PRACTICE : Practice. Baseband Communication.. Objectives To learn to use the software Simulink of MATLAB so as to analyze baseband communication systems... Practical development... Unipolar NRZ signal The program Simulink can be used to determine the main characteristics of a unipolar NRZ signal. Design a random generator for this kind of signals by using the block Bernoulli Binary Generator, available in the Communications Blockset (Comm Sources/Random Data Sources). Add a sample & hold circuit (Simulink/Discrete/ Hold) which will convert the data digits (ones or zeros) supplied by the binary generator in square waves with certain duration. In order to attain this, the parameter sample time of the block generator must be modified, which indicates the time between consecutive generated symbols. In this case, we are going to keep the default value of second. Net, set the sample time parameter of the block zero-order hold to the value, for eample, /6 s. We are going to add an channel (Communications Blockset/Channels/ Channel) to check the effect of noise over the transmitted signal. The mode of the channel will be set to signal to noise ratio (SNR), with a SNR value of db, for an input signal power (watts) of.5 W. This last value has not been established arbitrarily, but it is due to the input signal waveform. This signal is a binary wave with levels of volt or volts, each value being held during second. Then, if we consider that the probability of sing zeros or ones is the same and equal to.5, the mean power of this signal, the input signal to the channel, will be.5 W. B-FFT Spectrum Bernoulli Binary Bernoulli Binary Generator Hold Channel Discrete-Time Eye Diagram After that, we are going to introduce the visualization instruments into the system: a scope (Simulink/Sinks/), a spectrum scope (Signal Processing /7
PRACTICE : Blockset/Signal Processing Sinks/Spectrum ) and an eye diagram scope (Communications Blockset/Comm Sinks/Discrete-Time Eye Diagram ). In the scope, set two visualization aes (Parameters/number of aes) and deactivate Limit data points to last in Data History, so as to allow the block to save every sample received to its inputs. In the Spectrum, set Buffer size to 4, Buffer overlap to 5 and Number of spectral averages to 6. Finally, in the eye diagram scope, set samples per symbol to 6, offset (samples) to 8, i.e. 6/, and traces displayed to. In this last case, we are only indicating the number of samples per symbol, that is, 6, in the same way as it was established in the sample & hold circuit. In addition, the given offset value is required in order to centre the eye diagram in the screen, as we will see. Lastly, set the simulation time to, seconds.. Observe the eye diagram and check as the amplitude values of the signal swing approimately between and, plus a noise component. Also observe the square appearance of such eye diagrams.. Observe, in the scope, the unipolar signal with and without noise. 3. Observe the spectrum of the unipolar NRZ signal. By default, the spectrum is represented in decibels. In the properties of the spectrum scope, in the tab Ais Properties, set Frequency range to [-Fs/... Fs/], Amplitude scaling to Magnitude-squared, Minimum Y-Limit to a value of -5 and Maimum Y-Limit to. Run the simulation again and check the new appearance of the spectrum of the unipolar NRZ signal. Observe as there is a spectrum peak at Hz, which perfectly agrees with what is theoretically predicted. Moreover, the obtained spectrum has the shape of a sinc-squared function with nulls at n Hz, for n =,,..., etc.... Polar NRZ signal Net, we are going to analyze polar NRZ signals. To that, we are going to modify slightly the previous implemented system in order to remove the dc component from the unipolar signal and make the new generated signal take the values volt, that is: s ( s ( polar unipolar () Therefore, we only have to multiply by two the output signal from the generator (unipolar) and remove one to it, then obtaining the polar version of that same signal. The new system which allows us to analyze polar signals is shown in the net figure, where an analog filter has also been added (Signal Processing Blockset/Filtering/Filter Design/Analog Filter Design), as well as a system which calculates the mean power of any signal. /7
PRACTICE : B-FFT cheby Spectrum Bernoulli Binary Bernoulli Binary Generator Gain Hold Constant Analog Filter Design Hold Channel Discrete-Time Eye Diagram Running Var Variance In u Display Mean Math Function The mean power of a random signal is given by: P P AC P DC ( ( ( ( ( ( m ( () where indicates time averaging, as follows: T / f ( lim f ( dt (3) T T T / and where and m are the variance and the mean squared of, respectively. This is the reason why we have included the blocks to calculate the mean power of the signal. The blocks Variance and Mean are available in Signal Processing Blockset/Statistics. In the analogue filter, set Design method to Chebyshev II, keep the default values for Filter type, Filter order and Stopband attenuation in db, and set the cutoff frequency (Stopband edge frequency (rads/sec)) to.75 rad/s, i.e..75 Hz. With the last action, we are looking for distorting the signal in order to yield intersymbol interference (ISI) between consecutive symbols.. Set the simulation time to, seconds. Observe the eye diagram which is distorted by noise and the ISI induced by the channel filtering. In order to check the effect of the channel filtering over the spectrum of the received signal, set the cutoff frequency of the filter to rad/s during one simulation and then change this value to,75 rad/s in the net one. Observe how the secondary lobes and the highest frequencies of the main lobe are strongly attenuated by the filter. In order to better observe this changes, set Amplitude scaling in the tab Ais Properties of the spectrum scope to db. Check as the peak at Hz has also disappeared in this case for polar signals, which should not surprise us since this kind of 3/7
PRACTICE : signals present a null dc value (m = ). In fact, if we calculate its mean power, we would have: P polar m. Being established the filter cutoff frequency to,75 rad/s, remove the noise component by setting SNR to db in the properties of the block Channel. Moreover, set the input signal power to W (the polar signal has the double of mean power as the unipolar signal, which can be checked during the simulation by means of the display connected to the output of the different blocks whose task is to calculate the mean power). Check in the eye diagram the effect induced only by the ISI. In order to compare the obtained results with the ideal case (without filter), set the cutoff frequency to rad/s again and run a new simulation. Observe as the ISI, when we filter beyond.75 Hz, reduces the eye opening greatly. 3. Observe, by means of a scope, the shape of the temporal waves for the filtered and non-filtered signals. To that, connect the filter s input signal to the first scope input, and its output signal to the second scope input. 4. What do you think it would happen if we set the cutoff frequency to.6 Hz? And if we established it to.5 Hz?..3. Transmission using sinc-shaped pulses We are going to modify the shape of the pulse to that of sinc-shaped ones. To do this, in the previous system we will add several blocks whose task will be generating this kind of pulses. A sinc-shaped pulse is given by: f sen[ ( t / T] sinc[ ( t t )/ T] ( t t )/ T ( (4) where T is the elapsed time between consecutive nulls of the waveform and t is the time instant when the signal maimum is yielded. We are going to use a MATLAB function to generate this wave and add a block Simulink/User-Defined Functions/MATLAB Fcn which will be responsible of calling the previously defined MATLAB function (see net figure about the new system scheme). This block is inted to call a function, which we are going to denote as fsinc, with three input parameters (the input to the block, which is specified with letter u, the time duration of the pulse or elapsed time between consecutive nulls of the waveform, which we will set to second, and the number of samples per symbol time, which will be fied to 6). Thus, the parameter MATLAB function in the block properties must be set to fsinc(u,,6). Moreover, we have to establish the parameter Output dimensions to [6,], i.e. the output of the block will be a column vector of 6 rows. In order to convert this signal in a temporal 4/7
PRACTICE : sequence we have to introduce a block Frame Conversion followed by a block Unbuffer. In this way, the 6 output values from the block MATLAB Fcn will be converted to 6 temporal samples correctly ordered in time at a sample rate given by the block Hold placed net. B-FFT Bernoulli Binary Bernoulli Binary Generator Gain Constant MATLAB Function MATLAB Fcn [6] To Frame Frame Conversion [6] Unbuffer Hold cheby Analog Filter Design Hold Channel Spectrum Discrete-Time Eye Diagram The first thing we have to do is creating a function which generates the sincshaped waveform according to the range of values indicated as input. This MATLAB function can be easily defined in the net way: function y=sinc() % % Calculates sinc() for the range of values given by vector N=length(); y=zeros(,n); for i=:n if (i)== y(i)=; else y(i)=sin((i))/(i); The comments introduced below the function declaration will be epanded in the command window whenever we eecute the net instruction: help sinc Notice that the function sinc really eists as an internal function belonging to the MATLAB libraries. However, we have defined our own function in order to help us to better understand the implementation of the system we are describing. The function sinc() is characterized by presenting a maimum equal to one at =, and being null at values = n, where n is a natural number different to zero. Unlike rectangular pulses, where the pulse amplitude is only depent on the current data bit ( or ) along the T seconds of the symbol time, the sinc-shaped waveforms have ideally an infinite duration with non-negligible values for both negative and positive time instants. Theoretically, we cannot create waveforms with non-zero values at negative time instants, since the system would be not causal. However, we can take advantage of the amplitude attenuation of sinc shapes along the time. It can be checked as, beyond the 5th null after the maimum value, the amplitude has been attenuated greatly (is less than the % of the peak value of the signal). Notice that being strict we should consider greater elapsed times previous to make this kind of simplifications. Since the function sinc is symmetric from its 5/7
PRACTICE : peak point, we have to consider all the values both on the left and on the right previous to the 5th null from this peak value. In the net figure is depicted a sequence of binary data, the two first sinc waves corresponding to the first two data bits, as well the sum of these two waves (bottom graph). Evidently, to obtain the total waveform we have to add all the sinc waves corresponding to all the binary data. As an important remark, we can observe as the peak of the pulse waveform is delayed about 5 seconds (eactly 4.5 seconds) with respect to its corresponding binary bit. This is necessary to obtain a causal system, as we have previously indicated. The depicted signals also give us some information about the characteristics of the MATLAB function which we have to design to generate the sinc waveforms. In first place, we will have to save the values of the previous 9 bits because the time duration of the sinc pulse is 3T, being T the duration of each symbol or data bit (in this eample, a second). That is, in the eample shown in the figure, the 3 th bit has logic value and we would have to generate the sinc wave beginning from that instant, but we would also require from the previous 9 bits values whose contributions would be taking into account in the current time instant. Thus, the influence of the first bit would disappear after this time instant, and so on for the consecutive bits..5.5 -.5 5 5 5 3.5 -.5 5 5 5 3.5 -.5-5 5 5 3 - - 5 5 5 3 Net we show a possible MATLAB code to generate the sinc waveforms after receiving a new data bit: function signal_out=fsinc(new_data,t,n) persistent data; M=3; if isempty(data) data=zeros(,m); 6/7
PRACTICE : data=[data(:m) new_data]; signal_out=zeros(,n); t=:t/n:t*(-/n); for i=:m =pi*((t-t*(i-m/-.5))/t); signal_out=signal_out+data(i)*sinc(); signal_out=signal_out.'; In first place, we have declared a persistent variable in order to save its value whenever we go out of the function and call it again. This variable data saves the input binary data introduced to the block every T seconds, then it is set to zero the first time the function is called. Net, a new data bit is introduced to the block, new_data, which is apped to the memory data, removing the oldest one which was introduced 3 time instants before. Then the N-sample waveform is obtained during the current T seconds by adding the different contributions corresponding to the different data, both the new data bit and the previous ones: f ( M i [ t T( i M i sinc T /.5)], t T being i = according to the binary data value ( or ). The inde i goes from the first introduced data (i = ) to the current data bit (i = M ). Observe as this function only calculates the waveform during the present time interval, then requiring from saving previous data values along a long time. Finally, the generated waveform is saved in the variable signal_out, which is returned by the function and coincides with the output vector from block MATLAB Fcn.. By using the sinc-shaped pulses generator, check the waveforms in the time domain before and after the filter. What do you think about the good similarity between them?. Check the frequency spectrum of the sinc signal before and after the baseband filter with cutoff frequency of.75 Hz. What is approimately the bandwidth of the original sinc signal? 3. Check the effect of the ISI when we use sinc pulses. What main feature would you remark with respect to what was observed for rectangular pulses? 7/7