Digital Signal Processor (DSP) based 1/f α noise generator

Transcription

1 Digital Signal Processor (DSP) based /f α noise generator R Mingesz, P Bara, Z Gingl and P Makra Department of Experimental Physics, University of Szeged, Hungary Dom ter 9, Szeged, H-6720 Hungary Keywords: /f noise, noise generator, DSP, digital filter ABSTRACT /f noise is present in several natural and artificial systems, and even though it was discovered several years ago, it is still not completely understood Due to the lack of an universal model, the main methods of investigating a system where /f noise is present are numerical simulations and real measurements The second method can lead to more adequate results, since it is free from numerical artifacts In the case of real measurements, we need reliable, wide-band noise generators Many ways of generating noise are known; most of them have several limitations on the frequency bandwidth or on spectral properties We wanted to create a device which is easy to use, which can generate any kind of /f α noise and whose bandwidth is wide enough to make our investigations We used a DSP (ADSP28) to numerically generate the desired noise, and a D/A converter to convert it to an analogue signal The noise generation algorithm was based on the known method of filtering a Gaussian white noise with a series of first-order digital filters We enhanced this method to get a better spectral shape and to compensate for the side effects of the digital-to-analogue conversion INTRODUCTION High-performance and reliable noise generators are useful to aid system analysis, system simulation, exploring time and frequency structure of different kinds of noises and many more Nowadays computer simulation is a widely used and accepted method to generate, use and analyse noise, since sophisticated random generation and other algorithms are available However, in real world applications it is often necessary to have an analogue noise source, like in system analysis and when the experiment has real components requiring analogue input signals Furthermore, the limited performance of numerical methods may degrade speed and accuracy, therefore noise generators with analogue output have many potential applications There are many ways to realise noise generators It is quite easy and straightforward to design a white noise generator with flat spectrum, however realising a high-performance, wide frequency range /f α noise source is still a challenge, mainly due to the absence of a simple generation method There are many ways to make a /f α noise generator [-3] One may use a real physical process, like semiconductor (eg MOSFET) noise, may apply cascaded or paralleled analogue filters to white noise or use the so-called mixed signal design (both analogue and digital), where a digital computer is used to make a pseudo-random sequence and a precision digital-to-analogue converter provides a real analogue signal All methods have some drawbacks, like limited frequency range, absence of easily scaled frequency range and even more probably the impossibility of easy setting of the desired power exponent α Digital methods have many advantages to address these points, but a general-purpose computer although it may be very fast - is not optimised for real-world applications, since it may be loaded by unpredictable network traffic, background processing, etc A very efficient solution is the use of a special-purpose digital signal processor (DSP) [3], which has very accurate timing and an extremely efficient processing core Advanced DSPs and development tools are widely available; therefore it is quite reasonable to use a DSP to build an efficient and flexible noise generator In this paper we show a realisation of a /f α noise source based on a fixed-point ADSP-28 DSP microcomputer [4] Our design incorporates a precision digital-to-analogue converter and an easily configurable software to aid simple setting of the frequency range and the exponent α The noise generator is capable of providing accurate /f α noise over more than four decades of frequency, allows manual or host computer (typically PC) controlled setting of parameters via its standard serial interface

2 2 /f α NOISE GENERATION USING DIGITAL FILTERS Digital filters model analogue filters and have very similar characteristics, only digital filters are used to transform data sequences that are discrete in time For a discrete data sequence x i, the output of a digital filter is given by Eq R C y i = D xi + D0 xi C0 yi () E E Fig Analogue filter versus digital filter On the right panel, we can see the realisation and the transfer characteristics of an analogue low-pass filter, while the left panel shows the same for the corresponding digital filter The curves come from theoretical calculations; the sampling frequency is 000 Hz The operation of the analogue and digital filters can be linked through the Bilinear Z transformation Without going into too much detail, we should like to note that since the operation range of the digital filter is limited to the 0 f<f s /2 frequency interval (as it follows from the sampling theorem), we can obtain the transfer function of the digital filter from the transfer function of its analogue counterpart by projecting the whole frequency axis onto the 0 f<f s /2 frequency interval, and substituting a transformed frequency into the analogue transfer function The frequency transformation is given by f s π f f ' = tg (2) π fs where f s denotes the sampling frequency If f f s, f This way, the transfer function of the digital filter proceeds as fs f a( f ) =, f f ' ' π 0 0 = tg (3) + i π f s f 0 ' wherein f 0 is the cut-off frequency of the digital filter The parameters of a first-order digital filter with a cut-off frequency f 0 and amplitude A 0 are l A0 C0 =, D0 = D = (4) + l + l π where: l = ctg, Ω Ω = f s s s f (5) 0 where Ω s is a normalised sampling frequency These parameters pertain to Eq We should like to note here that while digital signals are considered, the sampling frequency plays no role; only the ratio of the frequencies is important The concrete values of the sampling frequency and the cut-off frequencies of the filters come to the fore only when we consider a sampled signal, or a signal converted from the digital into the analogue domain If we pass the signal coming from a white noise source (random generator) through a first-order low-pass filter, we get a Lorentzian noise If we arrange filters whose cut-off frequencies are equally spaced on a logarithmic scale and whose F m /2

3 amplitude (A 0 ) is proportional to /f 05 in parallel, we obtain /f noise Analogously, we can produce /f α noise if the amplitude of the filters is proportional to /f α/2 White noise A, f A2, f3 A3, f3 + /f α noise Fig 2 /f α noise generator realised by means of first-order filters 0 0 /f 00 E Fig 3 A filter with a transfer function of /f power rule 3 PROBLEMS AND SOLUTIONS The filters are somewhat distanced from each other, which introduces undulation into the transfer function It is the distance of the filters that determines the extent of this undulation: the closer they are to each other, the weaker the undulation is On the other hand, if we use too many filters, the calculation load will increase On the basis of our experiments, it is possible to reach an undulation weaker than per cent with 5-2 filters per decade Digital-to-analogue conversion modifies the spectrum of the output signals This is because D/A converters produce step functions instead of Dirac δ impulses at their output Consequently, the spectrum of the output is weighted by a sin(x)/x function 0 00 f s E sin( π f / f s ) Fig 4 The weight function π f / fs At a frequency that is small compared to the sampling frequency the influence of this weight function is negligible, but if we want to exploit the full frequency range with the upper cut-off frequency of our noise, we have to take this effect

4 into consideration As an illustration, we note that at one tenth of the sampling frequency this effect makes a contribution of 5 per cent Due to the frequency transformation there is a sharp cut-off around f s /2, but again, this effect is significant only when we want to exploit the full frequency range given by the sampling frequency 0 00 E-3 Effect of frequency transformation f s /2 E Fig 5 The sharp cut-off due to the frequency transformation around f s /2 We can visualise the deviations from the /f power rule better if we multiply the amplitude transfer function by the square root of f (the /f power rule applies to the power spectral density, which is proportional to the square of the amplitude spectrum, thus the amplitude spectrum must conform to a /f 05 power rule) In the ideal case, this should yield a line on the logarithmic scale Of course, if we want to obtain an /f α noise in general, we must multiply the amplitude spectrum by /f α/ E-4 E Fig 6 Multiplying the amplitude transfer function by the square root of f, we can see that in the case of the first and the last filter, the transfer differs significantly from what is desirable The curves are based on theoretical calculations, f s =000 Hz, and there is one filter per decade In Fig 6 we can see that the deviation from the ideal power rule is rather large This significant deviation in the transfer function is caused by the fact that the filters at the beginning and at the end of the filter chain are in an asymmetric position On the basis of Fig 6, we can easily understand the implications of this The transfer function is the complex sum of the individual transfer functions of the filters in parallel This way, the transfer at the cut-off frequency of the first filter is given by a ( f ) X+ X 2 + X 3 + (6) where X=a (f ), X2=a (f 2 ) and X3=a (f 3 ) (see Fig 6), a (f) is the amplitude transfer function of the first filter, and f, f 2 and f 3 are the cut-off frequencies of the first, second and third filters, respectively At the cut-off frequency of the second filter, it is a ( f 2 ) X+ 2 X 2 + X 3 + (7) while at the cut-off of the third (middle) filter it is a ( f 3 ) X+ 2 X (8) Amplitude * f 05 [au] 0 05 X X X2 00 X3 E-4 E

5 We see that the transfer function is largest at the middle filter, while in the case of the filters at the sides the amplitude is reduced significantly To compensate for this effect, we should increase the amplitude transfer of the filters at the sides (additionally, we may also mind the other effects mentioned above) The problem is that modifying the transfer function of an individual filter will affect the whole transfer function, thus it is difficult to find the appropriate coefficients Our experiments show that it is possible to reach an accuracy of few per cent only modifying the amplitude transfer of the filters at the sides If we want to be more accurate, we may modify the transfer of the other filters, albeit to a relatively small extent The degree of compensation depends on the density of filters per decade and on the spectral exponent of the desired noise In the case of /f noise, the modifications in the filters must be nearly symmetric (of course, if we take into account the sin(x)/x weight and the effects of the frequency transformation, this symmetry no longer applies: the amplitude transfer of the filter with the largest cut-off frequency must be increased to a larger extent than that of the others We can see the effects of this modification in Fig 7; the amplitude transfer of the two filters at the sides we multiplied by 42 The undulation is due to the fact that there is only one filter per decade 5 Amplitude * f 05 [au] E-4 E Fig 7 Modifying the amplitude of the first and the last filters, the transfer characteristic is closer to the desired shape Our aim was to create an analogue output noise generator that is able to produce noises with /f α -shaped power spectral density over four decades In the following we will optimise the arrangement and amplitude of the filters for this device Of course, if we use this noise generation method in numerical simulations, we have more freedom in choosing the number and location of the filters We placed nine filters in the frequency range of Hz and 400Hz, at the sampling frequency of 000Hz Then we optimised the amplitude of the filters The result is illustrated in Fig 8 The reason for choosing this frequency range we shall explain in the next section As long as the generated noise is used only in simulations and it is not linked to real processes, it is only the ratio of these frequencies that matters Amplitude * f 05 [au] E-4 E Amplitude * f 05 [au] E-4 E Fig 8 The transfer characteristic of the nine combined filters On the right we can see that its undulation is within % The filters amplitudes are modified by 7, 06, 02,,,,, 086 and 8 respectively

6 The power spectral density and the distribution of the noise generated using this filter is visible in Fig 9 According to simulation the power spectral density is rather accurately /f-shaped over the range of four decades, and it has a Gaussian distribution (The parameters of the simulation: the sample length is 2 samples, probability densities are calculated from the average of 00 individual simulations and the sampling frequency is 000Hz) Power spectrum [au] E-3 E-4 E-5 E-6 E-7 E-8 E-9 E-0 /f E Probability X Fig 9 The spectrum and the time-domain distribution of the /f noise generator realised by digital filters If we would like to generate /f α instead of /f noise, the amplitude of the filters should be modified asymmetrically The filter characteristics for /f 05 and /f 5 noises are shown in Figs 0 and Amplitude * f 075 [au] E-4 E Frequency Fig 0 Filter configuration for /f 5 noise The amplitudes of the filters are modified by 337, 076, 037,,,, 0986, and 839 respectively Amplitude * f 025 [au] E-4 E Fig Filter configuration for /f 05 noise The amplitudes of the filters are modified by 33, 0, 04,,,, 0, 02 and 33 respectively

7 If we would like to generate noise with arbitrary spectral exponent, it is difficult to optimise the amplitude of all filters for each spectral exponent But we can have satisfying results (within an error of -5 per cent) if we modify only the first and the last filter If we modify only these two filters, increasing the number of the filters will reduce the undulation of the transfer characteristic, but the deviation from the ideal characteristic will increase at the sides of the frequency range Over two filters per decade, we have to modify more filters to get acceptable results Using numerical simulations, we get the best multiplying parameters for noises with an exponent α between 0 and 9 The results are shown in Fig 2 We can see a slight asymmetry between the two curves; this is because the filter with the highest cut-off is too close to the sampling frequency In applications where the sampling frequency is significantly higher than the highest cut-off frequency, the curves can be approximated as c α y, and c (2-α) y respectively, where c=78 and y=-0874 for the filter used above First filter Last filter Multiplier ,0 0,2 0,4 0,6 0,8,0,2,4,6,8 2,0 α Fig 2 The multiplier of the filters amplitude if we modify only the first and the last filter 4 HARDWARE IMPLEMENTATION PC RS 232 DSP D/A User interface Fig 3 The block diagram of our noise generator /f α noise The system is built around a high performance fixed point ADSP-28 from Analog Devices because it has easy algebraic assembly syntax, efficient core and integrated peripherals The standard serial RS-232 interface allows a host computer to control the DSP noise generator, but stand-alone operation is also possible, yielding a very compact size solution An optional user interface (eg pushbuttons and LCD display) may be used to control the system manually In our experiments the DSP was controlled by a PC running LabVIEW development environment [5] The DSP generated a white noise random number sequence based on a tested linear congruential method, and successively applied second order digital filters produced the /f α noise This sequence of random numbers were converted to the analogue domain by a 4-bit digital-to-analogue converter allowing maximum update rate of 300 khz Let us note that the use of a floating point DSP may provide more easy programming and higher performance at the expense of higher price, more power consumption and different development tools The block diagram of our device can be seen in Fig3

8 One of the major drawbacks to the fixed-point arithmetic is that there is a limit to its precision If the results of operations are too small, the relative error will be too large On the other hand, if the results are too large, there will be a risk of overflow or saturation So we have to be careful choosing the coefficients in order to exploit the available numeric range fully A further problem is that also the coefficients of the filters can be stored with a limited precision To utilise the available numerical range fully, we should have the signal y (see Eq ) as high as possible, yet at the same time it must not be too high because the signal may be saturated then The value of the signal y depends on the cut-off frequency and the amplitude of the filter; for each cut-off frequency we can find an amplitude or amplitude range where we can exploit the available numerical range best Our simulations have shown that the best amplitude is A b =C/f 05 This amplitude is ideal for generating /f noise, but if we should like to generate other types of coloured noises, we must either choose an amplitude different from the ideal or initialise every filter with his parameter and perform the weighting only before the summing The noise generator realised by means of this method is shown on Fig 4 White noise A b, f A b 2, f3 A b 3, f3 A/A b A/A b A/A b Fig 4 The scheme of a noise generator that uses a modified filter arrangement + /f α noise We studied the transfer function of the filter thus set up also by means of numerical simulations and have found that though the very first components are somewhat noisier, the deviation from the ideal frequency profile is not very significant 25 Theory Amplitude * f 05 [au] E Fig 5 The transfer function approximates the theoretical profile fairly well It may be important to know the boundaries of the frequency range in which we can generate the desired noise One of the major limiting factors is the sampling frequency We cannot place the filter with the highest cut-off too close to the half of the sampling frequency, because the distortion of the transfer function due to the frequency transformation would be too large then At a sampling frequency of khz an upper frequency limit of 400 Hz seems appropriate From the figures above, we can see that the frequency profile agrees with what is desired fairly well until around one tenth of the sampling frequency Because of the frequency transformation, it is difficult to expand this limit At the same time, there

9 is another limiting factor, due to the fact that we use a DSP with fixed-point arithmetic and thus we cannot store the parameters of the filters with arbitrary precision However, the precision of the value of C0 restricts the lower frequency limit, which is f s under this value, we can represent only the zero frequency We took into account these two limits in arranging the filters The fixed-point arithmetic of the DSP can also be implemented on computer, so we can test the precision of the system on computer and discover the errors that stem from an inappropriate arrangement in time In this way we obtained the noise spectrum in Fig 6 We can see that the fixed-point realisation did not reduce the accuracy of the noise generator E-3 Power spectrum [au] E-4 E-5 E-6 E-7 E-8 E-9 E-0 /f E Fig 6 The spectrum of the generated noise (simulation) Power spectrum [au] Fig 7 The spectrum of the /f noise generated by our device (the sampling frequency is 00 khz) Finally, we should like to outline the algorithm that runs on the DSP The noise generation routine is included in an interrupt routine, where the interrupts come from a clock signal which can either be an internal clock signal from the DSP or an external signal At the beginning of the routine we implement a linear congruence method, which yields a 32- bit integer The cycle length of the random generator is determined by these 32 bits If needed, we can apply a different random generator method to increase the cycle length After the random generator, we implement the individual filters, and then the weighted sum of the output of these filters Finally, we convert the data into a format compatible with the D/A converter, and convert it to an analogue signal The number of instructions that must be executed during one instance of sampling is about 20 The DSP we applied is a 40-MHz DSP: it can execute 40 million instructions per second (even multiplication is executed in one clock period) With this DSP, we can achieve a sampling frequency of 300 khz We plan to use a 60-MHz DSP (ADSP-29) in the future, with which the maximal sampling frequency can reach 3 MHz

10 5 CONCLUSION A widely used technique of adding filtered white noise sources was used to realise an efficient mixed-signal circuit to provide /f α noise with tunable power exponent α Our compact design incorporates a high-performance fixed-point DSP microcomputer, which generates the random numbers, makes the digital filtering, and drives a digital-to-analogue converter to translate the generated signal into a real-world analogue signal The system allows stand-alone, PCcontrolled or manually controlled operation, selection of power exponent α between 0 and 2 in steps of 0 /f α spectrum is valid over more than four decades of frequency, which range can be shifted from a maximum frequency of a few hundred khz down to any lower frequency Depending on the DSP clock frequency and digital-to-analogue converter update rate a few MHz sampling rate is possible Our further development plans include implementation of more efficient coding, higher speed and accuracy and development of an easy-to-use manual user interface Further information (data and program samples) will be available on our homepage ( ACKNOWLEDGEMENTS Our research has been supported by OTKA, under grant T We should also like to convey our thanks to SMD Technology Ltd (representative of Analog Devices Inc) and Cobra Control Ltd (representative of National Instruments) for providing us with the development environments we used We are also indebted to Sándor Nagy for his valuable contribution REFERENCES Z Gingl, LB Kiss, R Vajtai, "/f k noise generated by scaled Brownian motion", Solid State Comm, 7, , ML Meade, "Time- and frequency-domain models for fractional noises", Proceedings of the 0 th International Conference on Noise in Physical Systems, A Ambrózi ed, , Akadémiai Kiadó, Budapest, Robin Whittle, "DSP generation of pink (/f) noise", 4 Analog Devices, 5 National Instruments LabVIEW,