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.
An Efficient DSP Implementation of a Dynamic Convolution Using Principal Comp...a3labdsp
In the recent years, several techniques have been proposed in the literature in order to attempt the emulation of nonlinear electro-acoustic devices, such as compressors, limiters, and pre-amplifiers. Among them, the dynamic convolution technique is one of the most common approaches used to perform a nonlinear convolution. In this paper, an efficient DSP implementation of a nonlinear system emulation based on the dynamic convolution technique and principal component analysis is proposed with the aim of lowering the required workload and the global memory usage. Several results are reported in order to show the effectiveness of the proposed approach, taking into consideration a guitar pre-amplifier as a particular case study and also introducing comparisons with the existing techniques of the state of the art.
An Efficient DSP Implementation of a Dynamic Convolution Using Principal Comp...a3labdsp
In the recent years, several techniques have been proposed in the literature in order to attempt the emulation of nonlinear electro-acoustic devices, such as compressors, limiters, and pre-amplifiers. Among them, the dynamic convolution technique is one of the most common approaches used to perform a nonlinear convolution. In this paper, an efficient DSP implementation of a nonlinear system emulation based on the dynamic convolution technique and principal component analysis is proposed with the aim of lowering the required workload and the global memory usage. Several results are reported in order to show the effectiveness of the proposed approach, taking into consideration a guitar pre-amplifier as a particular case study and also introducing comparisons with the existing techniques of the state of the art.
Real-Time Active Noise Cancellation with Simulink and Data Acquisition ToolboxIDES Editor
This paper presents the feasibility of implementing
single channel negative feedback Active Noise Cancellation
technique using adaptive filters in Real-time environment[1].
In order to establish the suitability and credibility of LMS
Algorithm for adaptive filtering in real world scenario, its
efficiency was tested beyond system based ideal simulations.
Within the MATLAB® software environment two different
methods were used to perform Real-time ANC namely
Simulink® and Data Acquisition ToolboxTM. Human voice is
used as test signal. For processing and performing adaptive
filtering, Block LMS Filter was utilised in Simulink and Error
Normalised Step Size algorithm was used in between input
and output of Signals by DAQ (Data Acquisition) toolbox
interface. A general method of using DAQ commands has been
employed which also allows for almost any kind of complex
real-time audio processing and is quite easy to follow.
Hardware Implementation of Adaptive Noise Cancellation over DSP Kit TMS320C6713CSCJournals
In noisy acoustic environment, audio signal in speech communication from mobile phone, moving car, train, aero plane, or over a noisy telephone channel is corrupted by additive random noise. The noise is unwanted signal and it is desirable to remove noise from original signal. Since noise is random process and varying at every instant of time, we need to estimate noise at every instant to remove it from original signal. There are many schemes for noise removal but most effective scheme to accomplish noise cancellation is to use adaptive filters. In this paper, we have carried out simulations for different adaptive algorithms (LMS, NLMS and RLS) and compared their performance for noise cancellation in noisy environment. Real time implementation of adaptive algorithm over DSP kit (TMS320C6713) is also presented in this paper. Performance of adaptive algorithm over hardware is also presented. Developed system incorporating best performance adaptive filter in any noisy environment can be used for noise cancellation.
Performance analysis of adaptive noise canceller for an ecg signalRaj Kumar Thenua
In numerous applications of signal processing, communications and biomedical we are faced with the necessity to remove noise and distortion from the signals. Adaptive filtering is one of the most important areas in digital signal processing to remove background noise and distortion. In last few years various adaptive algorithms are developed for noise cancellation. In this paper we present an implementation of LMS (Least Mean Square), NLMS (Normalized Least Mean Square) and RLS (Recursive Least Square) algorithms on MATLAB platform with the intention to compare their performance in noise cancellation. We simulate the adaptive filter in MATLAB with a noisy ECG signal and analyze the performance of algorithms in terms of MSE (Mean Squared Error), SNR Improvement, computational complexity and stability. The obtained results shows that RLS has the best performance but at the cost of large computational complexity and memory requirement.
PERFORMANCE ANALYIS OF LMS ADAPTIVE FIR FILTER AND RLS ADAPTIVE FIR FILTER FO...sipij
Interest in adaptive filters continues to grow as they begin to find practical real-time applications in areas
such as channel equalization, echo cancellation, noise cancellation and many other adaptive signal
processing applications. The key to successful adaptive signal processing understands the fundamental
properties of adaptive algorithms such as LMS, RLS etc. Adaptive filter is used for the cancellation of the
noise component which is overlap with undesired signal in the same frequency range. This paper presents
design, implementation and performance comparison of adaptive FIR filter using LMS and RMS
algorithms. MATLAB Simulink environment are used for simulations
Real-Time Active Noise Cancellation with Simulink and Data Acquisition ToolboxIDES Editor
This paper presents the feasibility of implementing
single channel negative feedback Active Noise Cancellation
technique using adaptive filters in Real-time environment[1].
In order to establish the suitability and credibility of LMS
Algorithm for adaptive filtering in real world scenario, its
efficiency was tested beyond system based ideal simulations.
Within the MATLAB® software environment two different
methods were used to perform Real-time ANC namely
Simulink® and Data Acquisition ToolboxTM. Human voice is
used as test signal. For processing and performing adaptive
filtering, Block LMS Filter was utilised in Simulink and Error
Normalised Step Size algorithm was used in between input
and output of Signals by DAQ (Data Acquisition) toolbox
interface. A general method of using DAQ commands has been
employed which also allows for almost any kind of complex
real-time audio processing and is quite easy to follow.
Hardware Implementation of Adaptive Noise Cancellation over DSP Kit TMS320C6713CSCJournals
In noisy acoustic environment, audio signal in speech communication from mobile phone, moving car, train, aero plane, or over a noisy telephone channel is corrupted by additive random noise. The noise is unwanted signal and it is desirable to remove noise from original signal. Since noise is random process and varying at every instant of time, we need to estimate noise at every instant to remove it from original signal. There are many schemes for noise removal but most effective scheme to accomplish noise cancellation is to use adaptive filters. In this paper, we have carried out simulations for different adaptive algorithms (LMS, NLMS and RLS) and compared their performance for noise cancellation in noisy environment. Real time implementation of adaptive algorithm over DSP kit (TMS320C6713) is also presented in this paper. Performance of adaptive algorithm over hardware is also presented. Developed system incorporating best performance adaptive filter in any noisy environment can be used for noise cancellation.
Performance analysis of adaptive noise canceller for an ecg signalRaj Kumar Thenua
In numerous applications of signal processing, communications and biomedical we are faced with the necessity to remove noise and distortion from the signals. Adaptive filtering is one of the most important areas in digital signal processing to remove background noise and distortion. In last few years various adaptive algorithms are developed for noise cancellation. In this paper we present an implementation of LMS (Least Mean Square), NLMS (Normalized Least Mean Square) and RLS (Recursive Least Square) algorithms on MATLAB platform with the intention to compare their performance in noise cancellation. We simulate the adaptive filter in MATLAB with a noisy ECG signal and analyze the performance of algorithms in terms of MSE (Mean Squared Error), SNR Improvement, computational complexity and stability. The obtained results shows that RLS has the best performance but at the cost of large computational complexity and memory requirement.
PERFORMANCE ANALYIS OF LMS ADAPTIVE FIR FILTER AND RLS ADAPTIVE FIR FILTER FO...sipij
Interest in adaptive filters continues to grow as they begin to find practical real-time applications in areas
such as channel equalization, echo cancellation, noise cancellation and many other adaptive signal
processing applications. The key to successful adaptive signal processing understands the fundamental
properties of adaptive algorithms such as LMS, RLS etc. Adaptive filter is used for the cancellation of the
noise component which is overlap with undesired signal in the same frequency range. This paper presents
design, implementation and performance comparison of adaptive FIR filter using LMS and RMS
algorithms. MATLAB Simulink environment are used for simulations
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.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
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.
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
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
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.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSE
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
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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
30. Radar Systems Course 30
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IEEE New Hampshire Section
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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
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
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 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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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
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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
53. Radar Systems Course 53
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IEEE New Hampshire Section
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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
54. Radar Systems Course 54
Review Signals, Systems & DSP 1/1/2010
IEEE New Hampshire Section
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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
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
Review Signals, Systems & DSP 1/1/2010
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
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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
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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
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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
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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
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IEEE New Hampshire Section
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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
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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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Butterfly for 2 Point DFT
[ ] [ ] [ ]1q0q0Q +=[ ]0q
[ ]1q [ ] [ ] [ ]1q0q1Q −=
[ ]kq [ ]kQ
1−
Now, Putting it all together…..
65. Radar Systems Course 65
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IEEE New Hampshire Section
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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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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
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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
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IEEE New Hampshire Section
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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
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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
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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
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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
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IEEE New Hampshire Section
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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
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−
75. Radar Systems Course 75
Review Signals, Systems & DSP 1/1/2010
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