ASU, EEE 558

Excerpt: ... ve two methods to generate fading channels respectively. For each channel model, carry out Monte Carlo simulations to estimate the bit error probability (BER) as a function of the signal-to-noise-ratio (SNR), for both modulation schemes. (Please note there is additive white Gaussian noise (AWGN).) You need to use the BER performance for the AWGN channel (i.e. no fading) as the benchmark. we consider two cases, namely the ideal channel condition may or may not be available. First consider the ideal channel state information is available. Then investigate the case where pilot symbols are inserted periodically to estimate the channel state so the channel estimates are "noisy". Plot the BER vs. SNR for both cases, as required in the above. Discuss the impact on the performance incurred by the rapidity of channel variation and the period of the pilot signals. Explain why it is so. Your project report should be typed; use single-column and give enough line-space (say 1.2) so I can make marks on your reports. Be com ...

Washington University in St. Louis, ESE 488

Excerpt: ... Gaussian Statistics Noise is frequently modeled as AWGN ( Additive White Gaussian Noise ). Additive Noise adds to the signal in the system. White Equal Power at all frequencies (Flat Power Spectral Density). Gaussian Instantaneous samples have a Gaussian Probability Density Function (PDF). The Probability Density Function (PDF) for a Gaussian process is: f x (x ) = e - ( x- )2 2 2 2 2 f x (x ) where is the mean and is the standard deviation. 1 2 2 1 2 2e - + In order to find the Probability that x is between x1 and x2, we integrate the PDF as follows: Pr ( x1 < x x2 ) = x2 f (x )dx x x1 Probability is always between 0 and 1 inclusive. PDF's always integrate to 1 (i.e., the Probability that x is between - and is 1. - f (x )dx = 1 x The area under the "tail" of the PDF is of great interest in a communication system. If we want to know the Probability that Signal + Noise exceeds a threshold, we need to integrate the PDF from the threshold to , i.e., the tail. D: ...

Sveriges lantbruksuniversitet, ENSC 805

Excerpt: ... ransmitted with equal probability over an additive white Gaussian noise (AWGN) channel. The receive signal model is thus r (t ) = s (t ) + n (t ) , where s (t ) represents the transmitted waveform (which is either s1 (t ) or s2 (t ) ), and n(t ) is the channel's AWGN, which as a power spectral density of N 0 . The receiver computes * r1 = r (t ) s1 (t )dt , 0 T and * r2 = r (t ) s2 (t )dt . 0 T (a) Assuming that s1 (t ) was transmitted (i.e. s (t ) = s1 (t ) ), determine the structures of r1 and r2 . Clearly specify the signal and the noise components in each of these two samples. For the noise components, determine their variances and covariance. (b) What choice(s) of f 2 (relative to f1 ) will force the covariance of the noise components in Part (a) to zero? What will happen to the signal components in this case? (c) For the condition in Part (b), determine the bit-error-probability, given that s1 (t ) was sent. 2 Question 2 Consider the three (low pass equivalent) waveforms shown in Fig. Q2. All th ...

W. Alabama, ECE 411

Excerpt: ... @G. Gong, E&CE 411, Spring 2008, Handout A2 1 Assignment 2 (Topic 3. Waveform Coding Techniques (Chapter 5 in the text) Part 1 1. For the binary communication system of Figure 1 in the note, the signals s0 (t) and s1 (t) are dened by s0 (t) = 4s(t) and s1 (t) = 2s(t) where s(t) is given by (4) in the note. Suppose that T = 2 and N0 = 4. (a) Find the error probabilities P (e|i), i = 0, 1, for the system with threshold = 7. (b) What is the minimax threshold for this system? (c) Find the average error probability. 2. Consider the binary signal set s0 (t) = As(t) and s1 (t) = (4At/T )s(t) where s(t) is dened by (4) in the note. Assume that there are to be used to transmit binary PCM signal over an AWGN channel with spectral density N0 /2. (a) Find the matched lter for this signal set. (b) Find the minimax (optimal) threshold and the resulting error probability. 3. The signal s(t) = et , 0 t T is corrupted by additive white gaussian noise with zero mean and power spectral density N0 /2 ...

East Los Angeles College, OA 214

Excerpt: ... Image Processing IB Paper 8 Part A Ognjen Arandjelovi http:/mi.eng.cam.ac.uk/~oa214/ Lecture Roadmap Lecture 1: Face geometry Geometric image transformations Lecture 2: Colour and brightness enhancement Lecture 3: Denoising and image filtering Image Denoising and Filtering Image Noise Sources Image noise may be produced by several sources: Quantization Photonic Thermal Electric Denoising To effectively perform denoising, we need to consider the following issues: Signal (uncorrupted image) model Typically piece-wise constant or linear Noise model (from the physics of image formation) Additive or multiplicative, Gaussian, white, salt and pepper Salt and Pepper Noise Gaussian Noise Modelling Noise Most often noise is additive: Observed pixel luminance True luminance Noise process Additive Gaussian Noise Example A clear original image was corrupted by additive white Gaussian noise : Original, uncorrupted image Additive Gaussian ...

Texas A&M, FTP 455

Excerpt: ... " '"1" Ifl",!"rr!, EE 455 Test # 2 Fall 2003 Given November 20, 2003. Closed-book,closed notes, one formula sheet allowed. Problems carry equal weight. 1. Consider binary signaling in an additive, white Gaussian noise channel using the two signals shownbelow. Design the optimum receiver (simplified as much as possible) and write an expression for the probability A ~ At~ , -t of error for this receiver. 1~2-~ -t Solution 1/2 !~ E = ) -z. , l\ -= ) 5. (t)~'l D -I -\" 2 = It: /2. 2/ 62. -:42 ~q -\ e -= t 2 -.e, -=L ~{' (.icY S2.(-t) -A/~ Jt l -L ~ 1/(+)\ tt)d-t -AI'f l 2. = \ A , S v (-t-)o.t -A -t 2- 7./ ~ t.f I c) :)c9p\ ~= S /,(\;-)tJ.tt ~A/Lf 1(2. I (D) '1-:;:. ~ [C; \ ~.1.:;) ~ l t >l ~ t =- A '"?-S / 2() ? ( e.,) -= e r f c. l2- GJ~q~ .2-) ;:~ ~2~_ ) f:> If c. (/K ) ) ,.u", :.~ (? C iF! r~ 1 2. You are to design a two-dimensional signal constellation having 5 signals to be used in an additive white Gaussian noise channel. Each signal's ...

Ill. Chicago, EECS 422

Excerpt: ... 2 2 2 u(t) : normalized current of voltage The problem of fast fading is that when the actual signal level at receiver is comparable to the noise level and SNR (signal-to-noise ratio) is below the max tolerable level. In order to understand the SNR issue, we assume that our radio channel is an additive white Gaussian noise (AWGN) channel. If the noise signal is n(t) then the actual received signal at time t is y (t ) = Au(t ) + n(t ) A : is the overall path loss and is assumed invariant with time, particularly true when the mobile moves within a small distance. n(t) is a complex phasor and is written as n(t ) = x n (t ) + jy n (t ) Then the mean noise power is 1 E [n (t )n* (t )] 1 2 2 = E [xn (t )]+ E [yn (t )] Pn = 2 2 2 Note that if xn(t) and yn(t) both have a standard deviation n (usually the case) Pn = n2 The SNR then is 2 2 A2 A2 signal power E A u (t ) 2 = SNR = = = E [u (t )] = noise power 2 Pn 2 Pn 2 Pn [ ] with the signal power out of a modulator normalized to 1. We can now in ...

Iowa State, EE 422

Excerpt: ... Due on 4pm, Tuesday, Mar. 30, 2004 Note: 1. This project carry 7.5 points to your final score. 2. You are required to do the programming using the Matlab software, and send the .m file (source code) to the instructor via email. Deduction of score or zero score for late submission. 3. Put the results in a report (one additional point awarded for preparation using Latex, or Word, unless full score is attained), and submit it before the deadline. 4. A sample Matlab file and a report format file are available on the course website and WebCT. 5. Absolutely no copying of each other's Matlab codes and report. The student(s) will receive Zero score and being reported to university if such behavior is found. Project #1 for Course EE422 Title: Error Probability Result for Binary Modulation Model: Assuming we have a receiver over a non-fading additive white Gaussian noise (AWGN) channel. The received signal at the ith bit interval is given by x(i) = d(i) + n(i), where n(i) is the noise element and d(i) is the data sym ...

Sanford-Brown Institute, AM 282

Excerpt: ... AM282, Section 3: Advanced Topics in Information Theory Brown University, Spring 2004 April 7, 2004 Handout #8 Midterm You have 3 hours to complete this midterm. You are allowed to use three pages (back and front) of notes. Please clearly indicate your reasoning in the space provided. Notice that the point allocation is not uniform. Name: Problem Points Earned Potential Points 1 25 2 15 3 15 4 20 5 25 Total 100 1 (1) Multiple Access Channel (25 points) (i) Consider the standard additive white Gaussian noise channel: Yi = Xi + Zi , where {Zi } is IID N (0, N ) and the input must satisfy the capacity? 1 n n i=1 (1) x2 P . What is i (ii) Now suppose that (1) is a multiple access channel, where Xi is the ith input from user 1 and Zi is the ith input from user 2. (Note: There is no noise.) User 1 1 has the same constraint as before ( n n x2 P ) and user 2 also has a power i=1 i 1 constraint, n n zi2 N . What rates (R1 , R2 ) are achievable? i=1 (Continued on next page.) 2 (iii) Now consider the mu ...

University of Illinois, Urbana Champaign, ECE 459

Excerpt: ... as Y = AX + b. Find A and b such that Y is zero mean with covariance matrix I. (Hint: Diagonalize .) 2. (15 pts total) Bounds on the Q function. Q(x) = x e-t /2 dt 2 2 (a) (8 pts) For x > 0 show that the following upper and lower bounds hold for the Q function: 1 1- 2 x e-x /2 e-x /2 Q(x) x 2 x 2 2 2 2 Hint: For the upper bound, write the integrand as a product of 1/t and te-t /2 , use integration by parts, and bound. For the lower bound, integrate by parts once more and bound. c V. V. Veeravalli, 2000 1 (b) (7 pts) As you know from your undergraduate communications course, the bit error probability for BPSK signaling in additive white Gaussian noise (AWGN) with PSD N0 /2 is given by: 2Eb Pe = Q N0 where Eb is the bit energy. Plot the error probabililty Pe (on a log scale) versus signal-to-noise ratio Eb /N0 (in dB) using Matlab or Mathematica. (You may need to use an appropriately modified version of the error function in these packages.) Consider Eb /N0 ranging from -5 dB to 15 dB. Also plot th ...

University of Illinois, Urbana Champaign, ECE 559

Excerpt: ... answered twice (both in class and at home) then the minimum of the two scores will be chosen. 6. The maximum score for any question is somewhat indicative of the fraction of the total time you might want to spend on that question, but do not be misled by that. The questions are posed in a way that they can be answered easily if seen in the correct light. 1 1. (7 points) It is known that the ML error probability is q when the two signal vectors s0 and s1 shown in Fig (a) below are transmitted over a channel disturbed by additive white Gaussian noise . Compute the ML error probability in terms of q, , l when the nine vectors indicated by s in Fig (b) are used as signals over the same channel. 2 2. (16 points total) A transmitted symbol X, equally likely to be 1, is received at n antennas: yi = x + zi , i = 1 . . . n. Here the noise (z1 , . . . , zn ) is independent of the transmitted symbol and is zero mean, jointly Gaussian. (a) (3 points) Suppose the covariance matrix of the jointly Gaussian ran ...