In previous problems, we have seen how a graphical vector representation of a signal can be used to see how the radar works. The radar, however, cannot use a graphical representation of the vectors for addition. How does the computer add phasors?
Until about 1960, radars did not add phasors as you have been doing. Instead, radars took the magnitude of the signals (known as magnitude detection) and threw away the phase information. At that time there was no way to effectively use the phase information.
About 1960, the first practical doppler radars were introduced. They used analog filters to add signals indirectly. These systems were expensive and tended to require a great deal of tuning of the analog doppler filters. Radar engineers of the time knew how to add vectors, but did not have any hardware available at the time that could affordably and reliably add the signal vectors directly.
The change that brought direct vector addition from theory to common practice was the introduction of the computer into radars for signal processing, which happened in about 1970. With a computer, almost any mathematical computations can be performed, subject to speed limitations of the computer. Although these computers could keep track of the magnitude and phase of signal samples and add them, there is a much faster technique, which you will develop in this problem.
We have seen vectors represented with magnitude and
phase. This is mathematically identical with a polar coordinate representation.
There is another representation of the vector in rectangular coordinates which is
much easier to use for calculation.
Rectangular coordinates just represent the vector in terms of its projection on the x-axis and its projection on the y-axis.
Any vector can be converted from its polar coordinate form to rectangular coordinates by a trigonometric transformation as shown in the figure.
Mathematically, the projections are:
x-axis: Magnitude*cos(Phase)
y-axis: Magnitude*sin(Phase)
The powerful features of the rectangular coordinate representation is that vector addition and multiplication become straightforward. To add two vectors in rectangular representation,
(x1, y1) + (x2, y2) = (x1+x2, y1+y2)
Multiplication of vectors in rectangular coordinates is only a little more difficult as
(x1, y1) * (x2, y2) = (x1*x2 - y1*y2, x1*y2 + x2*y1)
It's so easy a computer can do it! Because of the way that sinusoidal waves are converted to this form, we call the rectangular coordinates inphase and quadrature for the x-axis and y-axis. Inphase, which means the portion of the signal in phase with the reference sinusoid, is abbreviated I. Quadrature, which means the portion of the signal 90° out of phase with the reference sinusoid, is abbreviated Q. The signal representation is abbreviated as I/Q.
Generally, I/Q data refers to data having both magnitude and phase, and it typically refers to data collected as soon as analog data is converted to digital. When such I/Q data is available, it is possible to emulate the radar processing on a computer. This is very useful for determining problems in the radar during development.
We have programmend your computer to convert vectors between rectangular and polar coordinates and to show you the resultant. The vector adder calculator will help you with the following problems:
Question 1: Transform a vector of magnitude 3 and phase 60° into I/Q. Now transform a vector of magnitude 3 and phase 30° into I/Q. Compare the results from the two transformations. Why did this happen?
Question 2: Transform a vector of magnitude 3 and phase 36.8699° into I/Q. Add it to a vector of magnitude 4 and phase 126.8699°. What is the resultant vector in I/Q representation? What is the resultant magnitude? What relationship do you see between the magnitudes of the first vector, the second vector, and the resultant from adding the two vectors? Why is this relationship present?
Question 3: A resultant vector is represented in I,Q as (5,12). The vector v1 can be represented as (3,4). What is the I/Q representation for vector v2, such that v1+v2=resultant? What is the magnitude/phase representation?
Question 4: How many mathematical operations (adds and multiplies) are required to add two vectors in I/Q representation? How many mathematical operations to perform a multiply?
Question 5: Work out the mathematical operations to add two vectors, each of which is represented in amplitude / phase form. See if you can estimate the computational requirements for this operation. This problem will take significant time to complete. Even if you do not complete this problem, read the answer.