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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
Radar Systems Course 3Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 4Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 5Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 6Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Continuous Time Signal
( )tx
t0
( ) ( ) ( )( )( ) 532 t25tttx
300t12txt3cos79tsin100tx
−+−=
−=π−π=
Examples:
Radar Systems Course 7Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Continuous Time Signal
t0
( )tx ( )tx
t0
• Types of continuous time signals– Periodic or Non-periodic
Non-periodic
• • •• • •
( ) ( )txttx =Δ+
Periodic
Radar Systems Course 8Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Continuous Time Signal
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
Radar Systems Course 9Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 10Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Linear Time Invariant Systems (Delta Function)
• The impulse response is the response of the system when the input is
ContinuousLinear Time
InvariantSystem
( )tx ( )tyContinuousLinear Time
InvariantSystem
( )tδ ( )th
( )
( ) 1dtt
0t0t0
t
=δ
=∞≠
=δ
∫∞
∞−
Properties of Delta Function
( )tδ( )th
Radar Systems Course 11Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Linear Time Invariant Systems
ContinuousLinear Time
InvariantSystem
( )tx ( )tyContinuousLinear Time
InvariantSystem
( )tδ ( )th
Definition : Convolution of Two Functions
( ) ( ) ( ) ( ) ττ−τ≡∗ ∫∞
∞−
dtxxtxtx 2121
Reversedand
Shifted
Radar Systems Course 12Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Linear Time Invariant Systems
ContinuousLinear Time
InvariantSystem
( )tx ( )tyContinuousLinear Time
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
Radar Systems Course 13Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 14Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 15Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 16Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 17Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 18Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
−
Radar Systems Course 19Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
“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
Radar Systems Course 20Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 21Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 22Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 23Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 24Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 25Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Signal Sampling Issues
• Signal Reconstruction
• Elimination of “Aliasing”
( )FX
0
• • •
sF2sF
LPF ( )FXc
0
B2B2Fs >
( )FX
0• • •
sF sF3sF2 sF4sF−• • •
Overlapping, Aliased Spectra
B2Fs <
Radar Systems Course 26Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 27Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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−
• • •
Radar Systems Course 28Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
● ● ● ● ● ●
● ● ● ● ● ●
Radar Systems Course 29Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 30Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Adapted from Proakis and Manolakis, Reference 1
Radar Systems Course 31Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 32Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Convolution for Discrete Time Systems
( ) ( ) ( ) ττ−τ= ∫∞
∞−
dtxhty
ContinuousLinear Time
InvariantSystem
( )tx ( )tyContinuous-time
System
( )tx
Discrete-timeSystem
[ ]nx
DiscreteLinear Time
InvariantSystem
[ ]ny[ ]nx
[ ] [ ] [ ]knxkhnyk
−= ∑∞
−∞=
Radar Systems Course 33Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 34Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=
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
Radar Systems Course 35Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=
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
Radar Systems Course 36Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=
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
Radar Systems Course 37Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
No overlap –
= 0
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 38Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
One sample overlaps –
= (1x1) = 1
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 39Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
Two samples overlaps –
= (1x2)+(2x1) = 4
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 40Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
Three samples overlaps –
= (1x3)+(2x2)+(4x1) = 11
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 41Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
Three samples overlaps –
= (2x3)+(4x2)+(3x1) = 17
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 42Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
Three samples overlaps –
= (4x3)+(3x2)+(1x1) = 17
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 43Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
Two samples overlaps –
= (3x3)+(1x2) = 11
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 44Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
One sample overlaps –
= (1x3) = 3
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 45Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Graphical Implementation of Convolution
No overlap –
= 0
[ ] [ ] [ ] [ ] [ ]knhkxknxkhnykk
−=−= ∑∑∞
−∞=
∞
−∞=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
Radar Systems Course 46Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 47Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 48Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 49Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Time and Frequency Domains
Analysis
Synthesis
Fourier Transform
Inverse Fourier Transform
Time History Frequency Spectrum
Frequency DomainTime Domain
Radar Systems Course 50Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 51Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 52Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 53Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 54Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Adapted from Proakis and Manolakis, Reference 1
Radar Systems Course 55Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Continuous Signals and Systems
• Sampled Data and Discrete Time Systems
• Discrete Fourier Transform (DFT)– Calculation
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
Radar Systems Course 56Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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
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”
Radar Systems Course 57Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• 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
Radar Systems Course 58Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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”
( )2NO
⎟⎠⎞
⎜⎝⎛ Nlog
2NO 2
Radar Systems Course 59Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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/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 =
Radar Systems Course 60Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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 knN+=
[ ] [ ] ⎥⎦⎤
⎢⎣⎡ +=⎥⎦
⎤⎢⎣⎡ +=
2NkHkH
2NkGkG
( )( ) )k(HW2/NkHW
WWWWkN
)2/N(kN
kN
2/NN
kN
)2/N(kN
−=+
−==+
+
then
Radar Systems Course 61Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 62Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 63Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Butterfly for 2 Point DFT
[ ] [ ] [ ]1q0q0Q +=[ ]0q
[ ]1q [ ] [ ] [ ]1q0q1Q −=
[ ]kq [ ]kQ
1−
Now, Putting it all together…..
Radar Systems Course 64Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 65Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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/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 =
Radar Systems Course 66Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 67Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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
Radar Systems Course 68Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• 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
Radar Systems Course 69Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
−=
= ∑−
=
Radar Systems Course 70Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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−
π1ω−1ω−
π−
1ω 2ω2ω−2ω− 1ω 2ω
π−
π−π−
ππ
π
( )jweH
( )jweH
( )jweH
( )jweH
Radar Systems Course 71Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• 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
Radar Systems Course 72Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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−ωω
=ω
ωω
=−−
==ω −−ω−
ω−−
=
ω−∑
2/sin2/Msin
W
2/sin2/Msine
e1e1eW 2/1Mj
j
Mj1M
0n
nj
Radar Systems Course 73Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 74Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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−
Radar Systems Course 75Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
)
Radar Systems Course 76Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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
Radar Systems Course 77Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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., 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
Radar Systems Course 78Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE AES Society
Acknowledgements
• Dr Dimitris Manolakis• Dr. Stephen C. Pohlig• Dr William S. Song
Radar Systems Course 79Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire SectionIEEE 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