ANSWERS TO SELECTED EXCERCISES, CH 18-27
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1 Eercise 8. The moifie version of pr7_.m for he 5-55 banpass filer using a n orer Buerworh an is oupu graph. To moify he scrip for he 5-5 ban change he values in he buer comman below i.e.: [b a]=buer[5/500 5/500]; % Eercise 8. % Specrum srae=000; p=5; range=p/; =0:/srae:0.6; f=srae*0:range/p; =sin*pi*50*+sin*pi*0*; y=+rannlengh; % sample rae % poins ^n for he FFT % range for he specral plo % ime ais % frequency ais % signal + noise in mv % aiion o implemen n orer Buerworh filer [b a]=buer[5/500 55/500]; y=filerbay; figure ploy ile'time Series' label'ime s' ylabel'ampliue mv' Y=ffyp; Pyy=Y.*conjY/p; figure plofpyy:lenghf; ile'powerspecrum' label'frequency Hz' ylabel'power mv' % plo signal % o a 5 p FFT % Power specrum % Plo resul Noe ha as compare o he unfilere resul in Fig. 7. he power ousie he 5-55Hz ban is now reuce.
2 Eercise 9. A general epression for he relaionship is m n = n m n + n n This can be rewrien: Cancelling n gives m n = n n m n + n n = n n n n { + + n } + n n m n = n { + + n + n } This resul can be wrien as: m n = n n i= i i.e. he epression for he esimae mean of n observaions of i.
3 Eercise 9. The equaion 9.7 o compue he new variance combine form he wo s an s is: s s s Wih a bi of algebra we ge s = s s s + s This will be smaller han eiher s or s To emonsrae his le s ask if he saemen s s s +s < s is rue. Muliply boh sies wih s + s an he quesion is now if: s s < s + s s which mus be rue if s is no zero. The same proceure can be followe o show ha s is smaller han s. Thus for nonzero variance values s an s heir combine effec i.e. s will resul in a reucion of he variance. Noe ha he relaionship beween s an s is he same as for a resuling resisance R of wo parallel resisors R an R. Also in his case aing a resisor R in parallel o any oher resisance R resuls in a reucion of he combine resisance R ha can be compue wih: R R R
4 Eercise 9. See also Secion 9.. Fin he bes fi for y i= a i by minimizing N i= N i= he Error = [y i y i ] = [a i y i ] This will be accomplishe by seing is parial erivaive wih respec o coefficien a o zero: Error a N = [a i y i i ] i= = 0 Wih a bi of algebra we ge: N a i i= N i= i y i = 0 Solving for a gives: N a = i= iy i N i i=
5 Eercise 0. Hin: Review Secion 0. an compare his wih he eamples in Appeni..
6 Eercise 0. We efine he firing hreshol a 0.5: ep CrT = 0.5 where C is he convergence raio r is he rae of firing an T he epoch over which we observe he response. We se T equal o one for convenience. Ne we re-arrange an ake he log: log[ep Cr] = log Now we can eermine he sensor s firing rae require o reach he hreshol a he miral cell level: r 0.7/C A plo of his relaionship for values beween an is shown here. This plo emonsraes ha a higher levels of convergence he miral cells may respon while he aciviy a he sensory level may be harly observable!
7 Eercise. For he power of he scaling signal compue i i Ds For he average ampliue of he scaling signal compue i i Ds For he power of he wavele signal compue i i Dw For he average ampliue of he wavele signal compue 0 i i Dw
8 Eercise. The four coefficiens of he Daubechies wavele applie o n- n- n- n are: In orer o analyze is effec as a filer following he reasoning in Secion. we are inerese in he reverse version of he wavele. The ransfer funcion for he reverse wavele is herefore: z z z z X z Y z H Applying he MATLAB freqz comman for boh he wavele an scaling signals shows ha he wavele is inee a high pass filer an he scaling a low pass filer. Noe ha we use he fliplr comman o reverse he wavele coefficiens clear;close all b=[-sqr/*sqr -+sqr/*sqr +sqr/*sqr --sqr/*sqr]; a=[]; freqzfliplrba bs=[+sqr/*sqr +sqr/*sqr -sqr/*sqr -sqr/*sqr]; figure;freqzfliplrbsa
9 Resul for he wavele a high pass characerisic Resul for he scaling a low pass characerisic
10 Eercise. We moify pr_ by removing he par on iniial coniion an replacing i wih a loop for a range of inpus e.g.: u=0:.: This moificaion resuls in a series of responses ha we can employ o eermine he spikes generae over he simulaion inerval. Ne we coun hese spikes by eermining maima in he poenial v an show he I/O relaionship by ploing he eece spikes agains he inpu. % Eercise. % Eciable Sysems: D on a circle clear; close all; % parameers =.0; I=0; % curren e.g 0 or 0. u=0.9; =0; n=0000; figure;hol; coun=0; for u=0:.:; coun=coun+; incoun=u; phi=pi; for i=:n; % loop o simulae he oupu _phi=u+i+sinphii*; phii+=phii+_phi; % upae phase phi vi=cosphii; % Membrane Poenial=projecion of phi on X-ais i+=i+; plo:nv'k' fcoun=0; for i=:lenghv-; if vi>vi-&vi>vi+; fcoun=fcoun+; ploivi'r+' ile'superimpose response for ifferen inpu. Zoom o check eecion re +' label'time' ylabel'ampliue' figure; ploinf'r' ile'eciable sysem: u+sinphi. I/O relaionship' label'inpu AU' ylabel'oupuspikes/s'
11
12 Eercise. Use he hir orer erm of he Volerra series o compue he response o C : C C C h C H y H C h C H In his case we fin a scaling ha is proporional wih C [In conras he scaling for a linear sysem is equal o C].
13 Eercise.5 % Eercise.5 clearclcclose all se0'efaulaesfonname''courier new' se0'efaulaesfonsize' %% a plo he UIR n = -:8; h = zerossizen; h = ; h = ; h5 = ; h6 = ; h7 = ; h8 = 0.5; h9 = 0.5; figuresemnh'linewih' ais igh bo off gri on label'n' ylabel'value' ile'uir of L componen' %% b plo he analyic secon-orer Volerra kernel [nn nn] = meshgrinn; h = zerossizenn; for i = :numelnn hi = hn==nni.*hn==nni; % given pairs of impulses a imes nn an nn his is wha he oupu % of he squarer shoul look like: muliplicaion of he nn impulse % response by he nn impulse response. We can ONLY compue he % paire-pulse impulse response in his manner if we KNOW our sysem % beforehan. en fig_h = figure; meshnnnnh; ais igh bo off gri on label'n' ylabel'n' zlabel'value' ile'orer Volerra kernel' a_h = gca; %% c se up he cascae an ake ev from superposiion: yy 0.5*:en *:en-5 + :en- + *:en *5:en- + *6:en- + 7: zz yy.^; o zzyy;
14 % remember o have a 6-zero buffer o your signal i.e. ime 0 is locae % a ine 7 % you have he following ifferences in elt o es: elt = 0:7; % oesn' maer which impulse comes firs = -6:9; % now le's go hrough an es hem using eviaion from superposiion as % we o no know eacly he form our sysem akes fig_h = figure; fig_h = copyobja_hfig_h; hol all for i = :numelelt = ==0; % iniial pulse = ==elti; % elaye pulse s = + ; % summe pulses y = o; % response o jus he firs pulse y = o; % response o jus he secon pulse y = o + ; % response o boh pulses % now ake your eviaion from superposiion o ge H H = y - y - y/; % now we evaluae he kernel across all lags for which elt is % rue hus giving us he paire-impulse response o plo along he % iagonal = 0:numelH-; au = <= numelh--elti ; au = au + elti; H = Hnumel-numelau+: en % now we can plo ploauauh'linewih''color'[ 0.5 ] hol all ploauauh'linewih''color'[ 0.5 ] % symmery! hol all ais igh ile'orer Volerra kernel mesh = heoreical pink = ev from superpos' a_h = gca; % resore efauls o avoi freaking ou over all he monospace ypeface se0'efaulaesfonname''arial' se0'efaulaesfonsize'0
15 Eercise 5. Using Equaion 5.7 for G changing he inegraion variable by ropping is subscrip: k k g k G ; ; Using Equaion 5. for G : ; k k k G If we now ge o he epecaion of heir prouc i.e. G G we will ge wo erms: k k an k k Boh erms inclue an o prouc of he inpu an herefore boh erms in he prouc s epecaion will evaluae o zero see Appeni 5.. Hence operaors G an G are orhogonal.
16 Eercise 7. a. See MATLAB scrip below b. The MATLAB scrip below can be aape in a sraighforwar manner o accomplish ask b c. See MATLAB scrip below. See MATLAB scrip below e. The logisic equaion clearly shows nonlinear behavior. In general his behavior canno be capure in he linear auocorrelaion funcion. The reurn plo is a beer ool o epic he relaionship wihin he samples of he signal generae by he logisic equaion. The chaoic regime of he equaion shows srong epenence on he iniial coniion. MATLAB scrip for Eercise 7. a c an % Eercise 7. % Logisic equaion Equaion 7. clear; close all; % parameers n=00; a=; % Also es.5 an.5 See Figure 7. for % he relaionship beween parameer a an behavior; p=e-6; % small perurbaion of he iniial coniion % reference rajecories in mari k=0; for ic=.:0.05:.9; k=k+; k=ic; for i=:n; ki=a*ki-*-ki-; % perurbe rajecories in mari Eercise 7.a k=0; for ic=.:0.05:.9; k=k+; k=ic+p; for i=:n; ki=a*ki-*-ki-; figure;hol; X=-; % compue he ifference rajecories plox' ['perurbaion iniial coniion: '] meanx:
17 ['eviaion afer 00 seps: '] meanx:00 ['raio ifference afer 00 seps / variaion iniial coniion: '] raio=meanx:00/meanx: ile'logisic equaion - sensiiviy o iniial coniion e-6' label'sample #' ylabel'ifference rajecories of a-' % correlaion of one of rajecory Eercise 7.c c=cov:'coef'; figure; ploc'k' ile'auocorrelaion' label'lag' ylabel'correlaion Coef' % reurn plo of one of rajecory Eercise 7. figure;hol; for j=:n ploj-j'k.' ile'reurn plo' label'i' ylabel'i+'
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