SIGNAL CLASSIFICATION BY DISCRETE FOURIER TRANSFORM Pauli Lallo Email:pauli.lallo@mail.wwnet.fi ABSTRACT This paper presents a signal classification method using Discrete Fourier Transform (DFT). In digital data transmission over telecommunication networks and radio channels digital modulation schemes are used. In information warfare these waveforms are detected, classified and identified. We study here how DFT-methods can be used for classification purposes. We will analyze transmission baud rates, modulation methods (MFSK, MPSK, QAM), transmitted symbols and bit rates using different sampling frequencies and symbol times (numbers of samples) with DFT. Our simulation results gave some ideas for a new data modem design using DFT and high speed Digital Signal Processing (DSP). We have developed an adaptive modem prototype and an algorithm based on DFT. We have tested new adaptive waveforms and we had seamless data transmission through different public analog and digital telecommunication networks including gsm network with much higher bit rates than with any standard ITU-T modems. INTRODUCTION In signal classification we investigate different carrier frequencies (Hz), modulation rates (symbol/s) and data rates (bit/s) used in data transmission with modems. Digital modulation characteristics include modulation method parameters like amplitudes, phases and carrier frequencies used. Also we are interested in time base elements as symbol time, bit time, and sample time. These time elements are used in the Fourier analysis of the signal. We get the analogue values for frequency, amplitude and phase using the discrete Fourier transform. To investigate unknown signals we have to synchronize our detection to the bit stream. Then we get the carrier frequency, we find the symbol rate and we investigate the modulation method with some algorithms. With DFT [1]-[2] we can focus our investigations to interesting signal frequencies. We use different sampling frequencies and symbol times (number of samples) in DFT to get the amplitude, frequency and phase of the unknown signal. We have designed a set of calculator type simulators [3] to investigate signals as discontinous functions with DFT. First we look at the standard modems of ITU-T [4]. ITU-T has recommended following modulation methods for data modems since 1964: 2-FSK, 4-DPSK, 8-DPSK, 16-QAM and 32-QAM. Modem development is seen in modulation, bit rate, and symbol rate. Each symbol represents several bits from 2 to 5. Symbol rates are 300, 600, 1200, 1600 and 2400 Bd. The bit rates have developed from 300 bit/s to 33600 bit/s in 30 years. Carrier frequencies of most modems are 1800 Hz but a pair of 1300 and 2100 or 1200 and 2400 and also 1700 Hz alone is used. The detection of frequency is based on the useful frequency band of the channel and the calculation of DFT of the received signal which is sampled. We find that we need only a limited number of samples [6]. ANALYSIS Discrete Fourier Transform We will analyze first the carrier frequency using Discrete Fourier Transform with different DFT frequencies m ü(f), Formula (1). We calculate out the modulation method with proper sampling frequencies ü(f), numbers of samples n, and symbol time n ü(t). DFT calculations will give us then the amplitudes and the phases of the known frequencies of the received signal constellation Sx. Formula (1) Carrier frequency First we examine the number of samples needed for carrier frequency estimation. Figure 1 shows DFT as a filter. Figure 1 DFT as filter 1
Figure 1 show the signal estimation capabilities of some DFT filters with 26, 40, 80 and 160 samples. Sample frequency 16 kbit/s is used although any frequency can be selected. The received bandwidth varies between 800 Hz and 200 Hz depending on the number of samples. The filter performance is best with large number of samples (long symbol times). In practice we may need high sampling frequencies in signal hunting. We can optimize the DFT parameters according to the selected sampling frequency. Modulation (symbol) rate If we analyze the symbol rate for the signal we have to know the carrier frequency. Figures 2-3 show how the carrier frequencies are found with DFT. In figure 2 we have about 300 samples of an unknown signal sampled at 16 kbit/s. It has some noise or other inaccuracies. In figure 3 we have calculated its DFT using 160 samples. We can use any number of samples with DFT which is not possible with Fast Fourier Transform (FFT). Thus we can adapt our investigations to any symbol rate. DFT gives us now the frequency response of the unknown signal in 100 Hz steps. We found in figure 3 that there are at least two main carrier frequencies at 900 and 2800 Hz when DFT threshold value 60 is used. Figure 2 Unknown analog signal Modulation method To analyze the modulation method of the signal with a given carrier frequency and symbol time we used DFT and calculated the amplitude and phase values of a simulated signal. Table 1 shows the results of the analysis with DFT using only 26 samples. We calculated the amplitude and phase values of the known two carrier frequency using Formula (1) and Euler's formula. Table 1 shows the phase jitter of the received signal vs the received 8PSK signal mean symbol phase values both at 615 Hz and at 1230 Hz carrier. The eight mean phase values of the phase were found easy because the phase jitter was low < 12 degrees in all cases. Thus we classified this signal as 2FSK8PSK. 615 Hz mean 23,73 23,73 63,11 101,34 155,68 202,30 242,76 282,27 333,33 Jitter dgr 3,48 3,48 3,62 6,69 3,69 3,53 3,95 6,33 4,56 1230 Hz mean 22,01 22,01 70,48 112,57 161,03 200,57 249,18 292,25 337,71 Jitter dgr 11,88 11,88 6,44 8,24 10,56 10,49 5,48 7,12 9,10 Table 1 Results of a 8PSK detection with DFT Effect of sample frequency in identification Figure 3 DFT of the unknown signal We found that the sample frequency in DFT simulations can be adjusted to the received signal. Thus we could adapt our DFT receiver to the proper symbol rate and time. Using the right number of samples we could calculate the phase and amplitude most accurately for any signal. Discussion of unknown signal If we do not have any standard modulation method but an unknown signal to investigate, we can try the correlation method. We calculate the correlation of our received unknown signal with all signals in our signal library. We get the estimate for the best fit. If we have a large library for detected candidate signals they can be calculated as easy as the standard signals. 2
Figure 4 FH signal simulation symbol time. 4. The calculated phase and amplitude distributions are used for modulation analysis for each carrier frequency. DFTvalue Threshold Received N=160 50,00 khz Received OUT 3,40 100,00 8,73 200,00 3,58 300,00 7,75 400,00 10,67 500,00 3,45 600,00 6,85 700,00 2,92 800,00 24,78 900,00 75,89 76,00 1000,00 3,36 1100,00 5,36 1200,00 73,93 74,00 1300,00 6,24 1400,00 7,02 1500,00 1,81 1600,00 2,19 1700,00 Table 2 Simulation results of a FH signal Transmitted khz IN 1000,00 1300,00 Frequency hopping (FH) In figure 4 we have simulated a FH signal. Table 2 presents simulation results of this FH signal which corresponds a multi carrier signal in our simulations. The detection of the signal is calculated with DFT using 160 samples to get 100 Hz channel spacing with 16 kbit/s sampling (data transmission). This simulation result represents at the same time a FH signal with 100 khz channel spacing if we use 16 MHz sampling. In the first step with threshold value we can select the proper signal level (DFTvalue) of possible transmitted signal elements and carriers. Table 2 shows that with the threshold value 50 we found two signals in a 160 sample block. In FH there is a hop between 1 MHz and 1.3 MHz. Modulation analysis with DFT goes in the same way as earlier with data modems using discontinous functions. Adaptive modem Shannon stated as early as in 1940 that by sufficiently complicated encoding systems it is possible to transmit binary digits at a rate C with as small a frequency of errors as desired [7] p. 16. We investigated this channel capacity of bandlimited channels and compared PCM and DM systems at 2300 Hz bandwidth. We calculated their granular S/N-ratio and then we approximated the channel capacity with Shannon's Formula (9). The results are presented in Figure 5. Both PCM and DM systems should give over 20 000 bit/s channel capacities in this band-limited case. The present modems are at the Shannon's level already. In time presentation we hardly can see the signal properties necessary to classification, figure 4. Thus we make the following steps: 1. Fourier analysis (DFT) to get the carrier frequencies m üf involved. 2. Symbol rate analysis 1/(nüt) to get the symbol time used. 3. Amplitude and phase calculations of discontinous functions using Euler's formulae in developing Formula (1) with the proper number of samples n adapted to the Figure 5 Shannon's capacity limit vs S/N-ratio [7] Our interest is in developing Shannon's sufficiently complicated encoding system. In 1996-1998 our small 3
group (6 persons) worked out a system which we call the adaptive modem. We present here for the first time some waveforms made by our prototype in the tests, Figures 6-9. They all use 32 samples per symbol at a rate 45000/s. Figure 6 Simple 2-PSK 1406 bit/s success in transmissions over the basic telephone network at 26 000 bit/s which was possible also over gsm radio connections. Shannon's formula, Figure 5 tells us that over a band-limited channel B=3100 Hz and at a low signal to noise level S/N=30 db we should get nearly 30 000 bit/s data rates. Figure 8 40-QAM 56250 bit/s Figure 7 16-QAM 22500 bit/s Figure 9 60-QAM 84375 bit/s This system is based on the digital signal processing (DSP), analog-to-digital and digital-to-analog converters, assembly language algorithms, Digital Fourier Transform theory, and an advanced interface between personal computer (PC) and the telecommunications network. This is an invention which is a subject of another study. The field tests over band-limited channels are just now in the beginning of 1999 in an early state so we can only tell the We know that standard V.34 modem communicates at 33.6 kbit/s so we are at the Shannon's limit already. Our adaptive modem can work at Shannon's limit over different channels for example over band-limited radio channels adapting the encoding algorithm to the required bandwidth or other parameters. 4
The adaptive performance is in the program algorithm of the modem. Thus we can change on the fly most parameters of the data communication including modulation method, number of carrier frequencies, symbol rate, symbol time, sample frequency and number of samples in reception etc. Adaptation, distortion correction mechanism, level adjustment, and synchronization are the subjects of another study. They are essential parts of the complicated encoding system which we call the adaptive modem. This adaptive modem prototype is already capable to transmit and receive bit rates over 100 000 bit/s. The useful bandwidth of the prototype adaptive modem is not limited by the receiver A/D but by the channel used. SUMMARY The main result in this simulation study are: 1. The use of Excel or other worksheets in simulations is a robust method of signal, symbol or bit simulations. 2. DFT is not generally used because of more complex and time consuming calculation than FFT. In practical applications FFT method is used for signal analysis. Discrete Fourier Transform is capable of signal analysis with the present personal computer (PC) capabilities (processor speed, memory capacity, COTS programs, etc.) with limited number of samples in symbol. 3. Simulations are possible with most mathematical programs. The choice is made by the individual programmer based on the own knowledge and preference. 4. Signal classification is a complex process that requires a) frequency analysis b) symbol rate analysis c) modulations analysis. 5. DFT is capable of doing not only the analysis needed in signal classification but also real time data waform encoding. The variables that we have used are a) Sample frequency. b) Symbol rate and number of samples in DFT. c) Signal frequency. d) Number of amplitudes and phases of each carrier. After our simulation studies 1993-1996 we started a new investigation of the adaptive modem. We developed an algorithm and an interface during 1997-1998. This is based on the results of the Excel simulations presented here. In 1998 we build a new data modem prototype based on the simulated enchanced Discrete Fourier Transform. After the successful field tests with adaptive waveforms we put our patent pending on 17.11.1998. According to our simulation results and some prototype tests last year we have made with these new adaptive waveforms seamless data transmission through different public analog and digital telecommunication networks with much higher bit rates than with standard ITU-T modems. The new waveforms have promising high bit rates over band-limited channels or radio channels which are not possible with standard modems. REFERENCES [1] Råde, L., Westergren, B., Beta Mathematics Handbook, Lund, Studentlitteratur ab, 1990. [2] Lallo, P., Investigation of Data Transmission over an Adaptive Delta Modulated Voice Channel by Simulations using a Spreadsheet Program, pp. 554-559, IEEE Military Communications Conference November 2-5, 1997., Proceedings of MILCOM 97, Monterey, CA 1997. [3] Feldman, P. M., Discrete-Event Simulation for Communication Networks and Link, 53 pages, IEEE Military Communications Conference 18-21 October, MILCOM 98, Boston, MA 1998. [4] CCITT, IX th Plenary Assembly, Melbourne 14-25 November 1988, Blue Book, ITU, Geneva, 1989. [5] Harris, F. J., The Discrete Fourier Transform Applied to Time Domain Signal Processing, pp. 13-22, IEEE Communications Magazine, Vol. 20, No. 3, May 1982. [6] Proakis, J.G., Digital Communications, McGraw-Hill Book Company, New York, 1989. [7] Shannon, C. E., Communication in the Presence of Noise, pp. 10-21, Proceedings of the I.R.E. Vol. 37 January 1949, Original manuscript received by the I.R.E., New York, July 23, 1940. 5