ELT COMMUNICATION THEORY

Transcription

1 ELT COMMUNICATION THEORY Matlab Exercise #5 Carrier mdulated digital transmissin: Transmitter and receiver structures QAM signals, up/dwncnversin, timing and phase synchrnizatin, and symbl detectin 1 SYSTEM MODEL AND GENERATION OF QAM SYMBOLS 1.1 SYSTEM MODEL In this exercise, we create a carrier mdulated QAM transmissin system including a transmitter (TX), a receiver (RX), and a simple AWGN (Additive White Gaussian Nise) channel mdel (see Fig. 1). T add an additinal practical aspect, we assume that the TX and RX scillatr clcks are nt synchrnized which causes a phase errr in the detected in the receiver. T cmpensate this phase errr and t btain a crrect timing fr symbl sampling in the receiver, we include a specific training (als called as preamble, pilts, etc.) in the transmissin. Data TX Transmitted carrier-mdulated QAM signal AWGN channel Received nisy carrier-mdulated QAM signal RX Estimated data Training Figure 1: General system mdel We cnsider the TX and RX structures t be similar t the nes shwn in lecture ntes (see Fig. 2 and Fig. 3), except that we skip the bit-t-symbl and symbl-t-bit transfrmatin, and cnsider nly transmissin f (if yu like, it is trivial t include bits in the system later n). Here, bth the TX and RX structure are based n cmplex calculatins, but the same structures can als be implemented by using nly real-valued signals (i.e., I/Q-mdulatin with separate I and Q branches, see the lecture ntes fr mre details). As transmit and receive filters we use the Rt-Raised-Csine (RRC) filters (see the lecture slides). Tgether (in TX and RX) these filters fulfil the Nyquist criterin (i.e. n Inter-Symbl-Interference (ISI). Ntice that a single RRC des nt d this unlike with the cnventinal raised-csine filter.

2 j ct e Cmplex Transmit filter Cmplex Baseband signal gt () a () k j ct e Cmplex 2 Re st zt () xt () Figure 2: Transmitter (TX) structure. yt () rt () Receive filter 2 f( t) qt () Sampler q k Phase synchrnizatin q k Decicin aˆk Timing Training Clck-phase crrectin Figure 3: Receiver (RX) structure. 1.2 CONSIDERED SYSTEM PARAMETERS First let s define the general system parameters as fllws: alphabet_size = 16; % Number f in the QAM alphabet (e.g. 16 means 16-QAM). Valid alphabet % sizes are 4, 16, 64, 256, 1024,... (i.e. the pssible values are given % by the vectr 2.^(2:2:N), fr any N) SNR = 10; T = 1/10e6; fc = 75e6; r = 20; N per_pulse = 30; alfa = 0.25; % Signal-t-nise pwer rati [db] % Symbl time interval [s] % Carrier frequency % Oversampling factr (r samples per pulse) % Duratin f TX/RX-filters in numbers f % Rll-ff factr (excess bandwidth) Based n abve we can define sampling frequency and sampling time interval as Fs = r/t; Ts = 1/Fs; % Sampling frequency % Sampling time interval Then we define the number f t be transmitted. T acquire timing and phase synchrnizatin in the receiver, we add training (QAM- as well) in the beginning f the transmitted symbl frame (see Fig. 4). Ntice that training are predefined in the system, and thus knwn by the receiver, and they d nt carry any user infrmatin which reduces the spectral efficiency f the system. Training Data 940 samples samples in ttal (= 4.7 μs + 50 μs = 54.7 μs with T = 5 ns) Figure 4: Symbl frame t be transmitted

3 Based n Fig. 4, we define the number f training and data as N_data_ = 10000; % Number f data N_training_ = 940; % Number f training 1.3 GENERATION OF QAM SYMBOLS AND THE TRANSMITTED SYMBOL FRAME Generate the user data Create a QAM cnstellatin (help bsxfun) Scale the cnstellatin s that the expected average pwer f transmitted equals t ne Generate the specified number (N_data_) f randm data Plt the transmitted in cmplex plane (a cnstellatin) % Here qam_axis presents the symbl values in real/imaginary axis. S, %generally fr different alphabet/cnstellatin sizes (): qam_axis = -sqrt(alphabet_size)+1:2:sqrt(alphabet_size)-1; % Fr example, the abve results in % qam_axis = [-1 1]; % fr QPSK % qam_axis = [ ]; % fr 16-QAM % qam_axis = [ ]; % fr 64-QAM % generatin f a cmplex cnstellatin: alphabet = bsxfun(@plus,qam_axis',1j*qam_axis); %help bsxfun %%% equivalent t alphabet = repmat(qam_axis', 1, sqrt(alphabet_size)) + repmat(1j*qam_axis, sqrt(alphabet_size), 1); %%% alphabet = alphabet(:).'; % alphabet as a rw vectr % Scaling the cnstellatin, s that the mean pwer f a transmitted symbl % is ne (e.g., with QPSK this is 1/sqrt(2), and fr 16-QAM 1/sqrt(10)) alphabet_scaling_factr = 1/sqrt(mean(abs(alphabet).^2)); alphabet = alphabet*alphabet_scaling_factr; % Randm vectr f symbl indices (i.e., numbers between 1...alphabet_size) symbl_ind = randi(length(alphabet),1,n_data_); data_ = alphabet(symbl_ind); % Data t be transmitted plt(data_,'b') xlabel('re') xlabel('im') title('transmitted data ') Generate training (QAM) and create the transmitted symbl frame Use the abve-defined QAM cnstellatin als fr Generate the specified number (N_training_) f randm training % Generatin f training (similar t data ): training_ = alphabet(randi(length(alphabet),1,n_training_)); % Cncatenating the training and data t get the verall % transmitted : symbl_frame = [training_ data_];

4 2 TRANSMITTER STRUCTURE By fllwing the TX structure in Fig. 2, we generate a bandpass (carrier mdulated) QAM signal by based n the abve-defined system parameters and the given symbl frame. Implement the transit filter: Rt-Raised-Csine (RRC) and plt the pulse shape p = rcsdesign(alfa,n per_pulse,r,'sqrt'); plt(-n per_pulse*r/2*ts:ts:n per_pulse*r/2*ts,p,'b') hld n plt(-n per_pulse*r/2*ts:t:n per_pulse*r/2*ts,p(1:r:end),'r') xlabel('time [s]') xlabel('amplitude') title('transmit/receive RRC filter (pulse shape)') legend('pulse shape','ideal symbl-sampling lcatins') Filter the transmitted symbl frame Remember t upsample the symbl sequence rate t match with sampling rate f the filter/pulse: _upsampled = zers(size(1:r*length(symbl_frame))); % Zer vectr initilized fr Up-sampled symbl sequence _upsampled(1:r:r*length(symbl_frame)) = symbl_frame; % I.e. nw the up-sampled sequence lks like {a a a } x_lp = filter(p,1,_upsampled); % Transmitter filtering x_lp = x_lp(1+(length(p)-1)/2:end); % Filter delay crrectin %ntice the here x_lp is the cmplex-valued lwpass equivalent signal f the transmitted real-valued bandpass signal x_bp (generated in the next stage) Implement the upcnversin (mdulatin / frequency translatin t the carrier frequency f c) Define the time vectr fr the scillatr signal based n the reference clck time in the TX Generate the cmplex-expnential carrier signal using the abve-defined time vectr Multiply (= mix ) the carrier signal with the lw-pass equivalent QAM signal (x_lp) % Time vectr fr the TX scillatr signal: t_tx_scillatr = 0:Ts:Ts*(length(x_LP)-1); % TX scillatr signal: TX_scillatr_signal = exp(1j*2*pi*fc*t_tx_scillatr); % Carrier mdulatin / upcnversin (still cmplex valued): x_bp_cmplex = x_lp.*tx_scillatr_signal; T finalize the TX prcess (lwpass-t-bandpass transfrmatin) take the real part f the signal (and scale with sqrt(2)) % Taking the real value t finalize the lwpass-t-bandpass transfrmatin: x_bp = sqrt(2)*real(x_bp_cmplex); Plt the fllwing s The cmplex-valued lw-pass equivalent signal x_lp in time and frequency dmain The cmplex-valued bandpass signal x_bp_cmplex in frequency dmain The real-valued bandpass signal x_bp in time and frequency dmain % zm manually t see the signal better plt(t_tx_scillatr, abs(x_lp)) xlabel('time [s]') ylabel('amplitude (f a cmplex signal)') title('lwpass signal in time dmain') % zm manually t see the signal better plt(t_tx_scillatr, x_bp) %ntice n abs needed xlabel('time [s]')

5 ylabel('amplitude') title('bandpass signal in time dmain') NFFT = 2^14; %FFT size f = -Fs/2:1/(NFFT*Ts):Fs/2-1/(NFFT*Ts);%frequency vectr plt(f/1e6, fftshift(abs(fft(x_lp,nfft)))) xlabel('frequency [MHz]') ylabel('amplitude ') title('amplitude spectrum f the lwpass signal') plt(f/1e6, fftshift(abs(fft(x_bp_cmplex,nfft)))) %ntice n abs needed xlabel('frequency [MHz]') ylabel('amplitude') title('amplitude spectrum f the bandpass signal') plt(f/1e6, fftshift(abs(fft(x_bp,nfft)))) %ntice n abs needed xlabel('frequency [MHz]') ylabel('amplitude') title('amplitude spectrum f the bandpass signal') 3 CHANNEL MODEL Here we cnsider a simple AWGN channel mdel. Based n the earlier exercises, we create white randm nise, scale it with the prper scaling factr t btain the desired SNR, and then add it n tp f the transmitted signal (x_bp). Generate the nise n = randn(size(x_bp)); % White Gaussian randm nise P_x_BP = var(x_bp); % Signal pwer P_n = var(n); % Nise pwer % Defining nise scaling factr based n the desired SNR: nise_scaling_factr = sqrt(p_x_bp/p_n/10^(snr/10)*(r/(1+alfa))); Why r/(1+alfa) part? % Nisy signal y_bp = x_bp + nise_scaling_factr*n; Plt the amplitude spectrum f the nisy bandpass signal % use the previus spectrum s by just replacing the pltted variable 4 RECEIVER STRUCTURE Typically, due t many unknwn/uncertain parameters, mst f the cmplexity in a cmmunicatins system is fund n the RX side. Nw, by fllwing the RX structure in Fig. 3, we estimate the transmitted frm the received nisy bandpass QAM signal. 4.1 DOWNCONVERSION AND SIGNAL FILTERING Implement the dwncnversin (i.e. demdulatin back t the baseband) by using the same principle as in the upcnversin in TX, but remember t use the RX reference clck time: % Time vectr fr the RX scillatr signal: t_rx_scillatr = 0:Ts:Ts*(length(y_BP)-1); % RX scillatr signal (ntice the minus-sign cmpared t TX scillatr!!!):

6 RX_scillatr_signal = exp(-1j*2*pi*fc*t_rx_scillatr); % Carrier demdulatin / dwncnversin (signal becmes cmplex again) y_bp_dwncnverted = y_bp.*rx_scillatr_signal; Plt the amplitude spectrum f the dwncnverted signal % use the previus spectrum s by just replacing the pltted variable Filter the received signal with the receive filter (RRC similar t TX) X_LP_received = sqrt(2)*filter(p,1,y_bp_dwncnverted); % Receiver filtering X_LP_received = X_LP_received(1+(length(p)-1)/2:end); Filter delay crrectin Plt the amplitude spectrum f the dwncnverted and filtered signal % use the previus spectrum s by just replacing the pltted variable 4.2 SIGNAL SAMPLING Sample the and remve versampling % Sampling the received signal in rder t get symbl samples RX_symbl_frame = X_LP_received(1:r:end); Take user data and training in separate vectrs: RX_training_ = RX_symbl_frame(1:N_training_); RX_data_ = RX_symbl_frame(N_training_+1:end); Plt the received symbl samples in cmplex plane (cnstellatin) plt(rx_data_,'b') hld n plt(alphabet,'rs') hld ff xlabel('re') xlabel('im') title('received data with clck ffset (phase errr)') 4.3 OBTAINING SYMBOL DECISIONS AND CALCULATING THE SYMBOL ERROR RATE (SER) The final step is t make the symbl decisins. This is dne based n the minimum distance principle, where the symbl estimate is defined as that symbl f the alphabet, which minimizes the distance t the symbl sample. Calculate the Euclidian distance between each symbl sample and each alphabet symbl: alphabet_errr_matrix = abs(bsxfun(@minus,alphabet.',rx_data_)); % Nw, rws represent the alphabet symbl indices and clumns represent the % received symbl indices (e.g. the Euclidian distance between the 5th % received symbl and the 3rd symbl in the alphabet is given as % "alphabet_errr_matrix(3,5)" Find the indices f the alphabet, which have the minimum distance t the symbl samples: [~,estimated_symbl_ind] = min(alphabet_errr_matrix); In the end, we find ut which were received incrrectly and define the bserved Symbl Errr Rate (SER) % Finding ut which were estimated incrrecly: symbl_errrs =... estimated_symbl_ind ~= symbl_ind(1:length(estimated_symbl_ind));

7 % ntice that due t filtering transitins, we lse sme f the last % in the symbl frame. In practice we wuld simply cntinue taking % a few samples after the frame t try t get all the. Hwever, filter % transitins in the beginning and in end f the frame are always creating % nn-idealities t the transmissin (the same is als happening in % frequency dmain: e.g. cmpare data in the middle and in the edge f % the used band). % Symbl Errr Rate (SER) (0 means 0% f errrs, 1 means 100% f errrs) SER = mean(symbl_errrs) What's the wrst value fr the SER? Why? Try the abve cde by altering the SNR and the QAM cnstellatin size (QPSK, 64-QAM, etc.) 5 TIMING SYNCHRONIZATION AND PHASE CORRECTION This part f the exercise demnstrates cmmn errrs f transmissin systems: timing and phase errrs; and shws hw they can be crrected. Therefre, first we intrduce the errrs in the transmissin system and then synchrnize the in Sect. 5.1 and cmpensate the phase errrs in Sect We add an unknwn prpagatin delay t the AWGN channel mdel. In cntrast t, e.g., the RX filter delay which is knwn, the channel delay is randm and must be estimated in the RX. This is usually referred as timing synchrnizatin. In LTE nrmal peratin mde, the cyclic prefix duratin is 4.7 µs. Thus, delays less than this are assumed: delay = randi(940) % Delay in samples x_bp = [zers(1,delay), x_bp]; & Add delay zers NOTE: This shuld be added t the channel part befre the nise is added t X_BP (see Sect. 3). Lastly, t cnsider the fact that the scillatr clcks (in the frequency translatin with j t e c in the TX in Fig. 2 and with in the RX in Fig. 3) are nt synchrnized, we define separate reference clck times fr the TX and RX. Let s define the TX and RX clcks having randm ffsets unifrmly distributed between 0 1s. NOTE: Mdify the crrespnding system parameters (see Sect. 2 and Sect. 4.1). % Time vectr fr the RX scillatr signal: TX_clck_start_time = rand; % Clck start time in the TX scillatr RX_clck_start_time = rand; % Clck start time in the RX scillatr Then, the scillatr time vectrs are redefined t use these ffsets: t_tx_scillatr = TX_clck_start_time + (0:Ts:Ts*(length(x_LP)-1)); t_rx_scillatr = RX_clck_start_time + (0:Ts:Ts*(length(y_BP)-1)); e j ct 5.1 TIMING SYNCHRONIZATION We implement an estimatr fr the crrect symbl timing based n crss-crrelatin between the knwn training and the received QAM signal. NOTE: This shuld be perfrmed after the channel and RX filtering but befre the signal sampling (see Sect. 4).

8 T perfrm the crrelatin between the knwn training and the received QAM signal, we must upsample the training in rder match the sampling rates (like what we did in TX befre the TX-filtering): training_signal = zers(size(1:r*length(training_))); % Zer vectr initilized fr Up-sampled symbl sequence training_signal(1:r:r*length(training_)) = training_; % I.e. nw the up-sampled sequence lks like {a1 0 0 a2 0 0 a3 0 0 a4 0...} Calculate the crss crrelatin as functin time-delay between the received signal and the upsampled training (help xcrr) % Calculate the crss-crrelatin [crr_fun, delay] = xcrr(x_lp_received,training_signal); %crss-crrelatin Plt the amplitude f crss-crrelatin functin plt(delay,abs(crr_fun)) xlabel('delay [samples]') ylabel('crrelatin') title('crss-crrelatin between transmitted and received training ') Find the crrelatin peak t btain the timing delay (help max): % Find the sample index with the maximum crrelatin value: [~, max_ind] = max(abs(crr_fun)); %Estimated delay: The 1st signal sample shuld be taken at "timing_ind+1" timing_ind = delay(max_ind); Nw, the sampling can be perfrmed as RX_symbl_frame = X_LP_received(timing_ind+1:r:end); where after the training and data are separated, as abve. 5.2 PHASE ERROR ESTIMATION AND PHASE COMPENSATION Since we have a set f training, whse riginal phase is knwn by the receiver, we can estimate the phase errr caused by the ffset between TX and RX clcks. Basically, this is dne by cmparing the phases between the knwn training and received training. Plt the received symbl samples in cmplex plane (cnstellatin) befre phase errr cmpensatin t bserve the effects f the errr. % Just cpy here the same symbl pltting cmmands as abve. Withut ging t details here, we use the fllwing expressin t calculate the s called channel estimate (if yu are interested, yu can see the Appendix in the end f the exercise fr mre details): % Here RX_training_ and training_ shuld be rw-vectrs channel_estimate =... %with 3 dts the expressin cntinues n the next line RX_training_*training_'/nrm(training_)^2; %help nrm Then, the phase errr can be cmpensated by simply dividing the signal with the channel estimate as RX_data_ = RX_data_/channel_estimate; Again, plt the received symbl samples in cmplex plane (cnstellatin) t bserve the cmpensatin result. % Just cpy here the same symbl pltting cmmands as abve.

9 Actually, the abve channel cmpensatin prcedure wuld als cmpensate amplitude errrs (based n s called zer-frcing principle). 5.3 SYMBOL DECISIONS AND CALCULATING THE SYMBOL ERROR RATE (SER) Implement the remaining RX prcessing as abve. Cmpare the realizatin f randm channel delay t the estimated delay. What can yu say abut the accuracy? Hw the phase errr effects the received cnstellatin? Hw well is this errr cmpensated? APPENDIX (EXTRA MATERIAL) CHANNEL ESTIMATION PRINCIPLE Based n the AWGN channel, the received training symbl samples y (Nx1 vectr) can be mdeled as y hx n, where h is the channel cefficient (a cmplex scalar causing amplitude/phase errr), x (Nx1 vectr) is the transmitted (knwn) training, and n (Nx1 vectr) is the nise. Nw, we calculate a (zer-shift) crss-crrelatin between y and x as H H H H H x y x x x n x x x n, h h where H is the cmplex-cnjugate transpse. By nticing that is btained as H H ˆ x y x n h h 2 2 x x H 2 x x x the channel estimate ĥ where the last term defines the channel estimatin errr. Ntice that when the number f training (i.e. vectr length) increases, the channel estimatin errr decreases.,