Frequency response is H1 (zi, z2) = 1-a1 z]1 -a2z21 -a3zl1 z21 -a4z1 z21. cos cos

b. Frequency response is H1 (zi, z2) = 1-a1 z]1 -a2z21 -a3zl1 z21 -a4z1 z21. cos cos

a. Transfer function is

H ( w1, w2) = 1 -2cx w1 -2cx w2.

Chap. 2 Problems

2.14 a. Write the convolution of two sequences {1, 2, 3, 4} and {-1, 2, -1} as a Toeplitz matrix operating on a

3 x 1 vector and then as a Toeplitz matrix operating on a 4x1 vector.

b. Write the convolution of two periodic sequences {1, 2, 3, 4, . . . } and {-1,2, -1, 0, . . . }, each of period

4, as a circulant matrix operating on a 4 x 1 vector that represents the first sequence. 2.15 [Matrix trace and related formulas].

a. Show that for square matrices A and B, Tr[A] = Tr[AT] =

i= l L Tr[A + B]

Tr [A]+Tr [B], and Tr [AB] = Tr [BA] when X., are the eigenvalues of A.

{ n } aA . Then show that DA (AB) = BT, ( aa m, )Tr[Y]

b. Define DA (Y) � J_Tr [Y] � a

DA (ABA T_) = ABT + AB, and DA (A-1 BAC) = -(A-1 BACA-y + (CA-1 B)T. 2.16 Express the two-dimensional convolutions of Problems 2.S(a) as a doubly Toeplitz block matrix operating on a 6 x 1 vector obtained by column ordering of the x(m, n ) .

2.17 In the two-dimensional linear system of (2.8), the x (m, n) and y (m, n ) are of size M xN and are mapped into column ordered vectors x and y, respectively. Write this as a

matrix equation

y = '3f.x

and show '3e is an N x N block matrix of basic dimension M x M that satisfies the properties listed in Table P2.17.

Impu lse Response (and Covariance) Sequences and Corresponding Block Matrix Structures

TABLE P2. 1 7

Block matrix Spatially varying

Sequence

'ae, general Spatially invariant in m;

h (m, n ; m', n ' )

Toeplitz blocks Spatially invariant in n;

h (m - m ', n ; n ' )

Block Toeplitz Spatially invariant in m, n;

h(m, n - n '; m ')

Doubly Toeplitz Spatially invariant in m, n

h(m - m ', n - n ')

Block Toeplitz with circulant blocks Spatially invariant in m, n and periodic in n

and periodic in m

h (m modulo M, n)

h (m, n modulo N)

Block circulant with Toeplitz blocks

Spatially invariant and periodic in m, n

Doubly block circulant Separable, spatially varying

h (m modulo M, n modulo N)

Kronecker product H2@H1 Separable, spatially invariant h1 (m - m ') hz ( n-n ')

h1 (m, m ') hz (n, n ' )

Toeplitz Kronecker product H2@Hi,

H1, H2 Toeplitz Separable, spatially invariant, and periodic

h1 (m)h1 ( n) (m modulo M,

Circulant Kronecker

n modulo N)

product H2@ H 1, H1, H2 circulant

2.18 Show each of the following. a. A circulant matrix is Toeplitz, but the converse is not true.

46 Two-Dimensiona l Systems and Mathematical Preliminaries

Chap. 2 Chap. 2

c. The product of two Toeplitz matrices need not be Toeplitz. 2.19 Show each of the following. a. The covariance matrix of a sequence of uncorrelated random variables is diagonal.

b. The cross-covariance matrix of two mutually wide-sense stationary sequences is Toeplitz.

c. The covariance matrix of one period of a real stationary periodic random sequence is circulant.

2.20 In Table P2.17, if h (m, n; m ', n ') represents the covariance function of an M x N segment of a random field x (m, n), then show that the block matrix '3t represents the

covariance matrix of column-ordered vector x for each of the cases listed in that table.

2.21 Prove properties (2.9 7 ) through (2. 99) of SDFs. Show that (2.100) is the SDF of random fields whose covariance function is the separable function given by (2.84). 2.22 a. *Compute the entropies of several digital images from their histograms and compare them with the gray scale activity in the images. The gray scale activity may be represented by the variance of the image.

b. Show that for a given number of possible messages the entropy of a source is maximum if all the messages are equally likely.

c. Show that RD given by (2.118) is a monotonically nonincreasing function of D.

BIBLIOGRAPHY

Sections 2. 1-2.6 For fundamental concepts in linear systems, Fourier theory, Z-transforms and

related topics:

1. T. Kailath. Linear Systems. Englewood Cliffs, N.J.: Prentice-Hall, 1980.

2. A. V. Oppenheim and R. W. Schafer. Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1975.

3. A. Papoulis. Systems and Transforms with Applications in Optics. New York: McGraw-Hill, 1968.

4. J. W. Goodman. Introduction to Fourier Optics. New York: McGraw-Hill, 1968.

5. R. N. Bracewell. The Fourier Transform and Its Applications. New York: McGraw-Hill, 1965.

6. E. I. Jury. Theory and Application of the Z-Transform Method. New York: John Wiley, 1964.

Sections 2. 7, 2.8 For matrix theory results and their proofs:

7. R. Bellman. Introduction to Matrix Analysis, 2d ed. New York: McGraw-Hill, 1970.

8. G. A. Graybill. Introduction to Matrices with Applications in Statistics. Belmont, Calif.: Wadsworth, 1969.

* Problems marked with an asterisk require computer simulation or other experiments.

Chap. 2 Bibliography 47

Sections 2.9-2.1 3 For fundamentals of random processes, estimation theory, and information theory:

9. A. Papoulis. Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1965.

10. W. B. Davenport. Probability and Random Processes. New York: McGraw-Hill, 1970.

11. R. G. Gallager. Information Theory and Reliable Communication. New York: John Wiley, 1968.

12. C. E. Shannon and W. Weaver. The Mathematica/ Theory of Communication. Urbana: The University of Illinois Press, 1949.

48 Two-Dimensional Systems and Mathematical Preliminaries Chap. 2