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IEEE New Hampshire SectionRadar Systems Course 1Review Signals, Systems & DSP 1/1/2010 IEEE AES Society

Radar Systems Engineering Lecture 3

Review of Signals, Systems and Digital Signal Processing

Dr. Robert M. O’DonnellIEEE New Hampshire Section

Guest Lecturer

Radar Systems Course 2Review Signals, Systems & DSP 1/1/2010

IEEE New Hampshire SectionIEEE AES Society

Block Diagram of Radar SystemTransmitter

WaveformGeneration

PowerAmplifier

T / RSwitch

PropagationMedium

TargetRadarCross

Section

PulseCompressionReceiver Clutter Rejection

(Doppler Filtering)A / D

Converter

General Purpose Computer

Tracking

DataRecording

ParameterEstimation Detection

Signal Processor Computer

ThresholdingConsole /Displays

Antenna

ReceivedSignal

Time

Sign

alSt

reng

th

Application of Signals and Systems, and Digital Signal Processing Algorithms to the Received Radar Signals Result

in Optimum Target Detection



Reasons for Review Lecture

• Signals and systems, and digital signal processing are usually one semester advanced undergraduate courses for electrical engineering majors

• In no way will this 1+ hour lecture to justice to this large amount of material

• The lecture will present an overview of the material from these two courses that will be useful for understanding the overall Radar Systems Engineering course

– Goal of lecture-

Give non EE majors a quick view of material; they may wish to study in more depth to enhance their understanding of this course.

• UC Berkeley has an excellent, free, video Signals and Systems course (ECE 120) online at //webcast.berkeley.edu

– http://webcast.berkeley.edu/course_details.php?seriesid=1906978405– Given in Spring 2007

http://webcast.berkeley.edu/course_details.php?seriesid=1906978405



Signal Processing

• Signal processing is the manipulation, analysis and interpretation of signals.

• Signal processing includes:– Adaptive filtering / thresholding– Spectrum analysis– Pulse compression– Doppler filtering– Image enhancement– Adaptive antenna beam forming, and– A lot of other non-radar stuff ( Image processing, speech

processing, etc.

• It involves the collection, storage and transformation of data– Analog and digital signal processing– A lot of processing “horsepower”

is usually required



Outline

• Continuous Signals

• Sampled Data and Discrete Time Systems

• Discrete Fourier Transform (DFT)

• Fast Fourier Transform (FFT)

• Finite Impulse Response (FIR) Filters

• Weighting of Filters



Continuous Time Signal

( )tx

t0

( ) ( ) ( )( )( ) 532 t25tttx

300t12txt3cos79tsin100tx

−+−=

−=π−π=

Examples:




t0

( )tx ( )tx

t0

• Types of continuous time signals– Periodic or Non-periodic

Non-periodic

• • •• • •

( ) ( )txttx =Δ+

Periodic




t

• Types of continuous time signals– Periodic or Non-periodic– Real or Complex

Radar signals are complex

( )[ ]txRe

t0 0

( )[ ]txIm

• • • • • • • • • • • •

is a complex periodic signal( )tx



Continuous, Linear, Time Invariant Systems

ContinuousLinear Time

InvariantSystem

( )tx ( )ty

• Continuous– If and are continuous time functions, the

system is a continuous time system

• Linear– If the system satisfies

• Time Invariant– If a time shift in the input causes the same time shift in

the output

( )tx

( ) ( )[ ] == TtxTty

( )ty

( ) ( )[ ] ( ) ( )tytytxtxT 2121 β+α=β+α

Operator



Linear Time Invariant Systems (Delta Function)

• The impulse response is the response of the system when the input is


InvariantSystem

( )tx ( )tyContinuousLinear Time

InvariantSystem

( )tδ ( )th

( )

( ) 1dtt

0t0t0

t

=δ

=∞≠

=δ

∫∞

∞−

Properties of Delta Function

( )tδ( )th



Linear Time Invariant Systems


InvariantSystem


InvariantSystem

( )tδ ( )th

Definition : Convolution of Two Functions

( ) ( ) ( ) ( ) ττ−τ≡∗ ∫∞

∞−

dtxxtxtx 2121

Reversedand

Shifted



Linear Time Invariant Systems


InvariantSystem


InvariantSystem

( )tδ ( )th

( ) ( ) ( ) ( ) ( ) ττ−τ=∗= ∫∞

∞−

dthxthtxty

Convolution of and( )tx ( )th

• The output of any continuous time, linear, time-invariant (LTI) system is the convolution

of the input with the impulse response of the system

( )tx( )th



Why not Analog Sensors and Calculation Systems ?

Voltmeter

Torpedo Data Computer (1940s)

SlideRule

• Measurement Repeatability

• Environmental Sensitivity

• Size

• Complexity

• Cost

Disadvantages

Courtesy of US Navy

Courtesy of Hannes Grobe

Courtesy of oschene



Outline

• Continuous Signals and Systems

• Sampled Data and Discrete Time Systems– General properties– A/D Conversion– Sampling Theorem and Aliasing– Convolution of Discrete Time Signals– Fourier Properties of Signals

Continuous vs. Discrete Periodic vs. Aperiodic







Sampled Data Systems

• Digital signal processing deals with sampled data

• Digital processing differs from processing continuous (analog) signals

• Digital Samples are obtained with a “Sample and Hold”

(S/H) Amplifier followed by an “Analog-to-Digital”

(A/D) converter

– Sampling rate– Word length



Waveform Sampling

• Sampling converts a continuous signal into a sequence of numbers

• Radar signals are complex

Continuous-timeSystem

( )tx

Discrete-timeSystem

[ ]nx

A/D Converter



Outline










Ideal Analog to Digital (A/D) Converter

INV

A/DConverter

OUTVINV

12q2

VERROR=σ

OUTV

2VFS

2VFS−

q)V(P ERROR

q1

2q

2q

−INOUTERROR VVV −=

INV

2q

2q

−



“Non-Perfect Nature”

of A/D Converters

Output

InputOffset

Actual

Ideal

• Gain• Missing bits• Monotonicity• Offset• Nonlinearity• Missing bits

Input

Out

put

Missing Bit

Non-

Monotonic



Single Tone A/D Converter Testing

Frequency (MHz)

Pow

er L

evel

(dB

m)

0 2 4 6 8-100

-80

-60

-40

-20

0

Fundamental

HighestSpur

Spur Free Dynamic Range(SFDR)

For Ideal A/D S/N=6.02N + 1.76 dB



A/D Word Length

• A / D output is signed N bit integers – Twos complement arithmetic– Quantization noise power =

• Signal-to-noise ratio must fit within the word length:

– = maximum signal power (target, jamming, clutter)

– = thermal noise power in A / D input

– Typically, to reduce clipping (limiting)

• Required word length:

( ) ,N/S,SNR o2

oN

2S

4≈α

SNRlog10SNR 10DB =

o1L N12/1S2 <α>−

( ) 2.16/SNRL DB +>

12/1

A/D Saturation

Maximum Signal

Noise Quantization

Noise Signal

Head Room (~10 dB)

Foot Room (~10 dB)

ReceiverDynamic Range



Outline










Waveform Sampling

• Sampling converts a continuous signal into a sequence of numbers

• Radar signals are complex

Continuous-timeSystem

( )tx

Discrete-timeSystem

[ ]nx

A/D Converter



Sampling -

Overview

• Sampling Theorem constraint (a.k.a. Nyquist criterion) to prevent “aliasing”:

– For continuous aperiodic signals:

• Nyquist criterion:– Permits reconstruction via a low pass filtering– Eliminates Aliasing

=≥ ss FB2F Sampling Frequency



Signal Sampling Issues

• Signal Reconstruction

• Elimination of “Aliasing”

( )FX

0

• • •

sF2sF

LPF ( )FXc

0

B2B2Fs >

( )FX

0• • •

sF sF3sF2 sF4sF−• • •

Overlapping, Aliased Spectra

B2Fs <



The Sampling Theorem

• If is strictly band limited,

then, may be uniquely recovered from its samples if

The frequency is called the Nyquist frequency, and the minimum sampling frequency, , is called the Nyquist rate

)t(xc

BF0)F(X >= for

)t(xc [ ]nx

B2T2F

SS ≥

π=

B2FS =B



Spectrum of a Sampled Signal

• Sampling periodically replicates the spectrum– Fourier transform of a sampled signal is periodic

• If and are the spectra of and ( )FXc ( )FX

( ) ( )

( ) ( ) ( )

[ ] sF/nF2j

n

Ft2j

n

Ft2jcc

enx

dtenTttgFX

dtetxFX

π−∞

−∞=

∞

∞−

π−∞

−∞=

∞

∞−

π−

∑

∫ ∑

∫

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−δ=

=

( )txc [ ]nx

( )FXc

0

( )FX

0

• • •sF sF3sF2sF−

• • •



Distortion of a Signal Spectrum by “Aliasing”

• Assume band limited so that1

B

( )FXc

B− F

ST/1

B

( )FX

B− SFSF−

ST/1

2/FS

( )FX

SFSF− 2/FS−

)t(xc

Bf,0)f(X >=

• If is sampled with

• If is sampled with

B2FS ≥)t(xc

)t(xc

B2FS <Aliased parts of spectrum

for

F

F

No Aliasing

● ● ● ● ● ●

● ● ● ● ● ●



Effect of Sampling Rate on Frequency

00

t (sec)

1.0

0.5

- 5 5

0

)t(xc

22ctA

c )F2(AA2)F(X0A,e)t(xπ+

=>= −

Sampled Signal

Its Fourier TransformContinuous Signal

Its Fourier Transform

[ ] ( )T1F,eaaee)nT(xnx S

ATnnATnTAc ====== −−−

( ) [ ]S

2

2nj

n FF2,

acosa21a1enxX π=ω

+ω−−

==ω ω−∞

−∞=∑

( ) ( ) ( ) ==−= ∑∞

−∞=

FX̂FFXT1FX ccc l

l

( )2FFFXT S≤

2FF0 S>

)t(x̂c

InverseFourierTransformReconstructed Signal

Adapted from Proakis and Manolakis, Reference 1



Spectrum of Reconstructed SignalFrequency Spectrum

( )FXc

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Hz3FS =

Hz1FS =

Signal

)t(xc

[ ] ( )nTxnx c=

[ ] ( )nTxnx c=

t (sec)

t (sec)

t (sec)

sec31T =

sec1T =

0

1.0

0.5

1.0

1.0

0.5

0.5

0

0 0

0

0

5

- 5

- 5

- 5

5

5

0

0

0

0

0

0

1

1

1

2

2

2

2

- 2 4

- 2

2

2- 4

- 4 - 2 4

4

- 4

Continuous

Signal

Sampled

Signal

Sampled

Signal




Outline










Convolution for Discrete Time Systems

( ) ( ) ( ) ττ−τ= ∫∞

∞−

dtxhty


InvariantSystem

( )tx ( )tyContinuous-time

System

( )tx

Discrete-timeSystem

[ ]nx

DiscreteLinear Time

InvariantSystem

[ ]ny[ ]nx

[ ] [ ] [ ]knxkhnyk

−= ∑∞

−∞=



Graphical Implementation of Convolution

[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk

−=−= ∑∑∞

−∞=

∞

−∞=

Example:

0 1 2

1

23

[ ]=kh1 2 3 4 5

[ ]=kx 2

4

3

11

• Step 1 : Plot the sequences, and[ ]kx [ ]kh





−=−= ∑∑∞

−∞=

∞

−∞=

Example:

0 1 2

1

23

[ ]=kh1 2 3 4 5

[ ]=kx 2

4

3

11

• Step 2 : Take one of the sequences and time reverse it

[ ]=− kh-2 -1 0

12

3





−=−= ∑∑∞

−∞=

∞

−∞=

Example:

0 1 2

1

23

[ ]=kh1 2 3 4 5

[ ]=kx 2

4

3

11

• Step 3 : Shift by , yielding– a shift to the left– a shift to the right

[ ]kh −

[ ]=− kh-2 -1 0

12

3

n

0n >0n < [ ]=− knh

n-2,n-1,n

12

3





−=−= ∑∑∞

−∞=

∞

−∞=

Example:

0 1 2

1

23

[ ]=kh1 2 3 4 5

[ ]=kx 2

4

3

11

• Step 4 : For each value of ,multiply the sequences and ; and add products together for all values of to produce

[ ]knh −

[ ]=− kh-2 -1 0

12

3

nk

[ ]kx

[ ]ny




No overlap –

= 0


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− kh-2 -1 0

12

30nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− kh1

2

3

-2 -1 0 1 2 3 4 5

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




One sample overlaps –

= (1x1) = 1


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k1h-1 0 1

12

31nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k1h1

2

3

-1 0 1 2 3 4 5

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




Two samples overlaps –

= (1x2)+(2x1) = 4


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k2h0 1 2

12

32nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k2h1

2

3

0 1 2 3 4 5

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




Three samples overlaps –

= (1x3)+(2x2)+(4x1) = 11


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k3h 12

33nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k3h1

2

3

0 1 2 3 4 5

[ ]ny

1 2 3

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9





= (2x3)+(4x2)+(3x1) = 17


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k4h2 3 4

12

34nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k4h1

2

3

0 1 2 3 4 5 6

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9





= (4x3)+(3x2)+(1x1) = 17


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k5h3 4 5

12

35nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k5h 12

3

0 1 2 3 4 5 6

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




Two samples overlaps –

= (3x3)+(1x2) = 11


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k6h 12

36nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k6h1

2

3

0 1 2 3 4 5 6 7

[ ]ny

4 5 6

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




One sample overlaps –

= (1x3) = 3


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k7h5 6 7

12

37nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k7h1

2

3

0 1 2 3 4 5 6 7 8

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9




No overlap –

= 0


−=−= ∑∑∞

−∞=

∞

−∞=Example:

0 1 2

1

23

[ ]=kh [ ]=− k8h6 7 8

12

38nfor =

1 2 3 4 5

[ ]=kx 2

4

3

11

0 1 2 3 4 5

[ ]=kx2

4

3

11

[ ]=− k8h1

2

3

0 1 2 3 4 5 6 7 8

[ ]ny

10

15

5

0O

utpu

t of

Con

volu

tion

Sample Number

[ ]ny

0 1 2 3 4 5 6 7 8 9



Summary-

Linear Discrete Time Systems

• Any Linear and Time-Invariant (LTI) system can be completely described by its impulse response sequence

• The output of any LTI can be determined using the convolution summation

• The impulse response provides the basis for the analysis of an LTI system in the time-domain

• The frequency response function provides the basis for the analysis of an LTI system in the frequency-domain

[ ] [ ]nhnH→δ

[ ] [ ] [ ] ∞<<∞−−= ∑∞

−∞=

n,knxkhnyk

Adapted from MIT LL Lecture Series by D. Manolakis



Outline










Frequency Analysis of Signals

• Decomposition of signals into their frequency components– A series of sinusoids of complex exponentials

• The general nature of signals– Continuous or discrete– Aperiodic or periodic

• Radar echoes, from each transmitted pulse, are continuous

and aperiodic, and are usually transformed into discrete signals by an A/D converter before further processing

– Complex signals



Time and Frequency Domains

Analysis

Synthesis

Fourier Transform

Inverse Fourier Transform

Time History Frequency Spectrum

Frequency DomainTime Domain



Fourier Properties of Signals

• Continuous-Time Signals– Periodic Signals: Fourier Series– Aperiodic Signals: Fourier Transform

• Discrete-Time Signals– Periodic Signals: Fourier Series– Aperiodic Signals: Fourier Transform



Fourier Transform for Continuous-Time Aperiodic Signals

0 0

Adapted from Manolakis et al, Reference 1

Time DomainContinuous and Aperiodic Signals

Frequency DomainContinuous and Aperiodic Signals

)t(x )F(X

t

∫∞

∞−

π−= tde)t(x)F(X tF2j

∫∞

∞−

π= dFe)F(X)t(x tF2j

F



Fourier Properties of Signals

• Continuous-Time Signals– Periodic Signals: Fourier Series– Aperiodic Signals: Fourier Transform

• Discrete-Time Signals– Periodic Signals: Fourier Series– Aperiodic Signals: Fourier Transform



Fourier Transform for Discrete-Time Aperiodic Signals

Frequency DomainContinuous and Periodic Signals

Time DomainDiscrete and Aperiodic Signals

4− 2− 20 4 2− π −π 0 π 2πn

[ ]nx

ω

[ ]ωX

Adapted from Malolakis et al, Reference 1

[ ] ∫π

ω ωωπ

=2

nj de)(X21nx

[ ] nj

nenX)(X ω−

∞

−∞=∑=ω



Summary of Time to Frequency Domain Properties

01

P

FT

=

2( ) ( ) j FtX F x t e dt∞ − π

−∞= ∫

2( ) ( ) j Ftx t X F e dF∞ π

−∞= ∫

( )x t

Continuous and Aperiodic Continous and Aperiodic

( )X F

Discrete-

Time Signals

21

0

1 [ ]N j kn

Nk

nc x n e

N

π− −

=

= ∑21

0

[ ]N j kn

Nk

k

x n c eπ−

=

=∑

[ ]x n kc

N− N0

Discrete and Periodic Discrete and Periodic

n k

Continuous-

Time Signals

021 ( )P

j kF tk T

P

c x t e dtT

− π= ∫02( ) j kF t

kk

x t c e∞

π

=−∞

= ∑

( )x t

0

Time-Domain Frequency-Domain

Continuous and Periodic Discrete and Aperiodic

t 0

kc

FPT− PT

( ) [ ] j n

nX x n e

∞− ω

=−∞

ω = ∑

2

1[ ] ( )2

j nx n X e dω

π= ω ω

π∫

[ ]x n ( )X ω

4− 2− 204 2− π −π 0 π 2π

Continous and Periodic

n ω

Time-Domain Frequency-Domain

Ape

riodi

c Si

gnal

sFo

urie

r Tra

nsfo

rms

Perio

dic

Sign

als

Four

ier S

erie

s

Discrete and Aperiodic

0 t 0 F

N− N0




Outline



• Discrete Fourier Transform (DFT)– Calculation






Direct DFT Computation

• 1. evaluations of trigonometric functions• 2. real ( complex) multiplications• 3. real ( complex) additions• 4. A number of indexing and addressing operations

[ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]∑

∑

∑

−

=

−

=

π−−

=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ π

−⎟⎠⎞

⎜⎝⎛ π

−=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

=

=−≤≤=

1N

0nIRI

1N

0nIRR

N/nkj2knN

knN

1N

0n

nkN2cosnxnk

N2sinnxkX

nkN2sinnxnk

N2cosnxkX

eW1Nk0WnxkX

2N22N4

)1N(N −

2N)2N(N4 −

2N≈ ComplexMADS

MADSMultiply

AndDivides

Adapted from MIT LL Lecture Series by D. Manolakis

Aka “Twiddle Factor”



Outline









Fast Fourier Transform (FFT)

• An algorithm for each efficiently computing the Discrete Fourier

Transform (DFT) and its inverse

• DFT

MADS (Multiplies and Divides)

• FFT

MADS

• FFT algorithm Development -

Cooley / Tukey (1965) Gauss (1805)

• Many variations and efficiencies of the FFT algorithm exist– Decimation in Time (input -

bit reversed, output -

natural order) – Decimation in Frequency (input -

natural order, output -

bit reversed)

• The FFT calculation is broken down into a number of sequential stages, each stage consisting of a number of relatively small calculations called “Butterflies”

( )2NO

⎟⎠⎞

⎜⎝⎛ Nlog

2NO 2



Radix 2 Decimation in Time FFT Algorithm

• Divide DFT of size N into two interleaved DFTs, each of size N/2

– Example will be – Input to each DFT are even and odd s , respectively

• Solve each stage recursively, until the size of the stage’s DFT is 2.

[ ] [ ] [ ] N/nkj2knN

knN

1N

0n

N/nkj21N

0neW1Nk0WnxenxkX π−

−

=

π−−

=

=−≤≤== ∑∑

82N 3 ==[ ]nx

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] kl2/N

12N

0l

kN

kl2/N

12N

0l

k)1l2(N

12N

0l

klN

12N

0l

knN

Oddn

knN

Evenn

knN

1N

0n

WlhWWlgWlhWlg

WnxWnxWnxkX

∑∑∑∑

∑∑∑−

=

−

=

+

−

=

−

=

−

=

+=+=

+==

Even index and odd index terms of N/2 point DFT of

N/2 point DFT of [ ] [ ]kGlg =

[ ]nx [ ] [ ]kHlh =



Radix 2 Decimation in Time FFT Algorithm (continued)

• Using the periodicity of the complex exponentials:

• And the following properties of the “twiddle factors”:

• A block diagram of this computational flow is graphically illustrated in the next chart for an 8 point FFT

[ ] [ ] [ ]kHWkGkX knN+=

[ ] [ ] ⎥⎦⎤

⎢⎣⎡ +=⎥⎦

⎤⎢⎣⎡ +=

2NkHkH

2NkGkG

( )( ) )k(HW2/NkHW

WWWWkN

)2/N(kN

kN

2/NN

kN

)2/N(kN

−=+

−==+

+

then



8 Point Decimation in Time FFT Algorithm (After First Decimation)

4 –

PointDFT

4 –

PointDFT

[ ]0x

[ ]1x

[ ]2x

[ ]3x

[ ]4x

[ ]5x

[ ]6x

[ ]7x

[ ]0G

[ ]1G

[ ]2G

[ ]3G

[ ]0H

[ ]1H

[ ]2H

[ ]3H

[ ]0X08W[ ]1X

[ ]2X

[ ]3X

[ ]4X

[ ]5X

[ ]6X

[ ]7X78W

68W

58W

18W

48W

28W

38W



Decimation of 4 Point into two 2 point DFTs

• If N/2 is even, and may again be decimated

• This leads to:

[ ] [ ] [ ] [ ] nk2/N

12N

Oddn

nk2/N

12N

Evenn

nk2/N

12N

0nWngWngWngkG ∑∑∑

−−−

=

+==

[ ] [ ] [ ] nk4/N

12N

0n

k2/N

nk4/N

14N

0nW1n2gWWn2gkG ∑∑

−

=

−

=

++=

[ ]ng [ ]nh

2 –

PointDFT

2 –

PointDFT

[ ]0x

[ ]2x

[ ]4x

[ ]6x

[ ]0G

[ ]1G

[ ]2G

[ ]3G

04W

14W

24W

34W

2 –

PointDFT



Butterfly for 2 Point DFT

[ ] [ ] [ ]1q0q0Q +=[ ]0q

[ ]1q [ ] [ ] [ ]1q0q1Q −=

[ ]kq [ ]kQ

1−

Now, Putting it all together…..



Flow of 8-Point FFT (Radix 2 -

Decimation in Time Algorithm)

[ ]0x

08W

[ ]0X

18W

08W

08W

08W

08W

08W

08W

28W

28W

28W

38W

1−

1−

1−

1−

1−

1−

1− 1−

1−

1−

1−

1−

[ ]4x

[ ]2x

[ ]6x

[ ]1x

[ ]5x

[ ]3x

[ ]7x

[ ]1X

[ ]2X

[ ]3X

[ ]4X

[ ]5X

[ ]6X

[ ]7X



Basic FFT Computation Flow Graph

• Each “Butterfly”

takes 2 MADS (Multiplies and Adds)• Twiddle Factors (For 8 point FFT)

• 12 Butterflies implies 12 MADS vs. 64 MADS for 8 point DFT

• 512 point FFT more than 100 times faster than 512 DFT

( )( ) 2/j1eWjeW

2/j1eeW1eW4/j33

82/j2

8

4/j8/j218

008

−−==−==

−=====π−π−

π−π−−

N/nkj2nkN eW π−=

Checkover

“Butterfly”

“Twiddle”

Factor

1−b

a

rNW

bWaA rN+=

bWaB rN−=

nkr =



Computational Speed –

DFT vs. FFT

• Discrete Fourier Transform (O ~ N2)• Fast Fourier Transform (O ~ N log2 N)

Num

ber o

f Com

plex

Mul

tiplic

atio

ns

Number of points in Radix 2 FFT

LinesDrawn

Through Data

PointsDFT

FFT

104103102101101

103

105

107

109

Adapted from Lyons, Reference 2



Fast Fourier Transform (FFT) -

Summary

• Fast Fourier Transform (FFT) algorithms make possible the computation of DFT with O ((N/2) log2 N) MADS as opposed to O N2

MADS

• Many other implementations of the FFT exist:– Radix 2 decimation in frequency algorithm– Radar-Brenner algorithm– Bluestein’s algorithm– Prime Factor algorithm

• The details of FFT algorithms are important to the designers of real-time DSP systems in software or hardware

• An interesting history of FFT algorithms – Heideman, Johnson, and Burrus, “Gauss and the History of FFT,”

IEEE ASSP Magazine, Vol. 1, No. 4, pp. 14-21, October 1984



Outline









Finite and Infinite Response Filters

• Infinite Impulse Response (IIR) Filters– Output of filter depends on past time history– Example :

• Finite Impulse Response (FIR) Filters– Output depends on the finite past– Example: DFT

– Other examples:

( )∞−

[ ] [ ] [ ]1nyM

1MnxM1ny −

−+=

[ ] [ ] N/nkj21N

0nenxkX π−

−

=∑=

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]1,nx2,nxnyor

1xnxn,kaky1N

0n

−=

= ∑−

=



Four Basic Filter Types-

An Idealization

Ideal Low Pass Filter

Ideal Bandstop FilterIdeal Bandpass Filter

Ideal High Pass Filter

1

11

1

ω

ω

ω

ω

cω cωcω− cω−

π1ω−1ω−

π−

1ω 2ω2ω−2ω− 1ω 2ω

π−

π−π−

ππ

π

( )jweH

( )jweH

( )jweH

( )jweH



Outline









Windowing / Weighting of Filters

• If we take a square pulse, sample it M times, and calculate the Fourier transform of this uniform rectangular “window”:

• This is recognized as the sinc function which has 13 dB sidelobes

• If lower sidelobes are needed , at the cost of a widened pass band, one can multiply the elements of the pulse sequence with one of a number of weighting functions, which will adjust the sidelobes appropriately

( ) ( ) ( )( )

( ) ( )( ) π≤ω≤π−ωω

=ω

ωω

=−−

==ω −−ω−

ω−−

=

ω−∑

2/sin2/Msin

W

2/sin2/Msine

e1e1eW 2/1Mj

j

Mj1M

0n

nj



Commonly Used Window Functions

• Rectangular

• Bartlett (triangular)

• Hanning

• Hamming

• Blackman

[ ]=nw

[ ]=nw

[ ]=nw

[ ]=nw

[ ]=nw

,0Mn0,1 ≤≤

otherwise

,0Mn2/MM/n222/Mn0,M/n2

≤<−≤≤

otherwise

otherwise

otherwise

( ),0

Mn0,M/n2cos5.05.0 ≤≤π−

( ),0

Mn0,M/n2cos46.054.0 ≤≤π−

( ) ( ),0

Mn0,M/n4cos08.0M/n2cos5.042.0 ≤≤π+π−

otherwise



Comparison of Common Windows

Type of Window Peak Sidelobe Amplitude (dB)

Approximate Width of Main

LobeRectangular

Bartlett(triangular)

Hanning

Hamming

Blackman

( )1M/4 +π

M/8π

M/8π

M/8π

M/12π

13−

57−

31−

25−

41−



HammingRectangular

Comparison of Rectangular & Hamming Windows

10−

20−

30−

40−

50−

60−

20 lo

g 10|W

(ω|

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency (f = F/Fs

)



Summary

• A brief review of the prerequisite Signal & Systems, and Digital Signal Processing knowledge base for this radar course has been presented

– Viewers requiring a more in depth exposition of this material should consult the references at the end of the lecture

• The topics discussed were:– Continuous signals and systems– Sampled data and discrete time systems– Discrete Fourier Transform (DFT)– Fast Fourier Transform (FFT)– Finite Impulse Response (FIR) filters– Weighting of filters



References

1. Proakis, J, G. and Manolakis, D. G., Digital Signal Processing, Principles, Algorithms, and Applications, Prentice Hall, Upper Saddle River, NJ, 4th

Ed., 20072. Lyons, R. G., Understanding Digital Signal Processing, Prentice

Hall, Upper Saddle River, NJ, 2nd

Ed., 20043. Hsu, H. P., Signals and Systems, McGraw Hill, New York, 19954. Hayes, M. H., Digital Signal Processing, McGraw Hill, New York,

19995. Oppenheim, A. V. et al, Discrete Time Signal Processing, Prentice

Hall, Upper Saddle River, NJ, 2nd

Ed., 19996. Boulet, B., Fundamentals of Signals and Systems, Prentice Hall,

Upper Saddle River, NJ, 2nd

Ed., 20007. Richards, M. A., Fundamentals of Radar Signal Processing, McGraw

Hill, New York, 20058. Skolnik, M., Radar Handbook, McGraw Hill, New York, 2nd

Ed., 19909. Skolnik, M., Introduction to Radar Systems, McGraw Hill, New York,

3rd Ed., 2001



Acknowledgements

• Dr Dimitris Manolakis• Dr. Stephen C. Pohlig• Dr William S. Song



Homework Problems

• From Proakis and Manolakis, Reference 1– Problems 2.1, 2.17, 4.9a and b, 4.10 a and b, 6.1, 6.9 a and b,

8.1 and 8.8

• Or

• And from Hays, Reference 4– Problems 1.41, 1.49, 1.54, 1.59, 2.46, 2.57, 2.58, 3.27, 3.28,

3.34, 6.44, 6.45

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