Pulse compression for different types of radar signals.830629/... · 2015-06-30 · Pulse...

Thesis no: MCS-2012-09

Pulse compression for

different types of radar

signals.

This thesis is presented as of Bachelor Science

in Electrical Engineering with Emphasis on Telecommunication

Blekinge Institute of Technology

September 2012

Blekinge Institute of Technology School of Engineering

Department of Electrical Engineering

Supervisor: Mats Pettersson

Authors: Tomás Garnacho Aparicio & Alberto Trejo Roldán

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Abstract

The aim of our project is to compare pulse compression of different waveforms using

correlation of real signals or matched filter for analytic signals. In this thesis a set of programs is

developed for this comparison. The characteristic of the matched filter is that it processes the

highest possible Signal to Noise ratio (SNR) under the assumption of white noise. The

implementation of the algorithm is made in Matlab.

Radar signal processors are usually carried out over a specified range window. Returns from

all targets within the received window are collected and passed through the matched filter to

perform the pulse compression. Because of the recent development of digital signal processors

(DSPs), this process is often performed digitally. The aim of this work is to get better quality in the

radar images, including SAR (Synthetic Aperture Radar) and to explain the characteristics of each

waveform. The thesis also includes an appendix, where the implemented programs and program

code are attached. The work also aims at illustrative and didactic purposes. The programs have

been developed so as to be easily understood and therefore useful for engineering students.

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Gracias a todos aquellos que me han acompañado en este camino.

A mi madre.

“Si caes, levántate... si vuelves a caer, levántate otra vez... Y si vuelves a caer levántate

una y otra vez... Sólo en la lucha, la fe, la constancia y la fuerza de voluntad llegaremos al

triunfo...”

“If you fall, get up... If you fall back again, then get up again... and if you fall back once

more, get up again and again. In struggle, faith, perseverance and power are the ways to

achieve the victory..."

Tomás

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Firstly, I would like to thank to everyone who helped us with our thesis during our

stay as exchange students in Blekinge Tekniska Högskola.

I especially want to mention our project supervisor Mats Pettersson for his help and

support from the very beginning of the project. Furthermore I also want to give thanks to

all the classmates I met during my studies as well as all the friends that were not studying

with me but giving me their support, though. Additionally I am grateful for meeting so

many great persons during my stay, especially Tamara, and I want to thank them for the

help they were during my time in Sweden. Finally I want to thank my family, especially my

parents and my sister who gave me the possibility to study abroad and their endless support

and help.

Alberto

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Table of contents

Abstract ................................................................................................................................... 3

Table of contents .................................................................................................................... 9

List of Figures ....................................................................................................................... 11

List of Acronyms .................................................................................................................. 13

Chapter 1 Introduction .......................................................................................................... 15

Chapter 2 Matched Filter ...................................................................................................... 17

2.1. Fundamentals of the Matched Filter................................................................... 17

2.2. Range Resolution Properties .............................................................................. 20

Chapter 3 Analysis Matched Filter Response of Linear Frequency Modulated Waveforms

(chirp) ................................................................................................................................... 25

3.1. Basics of Linear FM Waveforms ....................................................................... 25

3.2. Frequency analysis of Linear FM Waveforms ................................................... 27

3.3. Pulse Compression Process ................................................................................ 33

Chapter 4 Cross-Correlation ................................................................................................. 37

Chapter 5 Analytic Signal .................................................................................................... 43

5.1. Definition of Hilbert transforms. ........................................................................ 43

5.2. Analytic Signal Application ............................................................................... 44

Chapter 6 Different types of radar signals ............................................................................ 53

6.1. Binary phase coded pulse ................................................................................... 53

6.2. Square pulse ....................................................................................................... 61

Chapter 7 Conclusions .......................................................................................................... 67

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References ............................................................................................................................ 69

Appendix .............................................................................................................................. 73

Program code ................................................................................................................ 73

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List of Figures

Figure 1.Matched Filter maximizes the signal peak to mean noise ratio. ................................................... 20

Figure 2. llustration of a square pulse and its corresponding spectra. ........................................................ 21

Figure 3. Square Pulse compressed ............................................................................................................ 22

Figure 4. Comparison between the ultimate resolution of a rectangular constant frequency and a chirp pulse ................................................................................................................................................. 23

Figure 5. Output pulse compression filter. ................................................................................................. 24

Figure 6. Real part and imaginary part of signal chirp ................................................................................ 26

Figure 7. Signal chirp in the frequency domain by the use of FFT ............................................................... 28

Figure 8. Graphical Representation of signal delay. ................................................................................... 29

Figure 9. Signal chirp delayed in the frequency domain by the use of FFT .................................................. 30

Figure 10. Ramp function to implement the delayed signal. ...................................................................... 31

Figure 11. Comparison of signal chirp and signal chirp delayed in the frequency domain .......................... 32

Figure 12. Comparison of signal chirp and signal chirp delayed in the time domain ................................... 33

Figure 13. Fast Convolution Processor ....................................................................................................... 34

Figure 14. Response in frequency after after the matched filter ................................................................ 35

Figure 15. Output of the matched filter ..................................................................................................... 36

Figure 16. Convolution of Reflected echo and Time reversed pulse ........................................................... 38

Figure 17. Convolution of Reflected echo and Time reversed pulse. Intermediate step ............................. 38

Figure 18. Convolution of Reflected echo and Time reversed pulse. Finished convolution ......................... 39

Figure 19. Matched filter by cross-correlation process ............................................................................... 40

Figure 20. Comparison of the responses of matched filter and cross-correlation process for a transmitted signal ................................................................................................................................................ 41

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Figure 21. Comparison of the envelopes of the responses of matched filter and cross-correlation process for a transmitted signal .................................................................................................................... 42

Figure 22. Real part and imaginary part of analytic signal. Orthogonal vectors .......................................... 44

Figure 23. Analytic signal property Hilbert ................................................................................................. 46

Figure 24. Diagram of the generation of the analytic signal ....................................................................... 47

Figure 25. Comparison between the original chirp signal and the recovered signal ................................... 49

Figure 26. Response of the recovered signal to the matched filter ............................................................. 50

Figure 27. Comparison between responses to the matched filter .............................................................. 51

Figure 28. Binary phase coded pulse .......................................................................................................... 54

Figure 29. Modulated binary phase coded pulse bit period. ...................................................................... 55

Figure 30. Detail of Binary phase coded pulse (BPSK signal)....................................................................... 56

Figure 31. Detail of modulated binary phase coded pulse. ......................................................................... 57

Figure 32.Comparison between original and delayed Binary phase coded pulses. ..................................... 58

Figure 33. Binary phase coded signal in the frequency domain by the use of FFT.. .................................... 59

Figure 34. Comparison of the responses to the different pulse compression processes with a Binary phase coded pulse. ..................................................................................................................................... 60

Figure 35. Ideal square pulse ..................................................................................................................... 61

Figure 36. Square pulse .............................................................................................................................. 62

Figure 37. Detail of Square pulse ............................................................................................................... 63

Figure 38. Comparison between original and delayed Square pulse .......................................................... 64

Figure 39. Comparison of the responses to the different pulse compression processes with a Square pulse ......................................................................................................................................................... 65

Figure 40. Response to the Matched filter with a Square pulse ................................................................. 66

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List of Acronyms

ADC - Analogical Digital Converter

CPM-Continuous Phase Modulation

DFT- Discrete Fourier Transform

FCP-Fast Convolution Processing

FFT- Fast Fourier Transform

IF - Intermediate Frequency

PRF- Pulse Repetition Frequency

RF-Radio Frequency

SAR-Synthetic Aperture Radar

SAW-Surface Acoustic Wave

SNR - Signal to Noise Relation

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Introduction

Chapter 1

Introduction

1930s the radar scientists were not aware of the concept of matched filter and pulse

compression. They were still learning from experience how to maximize the output signal-to-noise

ratio for the simple pulse waveforms that were used at that time.

In almost all conditions, usually met in practice, maximizing the output-peak-signal-to-

noise ratio of a radar receiver maximizes the detectability of a target. A linear filter that does this

transformation is called a matched filter under the assumption of white noise. Thus a matched

filter, or a close approximation to it, is the basis for the design of almost all radar receivers

waveforms that have to be pulse compressed. Methods for the detection of desired signals and

the rejection of undesired noise, clutter and interference in radar are called radar signal

processing. The processes of matched filter and pulse compression, described next, are an

important example of a radar signal processor.

Despite the fact that we will not consider the noise effect in this project, we will be able to

show the benefit of Matched Filtering. We also show Pulse Compression techniques based on

cross-correlation. Finally a transformation from real to complex signal is made by the Hilbert

transform. This transformation is important for many modern systems.

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Introduction

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Matched Filter

Chapter 2

Matched Filter

This chapter focuses on Matched filter derivation. The matched filter is the first step to

solve one fundamental problem in early radar systems, the ratio between detectability and

resolution. When we have developed the matched filter and tested its target resolution

properties, we will discuss the procedure of pulse compression.

2.1. Fundamentals of the Matched Filter

Matched filter is derived to maximize signal to noise ratio (SNR) and by that detectability

at the output of a filter, when signal and white noise are passed through the filter.

If we consider a radar system that transmits a signal ( ), we want to find a filter that

maximizes the instantaneous SNR at a determined frequency when the delayed replica of

( )plus additive white noise is present.

The receiver input can be represented by

( ) ( )

(2. 1. 1)

Where ( )is white noise. Let us look at the matched filter impulse response

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Matched Filter

( )

∫ ( ) ( ) ( )

(2. 1. 2)

Where ( ) is the Fourier transform of ( ) and ( ) is the matched filter

transformation.

At the time of the target to the signal at the filter output is

( )

∫ ( ) ( ) ( )

(2. 1. 3)

We consider a filter ( ) that gives the maximum output SNR at a predetermined delay

. Therefore, the SNR [1] is

(

)

| ( )|

( )̅̅ ̅̅ ̅̅ ̅̅

(2. 1. 4)

Where the noise is,

( )̅̅ ̅̅ ̅̅ ̅̅

∫ | ( )|

(2. 1. 5)

Substituting previously [1] we get

(

)

|∫ ( ) ( ) ( )

|

∫ | ( )|

(2. 1. 6)

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Matched Filter

And by the Schwartz inequality:

|∫ ( ) ( )

|

∫ | ( )|

∫ | ( )|

(2. 1.7)

Where equality holds if,

( ) ( )

(2. 1. 8)

Where * denotes complex conjugate and is an arbitrary constant. Applying the Schwarz

inequality to (2.1.6) we get

( ) ( )

(2. 1. 9)

and the maximum SNR

(

)

∫ | ( )|

(2. 1. 10)

Where E is the energy of the finite-time signal, ( ). Parseval’s theorem relates the

energy in the frequency domain and the energy in the time domain [2]

∫ ( )

∫

( )

(2. 1. 10)

Thus, we can draw the conclusion that the peak instantaneous SNR depends only on the

signal energy and input noise power, and is independent of the waveform utilized by the radar.

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Matched Filter

Figure 1. Matched Filter maximizes the signal peak to mean noise ratio.

The matched filter maximizes the peak-signal to mean noise ratio. For example, for a rectangular

pulse, the matched filter is a simple pass band filter.

Finally, we can define the impulse response for the matched filter taking the inverse

Fourier transform of (2. 1. 9)

( ) ( )

(2. 1. 11)

2.2. Range Resolution Properties

The range resolution is determined from the matched filter processing of the rectangular

pulse. If we consider the case that the transmitted signal consists of a constant frequency signal,

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Matched Filter

modulated by a rectangular pulse of width, . The sharp edges of the rectangular function in time

generate an infinite frequency spectrum, as follows

Figure 2. Illustration of a square pulse and its corresponding spectra.

From the frequency response we can see that the 3dB (50%) bandwidth is just

(2. 2. 1)

By the use of a matched filter with a rectangular pulse shown before (figure 2) we get

3𝑑𝐵

𝜏

𝜏

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Matched Filter

Figure 3. Square Pulse compressed.

The range resolution is determined by converting the time delay, , to the round trip time

required to achieve this delay

(2. 2. 2)

Here the number (2) refers to a monostatic radar with the same location on the

transmitter and the receiver antenna.

Using the relationship shown above, the range resolution, determined in terms of the

pulse width, can be rewritten in terms of the effective bandwidth of the signal,

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

x 10-6

-1000

-500

0

500

1000

time [s]

Am

plit

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Matched Filter

(2. 2. 3)

However, normally it is hard to transmit a high power pulse for short time duration. Using

chirp signal with the same duration, the matched filter generates a sinc function with a much

narrower peak, and hence a superior range resolution.

( ) ( ) ⁄

(2. 2. 4)

The range resolution is inversely proportional to the chirp bandwidth, . The targets need

to be separated by approximately the width of the output response at 3 . The effects of these

two forms of processing are shown in the following figure.

Figure 4. Comparison between the ultimate resolution of a rectangular constant frequency

and a chirp pulse (introduced in chapter 3).

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Matched Filter

The width of the sinc pulse is inversely proportional to the bandwidth of the

uncompressed pulse and the height is proportional to the product of the bandwidth and

uncompressed pulse width.

Figure 5. Output pulse compression filter.

τ

~1/B

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Analysis Matched Filter Response of Linear Frequency Modulated Waveforms (chirp)

Chapter 3

Analysis Matched Filter Response of Linear

Frequency Modulated Waveforms (chirp)

Preferably, most radar systems codes should permit long detection range and fine range

resolution. Therefore we have to transmit an extremely narrow pulse (high bandwidth) of

exceptionally high peak power if we use short pulses. However there are practical limits on the

peak power. To obtain long detection ranges for pulse delay ranging, very high power pulses must

be transmitted.

One solution to this dilemma is to use pulse compression. Which means, transmit

internally modulated pulses of sufficient bandwidth to provide the necessary average power at a

reasonable level of peak power (as we showed in chapter 2). Then, after reception, “compress”

the received echoes by decoding their modulation.

Linear frequency modulation (LFM) or often called chirp is the first and probably still the

most common method for transmitted pulse. It was developing during World War II, as can be

deduced from German, British and U. S patents (Cook and Bernfeld, 1967; Cook and Seibert 1988,

patent). The basic idea is to sweep the frequency band linearly during the pulse duration .

3.1. Basics of Linear FM Waveforms

The special case of the linear FM pulse should be noted explicitly. In order to develop a

general expression for the matched filter output when an LFM waveform is utilized, we will

consider the easier example, when the radar detects a stationary target. The transmitted signal is:

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(

)| |

(3. 1. 1)

Best known as LFM waveform or chirp.

Figure 6. Real part and imaginary part of signal chirp. The pulse length is 2 µseconds and the rate of

the frequency increase 3,75·1013 (1/s2). The bandwidth is then 75 MHz.

Where ⁄ .

Because of its similarity to the chirping of a bird, this method of coding was called “chirp”

by its developers and researchers. Since it was the first pulse compression technique, some people

still use the terms signal chirp and pulse compression synonymously. As an illustration a chirp

signal is show in figure 3.

0 2 4 6 8 10 12 14 16 18 20

x 10-7

-1

-0.5

0

0.5

1

Time [s]

Am

plit

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Real part of the Linear FM Waveform signal

0 2 4 6 8 10 12 14 16 18 20

x 10-7

-1

-0.5

0

0.5

1

Time [s]

Am

plit

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Imaginary part of the Linear FM Waveform signal

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The instantaneous frequency ( ) is obtained by differentiating the argument of the

exponential.

( )

( )

(3. 1. 2)

This instantaneous frequency is indeed a linear function of time. The frequency slope

has the dimensions .

3.2. Frequency analysis of Linear FM Waveforms

The analysis of waveforms and processors, their features, and the derivation of optimum

processing is partially based on Fourier transform theory. As in many other areas in signal

processing, it is easier to process the signal in the frequency domain than in the time domain.

Some aspects of the theory will be presented in this section.

Firstly, by the use of Fourier Transform

( )

∫ ( )

(3. 2. 1)

to transform the waveform ( ) to the frequency domain

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Figure 7. Signal chirp in the frequency domain by the use of FFT with = 62.5 MHz, = 75 MHz and = 2 µs.

But in reality we are not working with the Fourier transform. Our signal processing is

based on samples of the echoed signal ( ). Because we are using finite-duration sequences time

we have to work with the FFT in Matlab, which takes samples of the signal in use.

Since we are simulating the operation of a radar system we will introduce a delay in our

generated signal (sent signal), to simulate the received signal, which will be the same one we

transmitted but with a time delay. This is presented in figure 8.

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-20

-10

0

10

20

30

40

50

Frequency [Hz]

Magnitude (

dB

)

Spectrum Linear FM Waveform signal

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Figure 8. Graphical Representation of signal delay.

To create a delayed signal we work in the frequency domain, by using the time shifting

property of the Fourier Transform. For a given function ( ), the function ( ) is the delayed

signal.

( ( )) ∫ ( ) ∫ ( ) ( ) ( )

(3. 2. 2)

As we can see, working in the frequency domain makes it quite easy to apply a delay to a

signal because we just have to multiply by an exponential signal that includes the time delay. In

Matlab, with sampled signals, the delay is given by

[ ] ↔ ( )

(3. 2. 3)

Where ( ) is the Fourier transform of [ ].

We illustrate this in an example, where we assume a time delay (td) of 2µs to our simulated

signals, this refers to a distance

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3

(3. 2. 4)

in a monostatic radar system.

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Figure 9. Signal chirp delayed in the frequency domain by the use of FFT.

As we mentioned before, for getting the delay of the signal chirp we used the vector of

frequencies for the time shifting in Matlab, possible to see in the following figure

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-60

-40

-20

0

20

40

60

80

100

120

Frequency [Hz]

Magnitude (

dB

)Spectrum of the delayed Linear FM Waveform signal

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Figure 10. Ramp function to implement the delayed signal.

0 1 2 3 4 5 6

x 10-6

-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

8

Time [s]

Fre

quency [

Hz]

Time vs frequency

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Figure 11. Comparison of signal chirp and signal chirp delayed in the frequency domain.

In this plot we are not really able to see any difference, it shows the delay of our signal in

the time domain in a more intuitive way.

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-50

0

50

100

Frequency [Hz]

Magnitude (

dB

)

Comparison between Linear FM Waveform and delayed Linear FM Waveform

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-50

0

50

100

Frequency [Hz]

Magnitude (

dB

)

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Figure 12. Comparison of signal chirp and signal chirp delayed in the time domain.

3.3. Pulse Compression Process

If a chirp is transmits every echo, naturally, it has the same linear increase in frequency.

When the received echo passes through the filter, it introduces a temporal widening of the period

which increases linearly at exactly the same rate as the frequency of the echoes increases. Being

of progressively higher frequency, the trailing portions of an echo take less time to pass through

than the leading portion. Successive portions thus tend to bunch up. Consequently, when the

pulse emerges from the filter its amplitude is much greater and its width much less than at the

0 1 2 3 4 5 6

x 10-6

-1

-0.5

0

0.5

1

Time [s]

Am

plit

ude

Real part of the delayed Linear FM Waveform signal

0 1 2 3 4 5 6

x 10-6

-1

-0.5

0

0.5

1

Time [s]

Am

plit

ude

Imaginary part of the delayed Linear FM Waveform signal

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time it entered. The pulse has been compressed. In the practice, filtering may be done with analog

devices, such as an acoustical delay line, or digitally. Depending on the mechanization, the

frequency can either be increasing, as described here, or decreasing, in which case the delay

increases with frequency.

Linear FM pulse compression is accomplished by adding frequency modulation to a long

pulse at transmission, and by using a matched filter receiver in order to compress the received

signal. As a result, the matched filter output is compressed by a factor . Thus, by using

long pulses and wideband LFM modulation large compression ratios can be achieved.

Because of the recent advances in digital computer development, the matched filter is

often performed digitally using FFT. This digital implementation is called Fast Convolution

Processing (FCP) and can be implemented at base band. The fast convolution process is illustrated

here:

Figure 13. Fast Convolution Processor.

Since the matched filter is a linear time invariant system, its output can be described

mathematically by the convolution between its input and its impulse response,

( ) ( ) ( )

(3. 2. 1)

FFT

FFT of storedsignalc

hirp

multiplier Inv.

FFT

Input signal

MatchedFilter Output

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Where ( )is the input signal, ( ) is the matched filter impulse response, and the

operator symbolically represents convolution. From the Fourier transform properties we know,

{ ( ) ( )} ( ) ( )

(3. 2. 2)

And when both signals are sampled properly, the compressed signal ( ) can be

computed from

{ }

(3. 2. 3)

Where is the inverse FFT.

Figure 14. Response in frequency after the matched filter, with = 62.5 MHz, = 75 MHz and = 2µs.

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-40

-20

0

20

40

60

80

100

X: 6.2e+007

Y: 89.16

Frequency [Hz]

Magnitude (

dB

)

Interpolated output of the Matched filter

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As shown in figure 14, the IF frequency is situated in the middle of the spectrum which is

62.5 MHz for 89 dB (case of proposed simulation in Matlab), and the start frequency and the stop

frequency are respectively 25 MHz and 100 MHz. In the data we have simulated a delay of 2

microseconds. The time output is shown in figure 15 (i.e. the pulse compressed signal).

Figure 15. Output of the matched filter.

As shown the Figure 15, the delay distance-time of the target by the use of the IFFT in

Matlab. The target is placed at 4 microseconds or 600 meters of two way distance. The secondary

lobe is approximately 13 dB below the main lobe in the interpolated matched filter. Next, we will

have to consider if this difference is big enough or if we have to do more for a proper processing of

the signal.

3.8 3.9 4 4.1 4.2 4.3

x 10-6

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

Am

plit

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Interpolated output of the matched filter

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Cross-Correlation

Chapter 4

Cross-Correlation

All transmitted and received waves are real. However a matched filter is based on the

assumption of complex signals. Therefore in many systems the sampled real signal is used to

match the signal. In this case both the received and the reference signals will be real and therefore

a correlation used to match the signals. There will be a difference to the complex form, what we

will illustrate in this chapter.

The cross-correlation function measures the dependence of the values of one signal on

another signal, for continuous functions, and , the cross-correlation is defined as:

( )

∫ ( ) ( )

(4. 1)

Where is the observation time.

Since the output of the matched filter is the cross-correlation function of the received

signal and the transmitted signal, it is possible to implement the matched filter as a correlation

process based on the previous equation.

Similarly, for sampled signals the cross-correlation is defined as:

[ ]

∑ [ ] [ ]

3

(4. 2)

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Cross-Correlation

Where is the number of samples and [ ] is always a real valued signal.

We want to use the cross-correlation as another type of pulse compression. Here the

radar (simulated by our program) will take the cross correlation of two signals, the sent signal and

the received signal which is a delayed copy of the transmitted signal. Efficient implementations of

the cross-correlation processing based on the discrete Fourier transform are usually used by radar

systems. Analyzing the peaks in the cross-correlation signal, we could see a present object, its

location and the delay of the signal and therefore calculate how far away it is.

Thus, the matched filter is implemented by “convolving” the reflected echo with the “time

reversed” transmit pulse

Figure 16. Convolution of Reflected echo and Time reversed pulse.

The convolution process consists of a simple way, moving the digitized pulses in steps to

each other. When data overlaps, it multiplies samples and sums them up.

Figure 17. Convolution of Reflected echo and Time reversed pulse. Intermediate step.

For example, in the third sample, three samples overlap ( ) ( ) ( ) 3.

Therefore, as a result of applying the convolution; we will have two samples less than before.

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Cross-Correlation

Figure 18. Convolution of Reflected echo and Time reversed pulse. Finished convolution.

Now we will implement what we explained above in Matlab, to do so, we take advantage

of the predefined function “xcorr”, which is included in the Matlab’s library. When we want to

know how this functions works in Matlab, we realize, checking the “xcorr help” in Matlab, that

there are different types of applications for this function, however we will use the first one that

appears when we see the help.

>> help xcorr

XCORR Cross-correlation function estimates.

C = XCORR(A,B), where A and B are length M vectors (M>1), returns

the length 2*M-1 cross-correlation sequence C. If A and B are of

different length, the shortest one is zero-padded. C will be a

row vector if A is a row vector, and a column vector if A is a

column vector.

XCORR produces an estimate of the correlation between two random

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Cross-Correlation

(jointly stationary) sequences:

C(m) = E[A(n+m)*conj(B(n))] = E[A(n)*conj(B(n-m))]

It is also the deterministic correlation between two deterministic

signals.

As we have explained above, we apply the cross-correlation, to the sent signal (s) and to

the received signal (sdelayed), taking the real part of the chirp signal with we always work.

x=xcorr(real(s),real(sdelayed));

Figure 19. Matched filter by cross-correlation process.

3.8 3.9 4 4.1 4.2 4.3

x 10-6

0

100

200

300

400

500

600

700

800

900

1000

Time [s]

Am

plit

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Matched filter by cross-correlation process

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Cross-Correlation

Now, we have to corroborate the theory explained above, demonstrating that the result of

the cross correlation is the same that we obtained with the matched filters, we applied before to

the chirp pulse. To do that we will compare the two outputs graphics, we have generated in the

same plot.

Figure 20. Comparison of the responses of matched filter and cross-correlation process for a transmitted signal with = 75 MHz, = 2 µseconds , = 3,75·1013 (1/s2) and = 2 µseconds.

We are able to see that we obtain half of the power using the cross-correlation, due the

fact that we are using only the real parts of the signals in the correlation. The responses to both

processes seem different but we can see that both of them have the same envelope. To be able to

observe this in a better way, we will reduce the bandwidth of the signals to 5 Mhz instead of 75

MHz.

3.9 3.92 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 4.1

x 10-6

0

500

1000

1500

2000

Time [s]

Am

plit

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Matched filter with the original Linear FM Waveform signal

3.92 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 4.1

x 10-6

0

500

1000

Time [s]

Am

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Cross-Correlation

Figure 21. Comparison of the envelopes of the responses of matched filter and cross-correlation process for a transmitted signal with = 5 MHz, = 2 µseconds , = (1/s2) and

= 2 µseconds.

After this, we can say that we can work with the cross-correlation as another “type” of

matched filter for pulse compression.

0 1 2 3 4 5 6

x 10-6

0

500

1000

1500

2000

Time [s]

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1 2 3 4 5 6

x 10-6

0

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Analytic Signal

Chapter 5

Analytic Signal

All transmitted waves are real, however it was shown that it is convenient to transform

the real signal to a complex form. This complex form can then be matched, like we illustrated in

previous sections. To achieve this goal we will create an “analytic signal”, which is a common tool

in Mathematics and signal processing.

5.1. Definition of Hilbert transforms.

The Hilbert transform [ ( )] of a signal ( ) to construct the analytic signal is defined

[3] as

[ ( )] ( )

∫

( )

∫

( )

(5. 1. 1)

The Hilbert transform of ( ) is the convolution of ( ) with the signal ⁄ . It is the

response to ( ) of a linear time invariant filter (called Hilbert transformer), having impulse

response ⁄ . The Hilbert transform [ ( )] is denoted as ( ) in this point.

44

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Analytic Signal

5.2. Analytic Signal Application

The idea is that the imaginary component of the transmitted signal is orthogonal to the

real part. When we perform the analytic signal by the use of Hilbert transform we are discarding

the imaginary part at the same time, without loss of information. This makes certain attributes of

the signal more accessible and facilitates the derivation of modulation and demodulation

techniques. Consequently, the scalar product between the component real and imaginary part is

equal to zero.

The complex conjugate of an analytic signal contains only negative frequency components.

So, there is no loss of information or reversibility by discarding the imaginary component.

Obviously the real component of the complex conjugate is the same as the real component of the

analytic signal. But in this case, its extraction restores the suppressed positive frequency

components.

Figure 22. Real part and imaginary part of analytic signal. Orthogonal vectors

is the real part of the signal

𝑥𝑟(𝑡) 𝑥 (𝑡)

𝑥𝑟

𝑥

45

45

Analytic Signal

⁄

(5. 2. 1)

is the imaginary part of the signal

⁄

(5. 2. 2)

Due to the fact that they are orthogonal, we can carry on as following,

(5.2.3)

Analytic signals are often shifted in frequency (down-converted) towards 0 Hz, which

creates [non-symmetrical] negative frequency components. One motive is to allow low pass filters

with real coefficients to be used to limit the bandwidth of the signal. Another motivation is to

reduce the highest frequency, which reduces the minimum rate for alias-free sampling. A

frequency shift does not undermine the mathematical tractability of the complex signal

representation. So in this sense, the down-converted signal is still "analytic". However, restoring

the real-valued representation is no longer a simple matter of only extracting the real component.

Up-conversion is obviously required, and if the signal has been sampled (discrete-time),

interpolation (upsampling) might also be necessary to avoid aliasing.

As long as the manipulated function has no negative frequency components, we can get

the conversion from complex back to real, just discarding the imaginary part. In a few words, the

analytic signal corresponding to a real signal is a complex one, whose frequency spectrum is zero

for negative frequencies, and whose real part is equal to the original signal. The only difference

will be that in case we are discarding the negative frequencies, we are going to lose half of the

46

46

Analytic Signal

energy in our signal. The recovered signal from the real part of the analytic signal would be the

same signal than the original but with just half of its energy.

Mathematically, we could obtain the analytic signal in this way:

If the Fourier transform of a real signal ( ), is ( ), we use the positive frequencies

( ){

( )

( )

(5.2.1)

Figure 23. Analytic signal property Hilbert

And then, the analytic signal corresponding to the real ( ), is the inverse Fourier

transform of ( )

47

47

Analytic Signal

( ) { ( )}

(5.2.2)

We can also create the analytic signal ( ), from a real signal ( ), using the Hilbert

transform. Here we take advantage of the Hilbert function included in the Matlab’s library.

Figure 24. Diagram of the generation of the analytic signal.

Now we have created the analytic signal in this way:

( ) ( ) ( )

(5.2.3)

Where ( ) is the Hilbert’s transform of ( ).

As we can see, if we take the real part of the analytic signal that we have obtained, we will

have the original signal ( ).

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Analytic Signal

{ ( )} ( )

(5.2.4)

Now we will corroborate this mathematic theory implementing this system in Matlab with

a Chirp signal.

As said before, we will apply the predefined Matlab function to calculate this transform,

and if we write the command “help Hilbert” to understand how it works, we would see that:

>> help hilbert

HILBERT Discrete-time analytic signal via Hilbert transform.

X = HILBERT(Xr) computes the so-called discrete-time analytic

signal

X = Xr + i*Xi such that Xi is the Hilbert transform of real

vector Xr.

If the input Xr is complex, then only the real part is used:

Xr=real(Xr).

If Xr is a matrix, then HILBERT operates along the columns of

Xr.

Note that the Hilbert function produces the complete analytic signal, not just the

imaginary part. Thus, we will obtain the analytic signal only using the Hilbert predefined function

sanalytic = hilbert(real(sdelayed))

Now we already have a complex signal obtained from a real signal, and as we said before,

in radar communication we are used to work with real signals. Now, however, we will be able to

work with our original signal as a real signal, using the real part of the analytic signal generated.

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Analytic Signal

srecovered = real(sanalytic);

Here, we can see the comparison between the original signal and the signal that we have

obtained from the analytic signal.

Figure 25. Comparison between the original chirp signal and the recovered signal.

As we expected, we observe that the signals are identical. Furthermore, we want to know

if both signals work in the same way when we use the matched filter on them. For that we will

apply the same matched filter we designed for the chirp signal before and we get this result.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

x 10-6

-1

-0.5

0

0.5

1

Time [s]

Am

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Original Linear FM Waveform signal

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

x 10-6

-1

-0.5

0

0.5

1

Time [s]

Am

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Recovered Linear FM Waveform signal

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50

Analytic Signal

Figure 26. Response of the recovered signal to the matched filter.

Now we compare the response of both signals to the matched filter.

3.8 4 4.2

x 10-6

0

100

200

300

400

500

600

700

800

900

1000

Time [s]

Am

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Matched filter with the recovered Linear FM Waveform signal by its analytic signal

51

51

Analytic Signal

Figure 27. Comparison between responses to the matched filter.

As we can see, the response of both signals to the matched filter is exactly the same but in

the case of the recovered signal with half of the energy, due to the discard of the negative

frequencies implementing the analytic signal. In actual radar communication we would be able to

solve this problem with the calibration of our equipment because of the known situation and we

know that the loss of power does not mean a loss of the signal information. So it was right for our

work to apply the theory of the analytic signal to our program and now we are able to work with

complex signals as if they were real signals.

3.9 3.92 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 4.1

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3.9 3.92 3.94 3.96 3.98 4 4.02 4.04 4.06 4.08 4.1

x 10-6

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1000

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Matched filter with the recovered Linear FM Waveform signal by its analytic signal

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Analytic Signal

53

53

Different types of radar signals

Chapter 6


The next step is to investigate how the pulse compression works with different types of

radar signals. We have chosen binary phase coded pulse and square coded pulse. We will generate

both signals and apply them to the time delay, both matched filters (to the original signal and to

the recovered signal from the analytic signal) and the cross-correlation that we have implemented.

6.1. Binary phase coded pulse

A simple way to understand pulse compression with a matched filter is to consider the

binary phase-shift keying (BPSK) modulation technique. In this modulation the code is made up of

m chips which are either in-phase, 0⁰ (positive), or out-of-phase, 180⁰(negative), which is called

BPSK (Binary Phase Shift Keying). Each transmitted pulse is marked off into narrow (N) segments of

equal length (Tchip), and the radio frequency phase of certain segments is reversed (shifted by

180˚), according to a predetermined binary code. Received pulses are passed through a tapped

delay line having phase reversals in corresponding taps. The output pulse has the width of the

segments and it has sidelobes. Firstly, we explain barker coding. Barker codes are binary phase-

coding sequences of length N, which result in ambiguity functions with sidelobe levels, at Zero

Doppler, not higher than 1/N. However, only nine sequences are known. With Barker codes the

mainlobe to sidelobe ratio equals the pulse compression ratio, but the longest code is only 13

digits. The sidelobes can be eliminated by alternately transmitting complementary forms of the

four digit code and these can be chained to any length. Other binary codes can be made in

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54


practically any length, but their sidelobe characteristics, though reasonably good, are not quite so

desirable.

The principal limitation of phase coding is its sensitivity to Doppler. For an effective

scheme, the doppler shifts must be comparatively small or the uncompressed pulses reasonably

short, otherwise, performance deteriorates. Our code of generated signal is random. We have

selected a random modulated signal, from a uniform signal which interval is situated between 0

and 1. Then, we compare it with a round factor of .

Figure 28. Binary phase coded pulse with a bit rate equal to 20 nanoseconds, = 100 MHz and = 2µseconds.

0 1 2 3 4 5 6

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Binary Phase Coded signal

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55


Figure 29. Modulated binary phase coded pulse bit period.

We will get a detailed view by zoom, to be able to see the phase shifting in the signal.

0 1 2 3 4 5

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0.5

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1.5

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Original Digital Signal

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56


Figure 30. Detail of Binary phase coded pulse (BPSK signal) with a bit rate equal to 20 nanoseconds. (bit rate), = 100 MHz and = 2 µseconds.

1.7 1.75 1.8 1.85 1.9 1.95 2

x 10-6

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Binary Phase Coded signal

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57


Figure 31. Detail of modulated binary phase coded pulse.

Once we have generated a new type of radar signal, we will apply the same process to it

than to the chirp signal, to corroborate that our program works in the same way in case we

introduce different types of signals.

Now we apply a time delay to the Binary phase coded signal and then, we compare it with

the original signal.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-6

0

0.1

0.2

0.3

0.4

0.5

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0.8

0.9

1

Time (bit period)

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Original Digital Signal

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58


Figure 32. Comparison between original and delayed Binary phase coded pulses.

As can be observed in figure 34, the signal is delayed on time the 2µs as we defined when

we created the time delaying function.

This is the output matched filter where is 100 MHz, which is larger than the symbol rate

(50 MBaud or 50 MHz in this case).

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-0.5

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Real part of the delayed Binary Phase Coded signal

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x 10-6

-1

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Real part of the delayed Binary Phase Coded signal

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59


Figure 33. Binary phase coded signal in the frequency domain by the use of FFT with = 100 MHz, and symbol rate, = 50 MBaud.

The spectrum has a peak at the carrier frequency. It also has peaks at regular intervals

frequencies higher than the carrier frequency. There are no peaks at frequencies lower than the

carrier frequency. The distance between the peak of the first lobe in baseband and the first null in

the frequency domain is 50 MHz. As we have mentioned before, this is due to the symbol rate,

moreover, we did not use a raised cosine filter in this process.

After that, we will apply the different types of pulse compression we have implemented to

check how it affects to a Binary phase coded pulse. The cross correlation process or the process of

analytic signal and the Hilbert transform is the same, but the way to generate the transmitted and

the received signal are not necessarily the same. The pulse compression filter output by the use of

the three methods explained above, will be as following.

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 108

-500

-400

-300

-200

-100

0

100

Frequency [Hz]

Magnitude (

dB

)

Interpolated output of the Matched filter

60

60


Figure 34. Comparison of the responses to the different pulse compression processes with a Binary phase coded pulse.

We have used the BPSK code with the three methods explained above. The first plot

represents the matched filter process step by step, the second one is the matched filter process

through the “xcorr” function, which can we use in matlab. Finally, the last one represents the

recovered signal by the use of the Hilbert transforms and the analytic signal, as we have developed

on chapter five. We notice that the different pulse compression processes applied to a Binary

phase coded signal are the same, and the programs are working as we expected.

3.8 3.85 3.9 3.95 4 4.05 4.1 4.15

x 10-6

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1000

Time [s]

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Matched filter with the original Binary Phase Coded signal

3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2

x 10-6

500

1000

Time [s]

Am

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3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2

x 10-6

500

1000

Time [s]

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Matched filter with the recovered Binary Phase Coded signal by its analytic signal

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61


6.2. Square pulse

A square wave is a non-sinusoidal waveform which is used very often in signal processing.

An ideal square wave oscillates regularly between two levels. Here we show an illustration of the

ideal waveform for a square wave.

Figure 35. Ideal square pulse.

We called this ideal square wave because the switches between levels are instantaneous,

what is possible to see in this illustration. This feature makes it, due to its physical limitations,

impossible to generate an ideal square wave in analog and digital technology.

Knowing that we are not able to generate real and completely rectangular signals, because

due to the rise and fall of the signal always takes some time, we can implement an imperfect

square wave, which is a type of pulse, often used in radar systems. We will implement it, applying

a high frequency to a sinusoidal pulse in a finite period but it won't form a perfect square wave.

Afterwards, we implement our square pulse in Matlab, following the process described

above and using the same parameters as in the other example.

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62


Figure 36. Square pulse with = 25 MHz and = 2µseconds.

We will get a detailed view by zoom, to see how the signal really is.

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Square signal

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Figure 37. Detail of Square pulse with = 25 MHz and = 2µseconds.

Now we can follow the same procedure than with the Chirp and Binary phase coded

pulses, starting, as always, applying a time delay to the original signal that we have generated.

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05

x 10-6

-1

-0.8

-0.6

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Square signal

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64


Figure 38. Comparison between original and delayed Square pulse.

We observe that the time delay is correctly applied for a Square pulse, as well. Now, we

will apply the different types of pulse compression to the Square pulse.

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Real part of the delayed Square signal

0 1 2 3 4 5 6

x 10-6

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Real part of the delayed Square signal

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65


Figure 39. Comparison of the responses to the different pulse compression processes with a Square pulse.

0 1 2 3 4 5 6

x 10-6

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1000

2000

Time [s]

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Matched filter with the original Square signal

0 1 2 3 4 5 6

x 10-6

0

1000

2000

Time [s]

Am

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0 1 2 3 4 5 6

x 10-6

0

1000

2000

Time [s]

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Matched filter with the recovered Square signal by its analytic signal

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66


Figure 40. Response to the Matched filter with a Square pulse.

As we can see, the responses to every type of pulse compression, applied to a Square

pulse, are exactly the same. So, if we look at the test results with the different types of radar

signals, we can affirm that the simulations of pulse compression, which we implemented in

Matlab, work in the same way for every type of radar signal.

On the other hand, we are able to see that the signal is not compressed on time after the

pulse compression, so pulse compression is not an effective technique for this type of signal. A

better way to work with this kind of signal would be to work without any kind of pulse

compression and to use, for example, a low-price ship radar, working with = 10 GHz, which

sends shorter pulses ( = 0.1µs).

2.35 2.4 2.45 2.5 2.55 2.6 2.65

x 107

100

105

110

115

120

Frequency [Hz]

Magnitude (

dB

)

Interpolated output matched filter

67

67

Conclusions

Chapter 7

Conclusions

Pulse compression allows to use low transmitter power and still achieve the maximum

signal to noise ratio (SNR) and maximum resolution.

The drawbacks of applying pulse compression include:

Added complexity in the transmitter and the receiver

Blocking of the receiver when transmitting a long coded pulse (blind during the

time of the pulse).

These problems have been overcome with this thesis, as there is a wide range of benefits

that highly recommends to use the pulse compression in this area. For that reason we have used

matched filter as the technique for pulse compression in the development of this work.

An increase of effective radar power increases the radar system range capability. A long

pulse contains more power and this is why the effective pulse width achieved by a pulse

compression process leads to a wide range of advantages in resolution and detectability, due to

the increased SNR. In the project, we have worked with different types of radar signals using the

same technics of pulse compression for each of them, and obtaining the expected results of the

matched filter. The usefulness of pulse compression is determined by the width of the

uncompressed pulse and the bandwidth of the modulation within the uncompressed pulse. The

most impressive results come from broad uncompressed pulse widths that contain a modulated

pulse with a large bandwidth.

We have implemented different processors for pulses using matched filtering and

correlation processes. We have also created analytic signals from real signals to be able to use

matched filter processes. We have compared the results to demonstrate that all techniques we

have used work correctly and that they are effective for radar pulse compression.

68

68

Conclusions

It is especially important the information given in chapter 6, where different types of input

radar signals are used and the corresponding output signals are obtained either by applying the

matched filter process using cross correlation, or using analytic signal with the Hilbert transform.

With both processes the same results were obtained, but the way to perform the modulation and

demodulation process changes according to the different signals. The chirp signal oscillates in a

frequency sweep, but the binary phase coded signal is changing according to a random code in a

real case.

Despite of the fact that the programs developed are not a professional application, these

can be used as a didactic tool for other students to improve their knowledge in radar processing.

This would help to improve the understanding of different waveforms and their benefits or to

understand which process is the best in each case. To create a usable (user friendly) Matlab

program was the goal we wanted to reach with our project, and by the use of the pulse

compression and the matched filter described, it has been accomplished.

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69

Conclusions

References

[1] NadavLevanon, Eli Mozeson, “Radar Signals”,Jhon Wiley & Sons, 2004, Hoboken,

New Jersey

[2] Merrill I. Skolnik“Introduction to Radar Systems” Third Edition. McGraw-Hill Higher

Education. International Edition 2001.

[3] Frederick W. King, “Hilbert TransformsVolume1” University of Wisconsin, Eau

Claire, Encyclopedia of Mathematics and its Applications, May 2009

[4] John C. Curlander, Robert N. McDonough, “Synthetic Aperture Radar. Systems and

Signal Processing”. John Wiley & Sons, 1991

[5] Bassem R. Mahafza, “Radar Systems Analysis and Design Using Matlab”. Chapman &

Hall/CRC 2005. Taylor & Francis Group LLC

[6] MehrdadSoumekh, “Synthectic Aperture Radar, Signal Processing”, Jhon Wiley &

Sons, 1999

[7] David K. Barton, “Radar System Analysis and Modeling”Artech House, 2005.

[8] J. W. Arthur. “Moderm SAW-based pulse compression systems for radar applications”.

Electronic & Communication Engineering Journal. Volume 7. Dec 1195.

[9] George W. Stimson. “Introduction to Airbone Radar” Second Edition. George Stimson

III

[10] C. Gasquet, P. Witomski, “Fourier Analysis and Applications, Filtering, Numerical,

Computation, Wavelets” Springer-Verlag New York, 1999

[11] Hwei P. Hsu, “Análisis de Fourier”, Adison Wesley Iberoamericana, Wilmington,

Delaware, E.U.A, 1987.

[12] Alan V. Oppenheim, Ronald W. Schafer, “Discrete-Time Signal Processing”.

Prentice-Halls. A division of Simon & Schuster. The discrete Fourier transform and its

computation, 1989.

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Conclusions

[13] S. Lawrence Marple, Jr. ,“Computing the Discrete-Time Analytic” Signal via FFT”.

IEEE transactions on signal processing, Vol. 47, No. 9, September 1999.

[14] B. Boashash, "Estimating and Interpreting the Instantaneous Frequency of a Signal-

Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519-538, April

1992

[15] Merrill I. Skolnik“Radar Handbook” Second Edition”,McGraw-Hill, Chapter 20,

1990.

[16] Cook, C. E., M. Bernfeld, “Radar Signals: An introduction to Theory Application”,

Academic Press, New York, 1967

[17] Peyton Z. Peebles, Jr., “Radar principles”,Jhon Wiley & Sons, New York, 1998

[18] Ramon Nitzberg“Radar Signal Processing and Adaptative Systems” The Artech

House Radar Library. Norwood 1999.

[19] Ian G. Cumming, Frank Hay-Chee Wong., “Digital processing of synthetic aperture

radar data : algorithms and implementation”Artech House Remote Sensing Library,

2005.

[20] Toomay, J. C., “Radar Principles for the Non-Specialist”, New York, Van Nostrand

Reinhold, 1989

[21] J.L. Eaves, E.K. Reedy.“Principles of Modern Radar”, Van Nostrand Reinhold, New

York, 1987.

[22] Poularikas A. D. “The Hilbert Transform” The Handbook of Formulas and Tables

for Signal Processing. Ed. Alexander D. Poularikas, Boca Raton: CRC Press LLC,1999

[23] S. L. Hahn, “Comments on ‘A Tabulation of Hilbert Transforms for Electrical

Engineers”, IEEE Trans. on Commun., vol. 44, p. 768, July 1996.

[24] S. L. Hahn, “Hilbert transforms”, in The Transforms and Applications Handbook

(A.Poularakis, Ed.), Boca Raton FL: CRC Press, 1996, ch. 7.

[25] MIT Lincoln Laboratory. Radar Course

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Conclusions

[26] Cook and Bernfeld, 1967; Cook and Seibert 1988, patent

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Conclusions

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Conclusions

Appendix

Program code

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%% Pulse Compression for different kind of signals %%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%% 21th of May 2012 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%% Tomás Garnacho & Alberto Trejo %%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% %Data input

clc;close all;clear all t0=0; t1=2e-6; fs=1e9; Ts=1/fs; npoints=2000; fo=25e6;f1=30e6; % Start at 25 MHz, go up to 100MHz B=f1-fo%bandwith pulse chirp tau=t1-t0 k=B/tau n=1;%figure n

%% %Selection process of the signal

disp('[1]Square Pulse [2]Binary Phase Coded [3]Linear Frequency

Modulated '); signal=input('Choose signal: ');

while (signal~=1)&&(signal~=2)&&(signal~=3) disp('[1]Square Pulse [2]Binary Phase Coded [3]Linear Frequency

Modulated '); signal=input('Error: choose proper signal: '); end

disp('[0]No window [1]Hamming [2]Kaiser [3]Chebyshev'); winid=input('Choose window: ');

while (winid~=0)&&(winid~=1)&&(winid~=2)&&(winid~=3) disp('[0]No window [1]Hamming [2]Kaiser [3]Chebyshev'); winid=input('Error: choose proper window: '); end

if winid==0 wind='No window';

74

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Conclusions

elseif winid==1 wind='Hamming';

elseif winid==2 wind='Kaiser';

else wind='Chebyshev';

end

switch signal case 1 name='Square'; disp(sprintf('%s %s',name,wind)); tx=(t0:npoints-1)/fs; % 1 microsecond @ 1kHz sample rate v=sin(2*pi*fo*tx);

case 2 name='Binary Phase Coded'; disp(sprintf('%s %s',name,wind));

%This time variable is just for plot

% Enter the two Phase shifts - in Radians % Phase for 0 bit P1 = 0; % Phase for 1 bit P2 = pi;

% Frequency of Modulating Signal f = 100e6;

% The number of bits to send - Frame Length N = 100;

% Sampling rate - This will define the resoultion fs = 1e9;

% Time for one bit tbit=2e-8; t = (0:1/(fs-1):tbit);

% Generate a random bit stream bit_stream = round(rand(1,N));

% This time variable is just for plot time = [];

75

75

Conclusions

PSK_signal = []; Digital_signal = [];

for ii = 1: 1: length(bit_stream)

% The FSK Signal PSK_signal = [PSK_signal (bit_stream(ii)==0)*sin(2*pi*f*t +

P1)+... (bit_stream(ii)==1)*sin(2*pi*f*t + P2)];

% The Original Digital Signal Digital_signal = [Digital_signal (bit_stream(ii)==0)*... zeros(1,length(t)) + (bit_stream(ii)==1)*ones(1,length(t))];

time = [time t]; t = t + tbit;

end v=PSK_signal; v2=Digital_signal;

case 3 name='Linear FM Waveform'; disp(sprintf('%s %s',name,wind)); tx=(t0:npoints-1)/fs; % 1 microsecond @ 1kHz sample rate v=(exp(i*(2*pi*fo.*tx+pi*k.*tx.^2)));

otherwise disp('Undifined signal')

end

%% % Signal

t=(t0:3*npoints-1)/fs; s=[v zeros(1,2*npoints)]; figure(n); n=n+1; plot(t,s); xlabel('Time [s]');ylabel('Amplitude'),grid; title(sprintf('%s signal ',name));

t=(t0:3*npoints-1)/fs; s=[v zeros(1,2*npoints)]; figure(n); n=n+1; subplot(211) plot(t,real(s)); xlabel('Time [s]');ylabel('Amplitude'),grid;

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title(sprintf('Real part of the %s signal ',name)); subplot(212) plot(t,imag(s)); xlabel('Time [s]');ylabel('Amplitude'); title(sprintf('Imaginary part of the %s signal ',name));

%% %Original Digital Signal for PSK_signal

if (signal==2) s2=[v2 zeros(1,2*npoints)]; figure(n); n=n+1; plot(t,s2,'r','LineWidth',2); xlabel('Time (bit period)'); ylabel('Amplitude'); title('Original Digital Signal'); axis([0 t(end) -0.5 1.5]); grid on; else

end

%% %%Frequency domain chirp signal

om=(-fs/2):(fs)/(3*npoints):(fs/2)-((fs)/(3*npoints)); S=fft(s); figure(n); n=n+1; plot(om,fftshift(20*log10(abs(S)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); title(sprintf('Spectrum %s signal ', name));

%% %Signal delayed

td=2e-6; f=[0:floor((numel(s)-1)/2) -ceil((numel(s)-1)/2):-1]/(Ts*numel(s)); Sdelayed=S.*exp(-i*2*pi*f*td);

figure(n); n=n+1; plot(om,fftshift(20*log(abs(Sdelayed)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); title(sprintf('Spectrum of the delayed %s signal',name));

sdelayed=ifft(Sdelayed); figure(n); n=n+1; subplot(211);

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plot(t,real(sdelayed)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Real part of the delayed %s signal',name)); subplot(212); plot(t,imag(sdelayed)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Imaginary part of the delayed %s signal',name));

%% %Comparison of S & Sdelayed

figure(n); n=n+1; subplot(211) plot(om,fftshift(20*log(abs(S)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); axis([-5e8 5e8 -50 120]) title(sprintf('Comparison between %s and delayed %s',name,name)); subplot(212) plot(om,fftshift(20*log(abs(Sdelayed)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); axis([-5e8 5e8 -50 120])

figure(n); n=n+1; subplot(211) plot(t,real(s)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Real part of the delayed %s signal',name)); subplot(212); plot(t,real(sdelayed)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Real part of the delayed %s signal',name));

%% %Matched filter

H=conj(Sdelayed); So=S.*H;

%in the frequency domain figure(n); n=n+1; plot(om,fftshift(20*log10(abs(So)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); title('Interpolated output of the Matched filter');

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om2=(-fs/2):(fs)/(3*npoints):(fs/2)-((fs)/(3*npoints)); figure(n); n=n+1; plot(om2,fftshift(20*log10(abs(So)))),grid; xlabel('Frequency [Hz]');ylabel('Magnitude (dB)'); title('Interpolated output of the Matched filter');

%in the time domain so=ifft(So); figure(n); n=n+1; plot(t,abs(so)),grid; axis([3.7e-6 6e-6 0 2000]) xlabel('Time [s]');ylabel('Amplitude'); title('Interpolated output of the matched filter');

%% %Time vs Freq

figure(n); n=n+1; plot(t,f) xlabel('Time [s]');ylabel('Frequency [Hz]'); title('Time vs frequency');

%% %Correlated function

Npoints=2*npoints; tcorr=(t0:2*3*npoints-2)/fs; x=xcorr(real(s),real(sdelayed));%xcorr estimates the cross-correlation

sequence of a random process %double points due to correlation

figure(n); n=n+1; plot(tcorr,(x)),grid; xlabel('Time [s]');ylabel('Amplitude'); axis([3.7e-6 4.3e-6 0 max(x)]) title('Matched filter by cross-correlation process');

%% %Create analytic signal

sanalytic = hilbert(real(sdelayed));

figure(n); n=n+1; plot(t,real(sanalytic)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2])

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title(sprintf('Analytic %s signal',name));

figure(n); n=n+1; plot(t,imag(sanalytic)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Imaginary part of the analytic %s signal',name));

%% %Recovered signal and comparison with the original signal

srecovered = real(sanalytic);

figure(n); n=n+1; subplot(211); plot(t,real(sdelayed)); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Original %s signal',name)),grid; subplot(212); plot(t,srecovered); xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 -1.2 1.2]) title(sprintf('Recovered %s signal',name)),grid;

%% %Matched filter with the recovered signal

Srecovered=fft(srecovered);

H2=conj(Srecovered); So2=S.*H2;

so2=ifft(So2);

figure(n); n=n+1; plot(t,abs(so2)),grid; xlabel('Time [s]');ylabel('Amplitude'); axis([3.7e-6 4.3e-6 0 max(so2)]) title(sprintf('Matched filter with the recovered %s signal by its

analytic signal',name)); %% %No window

figure(n); n=n+1; subplot(311); plot(t,abs(so)),grid; xlabel('Time [s]');ylabel('Amplitude');

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title(sprintf('Matched filter with the original %s signal',name));

subplot(312); plot(tcorr,abs(x)),grid; xlabel('Time [s]');ylabel('Amplitude'); axis([0 6e-6 0 2000]) title('Matched filter by cross-correlation process');

subplot(313); plot(t,abs(so2)),grid; xlabel('Time [s]');ylabel('Amplitude'); title(sprintf('Matched filter with the recovered %s signal by its

analytic signal',name));

%% %Weightening %Window for cross-correlation function estimates.

if( winid == 0.) winc(1:2*3*npoints-1) = 1.; elseif(winid == 1.); winc = hamming(2*3*npoints-1)'; elseif( winid == 2.) winc = kaiser(2*3*npoints-1,pi)'; elseif(winid == 3.) winc = chebwin(2*3*npoints-1,60)'; end

%Window for the recovered signal by its analytic signal

if( winid == 0.) winp(1:3*npoints) = 1.; elseif(winid == 1.); winp = hamming(3*npoints)'; elseif( winid == 2.) winp = kaiser(3*npoints,pi)'; elseif(winid == 3.) winp = chebwin(3*npoints,60)'; end

x=x.*winc; so=so.*winp; so2=so2.*winp;

figure(n); n=n+1; subplot(311); plot(t,abs(so)),grid; axis([3.7e-6 4.3e-6 -10 2000]) xlabel('Time [s]');ylabel('Amplitude'); title(sprintf('Matched filter with the original %s signal',name));

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subplot(312); plot(tcorr,abs(x)),grid; axis([3.7e-6 4.3e-6 -10 2000]) xlabel('Time [s]');ylabel('Amplitude'); title('Matched filter by cross-correlation process');

subplot(313); plot(t,abs(so2)),grid; axis([3.7e-6 4.3e-6 -10 2000]) xlabel('Time [s]');ylabel('Amplitude'); title(sprintf('Matched filter with the recovered %s signal by its

analytic signal',name));

Pulse compression for different types of radar signals.830629/... · 2015-06-30 · Pulse...

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