1 radar signal processing

1

RADAR SignalProcessing

SOLO HERMELIN

Updated: 28.11.08http://www.solohermelin.com

http://www.solohermelin.com/

2

SOLO RADAR Signal Processing

Table of Contents

RADAR Signals

Waveform Hierarchy

RADAR TypesRadar Generic Procedures

Fourier Transform

Waveforms

Quadrature Form

Spectrum

Energy

Complex and Analytic Signals

Signal Duration and BandwidthComplex Representation of Bandpass Signals

Autocorrelation

Sampling and z-Transform

Nyquist-Shannon Sampling Theorem

3

SOLO RADAR Signal Processing

Table of Contents (continue – 1)

The Discrete Time Fourier Transform (DTFT)

The Discrete Fourier Transform (DFT)

Fast Fourier Transform (FFT)

Digital Filtering

Windowing

Doppler Frequency Shift

Coherent Pulse Doppler Radar

Signal Processing

Decision/Detection Theory

Search & Detect Mode

Acquisition Mode

References

4

SOLO

The transmitted RADAR RF Signal is:

( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=E0 – amplitude of the signal

f0 – RF frequency of the signal φ0 –phase of the signal (possible modulated)

The returned signal is delayed by the time that takes to signal to reach the target and toreturn back to the receiver. Since the electromagnetic waves travel with the speed of lightc (much greater then RADAR andTarget velocities), the received signal is delayed by

c

RRtd

21 +≅

The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

To retrieve the range (and range-rate) information from the received signal thetransmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.

ά < 1 represents the attenuation of the signal

RADAR Signal ProcessingRADAR Signals

5

SOLO

The received signal is:

( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

( ) ( ) tRRtRtRRtR ⋅+=⋅+= 222111 &

We want to compute the delay time td due to the time td1 it takes the EM-wave to reachthe target at a distance R1 (at t=0), from the transmitter, and to the time td2 it takes the EM-wave to return to the receiver, at a distance R2 (at t=0) from the target. 21 ddd ttt +=

According to the Special Theory of Relativitythe EM wave will travel with a constant velocity c (independent of the relative velocities ).21 & RR

The EM wave that reached the target at time t was send at td1 ,therefore

( ) ( ) 111111 ddd tcttRRttR ⋅=−⋅+=− ( )1

111 Rc

tRRttd

+⋅+=

In the same way the EM wave received from the target at time t was reflected at td2 , therefore

( ) ( ) 222222 ddd tcttRRttR ⋅=−⋅+=− ( )2

222 Rc

tRRttd

+⋅+=

RADAR Signal Processing

6

SOLO

The received signal is:

( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

21 ddd ttt += ( )1

111 Rc

tRRttd

+⋅+= ( )

2

222 Rc

tRRttd

+⋅+=

( ) ( )2

22

1

1121 Rc

tRR

Rc

tRRtttttttt ddd

+⋅+−

+⋅+−=−−=−

+

−+−+

+

−+−=−

2

2

2

2

1

1

1

1

2

1

2

1

Rc

Rt

Rc

Rc

Rc

Rt

Rc

Rctt d

From which:

or:

Since in most applications we canapproximate where they appear in the arguments of E0 (t-td), φ (t-td),however, because f0 is of order of 109 Hz=1 GHz, in radar applications, we must use:

cRR <<21,

1,2

2

1

1 ≈+−

+−

Rc

Rc

Rc

Rc

( )

−⋅

++

−⋅

+=

−⋅

−+

−⋅

−⋅≈− 2

.

201

.

1022

011

00 2

1

2

1

2

121

2

121

21

D

RalongFreqDoppler

DD

RalongFreqDoppler

Dd ttffttffc

Rt

c

Rf

c

Rt

c

Rfttf

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ˆˆˆ2cosˆ00 ϕπα

where 212

21

1212

021

01ˆˆˆ,,,ˆˆˆ,

2ˆ,2ˆ

dddddDDDDD tttc

Rt

c

Rtfff

c

Rff

c

Rff +=≈≈+=−≈−≈

Finally


Doppler Effect

7

SOLO

The received signal model:

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−≈ ϕπα 00 2cos

Delayed by two-way trip time

Scaled downAmplitude Possible phase

modulated

CorruptedBy noise

Dopplereffect

We want to estimate:

• delay td range c td/2

• amplitude reduction α

• Doppler frequency fD

• noise power n (relative to signal power)

• phase modulation φ

Return to Table of Content

8

SOLO Waveform Hierarchy

Radar Waveforms

CW Radars Pulsed Radars

FrequencyModulated CW

PhaseModulated CW

bi – phase & poly-phase

Linear FMCWSawtooth, or

Triangle

Nonlinear FMCWSinusoidal,

Multiple Frequency,Noise, Pseudorandom

Intra-pulse Modulation

Pulse-to-pulse Modulation,

Frequency AgilityStepped Frequency

FrequencyModulate Linear FM

Nonlinear FM

PhaseModulatedbi – phase poly-phase

Unmodulated CW

Multiple FrequencyFrequency

Shift Keying

Fixed Frequency

9

SOLO

( )tf

2

τ2

τ−

A

∞→t

2

τ+T2

τ−T

A

2

τ+−T2

τ−−T

A

t←∞−

T T

NONCOHERENT PULSESCOHERENT PULSES

( )tf

t

A

2

τ2

τ−T

AA

T T

A

22

τ+T2

2τ−T

A

T T

A

2

τ− 2

τ+T

TN

PULSED (UNCODED)

A Partial List of the Family of RADAR Waveforms

PRI – Pulse Repetition Interval PRF – Pulse Repetition Frequency

τ – Pulse Width [μsec]

PRF = 1/PRI

Pulse Duty Cycle = DC = τ / PRI = τ * PRF

Paverrage = DC * Ppeak

Pulse Waveform Parameters

Continuous Waves (CW)

Pulses• Coherent – Phase is predictable from pulse-to-pulse• Non-coherent – Phase from pulse-to-pulse is not predictable

Waveform Hierarchy

10

SOLO

( )tf

2

τ2

τ−

A

∞→t

2τ+T

2τ−T

A

2τ+−T

2τ−−T

A

t←∞−

T TA

t

A

t

A

LINEAR FM PULSECODED PULSE

T T

PULSED (INTRAPULSE CODING)

t

( )tf

A

2

τ2

τ−T

AA

T T

A

22

τ+T2

2τ−T

A

T T

A

2τ− 2

τ+T

TN

t

( )tf

A

2

τ2

τ−T

AA

T T

A

22

τ+T2

2τ−T

A

T T

A

2

τ− 2τ+T

TN

PHASE CODED PULSES HOPPED FREQUENCY PULSES

PULSED (INTERPULSE CODING)

t

( )tf

A

T

2/τ−

LOW PRFMEDIUM PRF

PULSED( )tf

T T T T

2/τ+

τ

HIGH PRF

TT T T

A Partial List of the Family of RADAR Waveforms (continue – 1)

Pulses

Waveform Hierarchy


11

SOLO RADAR Types

Frequency Modulated CW Radar Multi-Frequencies CW Radar

Step Frequency Pulse Radar Coherent Pulse Radar

Examples of CW and Pulse Radars


12

SOLO

Radar Generic Procedures: Matched Filters in RADAR Systems

• Transmits high frequency (f0) EM signal: ( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−≈ ϕπα 00 2cos

• Receives low power reflected EM signal that contains doppler information (f0 + fD):

• Down-converts to Intermediate Frequency (IF) signal (fIF + fD), Amplifies at Low Noise, and Automatically Controls the Gain (AGC) of the receiver:

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEGtE IFddDIFdIFIF +−+−⋅+−≈ ϕπα 2cos0

• Down-converts to Video Frequency (V) signal (fV + fD), (often using a Synchronous I,Q configuration), samples the video (A/D) for Digital Signal Processing.

• The Digital Signal Processing (DSP) performs Fast Fourier Transforms (FFT), to produce the Data Cube (Range, Doppler, Receiving Channels). Using the data DSP detects the potential targets, and computes the receiving delay td (Range), Doppler frequency (closing velocity), angular target position. According to the Radar policy, he will acquire the targets of interest, and will track them. Doing this he prevents unwanted signal (Clutter, ECM, …) to interfere with the target of interest received signals.

This presentation deals with some aspects of the Radar Digital Processing.

13

SOLO


14

Fourier Transform

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtftfF ωω exp:F

SOLO

Jean Baptiste JosephFourier

1768 - 1830

F (ω) is known as Fourier Integral or Fourier Transformand is in general complex

( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+=

Using the identities

( ) ( )tdtj δ

πωω =∫

+∞

∞− 2exp

we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1F=

( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )[ ]002

1

2exp

2expexp

2exp

++−=−=−=

−=

∫∫ ∫

∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

+∞

∞−

+∞

∞−

tftfdtfdd

tjf

dtjdjf

dtjF

ττδττπωτωτ

πωωττωτ

πωωω

( ) ( ){ } ( ) ( )∫+∞

∞−

==πωωωω

2exp:

dtjFFtf -1F

( ) ( ) ( ) ( )[ ]002

1 ++−=−∫+∞

∞−

tftfdtf ττδτ

If f (t) is continuous at t, i.e. f (t-0) = f (t+0)

This is true if (sufficient not necessary)f (t) and f ’ (t) are piecewise continue in every finite interval1

2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫+∞

∞−

dttf

15

( )atf −-1F

F ( ) ( )ωω ajF −exp

Fourier TransformSOLO( )tf

-1FF ( )ωFProperties of Fourier Transform (Summary)

Linearity 1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫+∞

∞−

F

Symmetry 2

( )tF-1F

F ( )ωπ −f2

Conjugate Functions3 ( )tf *

-1FF ( )ω−*F

Scaling4 ( )taf-1F

F

a

Fa

ω1

Derivatives5 ( ) ( )tftj n−-1F

F ( )ωω

Fd

dn

n

( )tftd

dn

n

-1FF ( ) ( )ωω Fj n

Convolution6

( ) ( )tftf 21-1F

F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫+∞

∞−

−= τττ dtfftftf 2121 :*-1F

F ( ) ( )ωω 21 FF

( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

= ωωωπ

dFFdttftf 2*

12*

1 2

1

Parseval’s Formula7

Shifting: for any a real 8( ) ( )tajtf exp

-1FF ( )aF −ω

Modulation9 ( ) ttf 0cos ω-1F

F( ) ( )[ ]002

1 ωωωω −++ FF

( ) ( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

−=−= ωωωπ

ωωωπ

dFFdFFdttftf 212121 2

1

2

1

16

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

(1) C.W.

( )2

cos00

0

tjtj eeAtAtf

ωω

ω−+==

0ω - carrier frequency

Frequency

( ) ( )∫+∞

∞−

= dtetfjF tjωωFourier Transform

( ) ( ) ( )00 22ωωδωωδω ++−= AA

jFFourier Transform

SOLO Fourier Transform of a Signal

17

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

(2) Single Pulse

( )

>≤≤−

=2/0

2/2/

τττ

t

tAtf

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( ) ( )( )2/

2/sin2/

2/ τωτωτω

τ

τ

ω AdteAjF tj == ∫−

Fourier Transform


18

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

>≤≤−

=2/0

2/2/cos 0

τττω

t

ttAtf

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( )

( )

( )

( )

( )

−

−

++

+

=

= ∫−

2

2sin

2

2sin

2

cos

0

0

0

0

2/

2/

0

τωω

τωω

τωω

τωωτ

ωωτ

τ

ω

A

dtetAjF tjFourier Transform


(3) Single Pulse Modulated at a frequency

0ω

ω

( )ωjF

0

τπω 2

0 +

2

τA

0ω

τπω 2

0 −τπω 2

0 +−

2

τA

0ω−

τπω 2

0 −−

τπω 2

20 +τπω 2

20 −


19

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

±±=>−≤−≤−+

=,2,1,0,2/0

2/2/cos 0

kkkTt

kTttAtf

rand

τττϕω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( )

( )

( )

( )

( )

−

−

++

+

=

= ∫−

2

2sin

2

2sin

2

cos

0

0

0

0

2/

2/

0

τωω

τωω

τωω

τωωτ

ωωτ

τ

ω

A

dtetAjF tj

Fourier Transform


(4) Train of Noncoherent Pulses (random starting pulses), modulated at a frequency 0ω

T - Pulse repetition interval (PRI)


20

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

( ) ( )( ) ( )( )[ ]

−++

+=

±±=>−≤−≤−

=

∑∞

=1000

0

coscos

2

2sin

cos

,2,1,0,2/0

2/2/cos

nPRPR

PR

PRseriesFourier

tntnn

n

tT

A

kkkTt

kTttAtf

ωωωωτω

τω

ωτ

τττω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


Fourier Transform


(5) Train of Coherent Pulses, of infinite length, modulated at a frequency 0ω


( ) ( ) ( ){

( ) ( ) ( ) ( )[ ]

+−+−+−−++

+

−+=

∑∞

=10000

00

2

2sin

2

nPRPRPRPR

PR

PR

nnnnn

n

T

AjF

ωωδωωδωωδωωδτω

τω

ωδωδτω

T/1 - Pulse repetition frequency (PRF)TPR /2πω =


21

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

( ) ( )( ) ( )( )[ ]

−++

+=

±±=>−≤−≤−

=

∑∞

=

≤≤−

1000

22

0

coscos

2

2sin

cos

2/,,2,1,0,2/0

2/2/cos

nPRPR

PR

PRNTt

NT

tntnn

n

tT

A

NkkkTt

kTttAtf

ωωωωτω

τω

ωτ

τττω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


Fourier Transform


(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω


( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

−−

−−

++−

+−

++

+

+

−+

−+

+++

++

++

+

=

∑

∑

∞

=

∞

=

10

0

0

0

0

0

10

0

0

0

0

0

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

nPR

PR

PR

PR

PR

PR

nPR

PR

PR

PR

PR

PR

TNn

TNn

TNn

TNn

n

n

TN

TN

TNn

TNn

TNn

TNn

n

n

TN

TN

T

AjF

ωωω

ωωω

ωωω

ωωω

τω

τω

ωω

ωω

ωωω

ωωω

ωωω

ωωω

τω

τω

ωω

ωωτω



22

Signal

( ) ( )

+=

±±=>−≤−≤−

= ∑∞

=11 cos

2

2sin

21,2,1,0,2/0

2/2/

nPR

PR

PRSeriesFourier

tnn

n

T

AkkkTt

kTtAtf ω

τω

τωτ

τττ

τ - pulse width0ω - carrier frequency

(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω



( ) ( )tAtf 03 cos ω=

t

A A

( )tf1

t

2

τ2

τ−T

A

T T

22

τ+T

22

τ−T

T T

2

τ− 2

τ+T

( )tf 2

t

TN

2/TN2/TN−

( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=

( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( )( )[ ]

−++

+=

±±=>−≤−≤−

=⋅⋅=

∑∞

=

≤≤−

1000

22

0

321

coscos

2

2sin

cos

2/,,2,1,0,2/0

2/2/cos

nPRPR

PR

PRNTt

NT

tntnn

n

tT

A

NkkkTt

kTttAtftftftf

ωωωωτω

τω

ωτ

τττω

( )

>≤≤−

=2/0

2/2/12 TNt

TNtTNtf ( ) ( )ttf 03 cos ω=


23

Range & Doppler Measurements in RADAR SystemsSOLORadar Waveforms and their Fourier Transforms

24

Range & Doppler Measurements in RADAR SystemsSOLORadar Waveforms and their Fourier Transforms


25

RADAR SignalsSOLO

Waveforms

( ) ( ) ( )[ ]tttats θω += 0cos

a (t) – nonnegative function that represents any amplitude modulation (AM)

θ (t) – phase angle associated with any frequency modulation (FM)

ω0 – nominal carrier angular frequency ω0 = 2 π f0

f0 – nominal carrier frequency

Transmitted Signal

( ) ( ) ( )[ ]{ }ttjtats θω += 0exp

Phasor (complex) Transmitted Signal


26

RADAR SignalsSOLO

Quadrature Form( ) ( ) ( )[ ]

( ) ( )[ ] ( ) ( ) ( )[ ] ( )tttattta

tttats

00

0

sinsincoscos

cos

ωθωθθω

−=+=

where: ( ) ( ) ( )[ ]( ) ( ) ( )[ ]ttats

ttats

Q

I

θθ

sin

cos

==

( ) ( ) ( ) ( ) ( )ttsttsts QI 00 sincos ωω −=

One other form: ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]tjtjtjtj eeta

tttats θωθωθω −−+ +=+= 00

2cos 0

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+= ( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:

Envelope of the signal

( ) ( ) tjetgts 0ω=

Phasor (complex) Transmitted Signal

Transmitted Signal


27

RADAR SignalsSOLO

Spectrum

Define the Fourier Transfer F

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtstsS ωω exp:F ( ) ( ){ } ( ) ( )∫+∞

∞−

==πωωωω

2exp:

dtjSSts 1-F

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+= ( ) ( ) ( )[ ]0*

02

1 ωωωωω −−+−= GGS-1FF

-1FF

( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:

( ) ( ) ( )[ ]tttats θω += 0cosInverse Fourier Transfer F -1

Envelope of the signalWe defined:


28

RADAR SignalsSOLO

Energy ( ) ( ) ( )[ ]tttats θω += 0cos

( ) ( ) ( )[ ]{ } ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≈++== dttadttttadttsEs2

022

2

122cos1

2

1: θω

Parseval’s Formula

Proof:

( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

= ωωωπ

dFFdttftf 2*

12*

1 2

1

( ) ( ) ( )∫+∞

∞−

−= dttjtfF ωω exp11

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

=−=−=πωωω

πωωω

πωωω

22exp

2exp 2

*

112*

2*

12*

1

dFF

ddttjtfFdt

dtjFtfdttftf

( ) ( ) ( )∫+∞

∞−

−=πωωω

2exp*

2

*

2

dtjFtf

If s (t) is real, than s (t) = s*(t) and

( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

=== ωωπ

dSdttsdttsEs

222

2

1:

29

RADAR SignalsSOLO

Energy (continue – 1) ( ) ( ) ( )[ ]tttats θω += 0cos

( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

=== ωωπ

dSdttsdttsEs

222

2

1:

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

−−−+−−−+

−−−−+−−=

−−+−−−+−=

−

−−

00

0000

0

*

0

*2

00

0

*

00

*

0

00

*

0

*

0

*

4

1

4

1

ϕϕ

ϕϕϕϕ

ωωωωωωωωωωωωωωωω

ωωωωωωωωωω

jj

jjjj

eGGeGG

GGGG

eGeGeGeGSS

For finite band (W << ω0 ) signals (see Figure)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )∫∫∫

∫∫∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

−+∞

∞−

=−−−−=−−

≈−−−=−−−

ωωωωωωωωωωωωω

ωωωωωωωωωω ϕϕ

dGGdGGdGG

deGGdeGG jj

*

0

*

00

*

0

2

0

*

0

*2

00 000

( ) ( ) gs EdGdSE2

1

2

1

2

1

2

1:

22 =≈= ∫∫+∞

∞−

+∞

∞−

ωωπ

ωωπ


30

RADAR SignalsSOLO

Complex and Analytic Signals

( ) ( ) ( )[ ]tttats θω += 0cosWe have the following definitions:

Real signal

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+=

( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=: Envelope of the signal

( ) ( ) ( )[ ] ( ) tjetgtjtjtats 00exp: ωθω =+= Complex Signal

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+= ( ) ( ) ( )[ ]0*

02

1 ωωωωω −−+−= GGS-1F

F

( ) ( ){ } ( ){ } ( )00 ωωω ω −=== GetgtsS tjFF

( ) ( ) ( ) ( ) ( )ωωωωω

ωωω SUS

GS 200

020 =

<>

≈−=

For Band limited signals

31

RADAR SignalsSOLO

Complex and Analytic Signals (continue – 1)

( ) ( ) ( )[ ] ( ) tjetgtjtjtats 00exp: ωθω =+=

Complex Signal

( ) ( ){ } ( ){ } ( )00 ωωω ω −=== GetgtsS tjFF

( ) ( ) ( ) ( ) ( )ωωωωω

ωωω SUS

GS 200

020 =

<>

≈−=

For Band limited signals

Analytic Signal

The Analytic Signal is a Complex Signal chosen that its spectrum if forced to be zero for ω<0.

( ) ( ) ( ) ( )[ ] ( )ωωωωω SsignSUS +== 12:~

( )

<−=>+

=01

00

01

:

ωωω

ωsign

( )[ ] ( )[ ] ( )t

jtsignU

πδωω +=+= −− 12 11 FF

The time function corresponding to the product of the spectrums of two time functions isgiven by the time convolution of the two functions

( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∫∫

+∞

∞−

+∞

∞−

−−

−+=

−

+−=== ξξ

ξπ

ξξπ

ξδξωωω dt

sjtsd

t

jtsSUS 2

~~ 11 FFts

32

RADAR SignalsSOLO

Analytic Signal

The Analytic Signal is a Complex Signal chosen that its spectrum if forced to be zero for ω<0.

( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∫∫

+∞

∞−

+∞

∞−

−−

−+=

−

+−=== ξξ

ξπ

ξξπ

ξδξωωω dt

sjtsd

t

jtsSUS 2

~~ 11 FFts

( ) ( ) ( ) ( ) ( )tsjtsdt

sjts ˆ~ +=

−+= ∫

+∞

∞−

ξξ

ξπ

ts

or

Complex and Analytic Signals (continue – 2)

From ( ) ( )[ ] ( ) ( ) ( )ωωωωω SjSSsignS ˆ1:~ +=+=

we have

( ) ( ) ( )( )

( )

<+=>−

=−=0

00

0ˆ

ωωωωω

ωωωSj

Sj

SsignjS

Assuming a Band Limited signal we can assume that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tsjtststsSUSS ˆ~2~ +=≈⇒=≈ ωωωω

where is the Hilbert Transform of s (t)( ) ( )∫

+∞

∞− −= ξ

ξξ

πd

t

sts

1:ˆ

(see “Hilbert Transformation” Presentation)


33

Signals

( ) ( )∫+∞

∞−

= fdefSts tfi π2

SOLO

Signal Duration and Bandwidth

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

∞+

∞−

∞+

∞−

−∞+

∞−

∞+

∞−

∞+

∞−

=

=

=

=

dffSfSdfdesfS

dfdefSsdfdefSsdss

tfi

tfitfi

ττ

τττττττ

π

ππ

2

22

( ) ( )∫+∞

∞−

= fdefSts tfi π2 ( ) ( ) ( )∫+∞

∞−

== fdefSfitd

tsdts tfi ππ 22'

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

+∞

∞−

+∞

∞−

−+∞

∞−

+∞

∞−

−+∞

∞−

=

−=

−=

−=

dffSfSfdfdesfSfi

dfdesfSfidfdefSfsidss

tfi

tfitfi

222

22

2'2

'2'2''

πττπ

ττπττπτττ

π

ππ

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSds 22 ττ

Parseval Theorem

From

From

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSfdtts2222

4' π

34

Signals

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =====dffS

fdfdfSd

fSi

dffS

fdtdetstfS

dffS

tdfdefStst

dffS

tdtstst

tdts

tdtst

t

fifi

22

2

2

2

22

2

2:

πππ

SOLO

Signal Duration and Bandwidth (continue – 1)

( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2Fourier

( ) ( )∫+∞

∞−

−−= tdetstifd

fSd tfi ππ 22( ) ( )∫

+∞

∞−

= fdefSfitd

tsd tfi ππ 22

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

−=

====tdts

tdtd

tsdtsi

tdts

tdfdefSfts

tdts

fdtdetsfSf

tdts

fdfSfSf

fdfS

fdfSf

f

fifi

22

2

2

2

22

2 2222

:

ππ ππππ

35

Signals

( ) ( ) ( ) ( ) ( )∫∫∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

=≤

dffSfdttstdttsdttstdtts

222222

2

2 4'4

1 π

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSdts22 τ

SOLO


0&0 == ftChange time and frequency scale to get

From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttgdttfdttgtf22

Choose ( ) ( ) ( ) ( ) ( )tstd

tsdtgtsttf ':& ===

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttsdttstdttstst22

''we obtain

( ) ( )∫+∞

∞−

dttstst 'Integrate by parts( )

=+=

→

==

sv

dtstsdu

dtsdv

stu '

'

( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

∞+

∞−

+∞

∞−

−−= dttststdttsstdttstst '' 2

0

2

( ) ( ) ( )∫∫

+∞

∞−

+∞

∞−

−= dttsdttstst 2

2

1'

( ) ( )∫∫+∞

∞−

+ ∞

∞−

= dffSfdtts2222

4' π

( )

( )

( )

( )

( )

( )

( )

( )∫

∫

∫

∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =≤dffS

dffSf

dtts

dttst

dtts

dffSf

dtts

dttst

2

222

2

2

2

222

2

244

4

1ππ

assume ( ) 0lim =→∞

tstt

36

SignalsSOLO


( )

( )

( )

( )

( )

( )

22

2

222

2

24

4

1

ft

dffS

dffSf

dtts

dttst

∆

∞+

∞−

+∞

∞−

∆

∞+

∞−

+∞

∞−

≤

∫

∫

∫

∫ π

Finally we obtain ( ) ( )ft ∆∆≤2

1

0&0 == ftChange time and frequency scale to get

Since Schwarz Inequality: becomes an equalityif and only if g (t) = k f (t), then for:

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttgdttfdttgtf22

( ) ( ) ( ) ( )tftsteAttd

sdtgeAts tt ααα αα 222:

22

−=−=−==⇒= −−

we have ( ) ( )ft ∆∆=2

1

37

Signals

t

t∆2

t

( ) 2ts

ff

f∆2

( ) 2fS

SOLO

Signal Duration and Bandwidth – Summary

then

( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2

( ) ( )

( )

2/1

2

22

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

tdts

tdtstt

t

( )

( )∫

∫∞+

∞−

+ ∞

∞−=tdts

tdtst

t2

2

:

Signal Duration Signal Median

( ) ( )

( )

2/1

2

2224

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

fdfS

fdfSff

f

π ( )

( )∫

∫∞+

∞−

+ ∞

∞−=fdfS

fdfSf

f2

22

:

π

Signal Bandwidth Frequency Median

Fourier

( ) ( )ft ∆∆≤2

1

38

Signal Duration and BandwidthSOLO

( )tf-1F

F ( )ωFRelationships from Parseval’s Formula

( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

= ωωωπ

dFFdttftf 2*

12*

1 2

1Parseval’s Formula7

Choose ( ) ( ) ( ) ( )tstjtftf m−== 21

( ) ( ),2,1,0

2

12

22 == ∫∫∞+

∞−

∞+

∞−

ndd

Sddttst

m

mm ω

ωω

π

( ) ( )tftj n−-1F

F ( )ωω

Fd

dn

n

and use 5a

Choose ( ) ( ) ( )n

n

td

tsdtftf == 21 and use 5b ( )tf

td

dn

n

-1FF ( ) ( )ωω Fj n

( ) ( ) ,2,1,02

1 22

2

== ∫∫∞+

∞−

∞+

∞−

ndSdttd

tsd mn

n

ωωωπ

Choosec

( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0

2* ==

= ∫∫

+∞

∞−

+∞

∞−

mndd

SdS

jdt

td

tsdtstj

m

mn

n

n

nmm ω

ωωωω

π

( ) ( )n

n

td

tsdtf =1

( ) ( ) ( )tstjtf m−=2


39

( ) ( ) ( )[ ]tttats θω += 0cos

SOLOComplex Representation of Bandpass Signals The majority of radar signals are narrow band signals, whose Fourier transform islimited to an angular-frequency bandwidth of W centered about a carrier angularfrequency of ±ω0.

Another form of s (t) is

( ) ( ) ( )( )

( ) ( ) ( )( )

( )

( ) ( ) ( ) ( )ttstts

tttatttats

QI

tsts QI

00

00

sincos

sinsincoscos

ωω

ωθωθ

−=

−=

sI (t) – in phase component sQ (t) – quadrature component

1

2

Define the signal complex envelope: ( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

Therefore:

( ) ( ) ( )[ ] ( )[ ]tstjtgts ReexpRe 0 == ω

( ) ( ) ( ) ( ) ( ) ( ) ( )tststjtgtjtgts *2

1

2

1exp

2

1exp

2

100 +=−+= ∗ ωω

or:

3

4

( ) ( ) ( )[ ]tjtjtats θω += 0expAnalytic (complex) signal


40

( ) ( ) ( )[ ]tttats θω += 0cos

SOLOAutocorrelation The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtstsRss ττ :

Real signal For

The Autocorrelation Function is defined as:

Properties of the Autocorrelation Function:

2 ( ) ( )ττ ssss RR =−

( ) ( ) ( ) ( ) ( ) ( )τττττ

ss

tt

ss RtdtststdtstsR =−=+=− ∫∫+∞

∞−

+=+∞

∞−

''''

1 ( ) ( ) ( ) ( ) ( ) sss EfdfSfStdtstsR === ∫∫+∞

∞−

+∞

∞−

*0 Es – signal energy

3

( ) ( ) ( ) ( ) ( ) ( ) 2222

2

20sss

EE

InequalitySchwarz

ss REtdtstdtstdtstsR

ss

==−≤−= ∫∫∫∞+

∞−

∞+

∞−

∞+

∞−

τττ

( ) ( )0ssss RR ≤τ

Autocorrelation is a mathematical tool for finding specific patterns, such as the presence of a known signal which has been buried under noise.

41

SOLOAutocorrelation (continue – 1(

The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtgtgRgg ττ *:

Signal complex envelope For


Properties of the Autocorrelation Function:

2 ( ) ( )ττ *gggg RR =−

( ) ( ) ( ) ( ) ( ) ( )τττττ

*''*'*'

gg

tt

gg RtdtgtgtdtgtgR =−=+=− ∫∫+∞

∞−

+=+∞

∞−

1 ( ) ( ) ( ) ( ) ( ) sgg EfdfGfGtdtgtgR 2**0 === ∫∫+∞

∞−

+∞

∞−

Es – signal energy

3

( ) ( ) ( ) ( ) ( ) ( ) 22

2

2

2

2

22

04** ggs

EE

InequalitySchwarz

gg REtdtgtdtgtdtgtgR

ss

==−≤−= ∫∫∫∞+

∞−

∞+

∞−

∞+

∞−

τττ

( ) ( )0gggg RR ≤τ

( ) ( ) ( )[ ]tjtatg θexp:=

42

SOLOAutocorrelation (continue – 2(

The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtgtgRgg ττ *:

Signal complex envelope For


3

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

∫ ∫∫ ∫

∫ ∫∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

=

+∞

∞−

+∞

∞−

∂∂+

∂∂=

−−∂∂==

∂∂=

0

111222

2

0

222111

1

0

212211

2

****

**00

gggg RR

gg

tdtgtgtdtgt

tgtdtgtgtdtgt

tg

tdtdtgtgtgtgRτ

τττ

ττ

( ) ( )0gggg RR ≤τ

( ) ( ) ( )[ ]tjtatg θexp:=

(continue – 1)Since Rgg (0) is a maximum of a continuous function at τ=0, we must have

( ) 002

==∂∂ ττ ggR

Therefore ( ) ( ) ( ) ( ) 0** =∂∂+

∂∂

∫∫+∞

∞−

+∞

∞−

tdtgt

tgtdtgt

tg


43

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

( ) ( ){ } ( ) σσ <==+∫

∞

−f

ts dtetftfsF0

L

SOLO

Sampling and z-Transform

( ) ( ){ } ( ) σδδ <−

==

−==−

∞

=

−∞

=∑∑ 0

1

1

00sT

n

sTn

n

T eeTnttsS LL

( ) ( ){ }( ) ( ) ( )

( ) ( ){ } ( ) ( )

<<−

=

=

−

==

−

∞+

∞−−−

∞

=

−∞

=

+∫

∑∑

0

00**

1

1

2

1 σσσξξπ

δ

δ

ξ

σ

σξ f

j

j

tsT

n

sTn

n

de

Fj

ttf

eTnfTntTnf

tfsF

L

LL

( )

( ) ( )( )

( )( )

( )

( )

( )( )

( )( )

( )

−=

−

−=

−=

∑∫

∑∫

∑

−−−

−−

Γ

−−

−−

Γ

−−

∞

=

−

tse

ofPoleststs

FofPoles

tsts

n

nsT

e

FResd

e

F

j

e

FResd

e

F

j

eTnf

sF

ξ

ξξ

ξ

ξξ

ξξξπ

ξξξπ

1

1

0

*

112

1

112

1

2

1

Poles of

( ) Tse ξ−−−1

1

Poles of

( )ξF

planes

Tnsn

πξ 2+=

ωj

ωσ j+

0=s

Laplace Transforms

The signal f (t) is sampled at a time period T.

1Γ2Γ

∞→R

∞→R

Poles of

( ) Tse ξ−−−1

1

Poles of

( )ξF

planeξ

Tnsn

πξ 2+=

ωj

ωσ j+

0=s

44

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

SOLO

Sampling and z-Transform (continue – 1)

( ) ( )( )

( )

( )

( ) ( ) ∑∑

∑∑

∞+

−∞=

∞+

−∞=−−→

∞+

−∞=−−

+→

+=−

−−

+=

−

+

−=

+

−

−−−=

−−=

−−

−−

nnTse

nts

T

njs

T

njs

e

ofPolests

T

njsF

TeT

Tn

jsF

T

njsF

eT

njs

e

FRessF

ts

n

ts

ππ

ππξξ

ξ

ξπξ

πξ

ξ

ξ

ξ

212

lim

2

1

2

lim1

1

2

21

1

*

Poles of

( )ξF

ωj

σ0=s

T

π2

T

π2

T

π2

Poles of

( )ξ*F plane

js ωσ +=


The poles of are given by( ) tse ξ−−−1

1

( ) ( )T

njsnjTsee n

njTs πξπξπξ 221 2 +=⇒=−−⇒==−−

( ) ∑+∞

−∞=

+=

n T

njsF

TsF

π21*

45

Fourier TransformSOLO

F F -1

frequency-B/2 B/2B

F F -1

-B/2 B/2

B

1/Ts-1/Ts frequency

Sample

Sampling a function at an interval Ts (in time domain)

Anti-aliasing filters is used to enforce band-limited assumption.

causes it to be replicated at 1/ Ts intervals in the other (frequency) domain.


46

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

SOLO


0=z

planez

Poles of

( )zF

C


The z-Transform is defined as:

( ){ } ( ) ( )( )

( ) ( )( )

−

−===

∑

∑

=

−

→

∞

=

−

=

iF

iF

iiF

Ts

FofPoles

T

F

n

n

ze

ze

F

zTnf

zFsFtf

ξξξ

ξ

ξξξξξ

1

0*

1

lim:Z

( ) ( )

<

>≥= ∫ −

00

,02

1 1

n

RzndzzzFjTnf

fCC

n

π

47



( ) ( ) ( )∑∑∞

=

−+∞

−∞=

=

+=

0

* 21

n

nsT

n

eTnfT

njsF

TsF

πWe found

For the δ (t) function we have:

( ) 1=∫+∞

∞−

dttδ ( ) ( ) ( )τδτ fdtttf =−∫+∞

∞−

The following series is a periodic function: ( ) ( )∑ −=n

Tnttd δ:

therefore it can be developed in a Fourier series:

( ) ( ) ∑∑

−=−=

n

n

n T

tnjCTnttd πδ 2exp:

where: ( )T

dtT

tnjt

TC

T

T

n

12exp

12/

2/

=

= ∫

+

−

πδ

Therefore we obtain the following identity:

( )∑∑ −=

−

nn

TntTT

tnj δπ2exp

Second Way

48

Fourier Transform

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtftfF νπνπ 2exp:2 F

( ) ( ) ( )∑∑∞

=

−+∞

−∞=

=

+=

0

* 21

n

nsT

n

eTnfT

njsF

TsF

π

( ) ( ){ } ( ) ( )∫+∞

∞−

== ννπνπνπ dtjFFtf 2exp2:2-1F

SOLOSampling and z-Transform (continue – 5)

We found

Using the definition of the Fourier Transform and it’s inverse:

we obtain ( ) ( ) ( )∫+∞

∞−

= ννπνπ dTnjFTnf 2exp2

( ) ( ) ( ) ( ) ( ) ( )∑∫∑∞

=

+∞

∞−

∞

=

−=−=0

111

0

* exp2exp2expnn

n sTndTnjFsTTnfsF ννπνπ

( ) ( ) ( )[ ]∫ ∑+∞

∞−

+∞

−∞=

−−== 111

* 2exp22 νννπνπνπ dTnjFjsFn

( ) ( ) ∑∫ ∑+∞

−∞=

+∞

∞−

+∞

−∞=

−=

−−==

nn T

nF

Td

T

n

TFjsF νπνννδνπνπ 2

1122 111

*

We recovered (with –n instead of n) ( ) ∑+∞

−∞=

+=

n T

njsF

TsF

π21*

Second Way (continue)

Making use of the identity: with 1/T instead of T

and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑

−−=−−

nn T

n

TTnj 11

12exp ννδννπ

( )∑∑ −=

−

nn

TntTT

tnj δπ2exp


49


Henry Nyquist1889 - 1976

http://en.wikipedia.org/wiki/Harry_Nyquist

Nyquist-Shannon Sampling Theorem

The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this band-limiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

Karl Küpfmüller 1887-1977

http://www.iec.ch/cgi-bin/tl_to_htm.pl?section=person&item=71

http://en.wikipedia.org/wiki/Image:Harry_Nyquist.jpg

50

Claude Elwood Shannon 1916 – 2001

http://en.wikipedia.org/wiki/Claude_E._Shannon

Fourier TransformSOLONyquist-Shannon Sampling Theorem

The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication").

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

Edmund Taylor Whittaker

1873 - 1956

Dennis Gabor 1900 - 1979

Vladimir Aleksandrovich Kotelnikov 1908 - 2005

John Macnaughten Whittaker

1905 - 1985

http://en.wikipedia.org/wiki/Image:Shannon.jpg

http://en.wikipedia.org/wiki/Image:Kotelnikov1.jpg

51

Fourier TransformSOLONyquist-Shannon Sampling Theorem (continue – 1)

• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.

• Start with a band limited signal s (t) ( )2

0 fBfforfS >≡

• Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz.

( ) ∑∞+

−∞=

−=

k sTkfjSfS

12* π

fjs π2=

( ) ( ) ( )

−= ∑

+∞

−∞=nsTnttsts δ* ( )

−= ∑

∞+

−∞=k sTjksSsS

π2*

L -1

L

FF -1

52

Fourier Transform

2

1

2

B

T

B

s

−<

SOLONyquist-Shannon Sampling Theorem (continue – 2)


BB

Ts

=

>

22

1

(Nyquist Sampling Rate)

• Complex signal band-limited to B/2 Hz requires B complex samples/second, or 2 B real samples/seconds (twice the highest frequency)

• Start with a band-limited signal f (t) ( )2

0 fBfforfF >≡ • Sample f (t) at a time period Ts,

replicates spectrum every 1/Ts Hz.

Nyquist-Shannon Sampling Theorem:


53

Fourier TransformSOLOThe Discrete Time Fourier Transform (DTFT)

• Start with a band limited signal s (t) ( )2

0 fBfforfS >≡

• Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz.

( )

−= ∑

∞+

−∞=k sTkfSfS

1*

( ) ( ) ( )

( ) ( )∑

∑∞+

−∞=

+∞

−∞=

−=

−=

nss

ns

TntTns

Tnttsts

δ

δ*

( ) ( )∫+∞

∞−

−= tdetsfS tfj π2 ( ) ( )∫+∞

∞−

= fdefSts tfj π2F

F -1

Continuous Fourier Transform

F

F -1

Discretization of a Continuous Signal ( ) ( )∫+∞

∞−

== fdefSTnts sTnfjs

π2

( ) ( ) ( )∑∑∞+

−∞=

−

=∞+

−∞=

− ==n

nf

fj

s

Tf

n

TnfjsDTFT

ss

s

s eTnseTnsfSπ

π2

1

2:

DTFT provides an approximation of the continuous-time Fourier transform.

Discrete Time Fourier Transform (DTFT)Define

54


The Discrete Time Fourier Transform (DTFT) (continue-1)


Discretization of a Continuous Signal ( ) ( )∫+∞

∞−

== fdefSTnts sTnfjs

π2

DTFT -1

DTFT

Discrete Time Fourier Transform (DTFT)

( ) ( ) ( )∑∑∞+

−∞=

−

=∞+

−∞=

− ==n

nf

fj

s

Tf

n

TnfjsDTFT

ss

s

s eTnseTnsfSπ

π2

1

2:

We can see that

( ) ( ) ( ) ( )∑∑∞+

−∞=

−

−∞+

−∞=

+−

===+n

DTFTnkj

nf

fj

sn

nf

fkfj

ssDTFT fSeeTnseTnsfkfS ss

s

1

222

πππ

The Discrete Time Fourier Transform SDTFT (fs) is periodic with period fs.Let compute

( ) ( )( )

( )( )

( )( )

( ) ( ) ( )[ ]( ) ( )∑ ∑

∑ ∫∫ ∑∫

∞+

−∞=

∞+

−∞=

=←≠←

+

−

−

∞+

−∞=

+

−

−

+

−

∞+

−∞=

−

+

−

=−

−=−=

==

ns

sn

nmnm

ss

f

fs

nmf

fj

s

n

f

f

nmf

fj

s

f

f n

nmf

fj

s

f

f

mf

fj

DTFT

TmsTnm

nmfTns

fnm

j

eTns

fdeTnsdfeTnsdfefS

s

s

s

s

s

s

s

s

s

s

s

s

1sin

2

10

2/

2/

2

2/

2/

22/

2/

22/

2/

2

π

π

π

π

πππ

( ) ( )∑+∞

−∞=

−=n

TnfjsDTFT

seTnsfS π2: ( ) ( )( )

( )

∫+

−

=s

s

s

T

T

nTfjDTFTss dfefSTTns

2/1

2/1

2π

55



Normalization of the frequency

DTFT -1

DTFT( ) ( )∑

+∞

−∞=

−=n

TnfjsDTFT

seTnsfS π2: ( ) ( )( )

( )

∫+

−

=s

s

s

T

T

nTfjDTFTss dfefSTTns

2/1

2/1

2π

( ) ( )[ ][ ]2/1,2/1

2/1,2/1

:

*

*

+−∈

+−∈=

f

TTf

Tff

ss

s

( ) ( )∑+∞

−∞=

−=n

nfjDTFT ensfS *2* : π

DTFT -1

DTFT ( ) ( )∫+

−

=2/1

2/1

*2 ** dfefSns nfjDTFT

π

Example ( ) 1,,1,002 −== − NneAns nfj π

( ) ( )( )

( )

( ) ( )

( ) ( )

( )

( )

( )[ ]( )[ ]

( ) ( )1*

0

0

*

*

**

**

*2

*21

0

*2*

0

0

0

00

00

0

0

0

*sin

*sin

1

1

−−−

−−

−−

−−−

−−−

−−

−−−

=

−−

−−=

−−=

−−== ∑

Nffj

ffj

Nffj

ffjffj

NffjNffj

ffj

NffjN

n

nffjDTFT

eff

NffA

e

e

ee

eeA

e

eAeAfS

π

π

π

ππ

ππ

π

ππ

ππ

|SDTFT(f*)|

Normalized Frequency

56



( ) ( )∑+∞

−∞=

−=n

nfjDTFT ensfS *2* : π

DTFT -1

DTFT ( ) ( )∫+

−

=2/1

2/1


π

Example ( )

≥==

=−

22&8,,00

21,,10,902

nn

nens

nfj

π

( )

≥==

=−

27&4,,00

26,,10,302

nn

nens

nfj

π

Frequency Resolution Increases with Observation Time N Ts

DTFT

DTFT


57

Fourier Transform

( ) ( )∑−

=

−=

1

0

2

:N

n

nkN

j

sDFT eTnskSπ

SOLOThe Discrete Fourier Transform (DFT)

Assume a periodic sequence, sampled at a time period Ts, such that s (n Ts) = s [(n+kN) Ts]

The Discrete Fourier Transform (DFT) requires an input function that is discrete and whose non-zero values have a limited (finite) duration.

Unlike the Discrete-time Fourier transform (DTFT), it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. Its inverse transform cannot reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions

For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:

58

Fourier Transform

( ) ( ) ( )∑∑−

=

−

=

−==

1

0

1

0

2

:N

n

nks

N

n

nkN

j

sDFT WTnseTnskSπ

SOLOThe Discrete Fourier Transform (DFT) (continue – 1)


where is a primitive N'th root of unityand is periodic

Nj

eWπ2

:−

=

n

Nm

Nj

n

Nj

Nmn

Nj

Nmn WeeeW =

=

=

−−+

−+

1

222 πππ

( )( )( )

( )( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

[ ]

( )( )( )

( )[ ]( )[ ]

N

N

N s

s

s

s

s

s

W

NNNNNNN

NNNNNNN

NN

NN

NN

S

DFT

DFT

DFT

DFT

DFT

TNs

TNs

Ts

Ts

Ts

WWWWW

WWWWW

WWWWW

WWWWW

WWWWW

NS

NS

S

S

S

⋅−⋅−

⋅⋅⋅

=

−−

−−−−−−−

−−−−−−−

−−

−−

−−

1

2

2

1

0

1

2

2

1

0

1121211101

1222221202

1222221202

1121211101

1020201000

[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix

59

Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 2)

nNmn WW =+

[ ] [ ] NH

NN IN

WW1=

Nj

eWπ2−

= 12

* −== WeW Nj

π

[ ]

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

=

−−−−−−−

−−−−−−−

−−

−−

−−

1121211101

1222221202

1222221202

1121211101

1020201000

NNNNNNN

NNNNNNN

NN

NN

NN

N

WWWWW

WWWWW

WWWWW

WWWWW

WWWWW

W

[ ] [ ]

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

==

−+−−+−−−−−−

−+−−+−−−−−−

+−+−−−

+−+−−−

+−+−−−

1112121110

2122222120

2122222120

1112121110

0102020100

*

NNNNNNN

NNNNNNN

NN

NN

NN

TN

HN

WWWWW

WWWWW

WWWWW

WWWWW

WWWWW

WW

Let multiply those two matrices

[ ] [ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( )

( )( ) ( )

( )

=

≠=−

−=−−

==

+++++=

−

−

−

−−

=

−

+−−−−

∑mkN

mkW

W

W

WW

WWWWWWWWWW

mk

mk

N

mk

NmkN

j

jmk

mNNkmjjkmkmkmk

HNN

01

1

1

1

1

1

0

111100

,

Where IN is the N x N identity matrix

60

Fourier Transform

( ) ( ) ( )∑∑−

=

−

=

−==

1

0

1

0

2

:N

n

nks

N

n

nkN

j

sDFT WTnseTnskSπ


For the sequence s (0), s (Ts),…,s [(N-1) Ts] we defined the Discrete Fourier Transform:

[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix

We found that

[ ] [ ] NH

NN IN

WW1= Where IN is the NxN identity matrix

Therefore the Inverse Discrete Fourier Transform (IDFT) is

[ ] NH

NN SWN

s1=

( ) ( ) ( )∑∑−

=

−

=

− ==1

0

21

0

11 N

n

nkN

j

DFT

N

k

nkDFTs ekS

NWkS

NTns

π

D.F.T.

I.D.F.T.

61


Second way to find the Inverse Discrete Fourier Transform (IDFT). Let compute:

( ) ( ) ( ) ( ) ( )∑ ∑∑∑∑

−

=

−

=

−−−

=

−

=

−−−

=

+==

1

0

1

0

21

0

1

0

21

0

2 N

n

N

k

rnkN

j

s

N

k

N

n

rnkN

j

s

N

k

rkN

j

DFT eTnseTnsekSπππ

( )

( )

( )

( )

( )

( )[ ] ( )[ ]( ) ( )

( )[ ]( )

( )[ ] ( )[ ]( ) ( )

( )[ ]( )

( )

( )( )[ ] ( )[ ]( ) ( )

≠−=−

=

−+

−

−+−

−

−

−

−=

−+

−

−+−

−

−=

−+

−−

−+−−=−

−=−

−

=−−

−−

−−

−−

−

=

−−

∑

Nmrn

NmrnN

rnN

jrnN

rnjrn

rnN

rnN

rn

rnN

rnN

jrnN

rnjrn

rnN

rn

rnN

jrnN

rnjrn

e

e

e

e

ern

Nj

rnj

rnN

j

Nrn

Nj

N

k

rnkN

j

0cossin

cossin

sin

sin

cossin

cossin

sin

sin

2sin

2cos1

2sin2cos1

1

1

1

1

2

2

2

2

1

0

2

ππππ

π

π

ππ

ππππ

ππ

ππππ

π

π

π

π

π

( ) ( )[ ] ,2,1,01

0

2

±±=+=∑−

=

+mTmNrsNekS s

N

k

rkN

j

DFT

π

62

Fourier Transform

( ) ( ) ( )∑∑−

=

−

=

−==

1

0

1

0

2

:N

n

nks

N

n

nkN

j

sDFT WTnseTnskSπ



where is a primitive N'th root of unityand is periodic

Nj

eWπ2

:−

=

n

Nm

Nj

n

Nj

Nmn

Nj

Nmn WeeeW =

=

=

−−+

−+

1

222 πππ

( )( )( )

( )( )

( )( )( )

( )[ ]( )[ ]

⋅−⋅−

⋅⋅⋅

=

−−

−−

−−

−−

−−

s

s

s

s

s

NN

NN

NN

NN

DFT

DFT

DFT

DFT

DFT

TNs

TNs

Ts

Ts

Ts

WWWWW

WWWWW

WWWWW

WWWWW

WWWWW

NS

NS

S

S

S

1

2

2

1

0

1

2

2

1

0

12210

23320

23420

12210

00000

63


The DFT ant Inverse DFT (IDFT) are given by

( ) ( )∑−

=

+=

1

0

21 N

k

nkN

j

DFTs ekSN

Tnsπ

( ) ( )∑−

=

−=

1

0

2

:N

n

nkN

j

sDFT eTnskSπ

IDFT

DFT

with the periodic properties

( )[ ] ( ),2,1,0 ±±=

=+m

TnsTmNns ss( ) ( )

,2,1,0 ±±==+

m

kSNmkS DFTDFT

The sequence s (0), s (Ts),…,s [(N-1) Ts] can be interpreted to be a sequence of finitelength, given for r = 0, 1,…,N-1, and zero otherwise or a periodic sequence, defined for all r.

64

Fourier Transform

( ) ( )∑−

=

−=

1

0

2

:N

n

nkN

j

sDFT eTnskSπ



( ) ( )∑−

=

+=

1

0

21 N

k

nkN

j

DFTs ekSN

Tnsπ

IDFT

DFT

( ) ( )∑+∞

−∞=

−=n

nfjDTFT ensfS *2* : π( ) ( )∫

+

−

=2/1

2/1


π

IDTFT

DTFT

The DTFT ant Inverse DTFT (IDTFT) where given by

We can see that DFT is a sampled version of DTFT by tacking:

( ) ( )[ ][ ]2/1,2/1

2/1,2/1

1,,1,0

*

*

+−∈

+−∈

−==⇒==

f

TTf

NkTN

kf

N

kfTf

ss

ss

( ) ( ) ( ) 1,,1,0:1

0

2

−====

−

=

−

∑ NkfSeTnskSsTN

kfDTFT

N

n

nkN

j

sDFT π

65

Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue –8)

We can see that DFT is a sampled version of DTFT :

( ) ( ) ( ) 1,,1,0:1

0

2

−====

−

=

−

∑ NkfSeTnskSsTN

kfDTFT

N

n

nkN

j

sDFT π

By changing f0 from 0.25 to 0.275 we move |SDTFT (f)| to the right, and since the samplingpoints didn’t change, we obtain different |SDFT (k)| values.

66


We can see that DFT is a sampled version of DTFT :

( ) ( ) ( ) 1,,1,0:1

0

2

−====

−

=

−

∑ NkfSeTnskSsTN

kfDTFT

N

n

nkN

j

sDFT π

Increase sampling density from N=20 to N=60.

67


Zero Padding

( ) ( )∑−

=

−=

1

0

2

:N

n

nkN

j

sDFT eTnskSπ


( ) ( )∑−

=

+=

1

0

21 N

k

nkN

j

DFTs ekSN

Tnsπ

IDFT

DFT

Let add to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:

( ) ( )

−+=−==

=1,,1,0

1,,1,0,

LNNk

NknkTnsTks ss

( ) ( )∑−

=

−=

1

0

2

:'L

m

mkL

j

sDFT eTksmSπ

Define: ( )∑−

=

−=

1

0

2N

n

mnL

j

s eTnsπ

( )∑−

=

−

=1

0

2N

n

L

Nmn

Nj

s eTnsπ

=

L

NmSDFT

68


Zero Padding (continue – 1)

We added to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:

( ) ( )

−+=−==

=1,,1,0

1,,1,0,

LNNk

NknkTnsTks ss

( ) ( ) ( )∑∑−

=

−−

=

−==

1

0

21

0

2

:'N

n

L

Nmn

Nj

s

L

m

mkL

j

sDFT eTnseTksmSππ

Define:

( )( )

( )∑ ∑∑ ∑−

=

−

=

−+−

=

−−

=

+==

1

0

1

0

21

0

21

0

2 11 N

k

N

k

L

Nmkn

Nj

DFT

N

n

L

Nmn

Nj

Tks

N

k

knN

j

DFT ekSN

eekSN

s

πππ

≠≠=

=

−

−

=

−

−=−

−=

−

−+

−−

−+

−−

−+

−+

−+

−+

−+

−

=

−+

∑

integer/0

integer/&/0

integer/&/1

sin

sin

1

1

1

2

2

1

0

2

notNLk

NLkNLkm

NLkNLkm

LN

mkN

LN

mk

e

ee

ee

e

e

e

ee

L

Nmk

N

Nj

L

Nmk

Nj

L

Nmk

Nj

L

Nmkj

L

Nmkj

L

Nmk

Nj

L

Nmkj

L

Nmk

Nj

L

NmkN

Nj

N

n

L

Nmkn

Nj

π

ππ

ππ

ππ

π

π

π

ππ

( ) ( )∑−

=

−

−+

−

−

=1

0

1

sin

sin1

'N

k

L

Nmk

N

Nj

DFTDFT

LN

mkN

LN

mk

ekSN

mSπ

ππ

69


Zero Padding (continue – 11)

We added to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:

( ) ( )

−+=−==

=1,,1,0

1,,1,0,

LNNk

NknkTnsTks ss

Define:

( ) ( ) ( )∑∑−

=

−

−+−

=

−

−

−

=

==

1

0

11

0

2

sin

sin1

:'N

k

LN

mkNN

j

DFTDFT

L

m

mkL

j

sDFT

L

Nmk

N

LN

mk

ekSNL

NmSeTksmS

π

πππ

We can see that S’ DFT has more points that S DFT by a factor of L/N, but it containsno more information because it uses only the N values s (nTs). If L/N is an integer then for the m=n L/N S’ DFT (m) = S DFT (n). Between thosepoints S’ DFT (m) is an interpolation of S DFT points, with the weight

≠≠=

=

−

−

−

−+

integer/0

integer/&/0

integer/&/1

sin

sin1

notNLk

NLkNLkm

NLkNLkm

LN

mkN

LN

mk

e L

Nmk

N

Nj

π

ππ

70


The Discrete Fourier Transform (DFT) (continue – 13)

Increase sampling density from N=20 to N=60.

0

0.5

1

0

60 - SAMPLE PULSE

Signal sample

Sig

nal a

mpl

itud

e

5 10 15 20 25 30 35 40 45 50 55 60

Zero Padding from n=21 to L=60.

DFT

DFT

DFT

71

SOLO

72

SOLOProperties of The Discrete Fourier Transform (DFT) (continue – 14)

( )mns − ( ) mkN

j

DFT ekSπ2−

Linearity 1 ( ) ( )nsns 2211 αα +

Shift of a Sequence2

3

4

5

Periodic Convolution

6

7

Conjugate

8

9

IDFTDFT ( ) ( )∑

−

=

−=

1

0

2

:N

n

nkN

j

DFT enskSπ

( ) ( )∑−

=

+=

1

0

21 N

k

nkN

j

DFT ekSN

nsπ

( ) ( )kSkS DFTDFT 2211 αα +

( ) ( )nsns 21 , Periodic Sequence(Period N)

( ) ( )kSkS DFTDFT 21 , DFT(Period N)

( ) nlN

jens

π2− ( )lkSDFT −

( ) ( )∑−

=−⋅

1

021

N

m

mnsms( ) ( )kSkS DFTDFT 21 ⋅

( ) ( )nsns 21 ⋅ ( ) ( )∑−

=−⋅

1

021

1 N

lDFTDFT lkSlS

N

( )ns∗ ( )kSDFT −∗

( )ns −∗ ( )kSDFT∗

Real & Imaginary ( )[ ]nsRe

( )[ ]nsImj

( ) ( ) ( )[ ] 2/kSkSkS DFTDFTeven −+= ∗

( ) ( ) ( )[ ] 2/kSkSkS DFTDFTodd −−= ∗

73

SOLOProperties of The Discrete Fourier Transform (DFT) (continue – 15)

( ) ( ) ( )[ ] 2/: nsnsnseven −+= ∗ ( )kSDFTReEven Part10

11

12 Symmetric Proprties(only when s (n) is real)

IDFTDFT ( ) ( )∑

−

=

−=

1

0

2

:N

n

nkN

j

DFT enskSπ

( ) ( )∑−

=

+=

1

0

21 N

k

nkN

j

DFT ekSN

nsπ

( ) ( )nsns 21 , Periodic Sequence(Period N)

( ) ( )kSkS DFTDFT 21 , DFT(Period N)

( )lkSDFT −

( ) ( )( )[ ] ( )[ ]( )[ ] ( )[ ]( ) ( )( ) ( )

−−∠=∠

−=

−−=

−=

−= ∗

kSkS

kSkS

kSmkSm

kSkS

kSkS

DFTDFT

DFTDFT

DFTDFT

DFTDFT

DFTDFT

II

ReRe

Odd Part ( ) ( ) ( )[ ] 2/: nsnsnsodd −−= ∗


74

Fourier TransformSOLOFast Fourier Transform (FFT)

John Wilder Tukey 1915 – 2000

http://en.wikipedia.org/wiki/John_Tukey

James W. Cooley1926 -

http://www.ieee.org/portal/pages/about/awards/bios/2002kilby.html

The Cooley-Tukey algorithm, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of smaller DFTs of sizes N1 and N2, recursively, in order to reduce the computation time to O(N log N) for highly-composite N (smooth numbers).

FFTs became popular after J. W. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm (first invented by Gauss) and describing how to perform it conveniently on a computer

http://en.wikipedia.org/wiki/Image:John_Tukey.jpg

75

Fourier TransformSOLOFast Fourier Transform (FFT)

The radix-2 DIT Algorithm

The radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley-Tukey algorithm, although highly optimized Cooley-Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage.

( ) ( ) ( )∑∑−

=

−

=

−==

1

0

1

0

2

:N

n

nks

N

n

nkN

j

sDFT WTnseTnskSπ


1,1, 22/12

*2

+==−====→= −−−− ππππ

jNjevenN

NNj

Nj

eWeWWeWeW

Suppose N is a power of 2; i.e. N=2L (L is integer). Since N is a even integer, let compute SDFT (k) by separate s (nTs) into two (N/2)-point sequences consisting of the even-numberedpoints (n=2r) and odd numbered points (n=2r+1).

( ) ( )( )

( ) ( )( )

( ) ( )( )( ) ( )( )

∑∑

∑∑−

=

−

=

−

=

+−

=

++=

++=

12/

0

212/

0

2

12/

0

1212/

0

2

122

122

N

n

kr

Nk

N

N

n

kr

N

N

n

krN

N

n

krNDFT

WrsWWrs

WrsWrskS

76



The radix-2 DIT Algorithm (continue – 1)

2/2/

2222

NN

jN

j

N WeeW ==

=

−− ππ

We divided the N-point DFT into two N/2-points DFTs.

( ) ( ) ( )( )( ) ( )( )

( )( )

( )

( )( )

( )

kH

N

n

krN

kN

kG

N

n

krN

N

n

krN

kN

N

n

krNDFT

WrsWWrs

WrsWWrskS

∑∑

∑∑−

=

−

=

−

=

−

=

++=

++=

12/

02/

12/

02/

12/

0

212/

0

2

122

122

Since

77





Reduction of an 8-points FFT to two4-points FFTs

A 2-points FFT(Butterfly)


78




Flow Diagram for an 8-points FFT

79



The radix-2 DIT Algorithm (continue – 2) ( ) ( ) kkj

kN

N

jNk

N eeW 12

2/ −==

= −− π

π


( ) ( ) ( ) ( )[ ]( )( ) ( )

( )

( )

∑∑−

=−

−

=

+

++=++=

12/

01

2/12/

0

2/ 2/2/N

n

knN

NkN

N

n

NnkN

knNDFT WWNnsnsWNnsWnskS

k

Since N/2 is an even integer (N=2L)

( ) ( ) ( )[ ]( )

( )( )

( )

( )

tgofFFTN

N

n

nlN

WW

N

N

n

nlN

ng

DFT WngWNnsnslkSNN

L

2/

12/

02/

2

12/

0

2 2/2

2/2 ∑∑−

=

=

=

−

=

=++==

( ) ( ) ( )[ ]( )

( )( )

( )

( )

thofFFTN

N

n

nlN

WW

N

N

n

nlN

nh

nNDFT WnhWWNnsnslkS

NN

L

2/

12/

02/

2

12/

0

2 2/2

2/12 ∑∑−

=

=

=

−

=

=+−=+=

80







A 2-points FFT(Butterfly)

81




Flow Diagram for an 8-points FFT

82

Fourier Transform

( ) ( ) 1,,1,0:1

0

2

−== ∑−

=

−NkeTnskS

N

n

nkN

j

sDFT π

8 64 24 64 8

16 256 64 256 24

32 1024 160 1024 64

64 4096 384 4096 160

128 16384 896 16384 384

SOLOFast Fourier Transform (FFT)

Arithmetic Operations for a Radix FFT versus DFT

For N = 2L we have L stages of Radix FFT and:

For N-point DFT we have:

For each row we have N complex additions and N complex multiplications, therefore for the N rows we have

Number of complex additions DFT = Number of complex multiplications DFT = NxN=N2

Number of complex additions FFT =N L=N log2 N

Number of complex additions FFT =N/2 (multiplications per stage) x L -1 =N/2 log2 (N/2)

Operation

Complex additions Complex multiplications

DFT DFTFFT FFTN=2L

Approximate number of Complex Arithmetic Operations Required for 2L-point DFT and FFT computations


83

Fourier TransformSOLODigital Filtering

Digital Filters can be partitioned in two distinct classes:

• Finite Impulse Response (FIR) filters that have an impulse response h (nT) of finite duration

( ) ( )

≥<−=

=Nnn

NnnhTnh

&,00

1,,1,0

• Infinite Impulse Response (IIR) filters that have an impulse response h (nT) of infinite duration

If s (n) is an input signal to the digital filter, then the output of the digital filter y (k) is related to the input by a relation of the type:

( ) ( ) ( )[ ] ( )[ ]( )[ ] ( )[ ]TMkybTkyb

TNksaTksaTksaTky

M

N

−−−−−−++−+=

1

1

1

10

If all the coefficients ai, bi are constants we can use the z transform to obtain:

( ) ( ) ( ) ( )zSzHzSzbzb

zazaazY

MM

NN ⋅=⋅

++++++= −−

−−

1

1

110

1

For a causal filter N ≤ M.

84

Fourier TransformSOLODigital Filtering (continue – 1)

( ) ( ) ( )zSzHzY ⋅=

( )N

N

NN

zbzb

zazaazH −−

−−

++++++=

1

1

110

1

If b1 = b2= … =bN =0 ( ) NN zazaazH −− +++= 1

10

This is Finite Impulse Filter (FIR) with ( ) −=

=otherways

Nnanh n

0

1,,1,0

If this is not the case we obtain the Infinite Impulse Filter (IIR) with

( ) Hn

nNN

NN rzzczcazbzb

zazaazH <++++=

++++++= −−

−−

−−

1101

1

110

1

where

( ) ( )zCzrzdzzHzj

c H

C

nn ∈∀<= ∫π2

1

85


( ) ( ) ( )zSzHzY ⋅= ( )N

N

NN

zbzb

zazaazH −−

−−

++++++=

1

1

110

1Rewrite this as:

( ) ( ) ( ) ( ) ( ) 00111

111 =−+−++−+− −

−−−−− YSaYbSazYbSazYbSaz NN

NNN

N

( ) ( ) ( ) ( )YbSazYbSazYbSazSaY NNN

NNN −+−++−+= −

−−−−−

111

111

0

Finally:

Transformation from Transfer Function to State-Space (Method 1)

86


( ) ( ) ( )zSzHzY ⋅= ( )N

N

NN

zbzb

zazaazH −−

−−

++++++=

1

1

110

1Rewrite this as:

( )( )( )( ) SazazazaW

WzbzbzbYN

NN

N

NN

NN

01

11

1

11

11 1

++++=

++++=−−−

−−

−−−−

−

Transformation from Transfer Function to State-Space (Method 2)

87


88


89


90



91

Fourier TransformSOLOWindowing

• Windowing is used for DFT data to reduce Doppler side lobes

• Windowing widen main lobe and this decreases Doppler resolution

• Windowing reduces the peak of the DFT producing a processing loss, PL

• Windowing causes a modest signal to noise (S/N) loss, called loss in peak gain, or LPG.

Windows are an overlay applied to a given time series to improve the spectral qualityof the data base.

92

Fourier TransformSOLOWindowing

Rectangular [ ] ≤≤

=otherwise

Mnnw

,0

0,1

Bartlett(triangular) [ ]

≤<−≤≤

=otherwise

MnMMn

MnMn

nw

,0

2/,/22

2/0,/2

Hanning

Hammming

[ ] ( ) ≤≤−

=otherwise

MnMnnw

,0

0,/2cos5.05.0 π

[ ] ( ) ≤≤−

=otherwise

MnMnnw

,0

0,/2cos46.054.0 π

Blackman [ ] ( ) ( ) ≤≤+−

=otherwise

MnMnMnnw

,0

0,/4sin08.0/2cos5.042.0 ππ

Julius Ferdinand von Hann (1839 -1921)

Richard Wesley Hamming (1915 –1998)

93

Fourier TransformSOLOWindowing (continue – 1)

cosine

[ ]

≤≤<

−−=

otherwise

MnM

Mnnw

,0

0&5.02/

2/

2

1exp

2

σσ

Lanczos[ ]

≤≤

−

=otherwise

MnM

nnw

,0

0,12

sinc

Gauss

[ ]

≤≤

=

−

=otherwise

MnM

n

M

nnw

,0

0,sin2

cosπππ

[ ]( )

≤≤

−−

=

otherwise

MnI

Mn

I

nw

,0

0,

12

1

0

2

0

α

αKaiser

α=2π

α=3π

94


Bartlett–Hann window

( )

38.0;42,0;62.0

1

2cos

2

1

1

210

210

===

−−−

−−=

aaa

N

na

N

naanw

π

Bartlett–Hann window; B=1.46 Low-resolution (high-dynamic-range) windows

Nuttall window, continuous first derivative

( )

012604.0;144232.0;487396,0;355768.0

1

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

πππ

Nuttall window, continuous first derivative; B=2.02

Blackman–Harris window

( )

01168.0;14128.0;48829,0;35875.0

1

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

πππ

Blackman–Nuttall window Blackman–Harris window, B=1.98

Blackman–Nuttall window, B=3.77

( )

0106411.0;1365995.0;4891775,0;3635819.0

1

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

πππ

95


Dolph-Chebyshev window

( ) ( )[ ]

( ) ( )[ ]( ) ( )4,3,2,10cosh

1cosh

1,,2,1,0,coshcosh

coscoscos

1

1

1

≈

=

−=

=

=

−

−

−

αβ

β

πβω

ω

α

N

NkN

Nk

N

W

WIDFTnw

k

k

The α parameter controls the side-lobe level via the formula:

Side-Lobe Level in dB = - 20 α

The Dolph-Chebyshev Window (or Dolph window) minimizes the Chebyshev norm of the side lobes for a given main lobe width 2 ωc:

( ) ( ){ }ωωω WWsidelobescwwww >=∞= ∑=∑ maxmin:min

1,1,

The Chebyshev norm is also called the L - infinity norm, uniform norm, minimax norm, or simply the maximum absolute value.

96


Comparison of Windows

97

Fourier Transform

SOLOWindowing (continue – 4)


WindowType

Peak Sidelobe

Amplitude (Relative)

Approximate Width of Mainlobe

Peak Approximation

Error20 log10δ

(dB)

Equivalent Kaiser

Windowβ

Transition Width

of EquivalentKaiser

Window

Rectangular -13 4π/(M+1) -21 0 1.81π/M

Bartlett -25 8π/M -25 1.33 2.37π/M

Hanning -31 8π/M -44 3.86 5.01π/M

Hamming -41 8π/M -53 4.86 6.27π/M

Blackman -57 12π/M -74 7.04 9.19π/M

98



99


Effect of Window in the Fourier Transform

• Good Effects

- Reduction of sidelobes

- Reduction of straddle loss

• Bad Effects

- Reduction in peak

- Widening of mainlobe

- Reduction in SNR

No Window

Hamming Window

∑−

=

1

0

21 N

nnw

N

21

0

1

0

2

1

∑

∑−

=

−

=

N

nn

N

nn

w

w

N

100



106Return to Table of Content

107

SOLO Doppler Frequency Shift

( )ωjF

2

NAτ

ω

TNπω 2

0 +

0ω−

TNπω 2

0 −

PRωω +− 0PRωω −− 0

TPR

πω 2=TPR

πω 2=

ω0

TNπω 2

0 +

0ω

TNπω 2

0 −

PRωω +0PRωω −0

TPR

πω 2=TPR

πω 2=

2

2sin

2 τω

τωτ

n

n

NA

PR

PR

( )

( )2

2sin

0

0

NT

NT

ωω

ωω

−

−

( )

( )2

2sin

2

2s in

20

0

NTn

NTn

n

n

NA

RP

RP

PR

PR

ωωω

ωωω

τω

τωτ

−−

−−

( )ωjF

( )02ωωδτ −NA

ω

0ω−PRωω +− 0PRωω −− 0

TPR

πω 2=TPR

πω 2=

ω0

PRωω +0PRωω −0

TPR

πω 2=TPR

πω 2=

2

2sin

2 τω

τωτ

n

n

NA

PR

PR

2

2sin

2 τω

τωτ

n

nNA

PR

PR

0ω PRωω 20 +PRωω 20 −PRωω 20 −−

PRωω 30 −−PRωω 40 −−

PRωω 20 +−

PRωω 30 +− PRωω 40 +−

Fourier Transform of an Infinite Train Pulses

Fourier Transform of an Finite Train Pulses of Lenght N

( )PR

PR

PR

NA ωωωδτω

τωτ

−−

0

2

2sin

2

( ) ( )tAtf 03 cos ω=

t

A A

( )tf1

t

2

τ2

τ−T

A

T T

22

τ+T

22

τ−T

T T

2

τ− 2

τ+T

( )tf 2

t

TN

2/TN2/TN−

( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=Train of Coherent Pulses,of finite length N T,modulated at a frequency 0ω

The pulse coherency is a necessary conditionto preserve the frequency information andto retrieve the Doppler of the returned signal.

Transmitted Train of Coherent Pulses

108

SOLO Doppler Frequency Shift

Fourier Transform of an Finite Train Pulses of Lenght N

2

NAτ

ω

TN

πω 20 +

0ω

TN

πω 20 −

PRωω+0PRωω−0

TPR

πω 2=TPR

πω 2=

2NAτ

ω

TN

πω 20 +

0ω

TN

πω 20 −

PRωω+0PRωω−0

TPR

πω 2=TPR

πω 2=

2

2sin

2 τω

τωτ

n

n

NA

PR

PR

( )

( )2

2sin

0

0

NT

NT

ωω

ωω

−

−

2NAτ

ω

TN

πω 20 +

0ω

TN

πω 20 −

PRωω+0PRωω−0

TPR

πω 2=TPR

πω 2=

πω

λ 2&

2P R

DopplerDopple r ftdRd

f <

−=

πω

λ 2&

2PR

Dopple rDopple r ftdRd

f >

−=

Fourier Transform of theTransmitted Signal

Fourier Transform of theReceiveded Signal

with Unambiguous Doppler

Fourier Transform of theReceiveded Signal

with Ambiguous Doppler

Received Train of Coherent Pulses

The bandwidth of a single pulse is usually several order of magnitude greater than theexpected doppler frequency shift 1/τ >> f doppler. To extract the Doppler frequency shift,the returns from many pulses over an observation time T must be frequency analyzed sothat the single pulse spectrum will separate into individual PRF lines with bandwidthsapproximately given by 1/T.

From the Figure we can seethat to obtain an unambiguousDoppler the following conditionmust be satisfied:

PRFc

td

Rdf

td

Rd

f PRMaxMaxdoppler =≤==

πω

λ 2

22 0

or02 f

PRFc

td

Rd

Max

≤


109

SOLO Coherent Pulse Doppler Radar

An idealized target doppler response will provide at IF Amplifier output the signal:

( ) ( )[ ] ( ) ( )[ ]tjtjdIFIF

dIFdIF eeA

tAts ωωωωωω +−+ +=+=2

cos

that has the spectrum:f

fIF+fd-fIF-fd

-fIF fIF

A2/4A2/4 |s|2

0

Because we used N coherent pulses ofwidth τ and with Pulse Repetition Time Tthe spectrum at the IF Amplifier output

f

-fd fd

A2/4A2/4|s|2

0

After the mixer and base-band filter:

( ) ( ) [ ]tjtjdd

dd eeA

tAts ωωω −+==2

cos

We can not distinguish between positive to negative doppler!!!

and after the mixer :

110


We can not distinguish between positive to negative doppler!!!

Split IF Signal:

( ) ( )[ ] ( ) ( )[ ]tjtjdIFIF

dIFdIF eeA

tAts ωωωωωω +−+ +=+=2

cos

( ) ( )[ ]

( ) ( )[ ]tAts

tA

ts

dIFQ

dIFI

ωω

ωω

+=

+=

sin2

cos2

Define a New Complex Signal:

( ) ( ) ( ) ( )[ ]tjQI

dIFeA

tsjtstg ωω +=+=2

ffIF+fd

fIF

A2/2|g|2

0

f

fd

A2/2|s|2

0

Combining the signals after the mixers

( ) tjd

deA

tg ω

2=

We now can distinguish between positive to negative doppler!!!

111


Split IF Signal:

( ) ( )[ ]

( ) ( )[ ]tAts

tA

ts

dIFQ

dIFI

ωω

ωω

+=

+=

sin2

cos2

Define a New Complex Signal:

( ) ( ) ( ) ( )[ ]tjQI

dIFeA

tsjtstg ωω +=+=2

ffd

A2/2|s|2

0

Combining the signals after the mixers

( ) tjd

deA

tg ω

2=

We now can distinguish between positive to negative doppler!!!

From the Figure we can see that in this case the doppler is unambiguous only if:

Tff PRd

1=<

Because we used N coherent pulses ofwidth τ and with Pulse Repetition Time Tthe spectrum after the mixer output is


112

SOLO Signal Processing

Collecting Pulsed Radar Data: 1 Pulse, Multiple Range-Gates Samples

• when using a coherent receiver, each range sample comprises one “I” sample and one “Q” sample, forming one complex number I+j Q.• Each range cells contains an echo from a different range interval.

• Also called Range-Bins, Range-Gates, Fast-Time Samples.

113


Collecting Pulsed Radar Data: Multiple Pulses

• when using a coherent receiver, each range sample comprises one “I” sample and one “Q” sample, forming one complex number I+j Q.• Repeat for multiple pulses in a “coherent processing interval” (CPI) or “dwell”

Sequence of samples for a fixed range bin represents echoes from same range interval over a period of time.

114


Perform FFT in Each Range Gate

After FFT a Range-DopplerMap is obtained for SignalProcessing

FFT

Run This

115


Perform FFT in Each Range Gate

Data-cube for Signal Processing

Repeat the Operation for each Receiver Channel (Σ,ΔAz,ΔEl,Γ for monopulse antenna or Σi,j for each element in an Electronic Scanned Antenna)

Range – Doppler Cells in Σ and ΔAz, ΔEl

FFT

FFT

FFT

FFT

Run This

116


Adaptive algorithms use additional data from the cube for weight estimation.

Datacube for Signal Processing

Standard radar signal processing algorithms correspond to operating in 1- or 2-D alongvarious axes of the data-cube

Space-Time Adaptive Processing:2-D joint adaptive weighting acrossantenna element and pulse number

Beamforming:1-D weighting acrossElectrical Scan Antennaelement number

Pulse Compression:1-D convolution alongthe range axis(“fast time”)

Synthetic Aperture Imaging:2-D matched filtering in slowand fast time

Doppler Processing:1-D filtering or spectralanalysis along the pulse axis(“slow time”)

Run This

117


Range – Doppler Cells in Σ and ΔAz, ΔEl

118


Generation of Σ , ΔAz, ΔEl Range – Doppler Maps

The Parameters defining the Range – Doppler Maps are:

Δ R – Map Range Resolution

Δ f – Map Doppler Resolution

RUnambiguous – Unambiguous Range

fUnambiguous – Unambiguous Doppler

Range – DopplerCell

Range – DopplerMap

f

fM

R

RN sunambiguousunambiguou

∆=

∆= &

Range Gates are therefore i = 1, 2, …, NNumber of Range-Doppler Cells = N x M

Doppler Gates are therefore j = 1, 2, …, M

Note: The Map Range & Doppler resolution (Δ R, Δ f) may change as function of Radar task (Search, Detection, Acquisition, Track). This is done by choosingthe Pulse Repetition Interval (PRI) and the number of pulses in a batch.

resolutionresolution ffRR ≥∆≥∆ &

119

SOLO Signal Processing Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 1)

The received signal from the scatter k is:

( ) ( )[ ] ( ) ( )ttTktttTkttfCts ddkdkr

kr

k ++≤≤++−= τθπ2cos

Ckr – amplitude of received signal

td (t) – round trip delay time given by ( )2/c

tRRtt kk

d

+=

θk – relative phase

The received signal is down-converted to base-band in order to extract the quadrature components. More precisely sk

r (t) is mixed with: ( ) [ ] τθπ +≤≤+= TktTktfCty kkk 2cos

After Low-Pass filtering the quadrature components of Σk, ΔAz k or ΔEl k signals are:

( ) ( )( ) ( )

==

tAtx

tAtx

kkQk

kkIk

ψψ

sin

cos

( ) ( )

+−≅−=

c

tR

c

Rfttft kkkdkk

2222 ππψ

The quadrature samples are given by:( ) ( )

+−≅=

c

tR

c

RfjAjAtX kkkkkkk

222expexp πψ

Ak - amplitude of Σk, ΔAz k or ΔEl k signals ψk - phase of Σk, ΔAz k or ΔEl k signals

( )

+−

+≅+=

c

tR

c

RfAj

c

tR

c

RfAxjxtX kk

kkkk

kkQkIkk

222sin

222cos ππ

120

SOLO Signal Processing Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 2)

The received signal from the scatter k is:

The energy of the received signal is given by: ( ) ( ) 2kkkk AtXtXP == ∗

( )

+−

+≅+=

c

tR

c

RfAj

c

tR

c

RfAxjxtX kk

kkkk

kkQkIkk

222sin

222cos ππ

where * is the complex conjugate.

Therefore:kk PA =


121

Decision/Detection TheorySOLO

Hypotheses

H0 – target is not present

H1 – target is present

Binary Detection

( )0Hp - probability that target is not present

( )1Hp - probability that target is present

( )zHp |0 - probability that target is not present and not declared (correct decision)

( )zHp |1 - probability that target is present and declared (correct decision)

Using Bayes’ rule: ( ) ( ) ( )∫=Z

dzzpzHpHp |00( ) ( ) ( )∫=

Z

dzzpzHpHp |11

( )zp - probability of the event Zz ⊂

Since p (z) > 0 the Decision rules are:

( ) ( )zHpzHp || 01 < - target is not declared (H0)

( ) ( )zHpzHp || 01 > - target is declared (H1) ( ) ( )zHpzHpH

H

|| 01

0

1

<>

122


Hypotheses H0 – target is not present H1 – target is present

Binary Detection

( )zHp |0 - probability that target is not present and not declared (correct decision)

( )zHp |1 - probability that target is present and declared (correct decision)

( )zp - probability of the event Zz ⊂

Decision rules are: ( ) ( )zHpzHpH

H

|| 01

0

1

<>

Using again Bayes’ rule:

( ) ( ) ( )( ) ( ) ( ) ( )

( )zp

HpHzpzHp

zp

HpHzpzHp

H

H

00

011

1

||

||

0

1

=<>=

( )0| Hzp - a priori probability that target is not present (H0)

( )1| Hzp - a priori probability that target is present (H1)

Since all probabilities arenon-negative

( )( )

( )( )1

0

0

1

0

1

|

|

Hp

Hp

Hzp

Hzp

H

H

<>

123


Hypotheses

( )1| Hzp - a priori probability density that target is present (likelihood of H1)

( )0| Hzp - a priori probability density that target is absent (likelihood of H0)

Detection Probabilities

( ) M

z

D PdzHzpPT

−== ∫∞

1| 1

( )∫∞

=Tz

FA dzHzpP 0|

( ) D

z

M PdzHzpPT

−== ∫∞−

1| 1

PD - probability of detection = probability that the target is present and declared

PFA - probability of false alarm = probability that the target is absent but declared

PM - probability of miss = probability that the target is present but not declared

T - detection threshold

DP

FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) T

Hzp

Hzp

T

T =0

1

|

|

H0 – target is not present H1 – target is present

Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

H

H

=<>=

1

0

0

1

0

1

|

|:Likelihood Ratio Test (LTR)

124


Hypotheses

Decision Criteria on Definition of the Threshold T

1. Bayes Criterion

DP

FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) T

Hzp

Hzp

T

T =0

1

|

|


Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

H

H

=<>=

1

0

0

1

0

1

|


The optimal choice that optimizes the Likelihood Ratio is ( )( )1

0

Hp

HpTBayes =

This choose assume knowledge of p (H0) and P (H1), that in general are not known a priori.

2. Maximum Likelihood Criterion

Since p (H0) and P (H1) are not known a priori, we choose TML = 1

( )1| Hzp( )0| Hzp

MP z

Tz

( )( ) 1

|

|

0

1 == ML

T

T THzp

Hzp

DP

FAP

125


Hypotheses

Decision Criteria on Definition of the Threshold T (continue)

3. Neyman-Pearson Criterion

DP

γ=FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) PN

T

T THzp

Hzp−=

0

1

|

|


Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

H

H

=<>=

1

0

0

1

0

1

|


Neyman and Pearson choose to optimizes the probability of detection PD

keeping the probability of false alarm PFA constant.

Egon Sharpe Pearson1895 - 1980

Jerzy Neyman1894 - 1981

( )∫∞

=T

TT

zzDz

dzHzpP 1|maxmax ( ) γ== ∫∞

Tz

FA dzHzpP 0|constrained to

Let use the Lagrange’s multiplier λ to add the constraint

( ) ( )

−+= ∫∫

∞∞

TT

TT

zzzz

dzHzpdzHzpG 01 ||maxmax γλ

Maximum is obtained for:

( ) ( ) 0|| 01 =+−=∂∂

HzpHzpz

GTT

T

λ( )( ) PN

T

T THzp

Hzp−==

0

1

|

|λ

zT is define by requiring that: ( ) γ== ∫∞

Tz

FA dzHzpP 0|


126

SOLO SEARCH & DETECT MODE During Search Mode the RADAR Seeker performs the following tasks:

• Slaves the Seeker Gimbals to the Designation Target direction (like in Slave Mode).

• Transmits the RF (by choosing the best waveform).

• Receives the returning RF.

• Compute the Σ Range-Doppler Map, chooses the Detection Threshold and policy.

• Perform Detections Clustering and compute Range and Doppler spread.

Note: Here is important to simulate the number of Batches that are needed to obtain the predefined probability of detection, the False Alarm Rate (FAR) and to resolve the differentdetections, i.e. the time necessary to perform this task.

• If a Detection is in the Target Designation (Uncertainty) Window we go to Acquisition Mode.

127

Target returns are the summation of signals (amplitude and phase) from all of the scattering centers within the radar resolution cell.

SOLO Target RCS

where

Nsc – number of scatters in the volume VResol

σk– Radar Cross Section of scatter k

Rk– Range to scatter k

The equivalent Radar Cross Section σTarget of the target in the resolution cell of volume VResol is:

2Nscatter i4

Target Resol 4i 1 iR

gV R

σσ η Σ

=

= = ∑24 N

scatter i

4i 1Resol iR

gR

V

ση Σ

=

= ∑ ( )2/4

2Resol τϕϕπ

cRV elaz=

gΣ (εAz,εEl) – antenna sum pattern ( gΣ(0,0)=1 )

R – Range to the center of the volume VResol

( ) ( )( )( )

∑=

Σ

+−=Σ

jiN

k

kkk

kElkAzproc

trver

RcvrXmtr

sc

cc

RRR

jgG

L

GGPji

,

12

k

kscatter

proc

Targ

3

20

2

Targ 22

2expR

,

L4,

πσεε

πλ

In the same way:

gΔ (εAz,εEl) – antenna difference pattern ( gΔ(0,0)=0 )

R G A AN TG EE S

DOPPLERFILTERS

Range-Doppler S cells

Detections

According to Range and Doppler of each scatter determine theRange-Doppler cell (i,j) for the scatter.

( ) ( )( )( )

∑=

∆

+−=∆

jiN

k

kkk

kElkAzElAzproc

trver

RcvrXmtr

sc

cc

RRR

jgG

L

GGPji

,

12

k

kscatter,

proc

Targ

3

20

2

Az/ElTarg 22

2expR

,

L4,

πσεε

πλ

128

SOLO SEARCH & DETECT MODE

According to the position of Target Uncertainty Window (TUW) versus Clutter chose the Range – Doppler magnitude (Runambiguous and funambiguous) by defining the Pulse Repetition Frequency (PRF) and the number of pulses in the batch, and choose resolution Δ R and Δ f.

Improvements

1. Change Range-Doppler cells indexes i,j tobring the Target Uncertainty Window inthe middle of the Range-Doppler Map

2. Choose on the Range-Doppler Map aarea that includes the Target UncertaintyWindow and perform Ground Cluttercomputations only for this area (we may addGround Clutter computations in Main Lobeand Altitude Line: Rk = hI).

Transmits the RF (by choosing the best waveform).

Computation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy

129

SOLO SEARCH & DETECT MODE

Computation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 1)

• Computation of Noise Threshold in each cell: ( ) ( ) ( ) BFTkjijijiN NoiseNoise 0,,, =Σ⋅Σ= ∗

• Computation of Clutter Power in CFAR Window cells (Cells in area around Target Uncertainty Window):

( ) ( ) ( ) ∗Σ⋅Σ= jijijiCCFAR

,,,

• Computation of Signal Power in Target Uncertainty Window cells:

( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,Window

yUncertaintTarget

• For each Range-Doppler Cell (i,j) perform the summation of complex signals for all the scatters in this cell:

∑∑∑===

∆=∆∆=∆Σ=Σjijiji N

kkEljiEl

N

kkAzjiAz

N

kkji

,,,

1,

1,

1, ,,

130

SOLO SEARCH & DETECT MODEComputation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 2).

DOPPLERWINDOW

R W A IN NG DE O W

R G A AN TG EE S

DOPPLERFILTERS

S cells

CFARWindow

R∆

f∆

Target Uncertainty

Window

( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=

n

j WindowCFARNoiseClutter jiji

niC ,,

1

Guard(Gap)

Window

• Computation of Clutter + Noise Threshold

• Coherent Detection:

( ) ( )( ) ( ) ClutterThjiNiCIf

ClutternoThjiNiCIf

NoiseClutter

NoiseClutter

⇒+>⇒+≤

+

+

1,

1,

( ) NoiseThNjiS +≥Window

yUncertaintTarget,

( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=

n

j WindowCFARNoiseClutter jiji

niC ,,

1

1. If no Clutter declare a Detection in the (i,j) cell of the Target Window if

ThNoise is chosen to assure a predefinedProbability of Detection pd and of False Alarm pFA

( ) NoiseClutterNoiseClutter ThCjiS ++ +≥Window

yUncertaintTarget,

2. If Clutter declare a Detection in the (i,j) cell of the Target Window if

ThNoise is chosen to assure a predefinedProbability of Detection pd and of False Alarm pFA

131


• Coherent Detection (M-out-of-N):How to Increase Probability of Detection and Reduce Probability of False Alarm:

Suppose that by Coherent Detection using one Range – Doppler Map we haveProbability of Detection pd and Probability of False Alarm pfa.

To Increase Probability of Detection to pD and Reduce Probability of False Alarmto pFA we use N consecutive batches (at different PRFs) , in each of them performing the Coherent Detection procedure. We declare a detection in the if we have at least M Detections for corresponding resolved Range-Doppler cells. In this way:

( ) ( )∑=

−−−

=N

Ml

lNd

ldD pp

lNl

NP 1

!!

!

( ) ( )∑=

−−−

=N

Ml

lNfa

lfaFA pp

lNl

NP 1

!!

!

Example: pd = 0.6, pfa = 10-3, N = 4, M = 2 gives pD = 0.82, pFA = 6 x10-6

Since we use different PRFs,to obtain correlation betweenDetections we must resolve theRange-Doppler ambiguities.

132


How to Increase Probability of Detection and Reduce Probability of False Alarm:

• Non-Coherent Detection:

To Increase Probability of Detection we use N consecutive batches, we compute thepower of each (i,j) cell, , in each Range-Doppler Map and we add (non-coherently) the powers of each corresponding (i,j) cell to obtaina non-coherent Range-Doppler Map. Now we perform the detection procedureas described before to declare a Detection.

( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,

133

SOLO SEARCH & DETECT MODEPerform Detections Clustering and compute Range and Doppler spread.

• Clustering

The Target signal may be spread in more then one Σ Range-Doppler cell. Clustering Process is to group the detections in the Σ Range-Doppler Map.

Group l parameters are mean and spread:

( )( )

( )( )∑

∑∑

∑==

il

ill

l

il

ill

l jiS

jiSii

jiS

jiSii

,

,&

,

, 2

2

( )( )

( )( )∑

∑∑

∑==

il

ill

l

il

ill

l jiS

jiSjj

jiS

jiSjj

,

,&

,

, 2

2

Range

Doppler

integer=∆+= mRiRmRlsunambiguoul

RiillRl

∆−= 22σ

integer=∆+= nfifnflsunambiguoul

fjjllf l

∆−= 22σ

If the spread of Target Range/Doppler spread σRl/ σRl are too high, we may remove theTarget detection assumption and declare the group l as Clutter.

lRadar

lf

f

cR

2=

ll fRadar

R f

c σσ2

=

134

SOLO SEARCH & DETECT MODEPerform Detections Clustering and compute Range and Doppler spread.

• Altitude Line and Main Lobe Clutter

The Interceptor altitude above ground hI is unknown (for simulation purposes we assume that the SeekerProcessor uses an estimation ĥI of hI). Thereforeis necessary to search for Altitude Line (Zero Doppler)and the Main Lobe Clutter in order to properly choosethe PRFs and the Σ Range-Doppler Map.

clutterdf _

( )RangeR

( )RangeR

Clutter

No Clutter

ClutterPower

ClutterPower

Main LobeClutter(MLC)

Altitude Return

λMV2

pMV θ

λcos

2

AAM e

Vcoscos

2 ψλ

pMV θ

λsin

2

pMV θ

λcos

2−

TargetRange

TargetDoppler

( ) ApA

I

e

h

ψθ cossin +

12

N1 2 M

Range-Doppler Map

• Check that the detection are from returns in the Main Lobe by comparing the signal power with the antenna Γuard power.

( ) ( ) ( ) ∗∗ Γ⋅Γ>Σ⋅Σ= jijijiS ,,,Window

yUncertaintTarget

If true the received signal is in the Main Lobe If not the received signal is in the Side Lobe and therefore rejected.


135

SOLO ACQUISITION MODE

During Acquisition Mode the RADAR Seeker performs the following tasks:

• Slaves the Seeker Gimbals to the Designated Target direction.• The Angular Tracker is initialized.

• Confirms that the Detection is steady and in the Designated Zone by solving the ambiguities in Range and Doppler by using a number of Batches with different PRFs (Pulse Repetition Frequency).

• The Angular Tracker uses the Δ Elevation and Δ Azimuth Maps, computes the Radar Errors in the Detected Range-Doppler cells, and controls the gimbals in the Track Mode, by closing the track loops.

• Compute the Σ and Δ Range-Doppler Maps.

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SOLO ACQUISITION MODE

In the Acquisition Mode the RADAR Seeker Signal Processor continue toPerform Detection in the Target Uncertainty Window of the Σ Range-Doppler Map asin Detection Mode, performing Detection cells Clustering. The Δ Elevation and Δ Azimuth Maps, are used to compute the Angular Radar Errors in the Detected Range-Doppler cells. For a cluster of l cells:

( ) ( )( ) ( )∑

Σ⋅Σ∆⋅Σ

= ∗

∗

lCluster ll

AzlldbAzAz

jiji

jiji

,,

,,Re

23θ

ε( ) ( )

( ) ( )∑

Σ⋅Σ∆⋅Σ

= ∗

∗

lCluster ll

EllldbElEl

jiji

jiji

,,

,,Re

23θ

ε


137

SOLO

References

J.V. DiFranco, W.I. Rubin, “RADAR Detection”, Artech House, 1981, Ch.5, pp.143-201

C.E. Cook, M. Bernfeld, “RADAR Signals An Introduction to Theory and Application”, Artech House, 1993

D. C. Schleher, “MTI and Pulsed Doppler RADAR”, Artech House, 1991, Appendix B

J. Minkoff, “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, Ch.5

M.A. Richards, ECE 6272, “Fundamentals of Signal Processing”, Georgia Institute of Technology, Spring 2000, Appendix A, Optimum and Sub-optimum Filters

W.B. Davenport,Jr., W.L. Root,”An Introduction to the Theory of Random Signals and Noise”, McGraw Hill, 1958, pp. 244-246

N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, Ch.5 & 6

Peeble, P.Z. Jr, “Radar Principles”, John Wiley & Sons, 1998


N. Levanon, E. Mozeson, “Radar Signals”, John Wiley & Sons, 2004


January 17, 2015 138

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

1 radar signal processing

Science

Transcript of 1 radar signal processing