What is a Costas array?
1 Introduction
To find the speed and distance of a moving object relative to an observer, a signal with a known frequency is sent from the observer towards the object. It reflects off the object, and the reflected signal has a new frequency, due to the Doppler effect. Knowing the frequencies of the emitted and reflected signals, the speed of the moving object relative to the observer can be determined. Knowing the time between emitting the signal and receiving the reflected signal, the distance of the moving object to the object can be determined. See [2, p. 74].
Let’s say we have a stationary observer, and an object is moving radially
away from the observer with a constant speed. One wants to find the
speed
Suppose that at times
The
Supposing that
If the emitted signals have frequencies
We assume that the set of frequencies is narrowband, so that approximately
Then the frequency of the signal received at time
Let’s use units such that the times
where
Certainly
We wish to find
For any
so
We want to figure out
2 Mathematics
Define a new sequence
The sequence
Let
Definition.
A Costas sequence of length
This can also be said in the following way.
For
Drakakis [1] gives a survey of Costas arrays.
3 Generalizations
Now that we have detached Costas sequences from the time-frequency setting in which they originate, we can investigate them as mathematical objects, and also investigate modifications of Costas sequences.
Let
Moran [3]
References
- [1] (2006) A review of Costas arrays. J. Appl. Math. 2006, pp. 1–32. Note: Article ID 26385 Cited by: §2.
- [2] (2004) Radar signals. John Wiley & Sons. Cited by: §1, §1.
- [3] (2001) Mathematics of radar. In Twentieth Century Harmonic Analysis - A Celebration, J. S. Byrnes (Ed.), pp. 295–328. Note: NATO Science Series, II. Mathematics, Physics and Chemistry - Vol. 33 Cited by: §3.