Design of Digital Filters


 Gyles McKinney
 1 years ago
 Views:
Transcription
1 Chapter 8 Desig of Digital Filters Cotets Overview Geeral Cosideratios Desig of FIR Filters Desig of liearphase FIR filters usig widows Timedelay i desired respose Sidelobes Hilbert trasform (9 phase shift) Desig of FIR filters by frequecy samplig Optimum equiripple liearphase FIR filters Compariso of FIR methods Desig of IIR Filters from Aalog Filters IIR filter desig by biliear trasformatio Frequecy Trasformatios Desig of Digital Filters Based o LeastSquares Method Summary
2 8. c J. Fessler, May 7, 4, 3:8 (studet versio) So far our treatmet of DSP has focused primarily o the aalysis of discretetime systems. Now we fially have the aalytical tools to begi to desig discretetime systems. All LTI systems ca be thought of as filters, so, at least for LTI systems, to desig a system meas to desig a digital filter. (The desig of oliear or timevaryig systems is geerally more complicated, ad ofte more case specific.) Goal: give desired magitude respose H d () phase respose H d () tolerace specificatios (how far from ideal?), we wat to choose the filter parameters N, M, {a k }, {b k } such that the system fuctio H(z) = where a =, yields a frequecy respose H() H d (). M k= b kz N k= a kz = g i(z z i) j (z p j), Ratioal H(z), so LTI system described by a costatcoefficiet differece equatio, so ca be implemeted with fiite # of adds, multiplies, ad delays. I other words, filter desig meas choosig the umber ad locatios of the zeros ad poles, or equivaletly the umber ad values of the filter coefficiets, ad thus H(z), h[], H(). Overview N =, FIR or allzero. liear passbad phase. N >, IIR. lower sidelobes for same umber of coefficiets. 8. Geeral Cosideratios Ideally would like N ad M or N + M to be as small as possible for miimal computatio / storage. Causal (for ow) Poles iside uit circle for stability 8.. Causality We will focus o desigig causal digital filters, sice those ca be implemeted i real time. Nocausal filter desig (e.g., for offlie applicatios) is much easier ad may of the same priciples apply ayway. A LTI system is causal iff iput/output relatioship: y[] depeds oly o curret ad past iput sigal values. impulse respose: h[] = for < system fuctio: umber of fiite zeros umber of fiite poles frequecy respose: What ca we say about H()? Fact: if h[] is causal, the PaleyWieer Theorem: H() caot be exactly zero over ay bad of frequecies. (Except i the trivial case where h[] =.) Furthermore, H() caot be flat (costat) over ay fiite bad. H R () ad H I () are Hilbert trasform pairs. Therefore they are ot idepedet. Hece magitude ad phase respose are iterdepedet. Thus those ideal filters with fiite bads of zero respose caot be implemeted with a causal filter. Istead, we must desig filters that approximate the desired frequecy respose H d ().
3 c J. Fessler, May 7, 4, 3:8 (studet versio) Characteristics of practical frequecyselective filters No perfectly flat regios Fact: sice causal filters caot have a bad of frequecies with zero respose, or ca they have ay bad of frequecies over which the frequecy respose is a costat. Proof by cotradictio. Suppose H() = c for, with correspodig impulse respose h[]. Now defie a ew filter g[] = h[] c δ[]. The certaily g[] is also causal. But G() = H() c = for, which is impossible if g[] is causal. Causal filters caot have a bad of frequecies with zero respose. Nor ca they have a ifiitely sharp trasitio betwee the passbad ad the stopbad. Nor ca they have perfectly flat passbad. So a typical realistic magitude respose looks like the followig. H() + δ δ Passbad Ripple PSfrag replacemets Passbad Trasitio Bad δ Stopbad Ripple Stopbad p s π Practical filter desig meas choosig δ, δ, c ad s, ad the desigig N, M, {a k }, {b k } to satisfy those requiremets. Ofte oe must iterate. Ofte we plot H() usig db, i.e., log H(), ad express the ripple i db as well. Note that stopbad ripple is ot defied peaktopeak, sice the highest magitude respose i the stopbad is more importat tha how wiggly the respose is i the stopbad. Example applicatio: CD digital crossover Separate woofer ad tweeter sigals digitally iside CD player, rather i aalog at speaker. Why is passbad ripple tolerable i this cotext (distortio)? Room acoustics act as aother filter. Speaker respose is ot perfectly flat. Keep filter ripple below these other effects. Now imagie that we have specified δ, δ, c ad s, ad we wish to desig N, M, {a k }, {b k } to satisfy those requiremets. As metioed above, we have two broad choices: FIR ad IIR. We focus first o FIR.
4 8.4 c J. Fessler, May 7, 4, 3:8 (studet versio) 8. Desig of FIR Filters A FIR filter of legth M is a LTI system with the followig differece equatio : y[] = M k= b k x[ k]. Note that the book chages the role of M here. Earlier, whe discussig ratioal system fuctios, M was the umber of zeros. Now M is the umber of ozero elemets of h[], which correspods to at most M zeros. (More precisely, we assume b M ad b, but some of the coefficiets i betwee could be zero.) The problem: give δ, δ, c, ad s, we wish to choose M ad {b k } M k= to achieve that goal. We focus o liearphase FIR filters, because if liear phase is ot eeded, the IIR is probably preferable ayway. We focus o lowpass filters, sice trasformatios ca be made to form highpass, badpass from lowpass, as discussed previously. Impulse respose Clearly for a FIR filter h[] = { b, =,..., M, otherwise. So rather tha writig everythig i terms of b k s, we write it directly i terms of the impulse respose h[]. I fact, for FIR filter desig we usually desig h[] directly, rather tha startig from a polezero plot. (A exceptio would be otch filters.) 8.. Symmetric ad atisymmetric FIR filters I focus o the symmetric case. System fuctio: H(z) = M = h[] z. How do we make a filter have liear phase? We previously aswered this i the polezero domai. Now we examie it i the time domai. A FIR filter has liear phase if h[] = h[m ], =,,..., M. Example. For M = 5: h[] = {b, b, b, b, b }. Such a FIR filter is called symmetric. Cautio: this is ot eve symmetry though i the sese we discussed previously. This is related, but ot exactly the same as circular symmetry. { } Example. For M = 3 ad h[] = /,, /. Does this filter have liear phase? Is it lowpass or highpass? H() = + e j + e j = e j [ ej + + e j ] = e j ( + cos ), so sice + cos, H() =, which is liear phase. So it works for this particular example, but why does the symmetry coditio esure liear phase i geeral? Cautio. At this poit the book switches from M k= to M k= apparetly. This icosistet with MATLAB, so there are M factors that appear frequetly i the MATLAB calls. I thik that MATLAB is cosistet ad the book makes a udesirable switch of covetio here.
5 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.5 Symmetric real FIR filters, h[] = h[m ], =,..., M, are liear phase. Proof. Suppose M is eve: H(z) = = = = = M = M/ = M/ = M/ = M/ = h[] z h[] z + h[] z + h[] z + M =M/ M/ = M/ = h[] z ] h[] [z + z (M ) M/ = z (M )/ = Thus the frequecy respose is H() = H(z) z=e j M/ = e j(m )/ h[m ] z (M ) h[] z (M ) ] h[] [z (M )/ + z ((M )/ ) = M/ = e j(m )/ = = e j(m )/ H r () M/ [ ( M H r () = h[] cos Phase respose: = (split sum) ( = M ) (symmetry of h[]) (combie) (split phase). ] h[] [e j((m )/ ) + e j((m )/ ) [ ( )] M h[] cos { M H() =, H r () > M + π, H r () <. )]. (Real sice h[] is real.) The case for M odd is similar, ad leads to the same phase respose but with a slightly differet H r ().
6 8.6 c J. Fessler, May 7, 4, 3:8 (studet versio) I fact, the odd M case is eve easier by otig that h[ + (M )/] is a eve fuctio, so its DTFT is real, so the DTFT of h[] is e j(m )/ times a real fuctio. This proof does ot work for M eve sice the (M )/ is ot a iteger so we caot use the shift property. What about polezero plot? (We wat to be able to recogize FIR liearphase filters from polezero plot.) From above, H(z) = z (M )/ M/ = h[] [ z (M )/ + z ((M )/ )] so H ( z ) = z (M )/ M/ = h[] [ z (M )/ + z ((M )/ )] = z M H(z). Thus H(z) = z (M ) H ( z ). So if q is a zero of H(z), the /q is also a zero of H(z). Furthermore, i the usual case where h[] is real, if q is a zero of H(z), the so is q. Example. Here are two polezero plots of such liearphase filters. Im(z) Im(z) r r Re(z) 4 Re(z) What is differece betwee this ad allpass filter? It was poles ad zeros i reciprocal relatioships for allpass filter. Now we kow coditios for FIR filters to be liear phase. How do we desig oe? Delays I cotiuous time, the delay property of the Laplace trasform is x a (t τ) L e sτ X a (s). How do we build a circuit that delays a sigal? Sice e sτ is ot a ratioal fuctio, so it caot be implemeted exactly usig RLC circuits. So eve though a time delay system is a LTI system, we caot build it usig RLC compoets! We ca make a approximatio, e.g., which is ratioal i s, so we ca desig such a RLC circuit. e sτ sτ + s τ Or, we ca use more mechaical approaches to delay like a tape loop. Picture. write, read, erase head. delay / tape velocity. Aother approach is to rely o sigal propagatio time dow a log wire, ad tap ito the wire at various places for various delays. What about i discrete time? We just eed a digital latch or buffer (flip flops) to hold the bits represetig a digital sigal value util the ext time poit.
7 c J. Fessler, May 7, 4, 3:8 (studet versio) Desig of liearphase FIR filters usig widows Perhaps the simplest approach to FIR filter desig is to take the ideal impulse respose h d [] ad trucate it, which meas multiplyig it by a rectagular widow, or more geerally, to multiply h d [] by some other widow fuctio, where h d [] = π H d () e j d. π π Typically h d [] will be ocausal or at least ofir. {, c, Example. As show previously, if H d () =, otherwise, We ca create a FIR filter by widowig the ideal respose: ( = rect ) c the h d [] = c π sic ( c π ). h[] = w[] h d [] = where the widow fuctio w[] is ozero oly for =,..., M. What is the effect o the frequecy respose? W() = { hd [] w[], =,..., M, otherwise, M = w[] e j ad by timedomai multiplicatio property of DTFT, aka the widowig theorem: where π deotes πperiodic covolutio. H() = W() π H d () = π π π H d (λ) W( λ) dλ, (8) I words, the ideal frequecy respose H d () is smeared out by the frequecy respose W() of the widow fuctio. What would the frequecy respose of the ideal widow be? W() = π δ() = impulse. Such a widow would cause o smearig of the ideal frequecy respose. However, the correspodig widow fuctio would be w[] =, which is ocausal ad ofir. So i practice we must make tradeoffs. Example. rectagular widow. w[] = {, =,...,, otherwise, So by the shift property of the DTFT: h d [] = sic ( ( ) ) ad h[] = So the resultig frequecy respose is H() = e j [ + 4 π cos()]. Picture This is oly a 3tap desig. Let us geeralize ext. k= with H d () = e j rect ( ) π, i.e., c = π/. { sic ( ( ),, sic π δ( πk), a Dirac ) } where sic ( ) = /π.64.
8 8.8 c J. Fessler, May 7, 4, 3:8 (studet versio) Example. rectagular widow. W() = M = w[] = {, =,..., M, otherwise, e j = = e j(m )/ W r (), where W r () = si(m/) si(/), M, =. I this case the ideal frequecy respose is smeared out by a siclike fuctio, because W r () M sic ( M π ). The fuctio si(m/) M si(/) is available i MATLAB as the diric fuctio: the Dirichlet or periodic sic fuctio. How much is the ideal frequecy respose smeared out? The width of the mai lobe of W() is 4π/M, because the first zeros of W r () are at = ±π/m. As M icreases, width of mai lobe W() decreases, so arrower trasitio bad. Example. Here is the case c = π/4 for various values of M. Rectagular Widow Widow Respose Magitude Respose.8 w[] W r () / M.6.4. H() π π. π π Rectagular Widow Widow Respose Magitude Respose.8 w[] W r () / M.6.4. H() π π. π π How did I make these figures? Usig H = freqz(h,, om) sice b = h for causal FIR filters.
9 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.9 Timedelay i desired respose The term e j(m )/ i W() above comes from the fact that the rectagular widow is ot cetered aroud =, but rather is timeshifted to be cetered aroud = (M )/. This phase term will cause additioal distortio of H d (), uless H d () is also phaseshifted to compesate. For a lowpass filter with cutoff c, widowed by a legthm widow fuctio, the appropriate desired respose is: H d () = { e j(m )/, c,, otherwise, = H d () = {, c,, otherwise, so that This is illustrated below. Note that from (8), H() = π π π H d (λ) W( λ) dλ = π h d [] = ( [ c π sic c M ]). π c e jλ(m )/ j( λ)(m )/ si(( λ) M/) e dλ c si(( λ) /) = e j(m )/ c si(( λ) M/) dλ. π c si(( λ) /) Thus there is a overall delay of (M )/ samples from such a legthm causal FIR filter. Without the phase term, the covolutio itegral is severely affected Audio applicatio, F s = 44kHz ad say M = 45. The delay i samples is (M )/ =. The time delay is M T = /44kHz =.5msec. The speed of soud i air is about 33meters / secod, so 3.3 meters away takes about msec. Thus a.5msec delay is well withi the tolerable rage for audio. Ideal Frequecy Respose.3.5 Ideal Impulse Respose H d () h d [] π π. Magitude Respose.3 Causal FIR Filter H() π π h[] Magitude Respose.3 Causal FIR Filter Shifted by (M )/ H().6 h[] π π.
10 8. c J. Fessler, May 7, 4, 3:8 (studet versio) Sidelobes The rectagular widow has high sidelobes i W(), ad the sidelobe amplitude is relatively uaffected by M. Sidelobes cause large passbad ripple, related to Gibbs pheomea. Sidelobes caused by abrupt discotiuity at edge of widow. Solutio: use some other widow fuctio. Examples: Bartlett, Blackma, Haig, Hammig, Kaiser, Laczos, Tukey. MATLAB s widow fuctio has 6 choices! All have lower sidelobes tha rectagular, hece less passbad ripple. Tradeoff? Wider trasitio bad for the same M compared to rectagular widow. Example. Haig widow: w[] = ( ) π cos. M What is the frequecy respose W()? Somewhat messy. How do we plot W()? W = freqz(w,, om) which i tur uses the DFT/FFT with N M via zero paddig. Cautio: this is the ha fuctio i MATLAB, ot the haig fuctio! Rectagular Widow 3 Norm. Widow Respose w[] M=9 W() / W() (db) π π Haig Widow 3 Norm. Widow Respose w[] M=9 W() / W() (db) π π Hammig Widow 3 Norm. Widow Respose w[] M=7 W() / W() (db) π π As M icreases, mai lobe width decreases. As M icreases, some of the sidelobe amplitudes decrease, but peak sidelobe amplitude remais approximately costat. Agai, the price is eed larger M tha for rectagular widow for same trasitio width. The various widows tradeoff mai lobe width with peak sidelobe amplitude. Haig mai lobe width is 8π/M, but peak sidelobe is 3dB compared with 3dB for rectagular.
11 c J. Fessler, May 7, 4, 3:8 (studet versio) 8. Example. Suppose we wat a FIR filter desig of a lowpass filter with H d () = widow with M = 5. The the desired impulse respose is: h d [] = 4 sic ( 4 ( (M )/) ). {, < π/4, otherwise, desiged usig the Haig I MATLAB, here is how we compute ad display the impulse respose ad the frequecy respose. % fig_wi_example.m M = 5; = [:(M)]; om = lispace(pi, pi, ); % for displayig frequecy respose oc = pi/4; % cutoff frequecy % desired impulse respose: hd = ilie( oc/pi * sic(oc/pi*((m)/)),, oc, M ); Hd = ilie( *(abs(om) < oc), om, oc ); h = hd(, oc, M).* ha(m) ; % Haig widow applied to ideal impulse resp. clf, subplot(3) stem(, h, filled ), stem_fix axis([ M .,.3]), xlabel, ylabel h[] title(spritf( Haig Lowpass, M=%d, M)) subplot(3) H = freqz(h,, om); plot(om, *log(abs(h)), , om, *log(max(hd(om,oc),eps)),  ) xlabel \omega, ylabel H(\omega) (db) title Magitude Respose axisy(8, ), xaxis_pi p p/4 p % savefig fig_wi_example.3 Haig Lowpass, M=5 Magitude Respose.5. h[].5..5 H() (db) π π/4 π If we are usatisfied with the width of the trasitio bad, or the sidelobe amplitude, or the passbad ripple, the what could we do? Icrease M, try other widow fuctios, ad/or try other filter desig methods. Filter desig by widowig i MATLAB The fir commad i MATLAB is its tool for widowbased FIR filter desig. I the precedig example, we could have simply typed h = fir(4,.5, ha(5), oscale ) to get exactly the same desig.
12 8. c J. Fessler, May 7, 4, 3:8 (studet versio) Example. Digital phaser or flage. Time varyig pole locatios i cascade of st ad dorder allpass filters, Sum output of allpass cascade with origial sigal, creatig timevaryig otches FIR differetiator H d () = j skim 8..6 Hilbert trasform (9 phase shift) H d () = j sg() skip 8..3 Desig of FIR filters by frequecy samplig For the frequecy samplig method of FIR filter desig, to desig a Mpoit FIR filter we specify the desired frequecy respose at a set of equallyspaced frequecy locatios: H d (), k =,..., M. = π M k I other words, we provide equally spaced samples over [, π). Picture Recall from the DTFT formula that if h d [] is ozero oly for =,..., M, the Thus, at the give frequecy locatios, we have H d () = M = h d [] e j. ( ) M π H d M k = h d [] e j π M k, k =,..., M. = This is the formula for the Mpoit DFT discussed i Ch. 5 (ad i EECS 6). So we ca determie h d [] from { ( H π d M k)} M by usig the iverse DFT formula (or h = ifft(h) i MATLAB): k= h d [] = M M k= ( ) π H d M k e j π M k. This will be the impulse respose of the FIR filter as desiged by the frequecy samplig method. If we wat( h d [] to be real, the H d () must be Hermitia symmetric, i.e., Hd () = H d( ) = H d (π ). So if we specify H π d M k) to be some value, we kow that Hd( π M k) ( = H d π π M k) ( = H π d M (M k)) (, so H π d M (M k)) is also specified. Thus, software such as MATLAB s fir commad oly requires ( H d () oly o the iterval [, π]. If h d [] is to be real, the it also follows that H d (π) = H π ) M d M must be real valued whe M is eve. If you choose H d () to be liear phase, the the desiged h[] will be liear phase. But you are ot required to choose H d () to be liear phase! The book discusses may further details, but the above big picture is sufficiet for this class.
13 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.3 Example. The treble boost revisited. Double the amplitude of all frequecy compoets above c = π/. % fig_freq_sample.m Hd = ilie( exp(i*om*(m)/).* ( + (abs(om) > pi/)), om, M ); M = 9; ok = [:(M)]/M * *pi; Hk = Hd(mod(ok+pi,*pi)pi, M); % trick: [pi,pi] specificatio of H(\omega) h = ifft(hk); h = reale(h, war ); % h = fir(m, [.5.5 ], [ ], boxcar(m) ); om = lispace(pi,pi,); clf, pl = 3; subplot(pl+), plot(om, abs(hd(om,m))) hold o, stem(ok(ok >= ), abs(hk(ok >= )), filled ), stem_fix, hold off xlabel \omega, ylabel H_d(\omega) axisy([.5]), xaxis_pi p p/ p subplot(pl+), stem(:(m), h, filled ), stem_fix, title(spritf( M=%d,M)) xlabel, ylabel h[], axis([ M ]) H = freqz(h,, om); subplot(pl+3), plot(om, abs(h), , om, abs(hd(om,m)), , ok, abs(hk), o ) xlabel \omega, ylabel H(\omega), axisy([.5]), xaxis_pi p p/ p % savefig fig_freq_sample.5.5 M=9.5 H d ().5 h[].5 H() π π/ π π π/ π I MATLAB, use h = fir(m, [.5.5 ], [ ], boxcar(m) ) to desig (almost) the above filter.
14 8.4 c J. Fessler, May 7, 4, 3:8 (studet versio) 8..4 Optimum equiripple liearphase FIR filters The widow method has a mior disadvatage, that it is difficult to precisely specify p ad s, sice these two result from the smearig. All we really specify is c, the cutoff. A ideal liearphase desig procedure would be as follows. Specify p, s, δ, δ, ad ru a algorithm that returs the miimum M that achieves that desig goal, as well as the impulse respose h[], =,..., M, where, for liear phase, h[] = h[m ]. To my kowledge, there is o such procedure that is guarateed to do this perfectly. However, we ca come close usig the followig iterative procedure. Choose M, ad fid the liearphase h[] whose frequecy respose is as close to H d () as possible. If it is ot close eough, the icrease M ad repeat. How ca we measure closeess of two frequecy respose fuctios? Pictures of H d () ad H(). Possible optios iclude the followig. π π π H d() H() d, average absolute error π π π H d() H() d, average squared error π π π H d() H() W() d, average weighted squared error [ ] π /p, π H d() H() p W() d weighted Lp error, p π max H d () H(), maximum error (p = ) max W(H d () H()), maximum weighted error (p = ) I this sectio, we focus o the last choice, the maximum weighted error betwee the desired respose ad the actual frequecy respose of a FIR filter. We wat to fid the FIR filter that miimizes this error. How to fid this? Not by brute force search or trial ad error, but by aalysis! If W() the Cosider the case of a lowpass filter. Note that H d () = E() = W(H d () H()) = W() H d () H(). { e j(m )/, c,, > c = e j(m )/ H dr () where H dr () = Also recall that for M eve ad h[] liear phase ad symmetric, So H r () is the sum of M/ cosies. The error ca be simplified as follows: H() = e j(m )/ H r (), where H r () = M/ = {, c,, > c. ( ( )) M h[] cos. E() = W() H d () H() = W() e j(m )/ H dr () e j(m )/ H r () = W() H dr () H r (). Thus we oly eed to cosider the real parts of the desired vs actual frequecy respose.
15 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.5 The logical approach to specifyig the error weightig fuctio W() is as follows: /δ, c pass bad W() =, c < < s trasitio bad /δ, s < < π stop bad. However, the effect of the weightig will be the same if we multiply W() by a costat, such as δ. So the followig weightig fuctio has the same effect: δ /δ, c W() =, c < < s, s < < π. A small δ value meas we wat very low sidelobes, ad are willig to sacrifice uiformity over the passbad, so less weight is give to the errors i the passbad. As a desiger you choose H dr (), the filter legth M, the ratio δ /δ, ad the passbad ad stopbads, i.e., c ad s. The procedure fids the impulse respose that miimizes the maximum weighted error betwee the desired respose ad the actual respose. Mathematically, the mimax or Chebyshev desig uses the followig criterio: mi {h[]} M/ = max E(). Usig the alteratio theorem, from the theory of Chebyshev approximatio, Parks ad McClella i 97 showed that a ecessary ad sufficiet coditio for a Mtap filter to be optimal (i the maximum weighted error sese) is that the error E() must reach its maximum at least M/ + poit over the itervals where W(). This theorem guaratees that there is a uique optimal filter. It also tells us that the error will alterate back ad forth, i.e., there will be ripples i passbad ad stopbad. The resultig filters are called equiripple because all ripples i passbad have the same peaktopeak amplitude, ad likewise for the stop bad. The actual procedure for fidig the best filter is iterative, ad it called the Remez exchage algorithm. It is implemeted by the remez fuctio i Matlab. The algorithm first guesses where the extremal frequecies are, ad the computes H r () from that, the fids ew estimates of the extremal frequecies ad iterates. Icreasig M reduces passbad ripple ad icreases stopbad atteuatio MATLAB s remezord commad gives approximatio to required M, based o (8..95): ˆM = D (δ, δ ) f f(δ, δ ) f +, where f = ( s p )/(π) ad D ad f(δ, δ ) are defied i text ( ). Aother formula is also give i text: log 3 (δ ˆM = δ ) f As trasitio bad f decreases, M icreases As ripples δ j decrease, M icreases This ca be explored graphically usig MATLAB s fdatool or filtdes.
16 8.6 c J. Fessler, May 7, 4, 3:8 (studet versio) Example. digital crossover for audio F s = 44kHz F pass = 4.4kHz p = πf pass /F s =.π F stop = 6.6kHz s = πf stop /F s =.3π 3 bads: pass, trasitio, stop M = 6 for ow; revisit shortly. % fig_remez_example.m M = 6; f = [..3 ]; Hdr = [ ]; W = [ ]; h = remez(m, f, Hdr, W); stem(:m, h, filled ) freqz(h,, ) Equiripple (zoom).5. M=6 h[].5..5 H() (db) H() π π.4. π.4 π Haig widow (zoom).5. h[].5..5 H() (db) H() π π.4. π.4 π
17 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.7 Further illustratios of the tradeoffs. The error has to go somewhere: passbad(s), stopbad(s), ad/or trasitio bad(s). 3 W() 3 W() 3 W() /π.4.5 /π.4.5 /π.374 H(), M=3.6 H(), M=3.78 H(), M= /π /π /π 3 W() 3 W() 3 W() /π.4.5 /π.4.55 /π H(), M=3 H(), M=7 H(), M= /π /π /π 8..7 Compariso of FIR methods skim
18 8.8 c J. Fessler, May 7, 4, 3:8 (studet versio) 8.3 Desig of IIR Filters from Aalog Filters Why IIR? With IIR desigs we ca get the same approximatio accuracy (of the magitude respose) as FIR but with a lower order filter. The tradeoff is oliear phase. Aalog filter desig is a mature field. There are well kow methods for selectig RLC combiatios to approximate some desired frequecy respose H d (F ). Geerally, the more (passive) compoets used, the closer oe ca approximate H d (F ), (to withi compoet toleraces). Essetially all aalog filters are IIR, sice the solutios to liear differetial equatios ivolve ifiiteduratio terms of the form t m e λt u(t). Oe way to desig IIR digital filters is to piggyback o this wealth of desig experiece for aalog filters. What do aalog filters look like? Ay RLC etwork is describe by a liear costat coefficiet differetial equatio of the form N k= α k d k dt k y a(t) = M k= β k d k dt k x a(t) with correspodig system fuctio (Laplace trasform of the impulse respose) M k= H a (s) = β ks k N k= α ks. k Each combiatio of N, M, {α k }, {β k } correspods to some arragemet of RLCs. Those implemetatio details are uimportat to us here. The frequecy respose of a geeral aalog filter is H a (F ) = H a (s). s=jπf Overview Desig N, M, {α k }, {β k } usig existig methods. Map from s plae to zplae somehow to get a k s ad b k s, i.e., a ratioal system fuctio correspodig to a discretetime liear costat coefficiet differece equatio. Ca we achieve liear phase with a IIR filter? We should be able to use our aalysis tools to aswer this. Recall that liear phase implies that H(z) = z N H ( z ), so if z is a pole of the system fuctio H(z), the z is also a pole. So ay fiite poles (i.e., other tha or ) would lead to istability. There are o causal stable IIR filters with liear phase. Thus we desig for the magitude respose, ad see what phase respose we get IIR by approximatio of derivatives skim s = z T 8.3. IIR by impulse ivariace skim h[] = h a (T ) z = e st
19 c J. Fessler, May 7, 4, 3:8 (studet versio) IIR filter desig by biliear trasformatio Suppose we have used existig aalog filter desig methods to desig a IIR aalog filter with system fuctio M k= H a (s) = β ks k N k= α ks. k For a samplig period T s, we ow make the biliear trasformatio s = T s z + z. This trasformatio ca be motivated by the trapezoidal formula for umerical itegratio. See text for the derivatio. Defie the discretetime system fuctio H(z) = H a (s) s= z. T s + z This trasformatio yields a ratioal system fuctio, i.e., a ratio of polyomials i z. This H(z) is a system fuctio whose frequecy respose is related to the frequecy respose of the aalog IIR filter. Example. Cosider a storder aalog filter with a sigle pole at s = α Picture where α >, with system fuctio H a (s) = α + s. How would you build this? Usig the followig RL voltage divider, where V out (s) = i(t) R R+sL V i(s) = H a (s) = R/L R/L+s. vi(t) + R vout(t) L Applyig the biliear trasformatio to the above (Laplace domai) system fuctio yields: + z H(z) = H a (s) s= z = = T s + z α( + z ) + = ( z ) T s = α + z T s + z α + /T s + z /T s α /T s + α z = α + /T s + z (α + /T s ) + (α /T s )z + z [ pz = α + /T s p + + /p ] pz, where p = αt ) s (, ). What is h[]? h[] = + αt α+/t s ( p δ[] +( + /p)p u[]. s Im(z) p Re(z) Where did zero at z = come from?
20 8. c J. Fessler, May 7, 4, 3:8 (studet versio) Solvig biliear trasformatio for z i terms of s yields: z = + st s st s, so zero at s = maps to z =. Note also that pole at s = α maps to z = p. I geeral, the (fiite) poles ad zeros follow the mappig. Poles (or zeros) at s = map to z =. Real poles ad zeros remai real, complexcojugate pairs remai pairs. What type of filters are H a (s) ad H(z)? Both are poor lowpass filters. This illustrates the method, but ot the power! The utility is for more sophisticated aalog IIR desigs. Picture : more sdomai poles ad correspodig zdomai poles The biliear trasformatio More geerally, if the system fuctio H a (s) is ratioal, i.e., H a (s) = g s p T s z + z p =... = T s so we will have oe root at z = +pts pt s ad aother oe at z =. See plot o ext page. real(s) > maps to z outside uit circle real(s) < maps to z iside uit circle s = jπf maps to z o uit circle i(s si) j (s pj), the for each term of the form s p we have ( pt s z = + st s st s, s = T s z + z. z = + jπf T s = r ejφ jπf T s r e jφ = ejφ, where r = + jπf T s ad φ = ( + jπf T s ) = arcta ( πf T s ). ( ) πf Thus z = e j, where = φ, so = ta Ts. Coversely, if z = e j the ) z +pt s pt s, z + s = e j T s + e j = e j/ e j/ T s e j/ + e = j si / j/ T s cos / = j ) ta( = F = ) ta(. T s πt s This is a oliear relatioship betwee the aalog frequecy ad the digital frequecy. It is called frequecy warpig. See plot o ext page. Because of this oliearity, a typical desig procedure goes as follows. Determie desired frequecy respose i terms of aalog frequecies, e.g., Hz. Covert the desired frequecies ito digital frequecies usig = πf/f s, yieldig p, s, etc. Map those digital frequecies ito aalog frequecies F = πt s ta(/). We ca use ay coveiet T s for this, e.g., T s =. Desig a aalog filter for those frequecies. Trasform aalog filter ito digital filter usig the biliear trasformatio with that same T s. This is all built ito commads such as MATLAB s cheby routie. Use MATLAB s filtdemo to experimet with various filter types. The Chebyshev Type I filters are allpole aalog filters with equiripple behavior i the passbad ad mootoic i the stopbad. { ( H a (F ) = + ε TN (F/F where T N (x) = cos N cos x ), x pass) cosh(n cosh x), x > is the Nth order Chebyshev polyomial. There is ripple i the passbad of amplitude /( + ε ) that is usercotrolled.
CooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationTagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper PartA
Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper PartA UitI. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationThe Binomial Multi Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi Sectio Trasforer Recall that a ultisectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationDescriptive statistics deals with the description or simple analysis of population or sample data.
Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationSolving DivideandConquer Recurrences
Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationAPPLICATION NOTE 30 DFT or FFT? A Comparison of Fourier Transform Techniques
APPLICATION NOTE 30 DFT or FFT? A Compariso of Fourier Trasform Techiques This applicatio ote ivestigates differeces i performace betwee the DFT (Discrete Fourier Trasform) ad the FFT(Fast Fourier Trasform)
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More information
Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationNEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,
NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationBasic Measurement Issues. Sampling Theory and AnalogtoDigital Conversion
Theory ad AalogtoDigital Coversio Itroductio/Defiitios Aalogtodigital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationDomain 1  Describe Cisco VoIP Implementations
Maual ONT (6428) 18004186789 Domai 1  Describe Cisco VoIP Implemetatios Advatages of VoIP Over Traditioal Switches Voice over IP etworks have may advatages over traditioal circuit switched voice etworks.
More information