ECE310 Digital Signal Processing

Problem Set V: Multidimensional Signals & Systems

1. Consider a medical CT (computerized tomography) scan which results in a set of twodimensional lateral image ìslicesî taken along a longitudinal axis. With reference to

(x; y; z) coordinates, x; y are the lateral coordinates, and z the longitudinal coordinate.

For simplicity assume the spacing between the slices has unit length.

We will model the resulting 3-D image as a signal g (~r; n) with coordinate system

~r 2 R

2

(the continuous lateral coordinates), and n 2 Z (the discrete longitudinal

coordinate). In what follows, do not separate ~r into x; y coordinates, or join it with n

to form a 3-D vector: keep ~r,n separate. Let the corresponding frequency coordinates

be denoted

~kxy; kz

, where ~kxy = (kx; ky) is kept separate from kz. Do not explicitly

write x; y; kx; ky in any of the following,

(a) Specify the domains of ~kxy and kz, respectively.

(b) Write the Fourier and inverse Fourier transforms.

(c) State Parsevalís theorem.

(d) Write the formulas for convolution in the spatial and wavenumber domains.

(e) Prove that convolution in the spatial domain leads to multiplication in the wavenumber domain.

(f) Let h (~r; n) denote the impulse response of an LTI system applied to such a 3-D

object. Write the condition for stability as an explicit mathematical expression

(i.e., do not simply write h 2 L

1

).

2. McClellan Transform Design Method for Multidimensional FIR Filters

The problem presents a technique for designing multidimensional FIR Ölters based on

1-D ìprototypesî. The approach described here can be used for 2-D, 3-D (or higher!)

Ölters, but we will stay with a 2-D example. The idea is to start with a 1-D FIR Ölter

“prototype”, H (z), and make a substitution z F (z1; z2) to obtain a 2-D Ölter:

G (z1; z2) = H (F (z1; z2))

The McClellan transform proposes the form of F () that we chose. Usually, the target

Ölter G should be zero-phase with real coe¢ cients, so we chose H to be zero-phase

with real coe¢ cients. The function F is also zero-phase with real coe¢ cients, and

it is usually designed to map “DC” to “DC” and high frequencies (say the points

(!1; !2) = (; )) to ! = . Thus, for example, if H is lowpass, so is G, if

H is bandpass, so is G. In addition, F is often chosen to achieve certain desirable

symmetries. For example, if F is nearly isotropic, so is G.