High Resolution Radar

Abstract:

The goal of this assignment is to give a detailed explanation on how a rectangular and

linear frequency modulated signals work, their properties, and behavior regarding autocorrelation,

spectrum and ambiguity.

Introduction:

As we are starting to discuss two out of many types of signals in electrical engineering, we

must first define how it is that we are going to study them. In order to study, deal and analyze

signals, we need to implement what is called Signal Processing. This area is a part of electrical

engineering, systems engineering, and applied mathematics, and we can use all the available tools

in order to perform operations on these signals. There are many signals that we can work with

such as sound, images, radio signals and more.

What is a signal? A signal is “a formal description of a phenomenon evolving over time or

space (Prandoni 1)”, and by signal processing we mean any kind of operation which modifies,

analyzes or manipulates the information that the signal contains.

Theory:

Rectangular – envelope monotone signal

The rectangular wave, is a non-sinusoidal waveform that is commonly seen in electronics

and signal processing, usually in switching circuits and it is usually used as a clock reference. But

before getting deeper into discussing this signal, we must first define what an envelope is. An

envelope detector is an electronic circuit that “takes a high frequency signal as input and provides

an output which is the envelope of the original signal”(wiki).

As we can see in Figure 1, we have a transmitted signal and its envelope is highlighted in

red (top of the signal).

Figure 1

Knowing this, we can now discuss the behavior of the rectangular monotone signal. First of

all, we must state the analytical expression of the signal, which is:

It is a simple step function, and it is define by being 1 or 0 depending on the limit that is stated

within the function. In this case, the limit is ½ being t an absolute value. Or similarly, the

rectangular can be also expressed as

The autocorrelation of a signal is a series of values obtained by summing the individual

values of the signal multiplied by the shifted version of itself. The process of doing autocorrelation

is exactly the same as computing the variance of the signal. Where the variance is defined as the

maximum value of the autocorrelation function, and also the variance is the measure of the power

in the signal.

Where Px=Rx(0)=Variance

The autocorrelation is derived from a very extensive algebraic and differential process, but after

doing all the summations and integrals of the functions we obtain:

x() = (E{an 2}/T) p(t +)p(t ) dt

-

So if the p(t) is the rectangular wave, we have:

p(t +)p(t ) dt=A2T[1-/T]; T

-

Therefore the autocorrelation is shown graphically and mathematically in figure 2:

Figure 2

x() = A2[1-/T]; T

x()

A2

-T 0 T

The frequency spectrum of a signal, is the representation of that signal but in the

frequency domain. The rectangular wave in the time domain is represented as follows in figure 3:

Figure 3

The rectangular signal

t0 T sec A

After applying the Fourier Transform which is defined as

We can obtain the spectrum of the signal, and it is shown in figure 4 which is known as the Sinc

function:

Figure 4

The Ambiguity function is used to characterize the range and the Doppler resolution of a

radar waveform (Li 265). The function returns the distortion of a returned pulse, which is reflected

from a moving target. The ambiguity function is given by:

So the ambiguity function for a simple rectangular pulse is:

Ψ(r,v) = [sin^2 pi*v(r’-|r|) / (pi*vr’)^2]*rect(r/2r’) (Nathanson 288)

And we find that for a rectangular wave, the ambiguity function is:

Figure 5

Linear FM signal

This signal is the linear frequency modulated pulse. This signal has de advantage of having

a larger bandwidth while keeping the pulse duration short and envelope constant (wiki). Radars

may use LFM waveforms so that both range and Doppler information can be measured (Mahafza

124).

The analytical expression of the linear FM signal is:

F(t)=cos[Wo*t + µt2/2] for –T/2<= t <=T/2 (Nathanson 501)

Where µ=Δw/T, the desired frequency slope at the dispersive delay output (the rate of frequency sweep)

T=the duration of the transit pulse envelope

Wo=transmit carrier frequency

The signal’s autocorrelation is given by:

X(r,v)= eJTPiV sin [π(kr+v)(r’-|r|)] / π(kr+v)r’

As mentioned before, the signal spectrum is obtained by taking the fourier transform of the

function. The linear FM in continuous time is shown below in figure 6

Figure 6

As we can see in the picture, the signal changes in frequency as time varies. This means that when

we take the FFT of the signal, we will obtain:

Figure 7

It is interesting to see that with the fourier transform, we can approximate |S(f)|with a rectangle.

Lastly, the ambiguity function for an LFM signal, if we consider the LFM complex envelope

signal defined by

S(t) = 1/√(T’) * Rect(1/T’) e^(j*pi*µ*t^2)

Therefore, after some algebra, the ambiguity function for an LFM signal is:

A(t,Fd) = | sin(π(Fd + βt/T)(T-|t|)) / Tπ(Fd+βt/T) | for –T<=t<=T (Richards 195)

The ambiguity function of the LFM signal yields the following plot:

Figure 8:

Conclusion:

Signal Processing is a very important tool in communications today. In this case, we can

implement these functions in radar signal processing in order to do detection, tracking or imaging.

The characteristics discussed in this paper help us understand better the signals behavior, and how

the analytical process yields important data that can be traduced into graphical results.

Works Cited

Li, Jian, and Stoica Petre. Mimo Radar Signal Processin. New Jersey: John Wiley & Sons, Inc. 2009.

Mahafza, Bassem. Radar Systems Analysis and Design using Matlab. Alabama: Chapman & Hall.

2000.

Mahafza, Bassem. Introduction to Radar Analysis. Alabama: CRC Press. 1998

Nathanson, Fred. Radar Design Principles. New York: McGraw-Hill Inc. 1969.

Prandoni, Paolo, and Vetterli Martin. Signal Processing for Communications. Florida: CRC Press.

2008.

Richards, Mark. Fundamentals of Radar Signal Processing. McGraw-Hill. 2005.

Wikipedia. 2010. Wikimedia Foundation Inc. April 4, 2010.www.wikipedia.com.