One easy method of solving (6.28) when we are given S ( w ) is to let H(w) = �.
6.4 One easy method of solving (6.28) when we are given S ( w ) is to let H(w) = �.
For K = 1 and S (w ) =(1 + p2) - 2p cos w, show that this algorithm will not yield a finite-order ARMA realization. Can this filter be causal or stable? 6.5 What are the necessary and sufficient conditions that an ARMA system be minimum phase? Find if the following filters are (i) causal and stable, (ii) minimum phase.
a. H(z) = l - 0.8z - 1
b. H(z) = (1 - z -1)/[l.81 - 0.9(z + z -1)]
c. H(z) = (1 - 0.2z - 1)/(l - 0.9z -1 ) 6.6 Following the Wiener-Doob decomposition method outlined in Section 6.8, show that a
1-D SDF S (z) can be factored to give H(z) � exp[ c+ (z)], c+ (z) � L� = 1 c(n)z -", K =
exp[c(O)], where {c(n),'v'n} are the Fourier series coefficients of logS(w). Show that H(z) is a minimum-phase filter.
6.7 Using the identity for a Hermitian Toeplitz matrix
[� where i b.
b. r(O) . ·
� [r(n), r(n - 1), . . . , r(l)f, prove the Levinson recursions.
6.8 Assume that R is real and positive definite. Then a. (k) and p. will also be real. Using the Levinson recursions, prove the following. a. IP" I < l,'v'n.
b. Given r(O) and any one of the sequences {r(n)}, {a (n)}, {p. }, the remaining se quences can be determined uniquely from it.
6.9 For the third-order AR model u (n) = O.lu(n - 1 ) + 0.782u (n - 2) + O.lu(n - 3) + E(n),
132 = 0.067716 find the partial correlations and determine if this system is stable. What are the first four
Sec. 6 Problems 227 Sec. 6 Problems 227
6.10 (Proof of noncausal MVR theorem) First show that the orthogonality condition for minimum-variance noncausal estimate u (n) yields
r(k) -L a(/)r(k - I) = (32B(k), I ,;,
'r/k
Solve this via the Fourier transform to arrive at (6.38) and show that S (z) = (32/A (z). Apply this to the relation S (z) = s. (z)IA (z)A (z - 1) to obtain (6.39). 6.11 Show that the parameters of the noncausal MVR of (6.43) for pth-order AR sequences defined by (6.3a) are given by
where a (0) � - 1 . 6.12 Show that for a stationary pth-order AR sequence of length N, its noncausal MVR can
be written as Hu = v + b, where H is an N x N symmetric banded Toeplitz matrix with 2p subdiagonals, and
the N x 1 vector b contains 2p nonzero terms involving the boundary values {u (l - k), u (N + k), 1 s k s p }. Show that this yields an orthogonal decomposition similar to (6.49). 6.13 For a zero mean Gaussian random sequence with covariance plnl, show that the optimum interpolator u (n) based on u (O) and u (N + 1) is given by
u (n) = [(pN+ l - n _ p-N - l + n)u (O) + (pn _ p-n)u (N + l))/(pN+ l _ p-N- 1)
Show that this reduces to the straight-line interpolator of (6.53) when p� 1 and a cubic polynomial interpolation formula when p = 1.
6.14 (Fast KL transform for a continuous random process) Let u (x) be a zero mean, stationary Gaussian random process whose covariance function is r(T) = exp(-0.'71 ), o. > 0. Let x t: [- L, L ] and define Uo � u ( - L ), u1 � u (L ). Show that u (x) has the orthogonal decomposition
u (x) = u0 (x) + ub (x),
-L s x S L
ao x sinh o.(L - x)
ai x - - sinh ( )_ cx(L + x)
sinh 2cxL such that ub (x), which is determined by the boundary values uo and u1 , is orthogonal to
- sinh 2o.L '
u0 (x). Moreover, the KL expansion of u0 (x) is given by the harmonic sinusoids
k = l , 3, 5 . . . , -L s x s L
. k-rrx sm 2L '
228 Image Representation by Stochastic Models Chap. 6
6.15 Determine whether the following filters are causal, semicausal or noncausal for a vertically scanned, left-to-right image.
a. H(z, , z2) = z, + zl1 + zl1 z21
b. H(z1 , z2) = 1 + z! 1 + z21 Z1 + z21 z2 1
c. H(z1 , z2) = (2 - Z1 - Z2 _, -1 )
4 _1 - Z1 - Z1 - Z2 - Z2 _1 Sketch their regions of support.
d. H(z1 , z2) =
6.16 Given the prediction error filters
A (z, , z2) = 1 - a, zl1 - a1 z21 - a3 zl 1 z2 1 - a4 z1 z2 1
(causal)
A (z1 , z2) = 1 - al (z1 + z!') - a1 z2 1 - aJ z21 (z1 + z!') (semicausal)
A (z1 , z2) = 1 - a, (z, + z! 1) - a1 (z2 + z2 1)
- a3 z21 (z1 + zl1) - a4 zl1 (z2 + z2') (noncausal) a. Assuming the prediction error has zero mean and variance �2, find the SDF of the
prediction error sequences if these are to be minimum variance prediction-error filters. b. What initial or boundary values are needed to solve the filter equations
A (z1 , z2) U(z1 , z2) = F(z1 , z2) over an N x N grid in each case?
6.17 The stability condition (6.78) is equivalent to the requirement that IH(z1 , z2)1 < oo,
lz1 I= 1, lz2
I = l.
a. Show that this immediately gives the general stability condition of (6.79) for any two-dimensional system, in particular for the noncausal systems.
b. For a semicausal system H(m, n) = 0 for n < 0, for every m. Show that this re
striction yields the stability conditions of (6. 8 0 ) .
c. For a causal system we need h(m, n) = O for n < O,Vm and h(m, n) = O for n= 0, m < 0. Show that this restriction yields the stability conditions of (6.81).
6.18 Assuming the prediction-error filters of Example 6.7 represent MVRs, find conditions on the predictor coefficients so that the associated MVRs are stable.