ENSC Discrete Time Systems. Project Outline. Semester

Transcription

1 ENSC 49 - iscrete Time Systems Prject Outline Semester Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding f Chapters, 4, and 5 f the LECTURE NOTES [1], and may be lt mre. It is strngly suggested that the prject be implemented in the Matlab envirnment.. Organizatin Yu are asked t wrk in grups f tw and submit ne cmmn reprt; refer t Sectin 6. Upn cmpletin f the reprt, yur grup needs t schedule a 30 minutes interview with the instructr, n later than August 4, Expectatins and Grading It is expected that the students can successfully simulate the fading channel specified in Part A f Sectin 5. Further marks will be given fr simulating the fading channel fr larger sequence lengths and/r slwer fading rates ; refer t Part B f Sectin 5. The quality f the simulated fading must be verified by cmparing the prperties f the cmputer simulated fading, with analytically expected results; refer t Part C f Sectin 5. ENSC-49 - Prject Outline Page 1 f 6

2 Yur mark in the prject is based n the amunt f wrk yu have dne, yur understanding, yur reprt, and yur perfrmance during the interview. 4. Fading Channel Mdel Let x() t and yt () be a pair f real-valued Gaussian prcesses that are statistically independent and identically distributed (iid). Bth prcesses have zer mean and an autcrrelatin functin [ ] [ ] R( τ ) = E xtxt () ( + τ) = E yt () yt ( + τ) = σ J( π f τ ), 0 where σ is the variance, J () 0 is the zer-th rder Bessel functin f the first kind (crrespnds t Matlab cmmand besselj(*)), and prcesses vary ver time. Let f (in Hz) is a parameter that determines hw fast these gt () = xt () + jyt () (1) be a cmplex Gaussian prcess cnstructed frm x() t and gt () is defined as yt (). The autcrrelatin functin f 1 * Rg( τ ) = E g ( t) g( t+ τ) = σ J0( π fτ ), () which is the same as R( τ ). In the mbile cmmunicatins literature, gt () represents the cmplex gain f a Rayleigh fading channel, with f being the ppler frequency (als knwn as the fade rate). The latter parameter is directly prprtinal t the velcity f the mbile device and the frequency f the transmitted signal. As an example, a velcity f 100 km/hr and a transmitted frequency f 1 GHz yields a ppler f apprximately 100 Hz. In general, as appraches zer, the fading generally becmes mre crrelated and mre smth. Cnversely, as f increases, the fading becmes less crrelated, fluctuatins in the signal ccur mre ften, and the signal appears mre randm. f ENSC-49 - Prject Outline Page f 6

3 In the mbile-cmmunicatins literature, the fading mdel in () is referred t as the Clark s mdel ([3] p. 809). Its Furier transfrm yields the fllwing pwer spectral density (psd) functin σ f f Sg( f) =I { Rg( τ )} = π f 1 ( f / f) 0 f > f. (3) It is interesting t nte that the psd appraches infinity as the frequency appraches the ppler frequency. Very ften during the design and perfrmance evaluatin f a mbile cmmunicatin system, the fading channel in (1) - (3) must first be simulated. Naturally, the simulated cmplex fading gain is a discrete-time prcess. Let g[ n ] be the value f g() t at t = nt, where T is the sampling perid. Then the autcrrelatin functin f the discrete-time randm prcess g[ n ] must be φ ( ) 1 * g[ m] = E g [ n] g[ n+ m] = σ J0 π mft. (4) Questin: What is the Furier transfrm f (3)? 5. Yur Tasks Part A Basic Simulatin The mst cmmn methd used t simulate fading gains is the filtering apprach. Here, a sequence cmplex white Gaussian nise gn [ ] wn [ ] f length K is fed t an FIR filter whse utput has the autcrrelatin prperties in (4). Yu are asked t use this methd t simulate fading gains at each f the fllwing ppler frequencies: 1. f T = 0.05,. f T = 0.03, and ENSC-49 - Prject Outline Page 3 f 6

4 3. f T = These fading rates crrespnds t fast fading scenari and the crrespnding gains can be generated relatively quickly. Fr each fade rate, (a) plt the impulse respnse j hn [ ] and the frequency respnse H( e ω ) f the FIR filter mentined abve, (b) explain why yu chse the particular filter length M and the input sequence length K, (c) prvide a sample plt f gn [ ] vs n (in db scale) and cmment n the signal fluctuatin (bth in terms f the magnitude f the swing and the frequency f a deep fade), (d) prvide a cntur plt f gn [ ] in the cmplex plain, (e) verify the crrectness f the simulated fading gains; see Part C belw fr details, (f) explain hw the cmputatinal cmplexity depends n the filter length M and the number f valid fading gain samples, N (see definitin belw), yu wish t btain. In additin, prvide a general descriptin f the structure f yur fading simulatr and explain why it is preferable ver ther alternatives. Nte Withut lss f generality, the sampling perid T can be set t 1. Similarly, yu can set the variance, σ, t unity. T make the cmparisn f results at different fade rates easier, yu may cnsider fixing the time-scale as well as the db-scale in the time-dmain plt in Part (c). The cmputatinal cmplexity is measured in terms f the number f cmplex multiplicatins. If yur FIR filter has a length M, then the first M-1 utput samples f the filter are invalid because they represent transient. The same is true fr the last M-1 utput samples. The abve suggests that if a sequence f K white Gaussian nise samples is fed t the FIR filter, then nly N = K M + 1 utput samples are valid. ENSC-49 - Prject Outline Page 4 f 6

5 It is recmmended that yu familiarize yurself with the cmmands (and help files assciated with): fft( ), ifft( ), fftshift( ), ifftshift( ), cnv( ) and xcrr( ). Part B - Advanced Simulatin Techniques Successful cmpletin f Part A will enable the student t receive a maximum f B+ in the prject (a lt depends n what happens in the interview t). If yu wish t btain a higher grade, yu may cnsider cmpleting this part. Tw prblems may arise with the filtering methd in Part A: a) an extremely lng simulated fading pattern is desired, i.e. K a very large number, and b) an extremely small fading rate is required. The difficulties lie in the cmputatinal demand required fr filtering a lng input sequence using a filter with many taps. If time permits, yu are encuraged t research in techniques t reslve these prblems. Nte that ur textbk [] des cntain sme pssible slutins. Yu just have t identify and implement them. Test the limits f yur slw-fading simulatr using input sequence length f K , f T Part C Verificatin f the Simulated Fading Gains After generatin f g[ n ], it is necessary t verify that it prtrays the characteristics expected f it. These include, but are nt necessarily limited t, its: mean, variance and pdf (first rder averages), autcrrelatin functin (secnd rder average) and pwer spectrum. ENSC-49 - Prject Outline Page 5 f 6

6 Nte that althugh it is desired that the simulated fading gains match the desired autcrrelatin functin fr all lags, it is mst imprtant that it match the shrt lags (i.e. the first few cycles in the autcrrelatin functin); hence, pay particular attentin t thse. 6. Reprt Yur Reprt shuld cntain the fllwing infrmatin: bjective, intrductin, and backgrund, simulatin mdels and methdlgy, and analyses, if any, the list f results in Part A, if applicable, a cncise descriptin f the advanced technique(s) yu used in Part B and the crrespnding results, and cnclusins. It is imperative that all plts in the reprt have the prper axis labels and scaling. References [1] P. H, ENSC 49 - iscrete Time Systems - Curse Ntes, 006. [] A.. V. Oppenheim and R.W. Schafer, with J.R. Buck, iscrete-time Signal Prcessing. Secnd Editin, Prentice-Hall [3] J. G. Prakis, igital Cmmunicatins, McGraw Hill, 4 th Ed., Aug ENSC-49 - Prject Outline Page 6 f 6