Lab 5: Digital Communication FSK Modem

Transcription

1 Lab 5: Digital Communication FSK Modem 1 Overview The goal of this lab is to understand a simple modem, the Frequency Shift Keying (FSK) Modem, referred to by the International Telecommunications Union (I.T.U.) as V.21. The V.21 modem communicates 1 s and 0 s by sending either a 1650 Hz tone or a 1850 Hz tone, respectively, for 1/300 second. Thus the overall data rate is 300 bits/second (one bit is sent in 1/300-th of a second). Even though 300 bps is quite slow compared with the theoretical maximum of 56 kilobits per second over a phone line, the V.21 format is still used in almost every modem call. This is due to the fact that receiving and decoding it is so simple. A V.21 modem call can be received without using difficult techniques such as equalizers, cancellers and matched filters. Furthermore, it can be received accurately even in the presence of a significant amount of noise. For these reasons, V.21 is used as an initial handshake between two modems, meaning that V.21 is a way to communicate some basic startup/control information between the two modems. You can hear the V.21 modem tones at home when your V.34, V.90, V.92 phone line modem or fax machine starts a phone call. V.21 is also used to transmit caller ID information over the phone line. 1.1 Transmitter Converting from Text String to Binary String An FSK modem is a digital communication system, so it transmits zeros and ones. In order to transmit messages in an alphabet, there must be a binary representation for each character in the alphabet. The standard for this representation is ASCII where eight bits are used to represent characters, numbers and special punctuation. For example, the upper-case character A is represented in ASCII with the number 65, which has an 8-bit binary form as ; lowercase b is 98 in ASCII, or Therefore, if we want to encode a message, such as Hello World, for the FSK modem, we must turn the eleven characters of the message into 0 s and 1 s. Since blanks are counted as characters, we would end up with 88 bits. Given a vector of bits, we could generate the appropriate sinusoids by agreeing to the convention that 1650 Hz is used for a 1, and 1850 Hz is used for 0. MATLAB has some useful functions for doing ASCII conversion and also for turning decimal numbers into bits. Check out help on the following functions: CHAR Convert numeric values to character array (string) ABS Convert character array (string) to numeric values DEC2BIN Convert decimal integer to a binary string BIN2DEC Convert binary string to decimal integer Run the script char(30:50), please refer to its help to understand the result. Also, try the following MATLAB expression abs([ ABC ; b01 ]), and note that the ASCII equivalents are returned in a matrix 1. To test your knowledge at this point, write a MATLAB expression that 1 In MATLAB, a string is actually an array of characters, so [ ABC ; b01 ] is a 2 3 array. Use size to verify this fact. 1/11

2 will give the binary representation for the message Hi!. Use the optional argument in dec2bin to get an 8-bit string for each character (with leading zeros). Notice that, when you convert a text string into binary format, you will get a column vector, each row is an 8-bit long binary string. You may want to use MATLAB function reshape to form your binary message into one string, so that it will be easier to generate your FSK signal in the next step. Be aware that the function reshape is based on column-wise operation. By convention, we send the most significant bit of byte first 2. For example, H corresponds to , and i So, the binary string sent out should be in order Generate an FSK Signal One part of the FSK encoder must take a binary bit stream and create a sinusoid for each bit. The duration of each sinusoid will determine the bit rate, e.g., 300 bits/second (bps) requires a duration of 1/300 sec. You must complete the function below and use it to generate the FSK signal. The functionality of this program is to covert a binary string into a FSK waveform consisting of dual tone sinusoids, given the ling rate and the bit duration. The only given script in the program is to convert the input binary string to a vector containing 0 s and 1 s. function xx = fsk_gen( bitstr, fs, bdur ) %FSK_GEN generate the FSK sinusoids at 1650 and 1850 Hz % 1650 encodes a "1", 1850 encodes a "0" % % bitstr = STRING of zeros and ones. % fs = ling rate % bdur = duration of the sinusoid for one bit % (this determine the "bit rate") % xx = synthesized FSK signal % inbits = abs(bitstr) - abs( 0 );%convert bit string to numbers %% %%%%%%%%%%%% put your code here %%%%%%%%%%%%%%%%%%% To test your function, input character H, use a duration of 50 millisec, even though that is only 20 bps, because then it will be easy to identify the tones on a spectrogram. Use a ling rate of 9000 les/second. Make a spectrogram of the generated FSK signal in order to show how the individual bits correspond to sinusoids. Use a window length that is the default of 128. Your spectrogram plot should look like Fig. 1. Notice those vertical bars on the spectrogram plot, and try to explain the cause of it. Conceptually, what could be the solution to this problem? 2 This is called the Big-Endian convention. As expected, there is a Little-Endian convention, too. The debate on choosing one over the other has been going on for decades. Check online materials for more details. 2/11

3 Figure 1. The spectrogram of the FSK signal when send letter H. 1.2 Receiver The receiver for V.21 must determine which of the two tones is present, and must make this decision every 1/300-th of a second. A block diagram of the FSK V.21 demodulator is given in Fig. 2. Each of the main sections will be described in more detail below. Figure 2: Block diagram of the FSK V.21 demodulator Mixing A basic operation that most modems need to perform is frequency shifting of the input signal. According to the frequency-shifting property of the Fourier Transform, this can be done by simply multiplying the input signal by a complex exponential. wt j2π fct () = xte () This effect can best be understood by thinking of x() t as a sum of complex exponentials and observing what happens to each individual frequency component. In this case, x() t is 1 1 xt = π ft = e + e 2 2 j2π f1t j2π f1t () cos(2 1 ). 3/11

4 When x() t is multiplied by the given complex exponential at frequency simply add and the resulting wt () is as follows: fc Hz, the exponents 1 j 2 π ( f1 f 1 c ) t j 2 π ( f1+ fc () ) t wt = e + e. 2 2 Note that the new frequencies in wt () are simply the old frequencies shifted down (to the left on the f axis) by f c, i.e., f1 fc and f1 fc. Also note that wt () is complex, so we no longer have the condition of complex conjugate symmetry between the two complex exponential components. This simply means that now our signal is really two signals: a real part signal and an imaginary part signal. If we want to filter this complex signal with a real filter, we simply filter the real part and the imaginary part separately with the same filter. In MATLAB, the filter() function does this very thing for you. The purpose of the filter is to remove the complex exponential with frequency f1 fc while leaving the other component whose frequency is f1 fc. Thus, the filter output should be of the form yt Ae π 2 ( 1 c ) () = j f f t, where A will depend on the gain of the filter in its passband. For V.21 FSK, we will choose f c so that the original frequencies of Hz and Hz in x() t are shifted to -100 Hz and +100 Hz respectively, and these are the frequencies passed by the filter. This is achieved by choosing f = c Discrete-Time Simulation The MATLAB implementation of the FSK V.21 modem is a simulation. This means that even though we seem to have continuous-time signals such as x() t and wt (), we actually use led versions in the MATLAB program. For this simulation, we will choose the input ling frequency to be f = 9000Hz. (1) This value is chosen to be rather high so that the input signal will appear to be continuous if we use the MATLAB plot() command. In Fig. 3 the continuous-time signals have been replaced with their led versions, e.g., x() t becomes x[ n ], wt () becomes wn [ ], and so on. The j2π fct complex exponential used in the mixer must also be converted from e to a discrete-time ˆc signal j n e ω. What is ˆc ω? Figure 3: FSK system demodulator system simulated as a discrete-time system at a ling rate of f. 4/11

5 1.2.3 Low Pass Filter Design The other thing that must be done in converting the block diagram of Fig. 2 to the simulation block diagram in Fig. 3 is to replace the analog filter by a digital filter. This filter must pass the band of frequencies equivalent to ±100 Hz, and it must remove the higher frequencies generated at the output of the mixer which are - f c = and - f c = Hz. Since the digital filter is typically used to filter signals that are led analog signals, it is important to know how the band edges of the digital filter can be expressed in terms of the desired analog cutoff frequencies. This can be done if the ling frequency f is known. For example, if we want to have a lowpass filter with an effective cutoff frequency of 1500 Hz, and we are using a digital filter running at f = 9000 Hz, then the digital filter must have its cutoff frequency 3 at ˆω = 2π (1500/9000) = 1 3 π. In the FSK V.21 system, the frequency shifting of the mixer will generate spectrum lines that must be removed by filtering. If we demand that these unwanted spectrum components must be reduced in magnitude by a factor of 100, then we have given the specifications on the stopband ripple. A reduction by a factor of 100 means that the stopband ripple must be less than 0.01, or - 40 db. The passband, on the other hand, must be made wide enough so that the desired frequency components will go through the LPF with little or no change. Since the LPF s frequency response will have a passband ripple, we will use a specification on the passband of 1 db, which forces the passband magnitude to lie between 0.89 and For detailed digital filter design, refer to Section 1.3 of this lab Frequency Estimation: Slicing Another basic operation of most modems is to measure the frequency of a received tone. This could be accomplished by optimal filtering algorithms such as matched filters designed to enhance the tones of interest. However, if you are guaranteed to be looking at only one tone at a given time and the noise is not severe (both of which are true for V.21), then there is a simpler method that can be employed to save computation when measuring frequency. Figure 4: Frequency estimation in a dual-frequency FSK system can be performed with a slicer. Slicing is defined as follows: sn = yny n (2) * [ ] [ ] [ 1]. 3 j ˆ Remember that the frequency response of a digital filter, H( e ω ), is a function of the frequency variable ˆω that runs from ˆω = π to ˆω = π. 5/11

6 It is a non-linear operation, but in the case where x[ n ] is a single cosine input, the filter output j ˆ0n will be of the form yn [ ] = Ae ω, where ˆ ω0 = m 200 π / f. In this case, the output of the slicer (2) reduces to sn Ae ω Ae ω j ˆ0 n * ˆ 0 ( 1) [ ] = ( ) ( j n ). After adding the exponents, the output simplifies to sn A e ω 2 j ˆ0 [ ] =. If the objective were to determine the frequency ˆω 0, then it is sufficient to take the imaginary part 2 j ˆ0 dn [ ] = Imsn { [ ]} = Im{ A e ω } dn = A ω 2 [ ] sin( ˆ0) and use dn [ ] to calculate the arcsin( ) to get an estimate of ˆω 0. However, the FSK V.21 system is even simpler than that, because we only need to decode two cases: a zero or a one. When ˆω 0 < 0 we have a 1, and when ˆω 0 > 0 we have a 0. In addition, the sign of ˆω 0 is the same as the 2 sign of A sin( ω ), so we only need to check the sign bit of dn [ ] to perform the decoding. ˆ0 As will be seen in the final implementation of this lab, the recovery of the V.21 signal reduces to discriminating between a +100 Hz tone and a -100 Hz tone. Taking the imaginary part of sn, [ ] the slicer output, will provide an easy way to determine whether a 1650 Hz or 1850 Hz tone was originally present. Thus we can define bn [ ] as an estimate of the bit that is represented by the slicer output at time n. 0 when dn [ ] 0 bn [ ] = (3) 1 when dn [ ] < 0 In MATLAB, you can use the find() function to implement (3) by creating a zero vector of the same length as dn [ ] and then changing all locations where dn [ ] < 0 to 1. Table 1: FSK Decoding Rule Note that yn [ 1] can easily be obtained from yn [ ] by with a delay of one le. In the interest of keeping the vectors the same size, the filter() command can be used to create yn [ 1] : y1 = filter([0 1],1,y) Decoding the Message Now we must extract the bit information from the output signal dn. [ ] Remember that the led waveform x[ n ] is a variable frequency sinusoid that can switch frequency every 30 6/11

7 les (assuming bit rate of 300 bps and ling rate of 9000 les/sec) and the sinusoid switches between 1650 Hz to represent a 1 and 1850 Hz to represent a 0. In our discrete-time simulation, the signal dn [ ] at the output of the slicer is either positive or negative at each le time, and we have just seen that the sign of this signal is an indicator of whether a 1 or a 0 is being encoded at any particular le time n. Now we should remember that if the bit rate is 300 bps and the ling rate is 9000 Hz, this means that each bit of the encoded message is actually represented by a group of 30 consecutive les. We can determine the sequence of bits by simply looking at the signal during each bit interval. Since each of these intervals is 30 les long, we can find the middle (where the answer is likely to most robust) by simply ling the sequence bn [ ] at a point that is offset by 15 les from the beginning of each bit interval. When there is noise in the communication channel, which is always true in real life, you don t want to base your result on only one le in each bit interval. One way to mitigate the noise effect is to use a majority vote scheme, that is, in one bit interval, if more than half of 30 les are negative, then it is more likely that the decoded bit is a 1. At last, we need to group the bits into 8-bit sets then convert them into ASCII symbols. This completes the decoding simulation. The MATLAB functions bin2dec and char can convert a bit stream to ASCII. Try the following example: bstr = [ ] char(bin2dec(char(bstr+abs( 0 )))) 1.3 Filter Design When you implement the FSK decoder, it will be necessary to have lowpass and (possibly) bandpass filters. We had previously encountered a similar filter design problem in the DTMF system, where several bandpass filters were needed. In that case, we used FIR filters that had acceptable passbands, but were actually quite poor in their stopband behavior. The objective of this part is to investigate two ways to get better filters GUI for Filter Design The process of filter design involves a trade off between the filter length L, the band edges, and the stopband and passband ripples. If we use a GUI we can manipulate these trade-offs directly on the screen and see immediately how the magnitude response changes as we try to satisfy the desired filter specs. Figure 5 shows the magnitude response of a digital FIR filter obtained with the MATLAB GUI called filtdemo. Decibels: One feature of the magnitude plot in Fig. 5 is that the vertical axis is logarithmic in j ˆ units called decibels (db), i.e., the quantity plotted is 20log 10 He ( ω ). When using a db-scale, the passband, which ideally has a magnitude of one, should be 0 db. In the stopband, we might j ˆ have a value like He ( ω ) = 0.01 which is -40 db. If we consider the case of a lowpass filter (LPF) such as Fig. 5, then we can describe the passband as extending from f = 0 to a cutoff frequency, f = f Hz; and the stopband is the region p 4 The Signal Processing Toolbox in MATLAB provides numerous functions that will perform filter design, but often this toolbox is not available. 7/11

8 f f s Hz. Even though the frequencies are given in hertz, the filter design GUI yields a digital filter. The ling frequency f controls the relationship between the given analog cutoff frequencies, of ˆω. f p and f s, and the filter s frequency response He ω j ˆ ( ) which is actually a function Question: Determine the values of ˆω for the passband and stopband edges in Fig Using the GUI The filter type is controlled by the drop-down menu in the upper right-hand corner of the GUI. There are several choices, but the only ones of interest for FIR design in this lab are REMEZ or KAISER. For given ripple specs, the REMEZ design will give a shorter filter, so it might be preferred, but either method is acceptable for this lab. The filter specifications can be entered into the GUI in two ways: either in the text boxes on the upper right part of the window, or by dragging the green bars that indicate the desired passband and stopband regions. The bars can be moved up and down to change the desired ripple sizes, but the ends of the bars can also be moved horizontally to change the desired bandedges. The filter order (which is L - 1) can be calculated automatically by the GUI, or the user can enter a specific value. This option is controlled by the radio buttons on the right side of the GUI Figure 5: The filtdemo MATLAB GUI for digital filter design. The magnitude is given in db (decibels). The frequency axis is labeled in hertz from DC to half the ling frequency. (Note that f = 6000 Hz in this figure.) 8/11

9 In Fig. 5, the value of M = 23 was entered by hand. Obviously, the stopband cutoff frequency f s must be greater than the passband cutoff frequency f p. If we try to make f s and f p nearly equal, then the filter length L would have to be very large, so it is customary to allow a transition region between the passband and stopband. Once the magnitude response has been manipulated into a desirable form, the filter coefficients can be extracted from the GUI by running the following in the command window. [bb,aa] = filtdemo( getfilt ); Since this is an FIR filter, only the vector bb is needed. It will contain all of the filter coefficients for the FIR filter. 2 Procedures 2.1 FSK Modem in VAB In this demonstration VAB part, you will hear what a two-tone FSK transmitter sounds like. The worksheet 2-Tone FSK Transmitter.Lst can be found on desktop of every computer in the lab, and it looks like that shown in Fig. 6. Every time you type in a letter, you will hear a pattern of tones repeating itself. Different letters will have different tone patterns. Based on the knowledge you have got from Section 1.1, explain the relationship between the letter you type and the tone pattern you hear. Figure 6. Two-tone FSK transmitter worksheet. 9/11

10 Suppose you are about to construct a 2-Tone FSK Receiver.Lst worksheet, you want to pair two computers up and form a two-tone FSK air modem just like you did on LAB 4 (DTMF modem). Suppose you have ideal bandpass filters for those two tones (their lengths are suitable), and noise effect can be ignored. Now, do you think your air modem will work properly? What could go wrong? Try to find the most critical issue we are not considered here. Explain why it is critical. 2.2 Implement FSK Modem in Matlab The main task of this lab is to construct a FSK modem implementation in Matlab. To recap, your simulation file should contain the following steps: 1) Define parameters, including ling frequency and bit duration. Define a message that you want to transmit, for example, you can use Hi! I am yourname@ud. 2) Load the filter coefficients obtained with the following specifications: f = 9000, f p = 200 Hz, f s = 1000 Hz, stopband ripples less than -40 db, and passband ripples less than 1 db. Use the GUI method to design your filter. 3) Convert your message into a binary string as described in Section ) Pass this binary string into your program fsk_gen() created in Section ) Add certain amount of Gaussian noise onto your signal. It is probably a good idea to begin your program development and initial testing with no noise. 6) Mix your signal as described in Section ) Lowpass filter the mixer output using the coefficients in step 2). 8) Use slicer to get the frequency estimation, and convert your signal to a binary string as described in Section ) Convert the binary string into a character based message as described in Section Compare the decoded message and the original message. Always remember, to simulate a complete system like this, you should create and test each single building block of your system before you put them together, to make sure that they all function correctly. So, you probably need to do some separate tests for the FSK generator, for the mixer and for the slicer etc. When your FSK modem is working, try the following exercises: A. Choose the magnitude of your noise to be 0.3. First, use middle bit scheme, run your simulation 20 times, count how many times you get your received message correct. Then, use majority vote scheme, repeat 20 simulations. Compare the results. Explain the difference. B. Change your LP filter f s = 800 Hz, keep everything else the same. Your will get a better filter in terms of transition time. Apply this filter to your communication system, observe the result. Try to explain why. (Hint: each bit interval has 30 les.) What can be inferred from this observation? Comments: A lot of practical issues in FSK modem, such as smooth phase and synchronization, are not covered to keep the lab in a manageable level. 10/11

11 3 Report Components Your report should answer the questions asked in both the VAB and MATLAB components of the laboratory. Please make sure that the questions are answered fully, along with any explanation or justification necessary, and that plots are appropriately labeled and referenced. 11/11