Radar 2009 a 3 review of signals, systems, and dsp

IEEE New Hampshire Section
Radar Systems Course 1
Review 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’Donnell
Guest Lecturer

Review Signals, Systems & DSP 1/1/2010
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Block Diagram of Radar System
Transmitter
Waveform
Generation
Power
Amplifier
T / R
Switch
Propagation
Medium
Target
Radar
Cross
Section
Pulse
Compression
Receiver
Clutter Rejection
(Doppler Filtering)
A / D
Converter
General Purpose Computer
Tracking
Data
Recording
Parameter
Estimation
Detection
Signal Processor Computer
ThresholdingConsole /
Displays
Antenna
Received
Signal
Time
Signal
Strength
Application of Signals and Systems, and Digital Signal
Processing Algorithms to the Received Radar Signals Result
in Optimum Target Detection

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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

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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

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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

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Continuous Time Signal
( )tx
t0
( ) ( ) ( )
( )
( ) 532
t25tttx
300t12tx
t3cos79tsin100tx
−
+−=
−=
π−π=
Examples:

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t0
( )tx ( )tx
t0
• Types of continuous time signals
– Periodic or Non-periodic
Non-periodic
• • •• • •
( ) ( )txttx =Δ+
Periodic

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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

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Continuous, Linear, Time Invariant
Systems
Continuous
Linear Time
Invariant
System
( )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

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Linear Time Invariant Systems
(Delta Function)
• The impulse response is the response of the system when
the input is
Continuous
Linear Time
Invariant
System
( )tx ( )ty
Continuous
Linear Time
Invariant
System
( )tδ ( )th
( )
( ) 1dtt
0t
0t0
t
=δ
=∞
≠
=δ
∫
∞
∞−
Properties of Delta Function
( )tδ
( )th

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Continuous
Linear Time
Invariant
System
( )tx ( )ty
Continuous
Linear Time
Invariant
System
( )tδ ( )th
Definition : Convolution of Two Functions
( ) ( ) ( ) ( ) ττ−τ≡∗ ∫
∞
∞−
dtxxtxtx 2121
Reversed
and
Shifted

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Continuous
Linear Time
Invariant
System
( )tx ( )ty
Continuous
Linear Time
Invariant
System
( )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

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Why not Analog Sensors and
Calculation Systems ?
Voltmeter
Torpedo Data Computer (1940s)
Slide
Rule
• Measurement Repeatability
• Environmental Sensitivity
• Size
• Complexity
• Cost
Disadvantages
Courtesy of US Navy
Courtesy of Hannes Grobe
Courtesy of oschene

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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

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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

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Waveform Sampling
• Sampling converts a continuous signal into a sequence of
numbers
• Radar signals are complex
Continuous-time
System
( )tx
Discrete-time
System
[ ]nx
A/D Converter

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Outline
– A/D Conversion

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Ideal Analog to Digital (A/D) Converter
INV
A/D
Converter OUTVINV
12
q2
VERROR
=σ
OUTV
2
VFS
2
VFS−
q)V(P ERROR
q
1
2
q
2
q
−
INOUTERROR VVV −=
INV
2
q
2
q
−

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“Non-Perfect Nature” of A/D Converters
Output
Input
Offset
Actual
Ideal
• Gain
• Missing bits
• Monotonicity
• Offset
• Nonlinearity
• Missing bits
Input
Output
Missing Bit
Non- Monotonic

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Single Tone A/D Converter Testing
Frequency (MHz)
PowerLevel(dBm)
0 2 4 6 8
-100
-80
-60
-40
-20
0
Fundamental
Highest
Spur
Spur Free Dynamic Range
(SFDR)
For Ideal A/D S/N=6.02N + 1.76 dB

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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 o
2
oN
2
S
4≈α
SNRlog10SNR 10DB =
o
1L
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)
Receiver
Dynamic Range

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Outline
– A/D Conversion

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Waveform Sampling
• Sampling converts a continuous signal into a sequence of
numbers
• Radar signals are complex
Continuous-time
System
( )tx
Discrete-time
System
[ ]nx
A/D Converter

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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

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Signal Sampling Issues
• Signal Reconstruction
• Elimination of “Aliasing”
( )FX
0
• • •
sF2sF
LPF ( )FXc
0
B2
B2Fs >
( )FX
0
•• •
sF sF3sF2 sF4sF−
•• •
Overlapping, Aliased Spectra
B2Fs <

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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
B2
T
2
F
S
S ≥
π
=
B2FS =
B

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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
Ft2j
cc
enx
dtenTttgFX
dtetxFX
π−
∞
−∞=
∞
∞−
π−
∞
−∞=
∞
∞−
π−
∑
∫ ∑
∫
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−δ=
=
( )txc [ ]nx
( )FXc
0
( )FX
0
•• •
sF sF3sF2sF−
•• •

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Distortion of a Signal Spectrum by “Aliasing”
• Assume band limited so that
1
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
● ● ● ● ● ●
● ● ● ● ● ●

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Effect of Sampling Rate on Frequency
0
0
t (sec)
1.0
0.5
- 5 5
0
)t(xc
22c
tA
c
)F2(A
A2
)F(X0A,e)t(x
π+
=>=
−
Sampled Signal
Its Fourier TransformContinuous Signal
Its Fourier Transform
[ ] ( ) T
1
F,eaaee)nT(xnx S
ATnnATnTA
c ====== −−−
( ) [ ]
S
2
2
nj
n F
F
2,
acosa21
a1
enxX π=ω
+ω−
−
==ω ω−
∞
−∞=
∑
( ) ( ) ( ) ==−= ∑
∞
−∞=
FXˆFFX
T
1
FX ccc l
l
( )
2
F
FFXT S
≤
2
F
F0 S
>
)t(xˆc
Inverse
Fourier
TransformReconstructed Signal
Adapted from Proakis and Manolakis, Reference 1

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Spectrum of Reconstructed Signal
Frequency Spectrum
( )FXc
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Hz3FS =
Hz1FS =
Signal
)t(xc
[ ] ( )nTxnx c=
[ ] ( )nTxnx c=
t (sec)
t (sec)
t (sec)
sec
3
1
T =
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

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Outline
– A/D Conversion

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Convolution for Discrete Time Systems
( ) ( ) ( ) ττ−τ= ∫
∞
∞−
dtxhty
Continuous
Linear Time
Invariant
System
( )tx ( )ty
Continuous-time
System
( )tx
Discrete-time
System
[ ]nx
Discrete
Linear Time
Invariant
System
[ ]ny[ ]nx
[ ] [ ] [ ]knxkhny
k
−= ∑
∞
−∞=

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Graphical Implementation of
Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
1 2 3 4 5
[ ]=kx 2
4
3
11
• Step 1 : Plot the sequences, and[ ]kx [ ]kh

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Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
1 2 3 4 5
[ ]=kx 2
4
3
11
• Step 2 : Take one of the sequences and time reverse it
[ ]=− kh
-2 -1 0
1
2
3

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Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
1 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
1
2
3
n
0n >
0n < [ ]=− knh
n-2,n-1,n
1
2
3

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Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
1 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
1
2
3
n
k
[ ]kx
[ ]ny

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Convolution
No overlap – = 0
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− kh
-2 -1 0
1
2
30nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− kh 1
2
3
-2 -1 0 1 2 3 4 5
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
One sample overlaps – = (1x1) = 1
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k1h
-1 0 1
1
2
31nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k1h 1
2
3
-1 0 1 2 3 4 5
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
Two samples overlaps – = (1x2)+(2x1) = 4
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k2h
0 1 2
1
2
32nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k2h 1
2
3
0 1 2 3 4 5
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
Three samples overlaps – = (1x3)+(2x2)+(4x1) = 11
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k3h 1
2
33nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k3h 1
2
3
0 1 2 3 4 5
[ ]ny
1 2 3
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k4h
2 3 4
1
2
34nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k4h 1
2
3
0 1 2 3 4 5 6
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k5h
3 4 5
1
2
35nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k5h 1
2
3
0 1 2 3 4 5 6
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
Two samples overlaps – = (3x3)+(1x2) = 11
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k6h 1
2
36nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k6h 1
2
3
0 1 2 3 4 5 6 7
[ ]ny
4 5 6
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
One sample overlaps – = (1x3) = 3
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k7h
5 6 7
1
2
37nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k7h 1
2
3
0 1 2 3 4 5 6 7 8
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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Convolution
No overlap – = 0
[ ] [ ] [ ] [ ] [ ]knhkxknxkhny
kk
−=−= ∑∑
∞
−∞=
∞
−∞=
Example:
0 1 2
1
2
3
[ ]=kh
[ ]=− k8h
6 7 8
1
2
38nfor =
1 2 3 4 5
[ ]=kx 2
4
3
11
0 1 2 3 4 5
[ ]=kx 2
4
3
11
[ ]=− k8h 1
2
3
0 1 2 3 4 5 6 7 8
[ ]ny
10
15
5
0
Outputof
Convolution
Sample Number
[ ]ny
0 1 2 3 4 5 6 7 8 9

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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
[ ] [ ]nhn
H
→δ
[ ] [ ] [ ] ∞<<∞−−= ∑
∞
−∞=
n,knxkhny
k
Adapted from MIT LL Lecture Series by D. Manolakis

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Outline
– A/D Conversion

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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

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Time and Frequency Domains
Analysis
Synthesis
Fourier Transform
Inverse Fourier Transform
Time History Frequency Spectrum
Frequency DomainTime Domain

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

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Fourier Transform for
Continuous-Time Aperiodic Signals
0 0
Adapted from Manolakis et al, Reference 1
Time Domain
Continuous and Aperiodic Signals
Frequency Domain
Continuous and Aperiodic Signals
)t(x )F(X
t
∫
∞
∞−
π−
= tde)t(x)F(X tF2j
∫
∞
∞−
π
= dFe)F(X)t(x tF2j
F

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Fourier Properties of Signals
• Continuous-Time Signals
• Discrete-Time Signals

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Fourier Transform for
Discrete-Time Aperiodic Signals
Frequency Domain
Continuous and Periodic Signals
Time Domain
Discrete 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
ωω
π
=
2
nj
de)(X
2
1
nx
[ ] nj
n
enX)(X ω−
∞
−∞=
∑=ω

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Summary of Time to Frequency Domain
Properties
0
1
P
F
T
=
2
( ) ( ) j Ft
X F x t e dt
∞
− π
−∞
=∫
2
( ) ( ) j Ft
x 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
N
k
n
c x n e
N
π− −
=
= ∑
21
0
[ ]
N j kn
N
k
k
x n c e
π−
=
=∑
[ ]x n kc
N− N0
Discrete and Periodic Discrete and Periodic
n k
Continuous- Time Signals
021
( )
P
j kF t
k
T
P
c x t e dt
T
− π
= ∫
02
( ) j kF t
k
k
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
n
X x n e
∞
− ω
=−∞
ω = ∑
2
1
[ ] ( )
2
j n
x 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
AperiodicSignals
FourierTransforms
PeriodicSignals
FourierSeries
Discrete and Aperiodic
0 t 0 F
N− N0

IEEE AES Society
Outline
Systems
– Calculation

IEEE AES Society
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
0n
IRI
1N
0n
IRR
N/nkj2kn
N
kn
N
1N
0n
nk
N
2
cosnxnk
N
2
sinnxkX
nk
N
2
sinnxnk
N
2
cosnxkX
eW1Nk0WnxkX
2
N2
2
N4
)1N(N −
2
N
)2N(N4 −
2
N≈ Complex
MADS
MADS
Multiply
And
Divides
Adapted from MIT LL Lecture Series by D. Manolakis
Aka “Twiddle Factor”

IEEE AES Society
Outline
Systems

IEEE AES Society
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”
( )2
NO
⎟
⎠
⎞
⎜
⎝
⎛
Nlog
2
N
O 2

IEEE AES Society
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/nkj2kn
N
kn
N
1N
0n
N/nkj2
1N
0n
eW1Nk0WnxenxkX π−
−
=
π−
−
=
=−≤≤== ∑∑
82N 3
==
[ ]nx
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] kl
2/N
1
2
N
0l
k
N
kl
2/N
1
2
N
0l
k)1l2(
N
1
2
N
0l
kl
N
1
2
N
0l
kn
N
Oddn
kn
N
Evenn
kn
N
1N
0n
WlhWWlgWlhWlg
WnxWnxWnxkX
∑∑∑∑
∑∑∑
−
=
−
=
+
−
=
−
=
−
=
+=+=
+==
Even index and odd index terms of N/2 point DFT of
N/2 point DFT of [ ] [ ]kGlg =
[ ]nx [ ] [ ]kHlh =

IEEE AES Society
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 kn
N+=
[ ] [ ] ⎥
⎦
⎤
⎢
⎣
⎡
+=⎥
⎦
⎤
⎢
⎣
⎡
+=
2
N
kHkH
2
N
kGkG
( )( ) )k(HW2/NkHW
WWWW
k
N
)2/N(k
N
k
N
2/N
N
k
N
)2/N(k
N
−=+
−==
+
+
then

IEEE AES Society
8 Point Decimation in Time FFT Algorithm
(After First Decimation)
4 – Point
DFT
4 – Point
DFT
[ ]0x
[ ]1x
[ ]2x
[ ]3x
[ ]4x
[ ]5x
[ ]6x
[ ]7x
[ ]0G
[ ]1G
[ ]2G
[ ]3G
[ ]0H
[ ]1H
[ ]2H
[ ]3H
[ ]0X
0
8W
[ ]1X
[ ]2X
[ ]3X
[ ]4X
[ ]5X
[ ]6X
[ ]7X
7
8W
6
8W
5
8W
1
8W
4
8W
2
8W
3
8W

IEEE AES Society
Decimation of 4 Point into two 2 point DFTs
• If N/2 is even, and may again be decimated
• This leads to:
[ ] [ ] [ ] [ ] nk
2/N
1
2
N
Oddn
nk
2/N
1
2
N
Evenn
nk
2/N
1
2
N
0n
WngWngWngkG ∑∑∑
−−−
=
+==
[ ] [ ] [ ] nk
4/N
1
2
N
0n
k
2/N
nk
4/N
1
4
N
0n
W1n2gWWn2gkG ∑∑
−
=
−
=
++=
[ ]ng [ ]nh
2 – Point
DFT
2 – Point
DFT
[ ]0x
[ ]2x
[ ]4x
[ ]6x
[ ]0G
[ ]1G
[ ]2G
[ ]3G
0
4W
1
4W
2
4W
3
4W
2 – Point
DFT

IEEE AES Society
Butterfly for 2 Point DFT
[ ] [ ] [ ]1q0q0Q +=[ ]0q
[ ]1q [ ] [ ] [ ]1q0q1Q −=
[ ]kq [ ]kQ
1−
Now, Putting it all together…..

IEEE AES Society
Flow of 8-Point FFT
(Radix 2 - Decimation in Time Algorithm)
[ ]0x
0
8W
[ ]0X
1
8W
0
8W
0
8W
0
8W
0
8W
0
8W
0
8W
2
8W
2
8W
2
8W
3
8W
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

IEEE AES Society
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/j1eeW1eW
4/j33
8
2/j2
8
4/j8/j21
8
00
8
−−==−==
−=====
π−π−
π−π−−
N/nkj2nk
N eW π−
=
Check
over
“Butterfly”
“Twiddle” Factor
1−
b
a
r
NW
bWaA r
N+=
bWaB r
N−=
nkr =

IEEE AES Society
Computational Speed – DFT vs. FFT
• Discrete Fourier Transform (O ~ N2)
• Fast Fourier Transform (O ~ N log2 N)
NumberofComplexMultiplications
Number of points in Radix 2 FFT
Lines
Drawn
Through
Data
PointsDFT
FFT
104103
102101
101
103
105
107
109
Adapted from Lyons, Reference 2

IEEE AES Society
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

IEEE AES Society
Outline
Systems

IEEE AES Society
Finite and Infinite Response Filters
• Infinite Impulse Response (IIR) Filters
– Output of filter depends on past time history
– Example :
– Output depends on the finite past
– Example: DFT
– Other examples:
( )∞−
[ ] [ ] [ ]1ny
M
1M
nx
M
1
ny −
−
+=
[ ] [ ] N/nkj2
1N
0n
enxkX π−
−
=
∑=
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]1,nx2,nxnyor
1xnxn,kaky
1N
0n
−=
= ∑
−
=

IEEE AES Society
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−
π
1ω−1ω−
π−
1ω 2ω2ω−2ω− 1ω 2ω
π−
π−π−
ππ
π
( )jw
eH
( )jw
eH
( )jw
eH
( )jw
eH

IEEE AES Society
Outline
Systems

IEEE AES Society
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−
ω
ω
=ω
ω
ω
=
−
−
==ω −−
ω−
ω−−
=
ω−
∑
2/sin
2/Msin
W
2/sin
2/Msin
e
e1
e1
eW 2/1Mj
j
Mj1M
0n
nj

IEEE AES Society
Commonly Used Window Functions
• Rectangular
• Bartlett (triangular)
• Hanning
• Hamming
• Blackman
[ ]=nw
[ ]=nw
[ ]=nw
[ ]=nw
[ ]=nw
,0
Mn0,1 ≤≤
otherwise
,0
Mn2/MM/n22
2/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

IEEE AES Society
Comparison of Common Windows
Type of Window Peak Sidelobe
Amplitude (dB)
Approximate
Width of Main
Lobe
Rectangular
Bartlett
(triangular)
Hanning
Hamming
Blackman
( )1M/4 +π
M/8π
M/8π
M/8π
M/12π
13−
57−
31−
25−
41−

IEEE AES Society
Hamming
Rectangular
Comparison of Rectangular & Hamming
Windows
10−
20−
30−
40−
50−
60−
20log10|W(ω|
0 0.1 0.2 0.3 0.4 0.5
Normalized frequency (f = F/Fs)

IEEE AES Society
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

IEEE AES Society
References
1. Proakis, J, G. and Manolakis, D. G., Digital Signal Processing,
Principles, Algorithms, and Applications, Prentice Hall, Upper
Saddle River, NJ, 4th Ed., 2007
2. Lyons, R. G., Understanding Digital Signal Processing, Prentice
Hall, Upper Saddle River, NJ, 2nd Ed., 2004
3. Hsu, H. P., Signals and Systems, McGraw Hill, New York, 1995
4. Hayes, M. H., Digital Signal Processing, McGraw Hill, New York,
1999
5. Oppenheim, A. V. et al, Discrete Time Signal Processing, Prentice
Hall, Upper Saddle River, NJ, 2nd Ed., 1999
6. Boulet, B., Fundamentals of Signals and Systems, Prentice Hall,
Upper Saddle River, NJ, 2nd Ed., 2000
7. Richards, M. A., Fundamentals of Radar Signal Processing, McGraw
Hill, New York, 2005
8. Skolnik, M., Radar Handbook, McGraw Hill, New York, 2nd Ed., 1990
9. Skolnik, M., Introduction to Radar Systems, McGraw Hill, New York,
3rd Ed., 2001

IEEE AES Society
Acknowledgements
• Dr Dimitris Manolakis
• Dr. Stephen C. Pohlig
• Dr William S. Song

IEEE AES Society
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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