Errata for Introduction to Statistical Signal Processing by R.M. Gray and L.D. Davisson

Transcript

1 Errt for Introduction to Sttisticl Signl Processing by R.M. Gry nd L.D. Dvisson Errors not cught in the corrected 2 edition re noted with n sterisk *. Updted October 2, 24 hnks to In Lee, Michel Gutmnn, Frédéric Vrins, André Isidio de Melo, Philippe Bonnet, Osmn Muso, Alex Rzumko, nd to the chmpion typo finder, Ron Aloysius. p. * prioriprobbility priori probbility p. 2 * generter genertor p. 49: he summtions in (2.42)-(2.43) should run from k to n, not from k to n s stted becuse the pmf is uniform on {,,..., n } nd not on {, 2,..., n}. As result the sum in (2.42) should evlute to (n )/2 nd not (n + )/2 nd the sum in (2.43) should evlute to (2n )(n )/6 nd not (n + )(2n + )/6. p. 5: Eq. (2.49) ( 2 m (2) k 2 p( p) k p p ) 3 p 2 nd not m (2) Eq. (2.5) nd not k k p. 6: In (2.68) m m (2). (2.74) is missing π, it nd not ( 2 k 2 p( p) k p p + ) 3 p 2 σ 2 p p 2 σ 2 2 p 2 2πσ 2 e (x m)2 /2σ 2 dx 2σ 2 e (x m)2 /2σ 2 dx

2 p. 62 As on the previous pge πs re missing. (2.75) nd not nd (2.76) nd not p. 63 Second line of (2.82): 2πσ 2 xe (x m)2 /2σ 2 dx m 2σ 2 xe (x m)2 /2σ 2 dx m 2πσ 2 (x m)2 e (x m)2 /2σ 2 dx σ 2 2σ 2 (x m)2 e (x m)2 /2σ 2 dx σ 2 (α m)/σ (α m)/σ 2π e u2 /2 dx 2π e u2 /2 du (the differentil du) p. 68: E(g) λ x F g(x)p(x) + ( λ) g(x)f(x) dx x F E(g) λ x Ω g(x)p(x) + ( λ) g(x)f(x) dx x Ω p. 75. In Problem 4 the comment In words: if the probbility of the symmetric difference of two events is smll, then the two events must hve pproximtely the sme probbility. belongs with Problem 2.5, not with 2.3. p. 94 In cption of Figure 3. Pr(f F ) P ({ω : ω F }) P (f (F )) Pr(f F ) P ({ω : f(ω) F }) P (f (F )) p. 3* he boldfce nottion is still found, but it is fr less common then it used to be. he boldfce nottion is still found, but it is fr less common thn it used to be. p. 25: P (X (F ) Y (F 2 ) P (X (F )) Y (F 2 )). 2

3 P (X (F ) Y (F 2 ) P (X (F ))P (Y (F 2 )). p. 3 Eq. (3.55) [ Λ σ 2 X ρσ X σ Y ρσ X σ Y σ 2 Y ], not [ σx Λ 2 ρσ X σ Y he second line of (3.58) nd not 2π det Λ e ρσ X σ Y σ Y 2 (x m X,y m Y )Λ (x m X,y m Y ) t 2π det Λ e 2 (x m X,y m Y )Λ (x m X,y m Y ) t p. 32 Second line, the men m Y X m Y + ρ(σ Y /σ X )(x m X ) nd not m Y X y m Y + ρ(σ Y /σ X )(x m X ) he finl line of (3.62) nd not n f X (x ) f Xl X,...,X l (x l x,..., x l ) l k f X (x ) f Xl X,...,X l (x l x,..., x l ) l (he upper limit of the sum n not k) p. 44 Eq. (3.99) P e ( ( ) (.5 Φ + Φ.5 )) ( Φ ). 2 σ W σ W 2σ W ] nd not P e 2 ( ( Φ (.5 ) ( + Φ.5 )) ( ) Φ. σ W σ W 2σ W 3

4 (remove extr left pren nd dd minus sign). p. 45 Just bove Section 3.2, f X Y (x y) f Y X (y x)f Y (y)/f X (x) should be f X Y (x y) f Y X (y x)f X (x)/f Y (y) p. 38, eqution bove (3.82): ( exp 2 (α m ) ) 2. σ 2 ( exp 2 (α m ) ) 2. σ p. 49 In line eqution following (3.5) M X (ju) F u/2π (f X ) L ju (f X ), tht is, the subscript of L needs minus sign. p. 5 Penultimte line of (3.2) { } ))2/2σ2 (2πσ 2 e (x (m+juσ2 dx e jum u2 σ 2 /2 ) /2 nd not { (the y 2 u 2 p. 62 Eq. (3.39) p Yn Y n,...,y (y n y n,..., y ) } ))2/2σ2 (2πσ 2 e (x (m+juσ2 dx e jum y2 σ 2 /2 ) /2 Pr(Y n y n Y l y l ; l,..., y n ) Pr(X n y n y n Y l y l ; l,..., y n ) Pr(X n y n y n X y, X i y i y i ; i 2, 3,..., n ), should red p Yn Y n,...,y (y n y n,..., y ) Pr(Y n y n Y l y l ; l,..., y n ) Pr(X n y n y n Y l y l ; l,..., n ) Pr(X n y n y n X y, X i y i y i ; i 2, 3,..., n ), tht is, the finl index in the penultimte line is n nd not y n. 4

5 p. 65 he sentence he discrete time, continuous lphbet cse of summing iid rndom vribles is hndled in virtully the sme mnner s the discrete time cse, with conditionl pdfs replcing conditionl pmfs. should red he discrete time, continuous lphbet cse of summing iid rndom vribles is hndled in virtully the sme mnner s the discrete time, discrete lphbet cse, with conditionl pdfs replcing conditionl pmfs. Eq. (3.49) k f Y,...,Y n (y,..., y n ) f Yl Y,...,Y l (y l y,..., y l ). l should red f Y,...,Y n (y,..., y n ) n f Yl Y,...,Y l (y l y,..., y l ). l ht is, the upper limit of the product n. Eq. (3.52) f Y,...,Y n (y,..., y n ) n f X (y i y i ) i should red f Y,...,Y n (y,..., y n ) n f X (y i y i ) tht is, the finl index on the left is n, not n. p. 68 Problem : X(r) r 2 X(r) r 2 Y (r) r /2 Y (r) r /2 p. 76, problem 43, lst line. B t β t. p. 84 Eighth line following (4.), r n (n) r (n) p. 85 Eq. (4.4) E(X) p X (x) x A i. E(X) x A xp X (x) 5

6 p. 28 Note tht if we tke g(x) x nd h(x, y), this generl form reduces to the previous form. Note tht if we tke g(x) nd h(x, y) y, this generl form reduces to the previous form. p. 2: f Y X (y x) f XY (x, y) f Y (y) ( ( )) x exp (/2)((x m X ) t (y m Y ) t )K mx U y m Y (2π) (k+m) det K U (2π)k det K X exp ( (x m X ) t K X (x m X)/2 ) (2π)m det K X / det K U ( exp (/2)((x m X ) t (y m Y ) t )K U ) + (x m X ) t K X (x m X) ( x mx y m Y ) f Y X (y x) f XY (x, y) f X (x) ( ( x exp (/2)((x m X ) t (y m Y ) t )K mx U y m Y (2π) (k+m) det K U )) (2π)k det K X exp ( (x m X ) t K X (x m X)/2 ) (2π)m det K U / det K X ( ( x exp (/2)((x m X ) t (y m Y ) t )K mx U y m Y ) + (/2)(x m X ) t K X (x m X) ) 6

7 Lst line: N k N n. p. 23 In the first displyed eqution following A direct proof of this result, E[(Y g(x)) 2 ] E[(Y E(Y X) + E(Y X) g(x)) 2 ] E[(Y E(Y X)) 2 ] 2E[(Y E(Y X))(E(Y X) g(x))] +E[(E(Y X) g(x)) 2 ]. E[(Y g(x)) 2 ] E[(Y E(Y X) + E(Y X) g(x)) 2 ] E[(Y E(Y X)) 2 ] +2E[(Y E(Y X))(E(Y X) g(x))] +E[(E(Y X) g(x)) 2 ]. tht is, the sign in front of the 2 on the penultimte line positive. p. 25 Second line from the bottom of the pge, K (X,Y ) E[((X t, Y t ) (m t X mt Y ))t ((X t, Y t ) (m t X mt Y ))] K (X,Y ) E[((X t, Y t ) (m t X, mt Y ))t ((X t, Y t ) (m t X, mt Y ))], tht is, the minus signs seprting the mens comms. p. 26 op line, K Y X E[(Y m Y )(Y m Y ) t ] K Y X E[(Y m Y )(X m X ) t ] p. 27 MMSE det(k(n) X ) det(k (n ) X ), () should red MMSE det(k(n+) X ) det(k (n) X ), (2) p. 28, four lines bove (2.64) nd one line below (4.64) b E(Y ) + AE(X) b E(Y ) AE(X) Eq. (4.64) MSE(A, b) r ( (Y AX b)(y AX b) t) r(k Y K Y X K X K XY ) 7

8 MSE(A, b) E [ r ( (Y AX b)(y AX b) t)] p. 29 op of the pge. r(k Y K Y X K X K XY ) MMSE(A, b) r ( (Y AX b)(y AX b) t) r ((Y m Y + A(X m X ) b + m Y + Am X ) (Y m Y + A(X m X ) b + m Y + Am X ) t) r ( K Y AK XY K Y X A t + AK X A t) + (b m Y Am X ) t (b m Y Am X ) MMSE(A, b) E [ r ( (Y AX b)(y AX b) t)] E [r ((Y m Y A(X m X ) b + m Y Am X ) (Y m Y A(X m X ) b + m Y Am X ) t)] r ( K Y AK XY K Y X A t + AK X A t) + (b m Y + Am X ) t (b m Y + Am X ) (4.65) b m Y + Am X. b m Y Am X. In heorem 4.6, K Y X E[(Y m Y )(Y m Y ) t ] K Y X E[(Y m Y )(X m X ) t ]. p. 22, first line, b E(Y ) + AE(X) b E(Y ) AE(X) p In proof of Lemm 4.4, finl eqution on pge: F Yn (y) F Y (y + ɛ) Pr(Y y + ɛ nd Y n > y) Pr(Y n y nd Y > y + ɛ) Pr(Y y + ɛ nd Y n > y) Pr( Y Y n ɛ). 8

9 F Yn (y) F Y (y + ɛ) Pr(Y n y nd Y > y + ɛ) Pr(Y y + ɛ nd Y n > y) Pr(Y n y nd Y > y + ɛ) Pr( Y Y n ɛ). p. 26, Problem 9. his problem considers some useful properties of utocorreltion or covrince function. his problem considers some useful properties of utocorreltion or covrince functions for rel-vlued rndom processes. E(X 2 t R X (t, t) R X (, ) E(X 2 t ) R X (t, t) R X (, ) ( right pren is missing) p. 262, problem2: Given two rndom processes {X t ; t } nd {X t ; t } should red Given two rndom processes {X t ; t } nd {Y t ; t } p. 267, Problem 4.37, the upper cse G k lower cse g k. p. 27, Problem 47. wo other rndom processes wo other rndom processes p. 273, Problem 4.57(c), E[X n ] E[X N ]. p. 289, the K Y (k, j)e should red σ 2 k j r2(min(k,j)+) r. r 2 p. 3 In severl formuls the index k n: N R X (k) < X n Xn k > lim X n Xn k N N k N R X (k) < X n Xn k > lim X n Xn k N N n N R X (k) lim X n (ω)x N N n k(ω), k 9

10 N R X (k) lim X n (ω)x N N n k(ω), n N P X R X () lim X n 2, N N k N P X R X () lim X n 2, N N n p. 33, the lst line in the top eqution should red nd not lim N N k (N ) lim N N N k (N ) ( k N )R X(k)e i2πfk tht is, the extr /N removed. p. 38 In Lemm 5., item 3, Eq. (5.6.2), ( k N )R X(k)e i2πfk E( X ) X. E(X) X. p. 39 Continuing the corrections of p. 38: in the proof of item 3 of the lemm, E( X ) E( X ) X X. E(X) E(X ) X X.

11 A relted error in the proof of item 5 is the presence of superfluous line in the sequence of inequlities. In prticulr, E[X n Y m XY ] E[X n Y m XY m + XY m XY ] E[(X n X)Y m] + E[X(Y m Y ) ] E[(X n X)Y m] + E[X(Y m Y ) ] E[ (X n X)Y m ] + E[ X(Y m Y ) ] X n X Y m + X Y m Y X n X Y m Y + Y + X Y m Y X n X ( Y m Y + Y ) + X Y m Y E[X n Y m XY ] E[X n Y m XY m + XY m XY ] p. 324: Replce E[(X n X)Y m] + E[X(Y m Y ) ] E[(X n X)Y m] + E[X(Y m Y ) ] X n X Y m + X Y m Y X n X Y m Y + Y + X Y m Y n X n X ( Y m Y + Y ) + X Y m Y ( n ) sin(π(t n )/ ) 2W x 2W π(t n )/ by n p. 325: Replce (5.) x (n ) sin(π(t n )/ ) π(t n )/ R X (t τ) n ( n ) sin(π(t n )/ ) R X 2W τ. π(t n )/ by R X (t τ) n R X (n τ) sin(π(t n )/ ). π(t n )/

12 by by p. 326: Replce the first eqution R X () R X () n n ( n ) sin(π(t n )/ ) R X 2W t. π(t n )/ R X (n t) Replce the penultimte eqution in the proof n,m:n mk n sin(π(t n )/ ). π(t n )/ sin(π(t n )/ ) sin(π(t m )/ ) π(t n )/ π(t m )/ sin(π(t n )/ ) sin(π( t (n k))) π(t n )/ π( t (n k)) sin(π( t k)) π( t k), n,m:n mk n sin(π(t n )/ ) sin(π(t m )/ ) π(t n )/ π(t m )/ sin(π(t n )/ ) sin(π(t + k n )/ ) π(t n )/ π(t + k n )/ sin(πk), πk Replce the finl eqution in the proof (just bove section 5.8.7) by lim N n N m N lim N n N m N sin(π(t n )/ ) R X ((n m) ) π(t n )/ sin(π(t n )/ ) R X ((n m) ) π(t n )/ 2 sin(π(t m )/ ) π(t m )/ R X (k) sin(π( t k)) π( t k) R X () sin(π(t m )/ ) π(t m )/ R X (k) sin(πk) πk R X ()

13 p. 327 In the equtions for R XN (t, s), the s severl times is incorrectly replced by t nd in the eqution preceeding (5.7), the subscript n should be m. hese re fixed by replcing [ N ] R XN (t, s) E[X N (t)xn(s)] E X n φ n (t) Xmφ m(t) so tht n m n E[X n Xm]φ n (t)φ m(t)] R X (t, s) lim N R X N (t, s) m λ N φ n (t)φ n(t) n λ n φ n (t)φ n(t). n Multiplying by φ-function nd integrting then yields by so tht R X (t, s)φ m (s) ds λ n φ n (t) n λ n φ n (t). [ N R XN (t, s) E[X N (t)xn(s)] E X n φ n (t) n m n E[X n Xm]φ n (t)φ m(s)] R X (t, s) lim N R X N (t, s) φ m (s)φ n(s) ds ] Xmφ m(s) m λ N φ n (t)φ n(s) n λ n φ n (t)φ n(s). n Multiplying by φ-function nd integrting then yields R X (t, s)φ m (s) ds λ n φ n (t) n λ m φ m (t). 3 φ m (s)φ n(s) ds

14 p. 328 In the middle of the pge, so tht λ n φ m(t)φ n (t) dt (λ n λ m) λ n φ m(t)φ n (t) dt λ m φ m(t)r X (t, s)φ n (s) ds dt ( ) φ n (s) R X (t, s)φ m(t) ds ( φ n (s) R X (s, t)φ m (t)) ds λ m φ n (t)φ m(t) dt φ n (t)φ m(t) dt. φ m(t)r X (t, s)φ n (s) ds dt ( ) φ n (s) R X (t, s)φ m(t)dt ds ( φ n (s) R X (s, t)φ m (t)dt) ds φ n (s)φ m(s) ds so tht replcing the dummy vrible s by t yields (λ n λ m) φ n (t)φ m(t) dt. p. 329 In the first eqution, first line, complex conjugte is missing: [( ) ( )] E[X n Xm] E X(t)φ n(t) dt X(s)φ m (s) ds [( E[X n Xm] E ) ( )] X(t)φ n(t) dt X (s)φ m (s) ds 4

15 In the lst eqution on the pge λ n is missing, should red φ n(t) n φ n(t) n R X (t, s)φ n (s) ds R X (t, s)φ n (s) ds φ n(t)φ n(t). n λ n φ n(t)φ n(t). p. 33: he missing λ n of p. 329 crries to this pge. he first eqution E[X N (t)x(t) ] E[X N (t)x(t) ] Ner the middle of the pge, chnge If R X hs Fourier series expnsion R X (τ) to If R X hs Fourier series expnsion R X (τ) n φ n(t)φ n(t), n λ n φ n(t)φ n(t), n n n b n e j2πnt/ b n e j2πnτ/ nd just below it chnge then the integrl eqution to be solved for the Krhunen Loeve expnsion is λφ(t) n b n e j2πnt/ φ(s) ds b n e j2πn(t s)/ φ(s) ds. n to 5

16 then the integrl eqution (5.7) to be solved for the Krhunen Loeve expnsion is λφ(t) n b n e j2πn(t s)/ φ(s) ds b n e j2πn(t s)/ φ(s) ds. n he subscripts in the text nd eqution t the bottom of the pge need repir, s do the relted subscripts t the top of p. 33. Chnge Guessing solution φ n (t) c n e j2πnt/ where c n is normlizing constnt, then c n e j2πnt/ c m e j2πmt/ dt c n c m so tht c n /. hen with the integrl eqution becomes λ ej2πnt/ φ n (t) ej2πnt/ n n b n b n e j2πnt/ b m ej2πnt/ t c n 2 δ N M j2πn(t s)/ ej2πns/ e e j2π(m n)s bm φ m (t), e j2π(n m)t/ dt which we lredy knew ws solution from the development for Fourier series. to Guessing solution φ m (t) c n e j2πmt/ where c m is normlizing constnt, then c n e j2πnt/ c me j2πmt/ dt c n c m t c n 2 δ n m ds ds e j2π(n m)t/ dt 6

17 so tht c n /. hen with the integrl eqution becomes λ ej2πmt/ φ m (t) ej2πmt/ n n b n b n e j2πnt/ b m ej2πmt/ j2πn(t s)/ ej2πms/ e e j2π(m n)s b m φ m (t), which we lredy knew ws solution from the development for Fourier series. p.334 he lst eqution on the pge, E( ɛ 2 n) E(ɛ 2 n) + { (h i g i ) E(X n Y n i ) } h j E(Y n j Y n i, i: n i K is missing right pren, it E( ɛ 2 n) E(ɛ 2 n) + { (h i g i ) E(X n Y n i ) i: n i K j: n j K j: n j K ds ds h j E(Y n j Y n i ) p. 35 A new problem begins t the top of the pge, it is missing number 5. he remining problems should ll be dvnced by one. p. 48, Problem 28 Find the power spectrl densities S X (f) nd S Y (f)? Find the power spectrl densities S X (f) nd S Y (f). p. 435, Problem A.23 (b) hs typo nd s result the integrl blows up. It replced by x dxe x dye y. p. 439 (B.7) k 2 q k k 2q ( q) 3 + ( q) 2 7 },

18 insted of k 2 q k k 2 ( q) + 3 ( q) 2 he second line of the next eqution insted of q q k 2 q k q k k 2 q k q k kq k q k kq k q k k 2 q k ( q) 2 q k k 2 q k ( q) 2 k he finl eqution of the proof insted of k 2 q k k k 2 q k k p. 442 (B.3) chnged from 2q ( q) + 3 ( q), 2 2 ( q) + 3 ( q), 2 2σ 2 e (x m)2 /2σ 2 dx e r2 σ dr 2σ 2 2π. 2π to 2πσ 2 e (x m)2 /2σ 2 dx e r2 /2 σ dr 2πσ 2 2πσ. 2πσ 2 p. 446: Uniform pmf. Ω Z n {,,..., n } nd p(k) /n; k Z n. men: (n + )/2 8

19 vrince: (2n + )(n + )n/6 ((n + )/2) 2. Uniform pmf. Ω Z n {,,..., n } nd p(k) /n; k Z n. men: (n )/2 vrince: (2n )(n )/6 [(n )/2] 2 (n 2 )/2. he vrince of the geometric pmf ( p)/p 2, not 2/p 2. p. 447, the men of the Uniform pdf (b+)/2 nd not (b )/2. he vrince of the Gmm pdf is 2 b nd not b. 9