8 PAM BER/SER Monte Carlo Simulation
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1 xercise.1 8 PAM BR/SR Monte Carlo Simulation - Simulate a 8 level PAM communication system and calculate bit and symbol error ratios (BR/SR). - Plot the calculated and simulated SR and BR curves. - Plot the theoretical SR and BR curves. - Calculate the average energy. - Plot the constellation and histogram of samples.
2 xercise.1 8 PAM BR/SR Monte Carlo Simulation Uniform random number generator Block Diagram I Gray-encode symbols S Modulator X Gaussian Noise + Y Gray-decoder Detector S_hat I_hat Compare symbols (I = I_hat)? Symbol rror Ratio Compare bits Bit rror Ratio
3 8 PAM System Signal constellation Gray encoding Symbols Bits in symbol -7d -5d -3d -1d 0 1d 3d 5d 7d Amplitude d : scaling Gray encoding: Slide 65 (Lecture_3_007_Hypothesisesting_SignalSpace.pdf)
4 Average transmitted signal energy = = M k k av dt t t s M 1 0 ) ( ) ( 1 φ S 0 (t) S 1 (t) S (t) S 3 (t) S 4 (t) S 5 (t) S 6 (t) S 7 (t) PAM System Derivation of standard deviation t t ( ) ) ( ) ( 8 1 d dt t t s k k av = = = = = φ 1 ) ( function Basis t φ t
5 8 PAM System Derivation of standard deviation ransmitted average energy per bit av av _ bit = = log M 3 av Average bit SNR SNR 1d 7d av _ bit av bit = = = = N0 3N0 3 σ σ Standard deviation used in the simulation σ = 7d SNR bit Note that some books use different notation av _ bit b b SNR = = = = bit N N No 0 0 bno
6 8 PAM System heoretical symbol error probability SP = (*(M-1)/M)*qfunc(sqrt((6*log(M)/(M^-1))*SNR_bit_abs)); M = Symbol error probabilities. 8 PAM theoretical simulation 10-1 Symbol rror Probability b/no [db] SP: Slide 63 (Lecture_3_007_Hypothesisesting_SignalSpace.pdf)
7 8 PAM System heoretical bit error probability approximation for low BP : BP = (1/log M) SP 10 0 Bit error probabilities. 8 PAM theoretical simulation 10-1 Bit rror Probability b/no [db] BP: Slide 68 (Lecture_3_007_Hypothesisesting_SignalSpace.pdf)
8 8 PAM System Matlab exercise -Download the file from the course web page and open with an editor SP 1 : transform SNR [db] to SNR [abs] SP : complete standard deviation for 8 PAM SP 3: generate N integer random numbers in the interval [0,7] and verify the distribution with a histogram Histogram of symbols
9 8 PAM System Matlab SP 4 : Add noise to symbols, use randn() SP 5: Calculate the symbol error ratio SP 6: Calculate the bit error ratio SP 7: Repeat the simulation for different SNR SP 8: Plot using the code in the file. Observe: - symbol error probabilities and compare to theoretical results - bit error probabilities and compare to theoretical results - constellation - histogram of the samples SP 9: Observe the average energy SP 10:Optional. Modulate using pammod() from the Communications oolbox
10 xercise BPSK / QPSK BR/SR Simulation -Simulate a communication system using BPSK/QPSK modulation and calculate bit and symbol error ratios. - Set BPSK modulation. - Plot the calculated and simulated SR and BR curves. - Plot the theoretical SR and BR curves. - Change modulation to QPSK - Plot the calculated and simulated SR and BR curves. - Plot the theoretical SR and BR curves. - Introduce a rotation in the channel and equalize the channel.
11 xercise BPSK / QPSK BR/SR Simulation Uniform random number generator Block Diagram I Gray-encode symbols S Modulator X Gaussian Noise + Y Gray-decoder Detector S_hat I_hat Compare symbols (I = I_hat)? Symbol rror Probability Compare bits Bit rror Probability
12 BPSK ransmitted average energy per bit av _ bit = av _ s SNR bit SNR b av _ bit av _ s bit = = = = = No N0 N0 σ σ heoretical Bit rror Probability for BPSK P = Q ( ) b SNR bit σ = SNR bit Symbol rror Probability = Bit rror Probability in BPSK (1bit=1symbol) P = Q ( ) s SNR bit
13 QPSK ransmitted average energy per bit av _ s av _ bit = = log M SNR bit SNR b av _ s av _ bit av _ s bit = = = = = No N0 N0 σ 4σ P = Q ( ) b SNR bit M = 4 heoretical Bit rror Probability for BPSK = QPSK σ = 4 SNR bit Approximation of heoretical Symbol rror Probability for M >= 4 P s = Q SNR bit sin π M
14 BPSK/QPSK Matlab exercise -Download the file from the course web page and open with an editor SP 1 : Set modulation to BPSK SP : Gray-encode symbols, use bingray() SP 3: Demodulate PSK by using pskdemod() SP 4: Plot using the code in the file. Observe: - symbol error probabilities and compare to theoretical results - bit error probabilities and compare to theoretical results SP 5: Change modulation to QPSK SP 6: Introduce a rotation in the channel SP 7: qualize the channel
15 BPSK/QPSK Matlab exercise SP 8 : Plot using the code in the file. Observe: - he symbols rotated - he symbols rotated + noise - he symbols after the equalizer
16 Bit rror Probability 10 0 Symbol/Bit error probabilities. BPSK BPSK SR theoretical SR simulation BR theoretical BR simulation Bit rror Probability Symbol/Bit error probabilities. QPSK QPSK SR theoretical SR simulation BR theoretical BR simulation b/no [db] 1.5 Rotated Signal Constellation with Noise QPSK b/no [db] Signal Constellation after qualization 1.5 QPSK imag 0 imag real real
17 xercise 3 System using Biorthogonal Signals -Simulate a communication system using biorthogonal waveforms and calculate bit and symbol error ratios. -Plot the calculated and simulated SR and BR curves. -Modify the simulator to use orthogonal waveforms. -Plot the calculated and simulated SR and BR curves.
18 xercise 3 System using Biorthogonal Signals S 0 (t) / S 1 (t) t / S (t) t / t S 3 (t) / t Symbols Signal constellation S 1 (t) 1 [ 0 1] S (t) 3 S 0 (t) 0 [-1 0] [ 1 0] S 3 (t) [ 0-1]
19 System using Biorthogonal Signals Average transmitted signal energy av _ s = ransmitted average energy per bit av _ s av _ bit = = log M SNR bit SNR b av _ s av _ bit av _ s bit = = = = = No N0 N0 σ 4σ σ = 4b / No
20 System using Biorthogonal Signals - Detector Signal constellation 1 Y Decision Areas 3 y Received symbol y1 0 Y1 Possible approach to make decisions if (y1 > abs(y), S = 0; elseif (y > abs(y1), S = 1; elseif (y1 <-abs(y), S = 3; else S = ;
21 Biorthogonal Signals Matlab exercise ask : Download the file from the course web page and open with an editor his code simulates biorthogonal signals and is ready to run. Compare the Symbol and Bit error probabilities obtained here to the results obtained in the QPSK exercise. SR / BR Symbol and bit error probabilities. Biorthogonal waveforms M = 4 SR BR Symbol/Bit rror Probability b/no [db]
22 Orthogonal. Signals Matlab exercise ask : he code provided simulates biorthogonal signals. Modify the code to simulate the orthogonal signals shown below (M = ). S 0 (t) / t / S 1 (t) t S 1 (t) Y [ 0 1] SP 1: Set M = S 0 (t) SP : Change standard deviation SP 3: Set modulation 0 [ 1 0] Y1 SP 4: Set the decision areas Signal constellation
23 Orthogonal. Signals Matlab exercise SP 5: Observe symbol and bit error rates SP 6: Plot theoretical bit error probabilities, for orthogonal and biorthogonal cases. Observe 3dB difference Symbol and bit error probabilities Symbol/Bit rror Probability SR BR BR heoretical Orthogonal BR heoretical Biorthogonal b/no [db]
24 xercise 4 Power Spectrum Density Consider a four-phase PSK represented by the following equivalent low pass signal: where n ( ) n ( ) u t = I g t n I takes one of the four possible values: 1 ( 1 j) n ± ± with equal probability. a. Determine and sketch the power spectrum density of u ( t) when: ( ) g t A 0 t = 0 otherwise b. Repeat a. when ( ) g t πt Asin 0 t = 0 otherwise c. Compare the spectra obtained in a. and b. in terms of the 3dB bandwidth and the bandwidth to the first spectral zero.
25 xercise 4 a). In PAM the I n coefficients are real but in PSK this is not the case since the signal space is not uni-dimensional. ( ) g t A 0 t = 0 otherwise ( ) ( π ft) ( π ft) F sin( π ftf ( f ) = A ) G( f A e π ft πf he power spectrum density is calculated as: 1 Φ uu ( f ) = G( f ) sin sin G f = A Φ f = A ( ) ( π ft) ( π ft) uu G sin(π ) jπ f Normalized Power spectrum density 10 0 Spectrum for Rectangular pulse Normalized frequency f b First Zero at 1/
26 xercise 4 b). ( ) g t πt Asin 0 t = 0 otherwise F G( f ) ( πf ) A cos = π 1 4 f e jπf he power spectrum density is calculated: ( ) ( ) 1 A Φ uu f = G f = First Zero at 1.5/ cos ( πf ) ( ) 1 4 f Normalized Power spectrum density 10 0 Spectrum for Sinusoid pulse Normalized frequency f b
27 xercise 4 c). he power spectrum for the rectangular pulse has narrower mainlobe but higher sidelobes Normalized Power spectrum density 10 0 Spectrum for Rectangular and sinusoid pulse Rectangular Sinusoid Normalized frequency f b he sinusoid pulse demands 50% more bandwidth compared to the rectangular pulse rectangular: sinusoid: ( π f 3 db ) ( π f 3 db ) ( π f ) sin cos ( 1 4 f ) = f 3 db = = f 3 db =
28 xercise 4 o determine the power spectrum density we used the Fourier transform of a proposed signal. But the Fourier transform is available only for deterministic signals. With random message signals we can not find the Fourier transform. Nevertheless we can determine the autocorrelation function of these signals from their statistical information. hen we find the power spectrum density from the Fourier transform of the autocorrelation function: Φ uu { ( )} ( f ) = I ψ τ Where ψ(τ) is the autocorrelation function * ψ ( τ ) = u ( t) u ( t + τ ) dt In Matlab we use xcorr(x) (crosscorrelation of vector x with itself)
29 xercise 4 ASK : Simulate a communication system for the given problem using Matlab and determine the Power Spectrum Density when g(t) is a rectangular and sinusoidal pulse.
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