that the samples are taken at a rate of at least twice the highest frequency present in the input signal (Nyquist theorem). Digitising is the process which makes a signal discrete in its values. In this process information is lost. The amount of loss depends on the number of possible output values.
During digitisation both amplitude and phase information are lost. The loss of phase information is described for a general signal by Cooper (1970) and for the specific case of a pulsar observation by Jenet & Anderson (1998) and Stairs et al. (2000). Our paper concentrates on the loss of the amplitude information by studying the signal-to-noise ratio of the digitised signal in comparison with the SNR of the incoming signal.
This study was performed as part of the calibration process of PuMa. PuMa is the new digital pulsar machine, constructed for the Westerbork Synthesis Radio Telescope (WSRT, Voûte et al. 2001). The changes of the SNR along the signal path of WSRT are described in Kouwenhoven & Voûte (2001). To test whether PuMa is performing up to its specifications, it is necessary to understand all changes of the SNR and especially the digitisation losses. Therefore, we provide a theoretical description of a digitiser and compare the test results of PuMa with this theory.
Section 2 describes the input signal we have used for our derivations and calculations. Section 3 describes a general digitiser and the settings of its free parameters, in particular for a 1, 1.5, 2, 4 and 8-bit digitiser. In Sect. 4 the effect of the digitisation is studied as a function of the incoming SNR, starting with the case of a 2-bit digitiser. We compare the results with measurements made with PuMa in Sect. 5. Finally, we discuss the difference between our theory and work presented in earlier papers (Sect. 6).
Two types of signal can be distinguished: raw voltage data ("undetected signal") and squared voltage data (power data or a "detected signal"). In the latter case the distribution of the signal can be or Gaussian. The effects of digitisation differ for all these cases and each will be described separately. In this section we first describe the signal itself.
Figure 1: Time series of a pulsar signal with a rectangular envelope a) and noise from the receiver and the background sky b). The addition of these signals is modulated Gaussian noise c). The dotted lines mark the boundaries between the on- and off-pulse. The mean (dashed line) of the input signal is constant over the whole interval. After squaring the mean increases during the on-pulse d). The distribution of the signal is changed by smoothing, but this does not affect the (relative) means e). All vertical scales are arbitrary. The strength of the pulsar signal is highly exaggerated compared to most real pulsar observations. | |
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We start with a generalised pulsed signal. The incoming signal x(t) has two components: a constant noise component
,
which is the sum of the sky background, the receiver noise etc. and secondly, the signal
from the pulsar, which is a periodic increase of the noise level. It is assumed that this signal has a rectangular envelope: zero during the intervals in which the pulsar signal is beamed away from us ("off-pulse") and with a constant power during the intervals W in which the pulse can be seen ("on-pulse"). An example is shown in Figs. 1a-c.
(1) |
Figure 2: Gaussian distribution function of the signal during the off-pulse (solid line) and the on-pulse (dashed line). In case of an undetected signal a) the distribution becomes wider during the on-pulse: the standard deviation increases from to . In case of a detected and smoothed signal b) the distribution shifts towards higher values by an amount . Below the plots the digitiser thresholds (vertical dashes) and output values X are indicated for the case of 2-bit digitiser. | |
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Both components are Gaussian, i.e. have a Gaussian distribution function f(x):
(2) |
The signal-to-noise ratio SNR is defined as the relative increase of the power during the on-pulse:
In the detection stage the signal is squared (Fig. 1d). After squaring, the signal has a non-zero mean. The power of the signal is no longer equal to the variance of the signal and the SNR definition given in Eq. (4) is no longer valid. Eq. (3) requires the absolute value of the off-pulse power ,
but this information is lost in the digitisation process, if the values of the thresholds are not recorded. Thus, we define the signal-to-noise ratio of a detected signal (SNR)
to be the difference between the mean level during the off- and on-pulse relative to the off-pulse noise:
After squaring the signal has a -distribution with one degree of freedom:
The signal is usually smoothed after squaring (Fig. 1e). There are several ways to smooth a signal. For an analogue signal the simplest way of smoothing is using a low-pass filter. This will remove all frequencies higher than a certain cut-off frequency from the signal. For a signal that is discrete in time (a sampled signal) the simplest way is to add two consecutive samples, thereby reducing the time resolution by a factor two. The distribution function of this smoothed signal is the auto-convolution of Eq. (9), which is a -distribution with two degrees of freedom. Analogously, adding k samples results in a -distribution with k degrees of freedom:
(10) |
If many samples are added ()
the distribution approaches a Gaussian (central limit theorem) with the same mean and standard deviation. We assume that the off-pulse mean has been subtracted,
so
.
During the on-pulse, the mean increases to
and the standard deviation from
to
:
the Gaussian distribution is shifted and has become wider. However, the widening can be neglected if enough smoothing has been applied: if the smoothing is made by adding k consecutive samples, the ratio of the standard deviation during the on- and off-pulse in case of a weak pulsar (
)
is given by
An n-bit digitiser has N = 2^{n} discrete output levels, except in the case of a so-called 1.5-bit digitiser, which has 3 output levels. The digitiser determines the output value by comparing the input signal level with N-1 threshold values
:
one boundary between each two
output levels. The output signal
can be described as:
(12) |
General-use digitisers make as few assumptions about the distribution of the signal that needs to be digitised as possible. If it is assumed that the signal has a symmetrical distribution around its mean
it is common to set the middle threshold at
and set the other thresholds symmetrically around it.
(13) |
(14) |
So, in case of ,
the thresholds and output values are
(19) |
n | N | X | X_{1}, X_{2} | D^{2} | ||||
() | () | () | () | () | ||||
1 | 2 | 1 | 1 | 0.404 | 0 | 0.798 | ||
1.5 | 3 | 0.6123 | 0, 1.3602 | 1 | 0.200 | 0.375 | 0.900 | |
2 (uniform) | 4 | 1 | 1 | 0.8846 | 0.119 | 0.547 | 0.938 | |
2 (equidist.) | 4 | 0.9957 | 1.0607 | 1 | 0.123 | 0.545 | 0.939 | |
2 (non-equi.) | 4 | 0.9674 | 0.5243, 1.5653 | 1 | 0.123 | 0.530 | 0.941 | |
4 | 16 | 0.33523 | 0.33718 | 1 | 0.0116 | 0.915 | 0.977 | |
8 | 256 | 0.030765 | 0.030766 | 1 | 0.999 | 1.000 |
A 1-bit digitiser has two output levels
(X_{1} = X/2) and one threshold. This threshold is set at the mean of the signal for symmetry reasons, i.e. at zero level for a voltage signal. The requirement of power conservation (17)
results in
,
so
We will discuss one example of a digitiser with non-equidistant output values and that is
the case of a non-equidistant 2-bit digitiser. This digitiser has three thresholds
and four output values
with
0 < X_{1} < X_{2}. Since there are now three free parameters, power conservation can be required over the intervals
(
not included) and
(the negative intervals give identical equations):
The settings of an equidistant 2-bit digitiser can be calculated in a similar way as for a non-equidistant digitiser with
X_{1} = X /2 and
X_{2} = 3 X/2 (Eq. (16)). Power conservation is now only demanded over the total interval
(Eq. (17)). The
numerical solution is:
Commonly, the values for the threshold and the output level are rounded-off:
The settings of a 1.5-bit digitiser can be calculated from the non-equidistant 2-bit digitiser by setting X_{1} = 0 and X_{2} = X. The requirement of power conservation leads to:
(28) |
In similar ways the threshold level and output values of a 4-bit and an 8-bit digitiser can be calculated. All results are summarised in Table 1.
In this section the behaviour of a digitiser as a function of the incoming SNR is studied and the degradation of the SNR for a weak signal is determined. First, we will discuss a general non-equidistant 2-bit digitiser. From this the result for a 1, a 1.5 and an equidistant 2-bit digitiser can easily be derived. After that, the other digitisers will be treated.
Figure 3: Digitised signal-to-noise ratio as a function of the incoming signal-to-noise ratio for several n-bit digitisers in the case of a) an undetected signal, b) a detected signal with a distribution with k degrees of freedom and c) a detected signal with a Gaussian distribution. The dashed line is the response of an ideal digitiser. | |
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To determine the digitiser efficiency factor
for an undetected signal, the power (i.e. the variance) during the on- and off-pulse after the digitisation should be calculated and compared with the power during the on- and off-pulse before digitisation. If the input signal is Gaussian and the thresholds and output values are set symmetrically, the mean of the digitised signal will be zero. The variance is calculated by integrating the square of the digitised values normalised with the Gaussian distribution function:
= | 0, | (30) | |
= | (31) |
If a Gaussian distributed signal with mean
and variance
is digitised by a non-equidistant 2-bit digitiser as described in the previous section, the mean remains
zero and
From Eq. (33) the digitiser efficiency factor for a weak pulsar can be approximated by:
These equations apply for all n-bit digitisers with : the equations for an equidistant 2-bit digitiser are obtained by substituting X_{1} = X/2 and X_{2} = 3X/2. A 1.5-bit digitiser is obtained after setting X_{1} = 0 and a 1-bit digitiser after X_{1} = X_{2}. All results are listed in Table 1.
From Eq. (34) it follows that the digitiser efficiency factor of a 1-bit digitiser in the case of an undetected signal is zero. The variance of the signal after 1-bit digitisation is equal to . If the variance of the incoming signal increases during the on-pulse, the variance of the digitised signal remains the same. The powers of the digitised signal during the on- and off-pulse are equal. A 1-bit digitiser with its threshold set at the mean of the input signal cannot discover whether the signal strength changes. Note that this digitiser can detect such a change when its threshold is set at another value, e.g. .
Nevertheless, with a digitiser with its threshold set at the mean of the input signal, it is possible to detect a dispersed signal. The process of coherent dedispersion uses the phase information that is stored in the crossings of the mean level to partly restore the original pulsed signal.
The digitiser efficiency factor
for a detected signal is calculated by determining the mean and the standard deviation of the signal after digitisation.
= | (36) | ||
= | (37) |
(38) |
With these settings the lower levels are hardly if ever used for small values of k. Assuming a Gaussian signal when determining the digitiser settings of -distributed signal with a small number of degrees of freedom does not produce an efficient digitiser. PuMa (see Sect. 5) checks if the lowest possible value of the signal (i.e. zero) is below the lowest threshold. If not, it will increase all thresholds such that the lowest threshold is at above the zero level (Voûte & van Haren 1999). PuMa calculates the total intensity data (Stokes I parameter) by performing a Fourier transform and adding the squares of the real and imaginary parts. The number of degrees of freedom is therefore at least 2. This special case is plotted in Fig. 3b with a dotted line. From this figure it is clear that in this case PuMa uses more efficient digitiser settings than the standard case (i.e. settings for a Gaussian signal).
During the on-pulse the Gaussian distribution function shifts from to . Here, it is neglected that the distribution function also becomes broader. We first discuss the non-equidistant 2-bit digitiser.
The on-pulse mean of the signal after digitisation is given by:
The signal-to-noise ratio can now be derived. This curve is plotted in Fig. 3c. Again, the efficiency for small values of the is better than in the previous case. However, for large values the is constant. Only the highest output value will occur and the digitiser is completely saturated.
For a small shift with respect to the threshold level (
), the digitiser efficiency factor is approximated by:
From these equations the efficiency functions for a 1, a 1.5 and all 2-bit digitisers can be derived. A 1-bit digitiser has no difficulty to detect a change in mean of the incoming signal. In general, a digitiser has less effect on the SNR of a detected signal than on the SNR of a non-detected signal: for symmetry reasons it is easier to find a mean in a detected digitised signal than to find a change of the variance in a non-detected digitised signal.
To determine the dynamic behaviour and the digitiser efficiency factor of a n-bit digitiser with n>2, the digitised mean of the detected signal should be calculated:
Figure 4: Ratio of the observed and expected digitised signal-to-noise ratio as a function of the incoming signal-to-noise ratio for several n-bit digitisers. The upper panel shows the case of an undetected signal, the lower panel shows the case of a detected signal. The dashed line indicates a ratio of unity. | |
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We have used these equations to verify the operation of PuMa, an almost completely digital pulsar machine, which is operational at the Westerbork Synthesis Radio Telescope (see Voûte et al. 2001 and for a more technical description: van Haren et al. 2000).
In the first (analogue) stage of PuMa an incoming band of 10 MHz is 12-bit digitised and sampled at 20 MHz (Nyquist rate). These samples are transferred to a cluster of digital signal processor (DSP) boards. These processors have two modes of operation.
Mode 0 is a baseband sampling mode. The data are 2, 4 or 8-bit sampled after a possible bandwidth reduction by a digital Finite Impulse Response (FIR) filter. PuMa uses FIR filters with symmetrical coefficients to preserve the phase of the signal. The resulting undetected data are suited for later off-line coherent dedispersion.
The second mode (Mode 1) is a digital filterbank mode, in which temporal resolution is converted to spectral resolution. Each 2n consecutive samples of both polarisations are complex Fourier Transformed to get n spectral channels ( within 10 MHz). The resulting complex data of both polarisations are (cross-)mul-tiplied to get the four Stokes parameters (I, Q, U and V). Consecutive data points can be added per channel to smooth the data and to reduce the effective sampling time. For each channel the resulting time series are 1, 2, 4 or 8-bit digitised and stored. The data from all channels can be added after incoherent dedispersion during the off-line data reduction.
We observed rectangular pulses with a known SNR, generated by an artificial pulsar: a noise generator, which can be modulated with a square wave. We performed test runs with PuMa in
Mode 0 with 2, 4 and 8-bit digitisation. Prior to each run the digitiser levels were set in a so-called pre-run. In this pre-run PuMa determined the variance of the signal without any modulation and set its thresholds equidistantly with
(42) |
In each run we observed pulses from the artificial pulsar with 11 different signal-to-noise ratios, ranging from 0.06 to 100. We calculated the observed by determining the on- and off-pulse variances (Eq. (5)). We compared these ratios with the expected values from the previous section. The off-pulse variance in the 4 and 8-bit case was not equal to the expected value of 1. This was probably due to a slight error in the setting of the thresholds during the pre-run. Therefore, we used the actual values of to determine the from Eq. (35).
The ratios of the observed and expected values are plotted in the upper panels of Fig. 4. The observed values are in excellent agreement with the expected values. Only the highest SNR-ratio of the 4-bit run differs by three times the error from unity. The error bars are mainly due to the uncertainties in the fit of the SNR of the artificial pulsar. Note that the errors within one run are not independent, since they all include the same error in the fit of the SNR of the artificial pulsar and the error in the off-pulse variance determination, which was done only once per run.
Since these observed values are equal to the expected ones, we conclude that there was no significant loss of SNR due to the 12-bit digitising in the first (analogue) stage of PuMa.
We performed similar Mode 1 runs for 1, 2, 4 and 8-bit digitisation. The incoming band of 10 MHz was split into 32 channels. The squared data were smoothed by adding 768 samples to get a Gaussian distribution. The thresholds were set by changing the scale and offset manually (see the PuManual, Voûte & van Haren 1999) until the thresholds were correct for at least one channel. The thresholds could not be set correctly for all channels at the same time, since they are set identically for all channels and the bandpass of the artificial pulsar was much less flat than the bandpass during a normal observation, e.g. due to aliasing effects in the analogue stage.
In each run we observed a signal from the artificial pulsar with 14 different levels, ranging from 0.16 to 14 after squaring and smoothing. Observing higher values was not useful, since the digitised signal was already saturated, i.e. in the flat part of the curves in Fig. 3c.
For each run we selected the channel for which the thresholds were set optimally and which where non-aliased. The expected values of the signal-to-noise ratio were calculated from Eqs. (39) and (41), where we used the actual setting of , which was slightly different from the value in Table 1 for the 4 and 8-bit runs.
Again, the observed values match the expected ones within the uncertainties of the test (lower panels of Fig. 4). The errors are mainly determined by the inaccuracy in the determination of from our observations.
Max | J & A | This paper | ||||||
equi | non | non | equi | non | ||||
0.9957 | 0.9816 | 0.9674 | 0.9957 | 0.9674 | ||||
X_{1} | 0.4979 | 0.4528 | 0.5419 | 0.5304 | 0.5243 | |||
X_{2} | 1.494 | 1.510 | 1.618 | 1.591 | 1.5653 |
The settings of a digitiser have been discussed before by other authors (Max 1976; Jenet & Anderson 1998; Stairs et al. 2000) as have been the effects of digitisation on the SNR (Cooper 1970; Jenet & Anderson 1998). Our work differs from these earlier works in a few respects.
Cooper (1970) describes the effect of the digitisation of two partly coherent signals. These signals z_{1} and z_{2} are Gaussian with zero mean and variance .
They can be written as
z_{1} = x + y_{1} and
z_{2} = x + y_{2}, where x is the coherent part, with mean zero and variance
and y_{1} and y_{2} are the incoherent parts. The correlation coefficient of the two signals is
.
Cooper derives that for a weakly correlated signal (
)
the correlation coefficient after digitising is:
These pulsar observations are described by Jenet & Anderson (1998). They derive the effect on the SNR due to the loss of coherency, i.e. the loss of phase information in the signal. This is different from our theory, which describes the effect on the SNR due to the loss of amplitude information. Jenet & Anderson can neglect this effect since they use thresholds that are adapted whenever the variance changes (e.g. during the on-pulse): their digitiser settings are always (also during the on-pulse) set optimally. However, these changing thresholds produce extra dips on either side of a dedispersed pulse (see their paper for explanation).
If one uses fixed thresholds and the undetected signal shows modulation at the point of digitisation (e.g. during a high frequency observation with a small bandwidth) our theory should be applied in addition to the Jenet & Anderson theory. However, their theory does not deal with the SNR loss of a detected signal (in case of incoherent dedisperion).
Our method of setting the digitiser thresholds and output values differs from that of Jenet & Anderson as well. They set their thresholds by demanding that the partial derivatives of D^{2} with
respect to the threshold and output values are zero, while we demand that the total derivative with respect to
is zero and that power is conserved. Besides, Jenet & Anderson use so-called power optimised levels: they define D^{2} as
(44) |
Our method for setting the digitiser thresholds also differs from the method described by Max (1976). He sets the thresholds and output values by minimising the square of the distortion (Eq. (18)) with respect to both and . His output values X_{i} are the centroids of the distribution function f(x) between and and his threshold values lie in the middle between X_{i-1} and X_{i}. As a result his output values are not equidistant. Because the value of X is just a scale factor that is not important for the SNR, we minimise the distortion only to and we set X by requiring power conservation.
Table 2 shows the best settings of an equidistant and a non-equidistant 2-bit digitiser calculated with three different methods (Max, Jenet & Anderson and our method). We conclude that the differences between the methods are small.
Acknowledgements
Some of the principles of this paper were developed after a course by Shri Kulkarni at the California Institute of Technology. We thank the staff of the Westerbork Synthesis Radio Telescope and especially R. Ramachandran and R. Strom for their support during our test observations. We thank B. Stappers of the University of Amsterdam for his contribution to the development of the PuMa reduction software and for his comments on the manuscript. The artificial pulsar was built by the astronomical department of the California Institute of Technology.