Computer Exercises in. Communication Theory SMS016

Transcription

1 Luleå Tekniska Universitet Avd. för Signalbehandling Jan-Jaap van de Beek Frank Sjöberg Computer Exercises in Communication Theory SMS016 November 2001 Computer Exercises to be carried out in groups of two students. Only complete and clear reports will be corrected and approved. Dealine: Thursday 6/

2 Introduction These computer exercises support the lectures and recommended exercises in Communication Theory -course, SMS016. The goal of these computer exercises is to gain insight in the behavior of optimal detection in the presence of ISI, and the suboptimal equalization techniques as linear and decision-feedback equalization. The use of these techniques for the transmission of images over time-dispersive s is illustrated. Simulations are performed in MATLAB. Written reports should include all code, plots and answers/discussions. Discrete time model of digital communication system We will consider transmission of data over a linear time invariant (LTI) using 2- dimensional signalling (e.g. QAM). A continuous-time model of the baseband representation of such a digital communication system is shown in Figure 1. pulse g (t) white Gaussian noise n p (t) matched filter h (t) (g h)( t) y(t) t = kt y k Figure 1: Continuous-time model of digital communication system. The complex data symbols are modulated on a pulse g(t) and transmitted over a time dispersive, band-limited linear Þlter h(t). The data is also corrupted by complex valued white Gaussian noise with average 2-dimensional power N 0 (N 0 /2 in each dimension). The receiver consists of a Þlter matched to (g h)(t) and a symbol rate sampler. We assume perfect synchronization in the sampling device. The data y k at the output of the sampler can be shown to be a sufficient statistic for the data, i.e., they contain all relevant information needed to detect optimally. We may describe this model by an equivalent discrete-time model. This model is not causal. Moreover, the noise in the output y k is generally not white due to the matched Þlter. We will therefore consider the system in Figure 1 in cascade with a linear whitening Þlter. This whitening Þlter is designed so that the noise at its output is white and so that the discrete time impulse response of the entire system becomes causal and minimum phase. We then obtain the equivalent discrete-time model as depicted in Figure 2 white Gaussian noise n k h k Figure 2: Discrete-time model of a digital communication system. In all computer exercises to follow, we consider this discrete-time communication system model, i.e. a discrete-time causal Þlter h [k] and additive white Gaussian noise. Optimal and suboptimal detection Due to the, the outputs of the system are corrupted by ISI. The effect of this ISI is taken care of optimally (in the sense of sequence error) by the ML sequence detector (MLSD). 2

3 The MLSD is implemented by means of the Viterbi algorithm, see Figure 3. h k n k MLSD ^ Figure 3: Discrete-time model of a digital communication system with ML sequence detection. However, the Viterbi algorithm is in many practical cases too complex to implement. Therefore, suboptimal methods may be used to address the effect of ISI. A simple method of suppressing ISI is to linearly Þlter the received (corrupted) data. The design of the linear equalizing Þlter is done according to some criterion of optimality. In Þgure 4 the received signal is detected after linear equalization. h k n k Equalizer ck z k ^xk Figure 4: Discrete-time model of a digital communication system with linear equalizer. Another suboptimal way of handling ISI is the use of a decision feedback Þlter. For high SNR values, this way of Þltering the received data in many cases approximates the performance obtained with the MLSD. For low SNRs, however, this method suffers from error propagation. In Þgure 5 the received signal is detected by decision feedback equalization. h k n k 0 ^ c k Feedback filter Figure 5: Discrete-time model of a digital communication system with decision feedback equalizer. 3

4 MATLAB functions and data The following MATLAB-functions serve as an aid in the exercises.. c=qam(n) Generates a vector c containing the symbol constellation for N-QAM signaling with unit average energy. x=source(c,n) Generates a (n,1)-vector x containing symbols randomly chosen from the vector s. y=(x,n0,h) Generates the received vector y as the transmitted vector x Þltered by the Þlter h and corrupted by additive complex Gaussian noise with average power n0. r=detect(y,c) Generates optimal decisions r for the received vector y containing disturbed symbols from a signal constellation c. r=code_image(x) Generates a symbol-vector r by coding an intensity image X with symbols taken from a 16 QAM constellation. X=decode_image(b,sx,sy) Generates a size sx*sy intensity image X from a symbol-vector b containing symbols taken from a 16 QAM constellation. d=viterbi(r,h) Detects the sequence r using the Viterbi algorithm and the impulse response h. The resulting detected vector is d. Antipodally signals only, i.e., s =[ 1 1]. imshow(x,n) Displays the intensity image X with a grayscale colormap of length n. Use this command with a value of 256 for n to display images decoded by decode_image. The following MATLAB-data Þles serve as an aid in the excercises. picture1.mat contains an ( )-pixel intensity image X. picture2.mat contains an ( )-pixel intensity image Y. satpic.mat contains a vector y with corrupted data. 4

5 ISI Channels The following three discrete time s will be considered through out the following exercises. h 1 =[1 0.5] h 2 =[ ] h 3 =[ ] 1. Calculate the matched Þlter bound, SNR MFB for both s (i.e., what is the energy of these s)? Remember to take this into consideration in the computer exercises that follows since they affect the SNR. You can normalize the s if you want to. 2. Calculate the frequency response H(f) of the s 1,2, and 3 by means of a points FFT (use the MATLAB-command fft). Plot these responses on a db-scale versus the frequency. 3. Simulate the transmission of data over a linear Þlter with AWGN by combining the MATLAB commands mentioned in the introduction. Choose a 2-dimensional signal constellation (e.g. 4-QAM). Generate a transmitted symbol-vector and the corresponding corrupted received symbol-vector for the impulse responses 1 and 2. Show plots of the received symbols in the 2-dimensional signal space for the s 1 and 2 and for SNR-values 10 db (the actual SNR in many practical systems) and 50 db (nearly noise-free SNR). 4. Write a MATLAB command err_s = ISIsim1(c,N,SNR,h) that simulates transmission and simple symbol-by-symbol detection over an LTI- with AWGN noise, for different SNRs. The parameter c is a vector containing the signal constellation, SNR is a vector containing the SNRs in db (e.g. [10:20]), N is the number of transmitted symbols for each SNR, i.e., the length of the transmitted data vector, and h is the vector containing the impulse response. The result err_s is a vector containing the estimates of the probability of error for symbol by symbol detection. Ex: err_s = ISIsim1([-1 1],10000,[0:2:10],[ ]); Simulate the BER performance for all three s with BPSK. Show the plots of the error-rate curves (logarithmic scale) versus SNR for SNR=[1 : 10]. Whichaffects the data worst? How does it compare with a ISI-free? Optimal detection As you could see in the previous exercise, ISI can be quite detrimental if not taken care of properly. In this part of the lab we will evaluate the performance of an optimal sequence detector by using the Viterbi detector. 1. Write a MATLAB command [err_v]=isisim2(c,n,snr,h) (by modifying the command ISIsim1 that was designed above) that performs simulations of a communication system using a maximum likelihood detection. The result err_v is a vector containing the estimates of the probability of error for the maximum likelihood sequence detector. Ex: err_v = ISIsim2([-1 1],500,[0:2:10],[ ]); Use the MATLAB-command Viterbi as described in the introduction. 5

6 2. Choose the signal constellation s =[ 1 1] (the command viterbi only works for this constellation) and perform a simulation for all three s. Show the plots of the error-rate curves (logarithmic scale) versus SNR for SNR=[1 : 10]. Whichaffects the data worst? Why? (compare the frequency characteristics.) The command viterbi calculates the ML sequence detection for a received sequence. This is a time consuming algorithm if it is applied on large sequences. Therefore, choose sequences of reasonable length (e.g. 100) and average over several results to get statistically reliable results. Since simulations may take some time, test your commands Þrst with small vectors. Linear equalization Linear equalizers are far less computational complex than the MLSD, but the performance is also lower as you will learn is this part. Since the Zero-forcing (ZF) equalizer can be seen as a special case of the minimum mean square error (MMSE) equalizer we only have to implement the MMSE-equalizer.. 1. Calculate the theoretical SNR-performance for an inþnitely long linear equalizer. Do this for s h 2 and h 3, for both a ZF-equalizer and an MMSE-equalizer. You can evaluate the integrals numerically by using MATLAB. 2. Write a MATLAB command eq=linear_eq(h,l,d,n0) that generates a linear equalizing Þlter based on the mean-square error criterion. The parameter h is a vector containing the impulse response, L is the length (order) of the equalizer, D is the delay, and n0 isthenoisevariance.theresulteq is a vector containing the L Þlter taps. Ex: eq = linear_eq([ ],5,3, n0); Test this command for the simple h 1 Þrst, for which you easily can calculate the result theoretically. Remember that if you set n0=0 you get the ZF-equalizer. You can also test the effect of your linear equalizer by convolving it with the : h_equalized = conv(eq, h); 3. Write a MATLAB command [err_zf err_mse]=isisim3(c,n,snr,h,l,d) that performs simulations of a communication system using a linear equalizer. The new parameters L and D represents the length of the equalizer and the delay, respectively. The result err_zf is now a vector containing an estimate of the probability of error for the zero-forcing linear equalizer, for each of the SNR-values speciþed, and err_mse is a similar vector for the mean-squared error linear equalizer. Ex: [err_zf err_mse] = ISIsim2(QAM(4),100000,[1:10],[ ],5,1); 4. Choose a 2-dimensional signal constellation and perform simulations with the s 1 and 2 using a 7-tap linear equalizer with delay D =2. Choose a suitable interval for the SNR, and simulate as long vectors as possible, to get statistically reliable results. Plot bit error curves (BER versus SNR) for your constellation choice and for the s h 2 and h What is a good choice of equalizer length? Evaluate the BER-performance on s h 2 and h 3 with some different lengths, e.g. L {3, 7, 15, 31, 101}. Is it always better with a longer equalizer? Is there any difference between ZF and MMSE? 6. Repeat the simulations for the constellation s =[ 1 1] and compare the results with the MLSD. What is the SNR-loss (in db) of the linear equalizer compared to the MLSD for the different s? 6

7 Decision feedback equalization While linear equalizers generally not can compete with MLSD, a decision feed back (DFE)- equalizer can in many cases perform almost as good as an MLSD with only a small increase in complexity compared to linear equalizers. However, when implementing a DFE in MATLAB we have to use loops that are known to be very slow. So we cannot perform as long simulations as for the linear equalizers. 1. Write a MATLAB command df=dfe(h,l) that generates a decision feedback equalizing Þlter. The parameter h is a vector containing the impulse response, and the parameter L is the length (order) of the equalizer. The result dfe is a vector containing the L Þlter taps. Ex: df=dfe([ ],3); 2. Write a MATLAB command r=equalize_dfe(y,df) that equalizes a received complex vector of data using the decision feedback equalizer, generated with the previous command. The parameter x is the vector containing the received complex data, and the parameter df is a vector containing the taps of the equalizing feedback Þlter. The result, r is the equalized data-vector. The experienced MATLAB-developer can implement this in C-code and compile it to MEX-Þle that MATLAB can execute. This can increase the speed by several factors of Write a MATLAB command err_dfe=isisim4(c,n,snr,h,l), that performs simulations for the decision feedback equalizer. The parameter h is the vector containing the impulse response, and L is the length (order) of the feedback Þlter. The result err_dfe is a vector containing an estimate of the probability of error for the decision feedback equalizer, for each of the SNR-values speciþed. Ex: err_dfe = ISIsim4(QAM(4),1000,[1:10],[ ],3); 4. Choose a 2-dimensional signal constellation and perform simulations for the s h 2 and h 3 using a 3 tap decision feedback equalizer. Choose a suitable interval for the SNR, and simulate as long vectors as possible, to get statistically reliable results. Plot bit error curves (BER versus SNR) for your constellation choice and for the s h 2 and h Repeat simulations for constellation s =[ 1 1] and compare the results with the MLSD and the linear equalizers. 7

8 Image transmission In this exercises we will simulate transmission of images. We will consider intensity images (grayscale images) only. MATLAB stores intensity images as a single matrix containing ßoating point values ranging from 0.0 to 1.0. Each element of the matrix corresponds to an image pixel. The elements in the matrix represent various intensities, or gray levels, where the intensity 0.0 represents black and the intensity 1.0 represents full intensity, or white. To display an intensity image we can use the MATLAB-command imshow. For example imshow(x,64) displays the intensity image X with 64 gray-levels. In the following exercises we will use 256 intensities, or gray levels. How do we transmit an intensity image over a digital communication? First, we create a vector by putting all columns of the intensity image (matrix) column-wise in one vector. An (N M)-intensity matrix is thus converted in a (NM 1)-vector. The second step is to code the vector using the symbols from a certain signal constellation. We will use a 16 QAM constellation to code the intensity data. First, the intensity value of a pixel is multiplied by 256 and rounded. The remaining value (ranging between 0 and 255) is then coded by two 16 QAM symbols, the Þrst one representing the 4 most signiþcant bits, and the second one the 4 least signiþcant bits. Both steps are carried out by the command Y=code_image(X). This command will generate a vector containing 2NM symbols from a 16 QAM constellation representing the NM pixels of the image. These 2NM symbols can now be transmitted over the communication. In order to do the inverse operation, i.e., to create an intensity image from a vector containing symbols from a 16 QAM constellation, we need to know the dimensions (number of rows and columns) of the image that is represented by the symbol vector. The command X=decode_image(b,sx,sy) performs this inverse operation. It generates an intensity image (matrix) X, from the symbol vector b, according to the number of rows sx andnumberof columns sy. The following sequence of commands loads a ( matrix) intensity image X, displays it, and converts it into a ( vector) sequence of 16 QAM symbols. load X imshow(x,256); b=code_image(x); The resulting vector b may be transmitted over an AWGN- and optimally detected at the receiver by: r=(b,0.001,1) d=decode(r,qam(16)) Finally, knowing that the original image is a matrix, we use Y=decode_image(d,50,100) imshow(y,256); in order to re-create the image and display it. This intensity image containing pixels results in the transmission of a ( )-vector. The equalization and detection may take time. Therefore, test your functions on small vectors (generated for example with source) before applying them on images. 8

9 1. Write a MATLAB command image_transmit(x,h,snr,l) that simulates the transmission of a intensity image as described in the introduction using a linear (choose either the zero-forcing or the mean squared error equalizer) and a decision feedback equalizer. Ex: image_transmit(x,[1.5],15,6); As a result this Þle must plot 4 images with the command imshow (see introduction): (a) Picture 1: The transmitted picture. (b) Picture 2: The received picture without equalization. (c) Picture 3: The received picture using a linear equalizer (choose either the zero-forcing or the mean squared error equalizer). (d) Picture 4: The received picture using the decision feedback equalizer. (The MATLAB command subplot is a convenient means of plotting four pictures in one!) 2. The Þles picure1.mat and picure2.mat contain intensity images. Simulate the transmission of images over the h 2 for5dband20dbsnr,usingalinearandadecision feedback equalizer,and generate the 4 plots as described above. Discuss your results. Does the decision feedback equalizer suffer from error-propagation? For which SNR-value? Simulation of the transmission takes time, so reduce the size of the images by e.g. X=X(1:2:100,1:2:100) (a data reduction with a factor 4) until the size of the image matches a satisfactory simulation time. 3. A satellite intensity image consisting of pixels is sent to earth. The Þle satpic.mat contains the ( ) vector y with the corrupted data from a 16 QAM signal constellation. The data has passed h 3. The SNR is unknown. Design a receiver Y=image_receive(r) that reproduces the image as good as possible. A suitable linear or decision-feedback equalizer should be part of this receiver. Motivate your equalizer choice. 9