Chapter 6. The Discrete Fourier Transform and The Fast Fourier Transform
|
|
- Joella Rice
- 5 years ago
- Views:
Transcription
1 Pusan ational Univrsity Chaptr 6. Th Discrt Fourir Transform and Th Fast Fourir Transform 6. Introduction Frquncy rsponss of discrt linar tim invariant systms ar rprsntd by Fourir transform or z-transforms. In this cas, frquncy rspons bcoms continuous function. For us in this cas, thr is an altrnativ rprsntation calld th discrt Fourir transform(dft) which is discrt in frquncy. Th DFT can b calculatd by computr. Th DFT of a circular convolution of two squncs is th product of thir DFT s In ral practical application, th FFT is usd. Digital Signal Procssing /9
2 Pusan ational Univrsity 6. Th Discrt Fourir Transform jt X ( j) x( t) dt For a squnc x n = x(nt) dfind for all intgr n, th z-transform is X ( z) x( n) z n (6.) On th unit circl, z = jωt, and z-transform bcoms th Fourir transform of th squnc, jnt X ( jt ) x( n) (6.) n For a finit-lngth squnc x n =,,,, (6.) can b writtn n (6.3) Both (6.) and (6.3) ar continuous functions of frquncy ω. To obtain discrt function of frquncy suitabl for computation, w rplac ω by Ω. Th DFT is dfind as follows: n ( ) X j T x( n) jn T n jnt X ( ) x( n) DFT x( n) (6.), (6.5) Digital Signal Procssing /9
3 Pusan ational Univrsity Th Ω is frquncy incrmnt in radians pr scond. n How do w choos th frquncy incrmnt Ω? Considr x n =,,,, as on priod of a priodic squnc x n = x nt. Th priod is T scond, which mans that th fundamntal frquncy, orfrquncy rsolution is givn by s (6.6) For brvity, and without loss of gnrality, w can lt T= so that (6.) and (6.6) bcom jn X ( ) x( n), and (6.7) and (6.8) n W shall prov that only distinct valus of X() can b computd. Th modulo function y modulo is dfind as whr r is th largst intgr th magnitud of which dos not xcd [y/]. T (( y)) y r (6.9) (( y)) y r y, if y,, if y r in gnral ( ) jn T X x( n) (6.) Digital Signal Procssing 3/9
4 Pusan ational Univrsity From th proprtis of th modulo function, th xponnt on th right hand sid of (6.7) can b writtn as jn jn (( )) r jn(( )) jnr jn(( )) ( X ( ) X (( )) (6.) bcaus K < in gnral, thr ar only distinct valus of X(). Thrfor, (6.7) may b writtn jn X ( ) x( n),,,,, (6.3) n This is a squnc of numbrs. From (6.) X ( ) X ( r) (6.) whr r is an intgr ranging from minus infinity to plus infinity, so th DFT can also considrd a priodic squnc of numbrs with priod. Th z-transform of a finit squnc x n =,,,, at points vnly spacd at ω = Ω on th unit circl in th z-plan is DFT x ( n ) X ( z jn ) x j ( n ) (6.5) z n () = r Digital Signal Procssing /9
5 Pusan ational Univrsity Ex 6.) Find th DFT of th finit squnc n sol) x( n) a, n,,,, a n jn j n X( ) a ( a ) n n j j j ( a ) a j j a a ( a cos jasin ) magnitud ; a M( ) ( a a cos ) phas ; asin ( ) tan a cos Th plot ovr on priod for =9, a=.5 => Digital Signal Procssing 5/9
6 Pusan ational Univrsity 6.3 Th Invrs Discrt Fourir Transform(IDFT) Th invrs discrt Fourir transform(idft) is usd to map th DFT bac into th squnc in tim domain. Considr first th DFT of a complx sinusoidal squnc of frquncy qω: DFT of x( n) jq n ~ X ( ) jn( q ) n j( q ) j( q ) ⅰ) If q is intgr = q = q r j( q ) j( qqr ), ~ ( ) n () n X ⅱ) If q is intgr ( q ), thn ~ j( q ) j( q ) X, ( ) Thrfor th DFT of ~ X ( ) x( n) jq n, if (( q)), if (( q)) (( q)), Digital Signal Procssing 6/9
7 If Pusan ational Univrsity Thorm 6. (Invrs DFT) n jn DFT x( n) X ( ) x( n), whr,,,, (6.) thn thr xists an invrs DFT such as ( ) ( ) jm DFT X x m X ( ), m,,, jm jn jm j( mn) proof jm x( m) X ( ) For convninc, th DFT pair ar oftn writtn DFT: IDFT: whr ) W X ( ) x( n) x( n) n n j( mn) X ( ) x( n) W n x( n) X ( ) W j n x(( m)) x( m) ( m n, ),,,, n, n,,, ; frquncy domain ; tim domain (( m)), n Digital Signal Procssing 7/9
8 Pusan ational Univrsity 6.. Linarity proof) x ( n) a x ( n) a x ( n) X ( ) a X ( ) a X ( ) 3 3 X ( ) x ( n) 3 3 n jn n a x ( n) a x ( n) jn jn ( ) ( ) n n a X ( ) a X ( ) jn a x n a x n Digital Signal Procssing 8/9
9 Pusan ational Univrsity 6.. Symmtry If x n, n =,,,, is a ral squnc with DFT X(), thn proof) Bcaus X X * ( ) ( ) X ( ) x( n) n jn( ) jn jn X ( ) x( n) x( n) n n n jn x( n) jn * jn * ( ) ( ) ( ) X x n X n Digital Signal Procssing 9/9
10 Pusan ational Univrsity W conclud from this proprty that Ral part : R{X()}=R{X(-)} Imaginary part : Im{X()}=-Im{X(-)} 3 Magnitud : M()=M(-) Phas : ( ) ( ) It is vidnt from th forgoing that obtain th DFT of a ral - point squnc, it suffics to comput ( + )/ or (/ + ) frquncy sampls for odd or vn, rspctivly. Th finit squnc x(n) can b considrd to b on priod of a priodic function with priod i) Evn function cas x x vn function M M n n ⅱ) Odd function cas x x odd function l n n Digital Signal Procssing /9
11 Pusan ational Univrsity 6..3 Circular Shifting If x n, n =,,,, is a finit squnc, thn th circular shift of this squnc by n sampls, say has maning only if w considr x n as on priod of priodic squnc x n = x n. W shift x n through n sampls and rcovr th shiftd finit squnc x s (n) by multiplying x (n n by th rctangular window squnc, n R ( n), othrwis if x ( n) x(( n n )) R ( n) s whr, X ~ ( ) DFT x ( n ) jn DFT x ( n) X ( ) R ( ) R s, ( ), othrwis ~ Digital Signal Procssing /9
12 proof) whr Pusan ational Univrsity DFT x( n n ) x( n n ) n jn jn j( nn) ( ) n x n n jn ( ) ( ) n jm DFT x n n x m mn whr, m=r- (considr priodic function) n n jm jm jm x( m) x( m) x( m) mn mn m jm j( r ) m n n x( m) x( r ) mn r n Digital Signal Procssing /9
13 Pusan ational Univrsity Bcaus is priodic x( r ) x( r) j( r ) jr j Also, ( ) jm jr x( m) x( r) mn r n Rplac th indx r by m n n jm jm jm x( m) x( m) x( m) mn m n m m jm x( m) X ( ) Thrfor n jn jm jn DFT x( n n ) x( m) X ( ) s xr () or X X jn ( ) ( ) mn whr X s is th priodic xtnsion of X s. ow X X R X R jn ( ) ( ) ( ) ( ) ( ) s s x( n) jq n n n jm jm jm x( m) x( m) x( m) mn mn m Digital Signal Procssing 3/9
14 Pusan ational Univrsity 6.. Circular Convolution Instad of a linar convolution of two squncs, w hav a circular convolution. A circular convolution of two -point squncs x(n) and h n is dfind as y( n) x( m) h(( n m)) R ( n) m x( n) nh(( n)) Ex 6.) Find, graphically, th circular convolutions of th two finit squncs, n, xn ( ), othrwis n a, n,,,, a.9, 5 hn ( ), othrwis Digital Signal Procssing /9
15 Pusan ational Univrsity y( n) x( m) h(( n m)) R ( n) m y() x() h() x() h( ) x() h( )..5.9 y() x() h() x() h() x() h( ).. y() x() h() x() h() x() h().8.8 Digital Signal Procssing 5/9
16 Pusan ational Univrsity Thorm 6.( Convolution Thorm for DFTs) If Y ( ) H ( ) X ( ) y( n) x( m) h(( n m)) R ( n) m h( m) x(( n m)) R ( n) m proof) W hav X ( ) x( m) m H ( ) h( l) l jm jl 이고 y(( n)) DFT Y ( ) Y ( ) jn Digital Signal Procssing 6/9
17 Pusan ational Univrsity Thn th priodic squncs y(( n)) DFT Y ( ) Y( ) m X ( ) H ( ) jn jn x( m) h( l) m l j( nml) x( m) h( l) jm jl jn m l x( m) h(( n m)) j( nml) (( nm)), l It follows that th -point squnc y( n) y(( n)) R ( n) m x( m) h(( n m)) R ( n) Digital Signal Procssing 7/9
18 Pusan ational Univrsity 6.7 Th Fast Fourir Transform Th amount of computation of th DFT of complx squnc f(nt) jnt F( ) f ( nt),,,, n (6.65) Whn a complx multiplication is carrid out, it rquirs four ral multiplication and two ral addition. A complx addition rquirs two ral additions. On opration: a complx multiplication plus a complx addition(that is, four ral multiplication and four ral additions) ( AB ( a jb)( c jd) ( ac bd) j( bc ad), Th calculation of nds oprations. If w us FFT and is a powr of, FFT mas log oprations Ex) = =, A B ( a c) j( b d) ) F ( ) FFT brings a saving of 99% oprations. DFT: 6 opration, FFT : opration Digital Signal Procssing 8/9
19 Pusan ational Univrsity To calculat FFT, data numbr should b a powr of. jn jnt Two approachs for FFT calculation X ( ) x( n) x( n) n n Dcimation in Tim : th squnc for DFT is succssivly dividd up into smallr squncs, and th DFTs of ths subsquncs ar combind in a crtain pattrn to yild th rquird DFT of th ntir squncs with th much fwr oprations. To rduc oprations, w us th symmtry and priodic charactristic of W n j( ) n Dcimation in Frquncy : th frquncy sampls of th DFT ar dcomposd into smallr and smallr subsquncs in a similar mannr. Th DFT of -point signal f(nt) n F fnw,,,, n whr W = jπ/ = jω on = b. Ex) If w considr th DFT of a squnc of /, thn Wwould b rplacd by W, and th summation limit would b /-. n / / j( ) n ( j( ) n) n W W T Digital Signal Procssing 9/9
20 Pusan ational Univrsity 6.7. Dcimation in Tim Th squnc f n is first dividd into two shortr intrwovn squncs, g n and h n g f, n,,, ( vn numbrd) n n h f, n,,, ( odd numbrd) n n F( ) f W,,,, n n 8points DFT f n is dividd as g n and h n points DFT n n n G g W n n( ),,,, H h W n n( ),,,, Digital Signal Procssing /9
21 Pusan ational Univrsity W can xprss th DFT of th ntir squnc F in trms of G and H n n (n) n n n n n F f W g W h W n n gnw W hnw n n G W H,,,, n (n) n n n ( F f W f W, j j / W W ) G and H hav priod /, hnc w may writ as follows: G W H, F G W H, n / ( / ) (whr, W ) / / / G g W g W W g W G G n ( ' ) ( ) n n ( n ) n n n n n n G G, as sam way, H H Digital Signal Procssing /9
22 Pusan ational Univrsity As th forgoing rduction is carrid out, G and H rquir on th ordr of (/) oprations ach, and oprations ar rquird for multiplying H by W and adding W H to G. This givs a total of approximatly + (/) oprations compard with of oprations for dirct valuation of th DFT. ( ) F G W H, G W H, G G G G W H W H W H 3 3W H3 G G G G W H W H 5 W H 6 W H Digital Signal Procssing /9
23 Pusan ational Univrsity To procd with this mthod, th / point squncs g n and h n ar both dcomposd into two / point squncs by taing th vn- and odd- numbrd sampls, as was don with f n Thus, p g, n,,, n n q n gn n,,,, r h, n,,, n n s n hn n,,,, By th sam rasoning as for th first stag, it follows that n n (n) n n n n n G g W p W q W P W Q, P W Q, Digital Signal Procssing 3/9
24 Pusan ational Univrsity whr P, Q, R, and S ar th / DFTs of p n, q n, r n, and s n, rspctivly. Th total numbr of oprations for F is now rducd to approximatly whr H R W S,, R W S (/) + + =(/) + n P p ( W ),, n n n n( ) n P p W p pw n n( ) n P p W p p W Digital Signal Procssing /9
25 Pusan ational Univrsity Total oprations G P W Q, P W Q, ( / ) H R W S,, R W S P Q W p pw p pw P QW G G W H W H G W H r r rw rw R R W S W S G W H R R W S W S 6 G G W H 6 W H Digital Signal Procssing 5/9
26 Pusan ational Univrsity P p n n W n f fw P W Q ( ),, G P W Q,, P W Q G W H f f W G P W Q G W H f 6 f W G P W Q G W H f 6 f W P W Q 6 G W H f 5 f W R W S G W H f 5 f W R W S G W H 5 f 3 7 f W R W S G W H 6 f 3 7 f W R W S 6 G W H Digital Signal Procssing 6/9
27 Pusan ational Univrsity Th procss nds whn th original -point squnc f n is dividd up into -point squncs with -point DFT computd for ach. Ths ar combind in an appropriat mannr to yild F. FFT Whn th DFT computation of th powr-of--lngth squnc has bn compltly rducd to a combination of complx multiplications and additions, th ultimat saving in computation is achivd approximatly to log oprations v v log v log For th 8-point xampl, 3 stps ar rquird to rach this stag, and th oprations ar combind as shown in Fig. 6.. Th rsulting numbr opration is about log = as compard with th = Using MATLAB X=fft(x, ) Digital Signal Procssing 7/9
28 Pusan ational Univrsity From Fig. 6., it is vidnt that th output squnc is in natural ordr F, F, F, F 3, F, F 5, F 6, F 7 -whras th input squnc is not. Th input squnc is f, f, f, f 6, f, f 5, f 3, f 7 is in bit rvrsd ordr. Using MATLAB X=fft(x, ) Digital Signal Procssing 8/9
29 Pusan ational Univrsity Exampl of FFT using MATLAB Fs = ; % Sampling frquncy T = /Fs; % Sampl tim L = ; % Lngth of signal t = (:L-)*T; % Tim vctor % Sum of a 5 Hz sinusoid and a Hz sinusoid x =.7*sin(*pi*5*t) + sin(*pi**t); y = x + *randn(siz(t)); % Sinusoids plus nois plot(fs*t(:5),y(:5)) titl('signal Corruptd with Zro-Man Random ois') xlabl('tim (millisconds)') Digital Signal Procssing 9/9
30 Pusan ational Univrsity FFT = ^nxtpow(l); % xt powr of from lngth of y Y = fft(y,fft)/l; f = Fs/*linspac(,,FFT/+); % Plot singl-sidd amplitud spctrum. plot(f,*abs(y(:fft/+))) titl('singl-sidd Amplitud Spctrum of y(t)') xlabl('frquncy (Hz)') ylabl(' Y(f) ') Digital Signal Procssing 3/9
31 Pusan ational Univrsity 6.7. Dcimation in Frquncy To illustrat th dcimation frquncy, th squnc is dividd into two squncs g n and h n, whr g n contains th first / sampls of f n, and h n contains th rmaining sampls. Thus g f, n,,, / n n h f, n,,, / n n / Thn th -point DFT of f n is / / / n n n n ( n / ) n n n n n n n n / n n F f W f W f W g W h W / n gn ( ) h nw,,,, n / ( j / ) / j ( W ) W / j Digital Signal Procssing 3/9
32 Pusan ational Univrsity ow divid th squnc F into th vn-numbrd and th odd-numbrd sampls (this accounts for th trm dcimation-in-frquncy) i) For th vn-numbrd frquncis / ( n n) n,,,, / n F g h W ii) For th odd-numbrd frquncis : / F ( g h ) W n n n / n () n n n ( gn hn) W ( W ),,, Digital Signal Procssing 3/9
33 Pusan ational Univrsity / ( n n) n,,,, / n F g h W Similarly, w can subdivid ach / point DFT into two /- point DFTs and continu until finally w hav -point DFTs Th complt procss is illustratd for = 8 in Fig Th output squnc is in bit-rvrsd ordr for input in natural ordr. Digital Signal Procssing 33/9
34 Pusan ational Univrsity 6.8 Us of th Discrt Fourir Transform for digital Filtr Dsign 6.8. Introduction Suppos that an th-ordr filtr is spcifid, with a dsird frquncy rspons H(jωT). Ta qually spacd sampls H, =,,, of this frquncy charactristic about th unit circl, or sampl at intrvals of Ω = π/t along th frquncy axis. Thr altrnativ approachs ar now fasibl.. Ta th IDFT with H = H() to obtain th finit puls rspons h(n), n =,,,. Thn, obtain th filtr output for a givn input squnc x n by linarly convolving x n with h n. y( n) x( n)* h( n). A scond mthod is to carry out th filtring ntirly in th frquncy domain by using th frquncy sampls, H. 3. Th trm frquncy sampling is gnrally applid to th third mthod, by which w s a continuous frquncy charactristic to approximat th dsird rspons H(jωT) by using th sampls H. H ( z) hn z n n Digital Signal Procssing 3/9
35 Pusan ational Univrsity 6.8. Filtring ntirly in th Frquncy Domain Us an FFT algorithm to ta an -point DFT of th discrt signal x(n) to b filtrd. Multiply th rsulting frquncy sampls X = X() by vnly spacd sampls H of th dsird filtr rspons H(jωT). 3 Th output Y = H X() is transformd to th tim domain by taing th IDFT via an FFT algorithm Digital Signal Procssing 35/9
36 Pusan ational Univrsity Frquncy Sampling This mthod uss th DFT to produc a finit puls rspons filtr that has a continuous frquncy rspons that is qual to th dsird frquncy rspons H(jωT) at th qually spacd frquncy sampling instancs Ω, =,,, and approximats th dsird rsponss btwn sampling instancs. Lt j H M Ta th IDFT of H to gt th unit puls rspons jn hn H, n,,, Stting a = jω for convninc, th transfr function of th filtr is H ( z) h z H a z n n n n n n n ( az ) H ( a z ) H n n a z j z z H H az az Digital Signal Procssing 36/9
37 Pusan ational Univrsity It is vidnt that th zros from z cancl th pols z = a, so w ar lft with whr, from zro z ( z ) z ( ) H ( z) H H z z a z z a ji i ai i i z a i i H( z) H ( z a ) / z, j j j j z z a j j z ( z a ) i i z z a This is similar to th moving avrag filtr and modifid comb filtr. z ( z ai ) T Digital Signal Procssing 37/9
38 Pusan ational Univrsity H() z H z a z To gt th frquncy rspons of th filtr, lt a = jω and T=. Th frquncy rspons is obtaind as follows: j j ( ) H ( jt ) H H j j j( ) j / This indicats that th frquncy rspons is th linar combination of th frquncy sampls H with frquncy intrpolation of th form sin ( ) / s( ) => Oscillation sin ( ) / j ( )/ sin ( ) / H j( )/ sin ( ) / j / j j / j( ) H H j ( ) sin ( ) / sin ( ) / sin ( ) / sin ( ) / Digital Signal Procssing 38/9
39 Pusan ational Univrsity 6.8. Filtr Dsign by th Frquncy Sampling Mthod This implis that th frquncy rspons sampls H =DFT[h n ] must satisfy th symmtry rquirmnt * H H M M l ⅰ) is odd numbr H,,,, ⅱ) is vn numbr H,,,, If th filtr is to b linar phas, thn h n must also satisfy th symmtry rquirmnt hn h n with constant phas dlay j j jn H M 일때 hn M, n,,, h * n (6.9) Digital Signal Procssing 39/9
40 Pusan ational Univrsity * * n n n DFT h h n n m * j( n) j( n) n * jm j( ) hm, ( m n ) * j( n) h H * * j( ) jn n jn h H M j j( ) * * j j ( H M M ) jn (6.95) (6.97) Digital Signal Procssing /9
41 Pusan ational Univrsity Using (6.9) and (6.97) j j( ) jn j jn M M j j( ) j ( ) If is substitutd for - ( ) ( ) ) Digital Signal Procssing /9
42 Pusan ational Univrsity Thrfor th unit puls rspons is givn by j jn hn M n whr n n hn ( ) M n j p M M ( ) M, for odd whr, p, for vn n n j j ( ) Digital Signal Procssing /9
43 Pusan ational Univrsity Thn, n n j ( ) j p (n ) hn M ( ) M cos. odd M M,,,, p (n ) hn M ( ) M cos Digital Signal Procssing 3/9
44 Pusan ational Univrsity. vn,,,, M M,,,, M p (n ) hn M ( ) M cos Also, j( ) / H M p j( ) / H M,,,,, (6.5),,, p (6.6) Digital Signal Procssing /9
45 Pusan ational Univrsity If is vn, H Using (6.8), H(z) is drivd as H ( z) z H j z p j( ) / p j( ) / z M M M j j( ) z z z p ( ) M cos( )( ) z z M H( z) z cos( ) z z (6.3) Sustituting Frquncy rspons is obtaind as ( ) T j T ( ) M cos cos T H j T z j T sin ( ) ( ) T sin p ( ) M cos( / ) cos / cost Digital Signal Procssing 5/9
46 Pusan ational Univrsity Exampl Ex 6.) Dsign linar phas lowpass digital filtr with gain approximatly unity in th passband and zro in th stopband. And f 8Hz f Hz, 5 Sol) c Bcaus th cut off frquncy is th dsird magnitud rspons ar To mt symmtry rquirmnts From (6.5)and (6.6), th rquird frquncy rspons is s s 8 rad / s, T 5 H = M j( ), =,,, p (6.5) H = M j( ), =,,, p (6.6) c M 6, M c,, 3,, 3, 3 j 5, H ( ), 3 j, 3, Digital Signal Procssing 6/9
47 Pusan ational Univrsity From (6.3) H() z 5 z.937 z z.756 z z 5 z.98( z ).937( z ) Digital Signal Procssing 7/9
48 Pusan ational Univrsity Frquncy rspons jt sin.5t H ( jt ) (cos cos T 5 sin.5t ( ).9686 cost.8763 cost Th situation can b improvd if, instad of spcifying that th rspons sampls go dirctly from unity in th passband to zro in th stopband, w allow in a transition rgion as follows: H H j / 5, j / 5.5, 3, j / 5.5, j / 5, 3, Digital Signal Procssing 8/9
49 Pusan ational Univrsity Th rsult of adding th transition sampls is shown in Fig. 6.8 Digital Signal Procssing 9/9
10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationProblem Set #2 Due: Friday April 20, 2018 at 5 PM.
1 EE102B Spring 2018 Signal Procssing and Linar Systms II Goldsmith Problm St #2 Du: Friday April 20, 2018 at 5 PM. 1. Non-idal sampling and rcovry of idal sampls by discrt-tim filtring 30 pts) Considr
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationEE140 Introduction to Communication Systems Lecture 2
EE40 Introduction to Communication Systms Lctur 2 Instructor: Prof. Xiliang Luo ShanghaiTch Univrsity, Spring 208 Architctur of a Digital Communication Systm Transmittr Sourc A/D convrtr Sourc ncodr Channl
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationProcdings of IC-IDC0 ( and (, ( ( and (, and (f ( and (, rspctivly. If two input signals ar compltly qual, phas spctra of two signals ar qual. That is
Procdings of IC-IDC0 EFFECTS OF STOCHASTIC PHASE SPECTRUM DIFFERECES O PHASE-OLY CORRELATIO FUCTIOS PART I: STATISTICALLY COSTAT PHASE SPECTRUM DIFFERECES FOR FREQUECY IDICES Shunsu Yamai, Jun Odagiri,
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationImage Filtering: Noise Removal, Sharpening, Deblurring. Yao Wang Polytechnic University, Brooklyn, NY11201
Imag Filtring: Nois Rmoval, Sharpning, Dblurring Yao Wang Polytchnic Univrsity, Brooklyn, NY http://wb.poly.du/~yao Outlin Nois rmoval by avraging iltr Nois rmoval by mdian iltr Sharpning Edg nhancmnt
More informationCapturing. Fig. 1: Transform. transform. of two time. series. series of the. Fig. 2:
Appndix: Nots on signal procssing Capturing th Spctrum: Transform analysis: Th discrt Fourir transform A digital spch signal such as th on shown in Fig. 1 is a squnc of numbrs. Fig. 1: Transform analysis
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationIntroduction to Medical Imaging. Lecture 4: Fourier Theory = = ( ) 2sin(2 ) Introduction
Introduction Introduction to Mdical aging Lctur 4: Fourir Thory Thory dvlopd by Josph Fourir (768-83) Th Fourir transform of a signal s() yilds its frquncy spctrum S(k) Klaus Mullr s() forward transform
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationAnnounce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationDiscrete Hilbert Transform. Numeric Algorithms
Volum 49, umbr 4, 8 485 Discrt Hilbrt Transform. umric Algorithms Ghorgh TODORA, Rodica HOLOEC and Ciprian IAKAB Abstract - Th Hilbrt and Fourir transforms ar tools usd for signal analysis in th tim/frquncy
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationTypes of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr Th tim-domain classification of an LTI digital transfr function squnc is basd on th lngth of its impuls rspons: - Finit impuls rspons (FIR) transfr function - Infinit impuls
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME Introduction to Finit Elmnt Analysis Chaptr 5 Two-Dimnsional Formulation Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systms Prof. Mark Fowlr ot St #21 D-T Signals: Rlation btwn DFT, DTFT, & CTFT 1/16 W can us th DFT to implmnt numrical FT procssing This nabls us to numrically analyz a signal to find
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationDiscrete-Time Signal Processing
Discrt-Tim Signal Procssing Hnry D. Pfistr March 3, 07 Th Discrt-Tim Fourir Transform. Dfinition Th discrt-tim Fourir transform DTFT) maps an apriodic discrt-tim signal x[n] to th frquncy-domain function
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationSinusoidal Response Notes
ECE 30 Sinusoidal Rspons Nots For BIBO Systms AStolp /29/3 Th sinusoidal rspons of a systm is th output whn th input is a sinusoidal (which starts at tim 0) Systm Sinusoidal Rspons stp input H( s) output
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationJNTU World JNTU World DSP NOTES PREPARED BY 1 Downloaded From JNTU World (http://(http:// )(http:// )JNTU World )
JTU World JTU World DSP OTES PREPARED BY Downloadd From JTU World (http://(http:// )JTU World JTU World JTU World DIGITAL SIGAL PROCESSIG A signal is dfind as any physical quantity that varis with tim,
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME 43 Introduction to Finit Elmnt Analysis Chaptr 3 Computr Implmntation of D FEM Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationA Sub-Optimal Log-Domain Decoding Algorithm for Non-Binary LDPC Codes
Procdings of th 9th WSEAS Intrnational Confrnc on APPLICATIONS of COMPUTER ENGINEERING A Sub-Optimal Log-Domain Dcoding Algorithm for Non-Binary LDPC Cods CHIRAG DADLANI and RANJAN BOSE Dpartmnt of Elctrical
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationAs the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.
7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationOptics and Non-Linear Optics I Non-linear Optics Tutorial Sheet November 2007
Optics and Non-Linar Optics I - 007 Non-linar Optics Tutorial Sht Novmbr 007 1. An altrnativ xponntial notion somtims usd in NLO is to writ Acos (") # 1 ( Ai" + A * $i" ). By using this notation and substituting
More informationDTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1
DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT
More informationDifference -Analytical Method of The One-Dimensional Convection-Diffusion Equation
Diffrnc -Analytical Mthod of Th On-Dimnsional Convction-Diffusion Equation Dalabav Umurdin Dpartmnt mathmatic modlling, Univrsity of orld Economy and Diplomacy, Uzbistan Abstract. An analytical diffrncing
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationECE Department Univ. of Maryland, College Park
EEE63 Part- Tr-basd Filtr Banks and Multirsolution Analysis ECE Dpartmnt Univ. of Maryland, Collg Park Updatd / by Prof. Min Wu. bb.ng.umd.du d slct EEE63); minwu@ng.umd.du md d M. Wu: EEE63 Advancd Signal
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More information5 Transform Analysis of LTI Systems
5 Transform Analysis of LTI Systms ² For an LTI systm with input x [n], output y [n], and impuls rspons h [n]: Fig. 48-F1 ² Nots: 1. y [n] = h [n] x [n]. 2. Y ( jω ) = H ( jω ) X ( jω ). 3. From th Convolution
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationTitle: Vibrational structure of electronic transition
Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum
More informationNumbering Systems Basic Building Blocks Scaling and Round-off Noise. Number Representation. Floating vs. Fixed point. DSP Design.
Numbring Systms Basic Building Blocks Scaling and Round-off Nois Numbr Rprsntation Viktor Öwall viktor.owall@it.lth.s Floating vs. Fixd point In floating point a valu is rprsntd by mantissa dtrmining th
More information[ ] [ ] DFT: Discrete Fourier Transform ( ) ( ) ( ) ( ) Congruence (Integer modulo m) N-point signal
Congrunc (Intgr modulo m) : Discrt Fourir Transform In this sction, all lttrs stand for intgrs. gcd ( nm, ) th gratst common divisor of n and m Lt d gcd(n,m) All th linar combinations r n+ s m of n and
More informationDigital Signal Processing, Fall 2006
Digital Signal Prossing, Fall 006 Ltur 7: Filtr Dsign Zhng-ua an Dpartmnt of Eltroni Systms Aalborg Univrsity, Dnmar t@om.aau. Cours at a glan MM Disrt-tim signals an systms Systm MM Fourir-omain rprsntation
More informationSymmetric centrosymmetric matrix vector multiplication
Linar Algbra and its Applications 320 (2000) 193 198 www.lsvir.com/locat/laa Symmtric cntrosymmtric matrix vctor multiplication A. Mlman 1 Dpartmnt of Mathmatics, Univrsity of San Francisco, San Francisco,
More informationComplex representation of continuous-time periodic signals
. Complx rprsntation of continuous-tim priodic signals Eulr s quation jwt cost jsint his is th famous Eulr s quation. Brtrand Russll and Richard Fynman both gav this quation plntiful prais with words such
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationChapter 6 Folding. Folding
Chaptr 6 Folding Wintr 1 Mokhtar Abolaz Folding Th folding transformation is usd to systmatically dtrmin th control circuits in DSP architctur whr multipl algorithm oprations ar tim-multiplxd to a singl
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationHomework #3. 1 x. dx. It therefore follows that a sum of the
Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationRandom Access Techniques: ALOHA (cont.)
Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision
More information