UNIVERSITETET I OSLO
Det matematisk-naturvitenskapelige fakultet
Exam in:
IN 350 — Space-Time Signal Processing
Day of exam:
13. December 1995
Examination hours:
9:00 – 13:00
This examination paper consists of 4 pages.
Appendices:
One
Permitted materials:
Calculator
Make sure that your copy of this examination paper
is complete before starting to answer the questions.
1
Miscellaneous (44%)
For all questions in problem 1:
Answer briefly using keywords and simple sketches. Maximum 10 lines per question.
1-a
What is the difference between plane waves and spherical waves?
1-b
What is the effect of dispersion on a propagating signal?
1-c
Explain how refraction can lead to multipath transmission in e.g. underwater acoustics.
1-d
What kind of information can be read from an aperture’s smoothing function? What does the
ideal aperture smoothing function look like when the goal is to make it as directive as possible?
Exam in IN 350, 13. December 1995
Page 2
1-e
Consider a finite continuous line aperture. How does aperture weighting affect the aperture
smoothing function?
1-f
What are grating lobes and how can one make sure that they do not appear?
1-g
What kind of information is found in an array’s coarray? What does the ideal coarray look
like?
1-h
What is delay-and-sum beamforming? What is the model behind this kind of beamforming?
1-i
Under what signal and noise conditions is conventional beamforming optimal?
1-j
Compare minimum variance and conventional beamforming. What is the difference in the
assumptions behind the two algorithms?
1-k
Compare minimum variance and conventional beamforming. What is the difference in performance between the two algorithms?
2
2-a
Aperture smoothing function (38%)
One-dimensional line aperture
Show that for a line aperture which lies from – D to D on the x-axis with weight w( x) = 1 ,
the aperture smoothing function is given by:
sin k x D
W (k x) = ----------------kx ⁄ 2
Exam in IN 350, 13. December 1995
Page 3
2-b
Find W (θ) where θ is the angle between the incoming wave and the broadside direction for
the aperture of a).
2-c
Compare the antenna in a), with an antenna with the same aperture, which has been divided
into two equal parts and weighted with opposite signs:
w( x)
1
-D
D
x
-1
Plot the aperture smoothing function for the two apertures
Assume an application where the problem is to detect whether a source lies exactly broadside to the antenna. Which antenna is best? Justify your answer.
2-d
Rectangular aperture
Find the aperture smoothing function, W ( k ) , for a two-dimensional rectangular aperture
with dimensions D x and D y .
2-e
Mainlobe/sidelobes for rectangular aperture
Draw a contour plot of W ( k ) for D x > D y . Be careful to specify the mainlobe and the locations of the largest sidelobes.
- What is the width of the mainlobe?
- What is the position of the first sidelobe and what is its value relative to the mainlobe?
2-f
Circular aperture
For a circular aperture with constant weight and radius R , the aperture smoothing function
is:
2πR
O(k xy) = ---------- J 1(k xy R).
k xy
Exam in IN 350, 13. December 1995
Page 4
(See appendix for properties of first order Bessel function of first kind)
Compare two apertures with the same area, where one is a quadratic aperture and the other a
circular aperture. What is the difference in mainlobes and sidelobes?
3
Array (18%)
3-a
Coarray
Find the coarray of the following two four-element arrays with unity weight on all elements.
d
d
3d
d
2d
d
3-b
Are the arrays redundant or nonredundant? Are they perfect or imperfect?
3-c
Array gain
Consider the last one of the two arrays in the figure above. Assume:
- that the four sensors are omnidirectional
- that the four sensors all receive the same signal
- that they receive a stochastic noise signal which is independent from sensor to sensor
Find the array gain for a conventional beamformer
Exam in IN 350, 13. December 1995
Page 5
Appendix: Properties of Bessel function
3
5
x x
x
Power series: J 1( x) = --- – ------ + ------ – …
2 16 96
Table 1: Zeroes and extremal values
J 1( x)
x
0
0
2.405
0.5191
3.832
0
5.520
-0.3403
7.016
0