This document provides an overview of signals, systems, and digital signal processing as they relate to radar systems engineering. It begins with introductions to continuous and discrete-time signals and systems. It then covers topics like sampling theory, the discrete Fourier transform, and finite impulse response filters. The goal is to give non-electrical engineering majors a brief introduction to relevant concepts from signals and systems courses to enhance their understanding of radar systems. Various signal processing techniques are applied to received radar signals to enable optimum target detection.
Describes Pulse Compression in Radar Systems.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
Tutorial Content
This tutorial provides a broad-based discussion of radar system, covering the following topics:
-Introduction to Radars in Military and Commercial Applications
-Radar System Block Diagram
-Radar Antennas (slotted waveguide array, planar array), Transmitter (magnetron, solid-state), Receiver, Pedestal and Radome
-Plot Extraction, Tracking Algorithms and Display
-Radar Range Equation, Detection Performance
-Wave Propagation and Radar Cross Section
-Emerging and Advanced Radar Systems (phased-array, multi-beam, multi-mode, FMCW, solid-state)
In the discussion, practical systems, technical specifications and data will be used to enhance learning.In addition, simulation results will also be used to present findings.
The objective of the tutorial session is to equip participants with solid understanding of radar systems for system level applications and prepare them for advanced and professional radar courses, projects and research.
This tutorial is designed and developed based on the following references:
[1] G. W. Stimson, Introduction to Airborne Radar Second Edition, Scitech Publishing, 1998.
[2] L. V. Blake, A Guide to Basic Pulse-Radar Maximum-Range Calculation, NRL Report 6930, 1969.
[3] K. H. Lee, Radar Systems for Nanyang Technological University, TBSS, 2014.
RADAR - RAdio Detection And Ranging
This is the Part 2 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
Describes Pulse Compression in Radar Systems.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
Tutorial Content
This tutorial provides a broad-based discussion of radar system, covering the following topics:
-Introduction to Radars in Military and Commercial Applications
-Radar System Block Diagram
-Radar Antennas (slotted waveguide array, planar array), Transmitter (magnetron, solid-state), Receiver, Pedestal and Radome
-Plot Extraction, Tracking Algorithms and Display
-Radar Range Equation, Detection Performance
-Wave Propagation and Radar Cross Section
-Emerging and Advanced Radar Systems (phased-array, multi-beam, multi-mode, FMCW, solid-state)
In the discussion, practical systems, technical specifications and data will be used to enhance learning.In addition, simulation results will also be used to present findings.
The objective of the tutorial session is to equip participants with solid understanding of radar systems for system level applications and prepare them for advanced and professional radar courses, projects and research.
This tutorial is designed and developed based on the following references:
[1] G. W. Stimson, Introduction to Airborne Radar Second Edition, Scitech Publishing, 1998.
[2] L. V. Blake, A Guide to Basic Pulse-Radar Maximum-Range Calculation, NRL Report 6930, 1969.
[3] K. H. Lee, Radar Systems for Nanyang Technological University, TBSS, 2014.
RADAR - RAdio Detection And Ranging
This is the Part 2 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
10 range and doppler measurements in radar systemsSolo Hermelin
Present method of Range and Doppler measurement in a RADAR system.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Recommend to view this presentation on my website in power point.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
A loop antenna has simple structure but its analysis is not easy to perform. Since a loop antenna is a dual pair of a dipole antenna, we can adopt the analysis of a dipole for a loop based on the duality theorem. By stacking a number of loops, we can increase the antenna gain and radiation resistance very easily.
Describes Signal Processing in Radar Systems,
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
I recommend to see the presentation on my website under RADAR Folder, Signal Processing Subfolder.
Wireless communications is a hot topic in technology today, driven by technologies like Wireless Networking, Cellular Telephony, Wireless Connectivity and Satellite Communications among others. Traditionally, wireless and RF communications has been one of the last bastions of analog engineering. With the advent of low cost digital, high speed integrated circuits, this too has become part of the digital domain. Although information transmitted today is largely digital high frequency signals whether digital or analog always behave like analog signals so having fundamental knowledge of this high frequency behavior is key.
10 range and doppler measurements in radar systemsSolo Hermelin
Present method of Range and Doppler measurement in a RADAR system.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Recommend to view this presentation on my website in power point.
RADAR - RAdio Detection And Ranging
This is the Part 1 of 2 of RADAR Introduction.
For comments please contact me at solo.hermelin@gmail.com.
For more presentation on different subjects visit my website at http://www.solohermelin.com.
Part of the Figures were not properly downloaded. I recommend viewing the presentation on my website under RADAR Folder.
A loop antenna has simple structure but its analysis is not easy to perform. Since a loop antenna is a dual pair of a dipole antenna, we can adopt the analysis of a dipole for a loop based on the duality theorem. By stacking a number of loops, we can increase the antenna gain and radiation resistance very easily.
Describes Signal Processing in Radar Systems,
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
I recommend to see the presentation on my website under RADAR Folder, Signal Processing Subfolder.
Wireless communications is a hot topic in technology today, driven by technologies like Wireless Networking, Cellular Telephony, Wireless Connectivity and Satellite Communications among others. Traditionally, wireless and RF communications has been one of the last bastions of analog engineering. With the advent of low cost digital, high speed integrated circuits, this too has become part of the digital domain. Although information transmitted today is largely digital high frequency signals whether digital or analog always behave like analog signals so having fundamental knowledge of this high frequency behavior is key.
Introduction to InSpec and 1.0 release updateAlex Pop
Contains an introduction to infrastructure and compliance tests as code and how InSpec can be used for this.
Agenda:
* Why infrastructure tests as code
* What is InSpec and how it works
* Core and custom resources
* What's new in InSpec 1.0 (released Sept 26, 2016)
* Documentation and installation
* Integrations
* Demo
* Chef Community Summit
Описание и структурирование бизнес-процессов в компании при внедрении корпор...Виктор Степанов
- Почему нужно описывать бизнес-процессы до автоматизации.
- Общие принципы описания бизнес-процессов.
- Как соблюсти баланс гибкости и регламентации.
- Как организовать проект внедрения информационной системы.
- Дальнейшее развитие и поддержка системы.
- Роль консультантов по управлению в проекте внедрения информационной системы.
Access the video from this presentation for free from
http://www.rohde-schwarz-usa.com/DebuggingEMISS_On-Demand.html
Overview:
Electromagnetic interference is increasingly becoming a problem in complex systems that must interoperate in both digital and RF domains. When failures due to EMI occur it is often difficult to track down the sources of such failures using standard test receivers and spectrum analyzers. The unique ability of real-time spectrum analysis and synchronous time domain signal acquisition to capture transient events can quickly reveals details about the sources of EMI.
What You Will Learn:
How to isolate and analyze sources of EMI using an oscilloscope
Measurement considerations for correlating time and frequency domains
Near field probing basics
Presented By:
Dave Rishavy, Product Manager Oscilloscopes, Rohde & Schwarz
Dave Rishavy has a BS in Electrical Engineering from Florida State University and an MBA from the University of Colorado. Prior to joining Rohde and Schwarz, Mr. Rishavy gained over 15 years of experience in the test and measurement field at Agilent Technologies. This included positions in a wide range of technical marketing areas such as application engineering, product marketing, marketing management and strategic product planning. While at Agilent, Dave led the marketing and industry segment teams for the Infiniium line of oscilloscopes as well as high end logic analysis.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Halogenation process of chemical process industries
Radar 2009 a 3 review of signals, systems, and dsp
1. 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
IEEE New Hampshire Section
Guest Lecturer
2. Radar Systems Course 2
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
3. Radar Systems Course 3
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
4. Radar Systems Course 4
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
5. Radar Systems Course 5
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
6. Radar Systems Course 6
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Continuous Time Signal
( )tx
t0
( ) ( ) ( )
( )
( ) 532
t25tttx
300t12tx
t3cos79tsin100tx
−
+−=
−=
π−π=
Examples:
7. Radar Systems Course 7
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Continuous Time Signal
t0
( )tx ( )tx
t0
• Types of continuous time signals
– Periodic or Non-periodic
Non-periodic
• • •• • •
( ) ( )txttx =Δ+
Periodic
8. Radar Systems Course 8
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
9. Radar Systems Course 9
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
10. Radar Systems Course 10
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
11. Radar Systems Course 11
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Linear Time Invariant Systems
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
12. Radar Systems Course 12
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Linear Time Invariant Systems
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
13. Radar Systems Course 13
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
14. Radar Systems Course 14
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
15. Radar Systems Course 15
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
16. Radar Systems Course 16
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
17. Radar Systems Course 17
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
18. Radar Systems Course 18
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
−
19. Radar Systems Course 19
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
“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
20. Radar Systems Course 20
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
21. Radar Systems Course 21
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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 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
22. Radar Systems Course 22
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
23. Radar Systems Course 23
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
24. Radar Systems Course 24
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
25. Radar Systems Course 25
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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 <
26. Radar Systems Course 26
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
B2
T
2
F
S
S ≥
π
=
B2FS =
B
27. Radar Systems Course 27
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
Ft2j
cc
enx
dtenTttgFX
dtetxFX
π−
∞
−∞=
∞
∞−
π−
∞
−∞=
∞
∞−
π−
∑
∫ ∑
∫
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−δ=
=
( )txc [ ]nx
( )FXc
0
( )FX
0
•• •
sF sF3sF2sF−
•• •
28. Radar Systems Course 28
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
● ● ● ● ● ●
● ● ● ● ● ●
29. Radar Systems Course 29
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
30. Radar Systems Course 30
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
Adapted from Proakis and Manolakis, Reference 1
31. Radar Systems Course 31
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
32. Radar Systems Course 32
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
−= ∑
∞
−∞=
33. Radar Systems Course 33
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
34. Radar Systems Course 34
Review Signals, Systems & DSP 1/1/2010
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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 2 : Take one of the sequences and time reverse it
[ ]=− kh
-2 -1 0
1
2
3
35. Radar Systems Course 35
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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 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
36. Radar Systems Course 36
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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 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
46. Radar Systems Course 46
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
47. Radar Systems Course 47
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Finite Impulse Response (FIR) Filters
• Weighting of Filters
48. Radar Systems Course 48
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
49. Radar Systems Course 49
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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Time and Frequency Domains
Analysis
Synthesis
Fourier Transform
Inverse Fourier Transform
Time History Frequency Spectrum
Frequency DomainTime Domain
50. Radar Systems Course 50
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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
51. Radar Systems Course 51
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IEEE New Hampshire Section
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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
52. Radar Systems Course 52
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
53. Radar Systems Course 53
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
54. Radar Systems Course 54
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
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
Adapted from Proakis and Manolakis, Reference 1
55. Radar Systems Course 55
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
56. Radar Systems Course 56
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IEEE New Hampshire Section
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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”
57. Radar Systems Course 57
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
58. Radar Systems Course 58
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
59. Radar Systems Course 59
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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 =
60. Radar Systems Course 60
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
61. Radar Systems Course 61
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
62. Radar Systems Course 62
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
63. Radar Systems Course 63
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IEEE New Hampshire Section
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Butterfly for 2 Point DFT
[ ] [ ] [ ]1q0q0Q +=[ ]0q
[ ]1q [ ] [ ] [ ]1q0q1Q −=
[ ]kq [ ]kQ
1−
Now, Putting it all together…..
65. Radar Systems Course 65
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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 =
66. Radar Systems Course 66
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IEEE New Hampshire Section
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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
67. Radar Systems Course 67
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
68. Radar Systems Course 68
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
69. Radar Systems Course 69
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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:
( )∞−
[ ] [ ] [ ]1ny
M
1M
nx
M
1
ny −
−
+=
[ ] [ ] N/nkj2
1N
0n
enxkX π−
−
=
∑=
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]1,nx2,nxnyor
1xnxn,kaky
1N
0n
−=
= ∑
−
=
70. Radar Systems Course 70
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
71. Radar Systems Course 71
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE 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
72. Radar Systems Course 72
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
74. Radar Systems Course 74
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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−
75. Radar Systems Course 75
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IEEE New Hampshire Section
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)
76. Radar Systems Course 76
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
77. Radar Systems Course 77
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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
78. Radar Systems Course 78
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
Acknowledgements
• Dr Dimitris Manolakis
• Dr. Stephen C. Pohlig
• Dr William S. Song
79. Radar Systems Course 79
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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