Design of Frequency Demodulator Using Goertzel Algorithm

Transcription

1 Design of Frequency Demodulator Using Goertzel Algorithm Rahul Shetty, Pavanalaxmi Abstract Far distance Communication between millions without a modulation is worthless, and Frequency modulation has many advantages. Frequency modulation with adequate data transfer capacity gives favorable position in crossing out actual happening noise. This project deals with the design of FM demodulator which includes FFT and Goertzel Demodulators using Xilinx system generator design. Xilinx system generator, a tool for Matlab environment plays a big role in implementing signal processing instead on DSP processors. This tool gives a programming HDL code and thus bit stream can be effortlessly produced. In this Thesis, a novel attempt is made to design reusable Goertzel blocks are designed for performing FM demodulation and its performance is evaluated with a typical Goertzel algorithm based system as well as with a system designed using FFT. The proposed system is designed using MATLAB Simulink, Xilinx Block sets and system generator, which enables model-based programming approach. Index Terms Communication, Demodulation, FFT, Goertzel, Processors, Simulation. I. INTRODUCTION Demodulators plays an important role in the field of digital signal processing and communications and hence it is used in various applications like Biomedical signal processing, digital spectral analysis for speech recognition, imaging and pattern recognition as well as signal manipulation using filters, Data compression, instrumentation, machine Inspection, terrestrial, axon prediction and multimedia, wireless communication systems, microwave, satellite, radio-over-fiber(rof), distribution antenna and radar systems, software-defined radio and many more applications. Amplitude, Frequency and phase are different kinds of modulation techniques available in communication systems. In this research Thesis, we motivated to do on FM techniques because it has better fidelity and noise immunity over AM. Slope Detector, Balanced Slope Detector, Foster- Seeley Phase Discriminator, Ratio Detector are direct methods and Phase Lock Loop is an indirect method for FM demodulation. Fast Fourier Transform (FFT) and Goertzel algorithm is approached in this Thesis for Demodulation of FM signals. Modulation is defined as the process by which some characteristics; usually amplitude, frequency or phase, of a carrier wave varies in accordance with the instantaneous value of modulating signal, or Modulation is a process of superimposing a low frequency signal on a high frequency carrier signal. The systems used to perform modulation are called as modulators and are considered as the most vital block of any communication system. Communication Systems employing these modulators are extensively used in a large number of applications such as radar, aerospace, naval/maritime communication, underwater communication, mobile communication and many more. Out of all the modulation techniques, frequency modulation (FM) is widely used because of its many advantages and its applications range from FM radio stations, on-board systems for fighter aircrafts, satellites and many more applications. Nowadays, digital signal processing plays a vital role in many real time applications in our day-to-day life and frequency demodulation techniques are recognized as basic elements needed for DSP. This provides a platform to design an algorithm for digital signal processors (DSPs) and Field Programmable Gate Arrays (FPGAs) based system design. The invention of digital signal processors (DSPs) and Field Programmable Gate Arrays (FPGAs) have spearheaded way for novel digital signal processing techniques and algorithms that are extremely used in large number of applications. A. Objective In this paper, a reusable Goertzel blocks are designed for performing FM demodulation and its performance is evaluated with a typical Goertzel algorithm based system as well as with a system designed using FFT. The proposed system is designed using MATLAB Simulink, Xilinx Block sets and system generator, which enables model-based programming approach. 680

2 II. MATHEMATICAL ANALYSIS OF FM DEMODULATOR Modem with respect to FM mainly has Transmitter and Receiver sections used to transmit and receive modulated signal. Transmitter part mainly has desired signal generator, Encoder, FM modulator and RF upconverter. Receiver consists of a FM demodulator, a decoder and decision logic. B. Frequency Demodulation Frequency demodulation is the reverse process of modulation which recovers the original modulating signal from the FM wave as shown in Fig 2, and so FM receiver must be sensitive to the frequency variations of the incoming signals. As FM signals can be wide or narrow band is made insensitive to the amplitude variations and this is achieved by having amplifier with a high gain, hence in this way any amplitude variations can be removed. A. Frequency Modulation (FM) Frequency modulation is one form of modulation where the instantaneous frequency f i (t) is varied linearly with the message signal m(t) and is given by expression f i t = f c + k f m(t) (1) where, f c is carrier frequency, k f expressed in hertz's per volt is frequency sensitivity of the modulator. Frequency modulation wave form is shown in Fig 1. Fig. 2. Waveform of FM demodulation signal Fig. 1. Waveform of FM signal General form of expression for Modulator wave is given as s(t)= A c. cos[θ(t)] where A c is carrier amplitude which is maintained constant overall and θ(t) is the angular argument which is varied accordingly with the message signal m(t). So θ(t) t value will be 2πfc + 2πkf m t. dt Hence FM wave in time domain can be expressed as t 0 s(t)=a c. cos[2πfc + 2πkf m t. dt] (2) 0 C. Classification of FM demodulators: FM demodulators can be classified into Direct methods: 1. Frequency discriminators/slope detector 2. Balanced Frequency discriminators/ Balanced slope detector 3. Zero crossing detectors Indirect methods: 1. Quadrature Demodulators 2. Phase lock loop (PLL) Indirect method known as PLL and Quadrature method is explained and designed here. D. Properties of FM signal Narrow band FM (NBFM): A NBFM is the FM signal with less bandwidth. The modulation index 'ß' is less as compared to 1 radian, Wide band FM (WBFM): The bandwidth of WBFM is much larger and the theoretical value tends to infinity. If the modulation index ß is larger, ideally FM wave contains the carrier and an infinite number of sidebands are located around the carrier. 681

3 DFT for an N point sequence is the complex computations that cause problems for high speed signal processing. E. FFT Based Frequency Demodulation Fig. 3. Block diagram of FFT based Frequency demodulator III GOERTZEL ALGORITHM BASED FREQUENCY DEMODULATION Goertzel algorithm is a digital signal processing technique most commonly used for single-tone and dual-tone multi frequency detection (DTMF) in telecommunication systems. In this Thesis, a novel attempt is made to use the algorithm for FM demodulation. It is a computationally efficient algorithm for computing a Discrete Fourier Transform (DFT) of sample sizes that are positive integer powers of 2. The DFT of a N point sequence of x(n), indexed by n =0, 1,2 N-1 is given by X N 1 kn (K) = n=0 x n. W N ; K = 0, 1 N- Where x n and X(K) are complex numbers. Assume x n has input data samples in time domain (n = 0, 1 N-1), N represents the number of input samples, W N represents twiddle factor is defined by, W N =e j 2π N = cos(2 π/n) jsin (2π/N), K denotes Index for a frequency Bin width of a discrete signals (k =0, 1 N-1), X K 2 = Xk re 2 + Xk Im 2, where Xk re and Xk Im are real and imaginary component of output data stream. Xk_index marks the index of the output data that can be calculated using N f in f s, where N denotes length of the transformation, f in is input frequency in Hz and f s is sampling frequency in Hz. Frequency can be estimated using k f s N. DFT (i.e. Discrete Fourier Transform) is one of the primary tools that are used for the frequency analysis of discrete time signals and to represent a discrete time sequence in frequency Domain using its spectrum samples. An N-point discrete Fourier transform (DFT) performs the conversion of time domain data into frequency domain data. The DFT pair equation of N point sequence of x(n) is given by, j (2π N )kn N 1 X(K)= n=0 x n. e 0 K N 1 (3) The more common and simplified notation for the DFT is written as, N 1 kn X(K)= n=0 x n. W N 0 K N 1 (4) Where W N represents the twiddle factor, 2π j 2π 2π W N = e N = cos j sin and N is number of point N N FFT. An important issue with the implementation of the Fig. 4. Block diagram of Goertzel based Frequency demodulator Goertzel Algorithm, shortly called GA is a Digital Signal Processing (DSP s) technique, for identifying frequency components of an input signal. Since it uses second order IIR filter, GA filter bank is necessary for identification of every individual frequency components. Single tone detection, Spectrum analysis, Dual-tone Multi-frequency (DTMF) for telecommunication systems are the application areas using Goertzel algorithm. The GA filter s transfer function is written as, X (K) = 1 W N k. Z Cos 2 π k N. Z 1 + Z 2 (5) The respective difference equation of this second order IIR system is, V k (n)= x (n) + A V k (n 1)- V k (n 2); 0 k N (6) X (k) = Y k n = V k (n) + W N k. V k (n 1) (7) V k (1) = V k (2) = 0 for each value of K. The V k n equation is iterated for all input samples until the last state variable V k N is obtained. The pre-computed coefficients are, A = 2cos (2 π k / N) (8) k = round (f / f s * N) + 1 (9) W N k = cos (2 π k / N) - j sin (2 π k / N) (10) Thereafter, Y k n only needs to be computed once, when n = N, where x (n) is current input sample, N represents 682

4 Goertzel Block Size is like the number of points in an equivalent FFT, k represents frequency index of targeted frequencies, f is targeted frequencies in Hz and f s is sampling frequency in Hz. The Magnitude squared is given as, X (K) 2 = Y k (n) = V k 2 (N) + V k 2 (N-1) 2cos (2 π k / N)V k (N). V k (N-1) (11) Magnitude of X(K) is needed to decode the detected frequencies. As N increases, V k (n 1) and V k (n 2) and X (K) 2 increases and also it takes longer time for detecting the frequencies. The direct form II realization of Goertzel algorithm to detect a single tone or frequency is shown in Fig. 3.5 The Goertzel algorithm computes the kth DFT coefficient of the input signal x[n] using a second-order filter. The kth DFT coefficient is produced after the filter has processed N samples: X [k] =y k [n] n=n. The key in an implementation is to run v k [n] for N samples and then evaluate y k [n]. The computation for v k [n] takes one add (x[n] - v k [n-2]) and one multiply-accumulate per sample. In DTMF detection, we are only concerned with the power of the kth coefficient: y k [n]. y k k [N] The value of N must be shorter than the samples in half of a DTMF signaling interval, N < 400, be large enough for good frequency resolution (N > 512), and meet the relative error specification. We used a conventional value of N= 205, because it is roughly half of 400 samples. Decision logic can be added to give a valid DTMF signal if the same two DTMF tones are detected in a row to add robustness against noise. The variable k mentioned is computed as K =round(f/fs*n)+1 (12) Where f denotes the input frequency tone to be detected, f s denotes the sampling frequency and N denotes the number of points. In order to detect eight different frequencies from the modulator generated with a frequency deviation of 1 KHz, frequency demodulator is designed using Goertzel algorithm and the parameters are required to detect those frequencies are tabulated above as shown in tabulation from 1 KHz to 10 KHz. Fig. 5. Direct Form II realization of Goertzel Algorithm The Goertzel algorithm is more efficient than the Fast Fourier Transform in computing an N-point DFT if less than N log 2 N DFT coefficients are required. In DTMF detection, we only need 8 of, for example, 205 DFT coefficients to detect the first harmonics of the 8 possible tones, and then apply decision logic to choose the strongest touch tone. Since DTMF signals do not have second harmonics, we could compute another 8 DFT coefficients to compute the second harmonics to detect the presence of speech. The value of frequency index k, cosine term A and parameter W N k with FFT points N=1024 and sampling frequency fs = 64kHz for detecting eight different various tones using Goertzel Algorithm the realization structure of direct form II given in Fig In FFT based realization, magnitude X[k] is computed for first sample to number of FFT points i.e. k=1:n, whereas in Goertzel algorithm based realization, the magnitude X[k] is computed only once for a particular value of frequency index K with the help of above k relation. The block diagram for FM demodulation using Goertzel Algorithm based realization is shown in Fig The Goertzel blocks are selected based on the frequency deviation f of the input FM signal from the modulator blocks designed. The input to the modulator block is 3-bit counter signal, based on the counter n-bit signal, i.e. 2 n combination of different frequency range is generated based on the frequency deviation specified on the f in the 683

5 modulator design. The magnitude X[k] from each Goertzel block output is computed and the decision logic block shown in Fig. 3.4 is used to find argmax k X(k), which is in turn used to find the frequency of the input FM signal and the output voltage is generated accordingly. An alternative architecture to the one shown in Fig. 3.5 is given in Fig. 3.4 wherein the same Goertzel block is reused to detect different frequencies and it is designed with the motive of reducing hardware resources on increasing the number of frequencies to be detected. A. Applications of FFT Algorithm Spectrum Analyzer for on-board satellite communication systems requires FFT to compute frequency spectrum of an input signals. A.1 Advantages Frequency resolution can be achieved exactly for desired input signals over FFT. Disadvantages of Goertzel algorithm is frequency index can compute only for known set of frequencies and not suitable for random noise input signals for unknown frequencies. IV IMPLEMENTATION OF FM MODULATOR IN XSG Xilinx system Generator block set integrated with Matlab Simulink is used in this project for implementation. Generation of Sine Wave From the basic block of system generator i.e. addsub, constant, mult, rom and shift blocks, we can generate sine wave of desired frequency as shown in the Fig. 6. It is the faster version of DFT; it can be applied to the number of samples in the signal is power of two. The number of complex multiplier is greatly reduced in FFT in the order of N 2 log 2 N from N log 2 N in DFT.More Computation time is required for sweeping all frequency components to compute the specified frequency of interest. Reordering of input signal i.e. Bit-Reversal order is required to perform FFT. Complexity has increased on increasing the transformation length. B. Applications of Goertzel Algorithm Goertzel Algorithm especially applicable in the field of single tone and DTMF (Dual Tone Multi-frequency) detection in touch-tone telephones to represent the digits corresponding to the user push buttons as well as it is suitable for computer applications such as voice mail, telephone banking, pager systems, application and interactive control applications such as conference calling and call forwarding. B.1 Advantages of Goertzel Algorithm Goertzel Algorithm is more suitable in DTMF applications which require only few spectral components for detecting frequencies instead of computing whole spectrum. In this area, Goertzel Algorithm is significantly faster and also it requires only few constants needs to compute. So, that it saves computation time, reducing hardware complexity and avoiding Complex algebra. Also, Goertzel Algorithm does not require reordering of data in input and output side. Fig. 6. Generation of Sine Wave in XSG For example, consider if we want to generate a sine wave of 'f d ' = 1KHz frequency we need to calculate constant ' C ' value as f d = 1KHz size of addsub block is 'n' bit=16 fs=200khz C = (2n 1) f d f s therefor C = (216 1) 1K = K So when the constant value is given as , and run the program it will generate 1 KHz frequency of saw tooth signal ranging from 0 to as shown in the Fig

6 Fig. 7. Simulation of Saw Tooth Signal of 1 KHz Frequency Fig. 9. Simulation of Sine wave Signal of 1 KHz Frequency. A. Generation of FM Signal Generation of FM signal is similar to generation of sine wave, which also include similar block sets as shown in Fig. 10. Fig. 8. Read only Memory Parameter Box Read only memory (ROM) block will convert saw tooth wave to sine wave and its parameter are to be set according to the Fig. 8. Depth = 216 Initial Vector Value 32767*sin(2*pi*[0:1/65535:1]) in order to generate sine wave When the saw tooth signal is given to ROM block, it will convert saw tooth wave to sine wave and shifting range by shift block with parameter 15 shift, will results in 1V p-p sine wave of 1 KHz frequency as shown in Fig. 9. Fig. 10. Generation of FM Wave in XSG Here 2 constants are necessary to calculate say 'C 1 ' and C2. For example f c = 20KHz size of addsub block is 'n' bit =16 fs=200khz Δf=10KHz C 1 = (2n 1) f c f s (13) C 2 = 2n 1 (f c + f) (2n 1) f d (14) f s f s 685

7 C 1 = (216 1) 20K 200K = 655, and C 2 = (20K+ 10k) (216 1) 20K 200K 200K = B.System Generator model for FFT based Frequency Demodulation Setting the constant value according to the above calculated value, FM signal is generated with carrier frequency f c = 20 KHz and Δf =10KHzwhich is as shown in Fig. 11. Fig. 11. Top Level block diagram for Frequency Modulator using System Generator Fig. 13. System Generator model design for Frequency demodulators The Xilinx Fast Fourier Transform 7.1 block implements an efficient algorithm for computing the Discrete Fourier Transform (DFT). The N-point (where, N = 2 m, m = 3 16) forward or inverse DFT (IDFT) is computed on a vector of N complex values represented using data width from 8 to 34, inclusive). The transform computation uses the Cooleyturkey decimation in time algorithm for the Burst I/O architectures, and Decimation in Frequency for the pipelined and Streaming I/O architectures. Fig. 12. Simulation of FM Signal in XSG C. Results Analysis of FFT 1) Input frequency = 2KHZ, with sampling frequency of 50 KHZ. Fig. 14. Simulation analysis for Peak detection using FFT 686

8 Done pin at in x - axis, the difference between done pin and next peak values of the FFT signal ( ) is ) Input frequency = 2KHZ, with sampling frequency of 50 KHZ Fig. 15. Simulation Analysis for Peak detection using FFT Done pin at in x - axis, the difference between done pin and next peak values of the FFT signal ( ) is which is five times of the input signal x(n) i..e 2 KHZ. Fig. 16. System Generator model design for Frequency demodulators using Goertzel Algorithm D. System Generator model for Goertzel based Frequency Demodulation The system generator model realization for FM demodulator module using Goertzel algorithm is depicted in Fig. 13. In order to design one Goertzel blocks, one complex multiplier, set of multipliers, delays and adders/sub tractors are required which is designed from system generator library. For computing argmax k X[k],seven set of M-code Block sets are used for performing simple comparator operations to finding frequency bin corresponding to input FM signal. For frequency estimation and final stage process has been carried out as same as FFT explained above. Fig. 17. System Generator model design for Goertzel for detecting a single frequency In Goertzel based demodulation, based on the magnitude a particular frequency is detected. If user designed Goertzel algorithm for a specified frequency, three important parameters are to be considered. One is frequency index, k, which is to be easily calculated if we know the frequency of the signal to be detected. In this design, frequency index is pre-calculated based on the known input and sampling frequency relation k. The X[k], magnitude is maximum only the particular frequency, other than that its value is negligible. So, based on the magnitude of the signal we can detect the particular block which is to be detected. Other two parameters are cosine terms and twiddle factors can be calculated based on the appropriate frequency detection. 687

9 In order to detect the frequencies, modulator is designed using FM techniques. Based on the counter limited signal, the number of frequencies is to be generated. The Goertzel blocks are designed and coefficients are pre-computed to detect various frequencies. Once if it is detected the appropriate frequency index values are considered as output. In this way, all the eight Goertzel blocks are detected one after another and corresponding frequency index is chosen. This logic is built using MATLAB M-code and it is embedded into the Xilinx blocks using if-else condition. After computing frequency index passing through self-built M-code block and it is multiplied with F s / N, to compute the frequency of the input signal to be detected. This process is continuous for detecting eight different frequencies to be detected. This can be extended to detect N number of frequencies. The frequencies are converted back into fullpledged demodulated signal. V EXPERIMENTAL RESULTS A. Simulation results for FFT based frequency demodulation The simulation waveforms of the FFT based frequency demodulator is shown in Fig. 18. The counter signal is given as input to the modulator and the modulator waveform as shown in fig. In order to show better visibility of waveforms to see clearly, a counter signal 2-bit is considered as input to the FM modulator. In Fig. 18, the first waveform is counter limited input signal fed to the Frequency modulator, second waveform is the frequency modulator from modulator design, third waveform is FFT, the magnitude of output X[k] from FFT blocks which has real and imaginary components to an applied input signal, followed by the value of k, corresponding to maximum values of X[k] based on the number of FFT points on FFT block. Frequency detection is done with the relation and the value of 'k' obtained from FFT block and frequency to voltage conversion based on the logic explained earlier. Fig. 18. Simulation results of FFT based Frequency Demodulator B. Simulation results for Goertzel block based Frequency Demodulation Similarly the simulation waveform results of the Goertzel based frequency demodulator is shown in Fig. 19. In Fig. 19, the first waveform is input signal fed to the Frequency modulator from external counter signal to the modulator block as input, second waveform is the frequency output, which is based on the counter bit signal the waveform is generated inside on the modulator block. For example, if you provide 2 bit counter as an input to the modulator, four steps from 0 to 3 is generated in the counter and that is given as a input to the modulator block. For every step, there is a generation of frequency with the frequency deviation. Here, frequency deviation is considered as 1 KHz. so, for the first step it will generate 1 KHz which is added to previous generated sine wave frequencies. The next consecutive four waveforms represent the Goertzel block output X[k] computed from 2nd order filter designed based on the parameter set into the filter block. The value of X[k] which is high only for specified detected Goertzel block and other Goertzel block magnitude value is very less compared to detected frequency energy. The next waveforms Goertzel frequency index 'k' value and next waveform is frequency detection, it is done based frequency index value and logic is implemented explained in previous chapter. Based on the frequency detection, it is converted into the corresponding voltage value. 688

10 [4] Bampi Sergio and Pablo Juan Brito Martinez, "Design of a Digital FM (DFM) Demodulator based on a All-Digital Phase- Locked Loop with filter order 2" PGMICRO Graduate Program on Microelectronics Federal University of Rio Grande do Sul, UFRGS. [5] P Sumathi, IEEE paper on A Frequency Demodulation (FM demodulation) Technique Based on Sliding Direct Fourier Transform (DFT) Phase Locking Scheme for FM Signals. Fig. 19. Simulation results of Goertzel block based Frequency Demodulator VI. CONCLUSION In this paper, design of frequency demodulators for Fast Fourier transform, Goertzel Block Algorithm methods are proposed. A detail about the performance evaluation and algorithm design development for the proposed work is reported in this paper. It is observed that Goertzel blocks simulation and algorithm design development is compared in terms of simulation results with standard FFT based demodulation techniques. It is observed that the proposed frequency demodulators worked satisfactorily for all above mentioned methods and the same can be employed for various industrial applications. In application, that requires minimum number of blocks for design and implement in hardware Goertzel demodulator is the best choice. Rahul Shetty, B.E in Electronics and Communication Engineering, Sahyadri College of Engineering and Management, Mangalore Pavanalaxmi, Assistant Professor, Dept. of E & C, Sahyadri College of Engineering and Management, Mangaluru VII. REFERENCES [1] Mrs. Mahmooda, M. Vinod Kumar Reddy, Sagar Nayakanti, paper titled "Implementation of Spectrum Analyzer using Goertzel Algorithm", 2013, International Journal of Scientific and Research Publications, Volume 3, Issue 3, ISSN , March [2] D. Divya, MRS. M. A. Asima begum, G. Kalyan, paper titled, "DTMF Signal Generation and Detection Using Effective DFT (Goertzel algorithm) Technique on FPGA", 2015, International journal of Science, Engineering and Technology Research (IJSETR), Volume 4, Issue 11, ISSN , November [3] Anis W R, "FM and FSK detection using a subtractor filter" a IEEE paper in Circuits of Electronics and Systems, 2005, ICECS 2005 on (Volume: 2), the 7th International IEEE Conference. 689