The document discusses the Fast Fourier Transform (FFT) algorithm. It begins by explaining that the Discrete Fourier Transform (DFT) and its inverse can be computed on a digital computer, but their computational requirements are high, scaling as O(N2) for a sequence of length N. It then describes how the FFT was discovered, providing an elegant way to compute the DFT much more efficiently in O(NlogN) time by taking advantage of symmetry and recursive decomposition. The derivation and computational steps of the FFT algorithm are outlined, showing how it decomposes the DFT computation into smaller DFTs.
Sampling is a Simple method to convert analog signal into discrete Signal by using any one of its three methods
if the sampling frequency is twice or greater than twice then sampled signal can be convert back into analog signal easily......
The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
Sampling is a Simple method to convert analog signal into discrete Signal by using any one of its three methods
if the sampling frequency is twice or greater than twice then sampled signal can be convert back into analog signal easily......
The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
fast-Fourier-transform-presentation and Fourier transform for wave
in
signal possessing for
physics and
geophysics
spectra analysis
periodic and non periodic wave
data sampling
The Nyquist frequency
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
fast-Fourier-transform-presentation and Fourier transform for wave
in
signal possessing for
physics and
geophysics
spectra analysis
periodic and non periodic wave
data sampling
The Nyquist frequency
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
FPGA Implementation of Large Area Efficient and Low Power Geortzel Algorithm ...IDES Editor
Spectrum analysis is very essential requirement in
instrumentation and communication signal interception
.Spectrum analysis is normally carried out by online or offline
FFT processing. But the FFT being highly mathematical
intensive, is not suitable for low area and low power
applications. Offline FFT processing can’t give the real time
spectrum estimation which is essential in communication
signal interception. Online FFT computation takes very high
resources, which makes the system costly and power hungry.
The Goertzel algorithm is a digital signal processing (DSP)
technique for identifying frequency components of a signal,
published by Dr. Gerald Goertzel in 1958. While the general
Fast Fourier transform (FFT) algorithm computes evenly
across the bandwidth of the incoming signal, the Goertzel
algorithm looks at specific, predetermined frequency. However
the implementation of Goertzel algorithm for spectrum
computation is not explored for FPGA implementation. The
FPGA being capable of offering high frequency data paths in
them become suitable for realizing high speed spectrum
analysis algorithms.
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Embedded systems increasingly employ digital, analog and RF signals all of which are tightly synchronized in time. Debugging these systems is challenging in that one needs to measure a number of different signals in one or more domains (time, digital, frequency) and with tight time synchronization. This session will discuss how a digital oscilloscope can be used to effectively debug these systems, and some of the instrumentation considerations that go along with this.
Design and Construction of Musical Lighting Water Fountainijtsrd
This paper presents the design and construction of a musical water fountain system driven by Arduino using Fourier Transform technology. Audio signals were taken from sources such as MP3 player, smart phone, laptop, etc and these signals were first amplified using LM386 IC and then transferred into an analog input of Arduino for FFT processing. To give out signals for DC pump motors, digital output pins was used. With various frequency band, turning on and off the DC motors and LED systems was created the lighting and water spray effects which should go well along with showed music. The response of this foundation is analysed for different audio frequency range and pump motor control was emphasised and, fountain model is designed for domestic usages. Kyi Phyu Soe | Win Zaw Hein "Design and Construction of Musical Lighting Water Fountain" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25254.pdfPaper URL: https://www.ijtsrd.com/engineering/electronics-and-communication-engineering/25254/design-and-construction-of-musical-lighting-water-fountain/kyi-phyu-soe
Performance Comparison of FFT, DHT and DCT Based OFDM Systems with BPSK as A ...paperpublications3
Abstract: Today, OFDM has grown to be the most popular communication system in high-speed communications. OFDM is becoming the chosen modulation technique for wireless communications. OFDM can provide large data rates with sufficient robustness to radio channel impairments. Different type of transform techniques such as Discrete Hartley transform (DHT), Discrete Cosine Transform (DCT), Fast Fourier Transform (FFT) are used to perform the modulation and demodulation operations for the OFDM system. In this paper these three transforms are used in OFDM and study the comparison between these different transforms techniques used in OFDM system based on Bit Error Rate (BER) performances.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
How to Get CNIC Information System with Paksim Ga.pptxdanishmna97
Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...
3 f3 3_fast_ fourier_transform
1. Fast Fourier Transform (FFT)
Elena Punskaya
www-sigproc.eng.cam.ac.uk/~op205
Some material adapted from courses by
Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner
55
2. Computations for Evaluating the DFT
• Discrete Fourier Transform and its Inverse can be
computed on a digital computer
• What are computational requirements?
56
3. Operation Count
How do we evaluate computational complexity?
count the number of real multiplications and
additions
by ¨real¨ we mean either fixed-point or floating point
operations depending on the specific hardware
subtraction is considered equivalent to additions
divisions are counted separately
Other operations such as loading from memory, storing in
memory, loop counting, indexing, etc are not counted
(depends on implementation/architecture) – present
overhead
57
4. Operation Count – Modern DSP
Modern DSP:
a real multiplication and addition is a single machine
cycle y=ax+b called MAC (multiply/accumulate)
maximum number of multiplications and additions must
be counted, not their sum
traditional claim ¨multiplication is much more timeconsuming than addition¨ becomes obsolete
58
5. Discrete Fourier Transform as a Vector Operation
Again, we have a sequence of sampled values x n and
transformed values Xp , given by
Let us denote:
Then:
x
59
6. Matrix Interpretation of the DFT
This Equation can be rewritten in matrix form:
x
complex
numbers
as follows:
X0
0
0
X1
0
1
N-1
2(N-1)
…
0
x0
N-1
x1
…
0
…
XN-1
…
…
…
=
…
2
…
…
0
(N-1) 2
N x N matrix
…
can be
computed
off-line
and
stored
x N-1
60
7. DFT Evaluation – Operation Count
Multiplication of a complex N x N matrix by a complex
N-dimensional vector.
No attempt to save operations
N2
complex multiplications (¨x¨s)
N(N-1) complex additions (¨+¨s)
First row and first column are 1 however
2
save 2N-1 ¨x¨s, (N-1) complex ¨x¨s & N(N-1) ¨+¨s
Complex ¨x¨: 4 real ¨x¨s and 2 real ¨+¨s;
complex ¨+¨ : 2 real ¨+¨s
2
Total: 4(N-1) real ¨x¨s & 4(N-0.5)(N-1) real ¨+¨s
61
8. DFT Computational Complexity
Thus, for the input sequence of length N the number of
arithmetic operations in direct computation of DFT is
proportional to N 2
.
For N=1000, about a million operations are needed!
In 1960s such a number was considered prohibitive
in most applications.
62
9. Discovery of the Fast Fourier Transform (FFT)
When in 1965 Cooley and Tukey ¨first¨ announced
discovery of Fast Fourier Transform (FFT) in 1965 it
revolutionised Digital Signal Processing.
They were actually 150 years late – the principle of the
FFT was later discovered in obscure section of one of
Gauss’ (as in Gaussian) own notebooks in 1806.
Their paper is the most cited mathematical paper ever
written.
63
10. The Fast Fourier Transform (FFT)
The FFT is a highly elegant and efficient algorithm,
which is still one of the most used algorithms in speech
processing, communications, frequency estimation, etc
– one of the most highly developed area of DSP.
There are many different types and variations.
They are just computational schemes for computing the
DFT – not new transforms!
Here we consider the most basic radix-2 algorithm
which requires N to be a power of 2.
64
11. FFT Derivation 1
Lets take the basic DFT equation:
Now, split the summation into two parts: one for even n and one for odd n:
Take
outside
65
13. FFT Derivation 3
Now, take the same equation again where we split the summation
into two parts: one for even n and one for odd n:
And evaluate at frequencies p+N/2
67
25. FFT Derivation Summary
• The FFT derivation relies on redundancy in the calculation
of the basic DFT
• A recursive algorithm is derived that repeatedly rearranges
the problem into two simpler problems of half the size
• Hence the basic algorithm operates on signals of length a
power of 2, i.e.
(for some integer M)
• At the bottom of the tree
we have the classic FFT
`butterfly’ structure
79
26. Computational Load of full FFT algorithm
The type of FFT we have considered, where N = 2M, is
called a radix-2 FFT.
It has M = log2 N stages, each using N / 2 butterflies
Complex multiplication requires 4 real multiplications
and 2 real additions
Complex addition/subtraction requires 2 real additions
Thus, butterfly requires 10 real operations.
Hence the radix-2 N-point FFT requires 10( N / 2 )log2 N
real operations compared to about 8N2 real operations
80
for the DFT.
27. Computational Load of full FFT algorithm
The radix-2 N-point FFT requires 10( N / 2 )log2 N real
operations compared to about 8N2 real operations for the
DFT.
This is a huge speed-up in typical applications, where N is
128 – 4096:
Direct DFT
FFT
81
29. The Inverse FFT (IFFT)
The IDFT is different from the DFT:
• it uses positive power of
instead of negative ones
• There is an additional division of each output value by N
Any FFT algorithm can be modified to perform the IDFT by
• using positive powers instead of negatives
• Multiplying each component of the output by 1 / N
Hence the algorithm is the same but computational load
increases due to N extra multiplications
83
30. Many types of FFT
The form of FFT we have described is called “decimation in time”; there
is a form called “decimation in frequency” (but it has no advantages).
The "radix 2" FFT must have length N a power of 2. Slightly more efficient
is the "radix 4" FFT, in which 2-input 2-output butterflies are replaced by 4input 4-output units. The transform length must then be a power of 4
(more restrictive).
A completely different type of algorithm, the Winograd Fourier Transform
Algorithm (WFTA), can be used for FFT lengths equal to the product of a
number of mutually prime factors (e.g. 9*7*5 = 315 or 5*16 = 80). The
WFTA uses fewer multipliers, but more adders, than a similar-length FFT.
Efficient algorithms exist for FFTing real (not complex) data at about 60%
the effort of the same-sized complex-data FFT.
The Discrete Cosine and Sine Transforms (DCT and DST) are
similar real-signal algorithms used in image coding.
84
31. Applications of the FFT
There FFT is surely the most widely used signal processing
algorithm of all. It is the basic building block for a large
percentage of algorithms in current usage
Specific examples include:
• Spectrum analysis – used for analysing and detecting
signals
• Coding – audio and speech signals are often coded in the
frequency domain using FFT variants (MP3, …)
• Another recent application is in a modulation scheme called
OFDM, which is used for digital TV broadcasting (DVB) and
digital radio (audio) broadcasting (DAB).
• Background noise reduction for mobile telephony, speech
and audio signals is often implemented in the frequency
85
domain using FFTs
32. Practical Spectral Analysis
• Say, we have a symphony recording over 40 minutes
long – about 2500 seconds
• Compact-disk recordings are sampled at 44.1 kHz and
are in stereo, but say we combine both channels into a
single one by summing them at each time point
• Are we to compute the DFT of a sequence one hundred
million samples long?
a) unrealistic – speed and storage
requirements are too high
b) useless – we will get a very much noiselike
wide-band spectrum covering the range from 20
Hz to 20 kHz at high resolution including all
notes of all instruments with their harmonics 86
33. Short-Time Spectral Analysis
• We actually want a sequence of short DFTs, each
showing the spectrum of a signal of relatively short
interval
• If violin plays note E during this interval – spectrum
would exhibit energies at not the frequency of note E
(329 Hz) and characteristic harmonics of the violin
• If the interval is short enough – we may be able to
track note-to-note changes
• If the frequency resolution is high enough – we will be
able to identify musical instruments playing
Remember: our – ear – excellent spectrum analyser?
87
34. Short-Time Spectral Analysis – Computational Load
• If we take a time interval of about 93 milliseconds
(signal of length 4096) the frequency resolution will
be 11 Hz (adequate)
• The number of distinct intervals in the symphony is
40 x 60/0.093 - about 26,000
• We may choose to be conservative and overlap the
intervals (say 50%)
• In total, need to compute 52,000 FFTs of length 4096
itself
the entire computation will be done in
considerably less time than the music
Short-time spectral analysis also use windowing
this later)
(more on
88
35. Case Study: Spectral analysis of a Musical Signal
Sample rate is
10.025 kHz
(T=1/10,025 s)
Extract a short segment
Load this into Matlab as a vector x
Take an FFT, N=512
X=fft(x(1:512));
Looks almost
periodic over short time
interval
89
36. Case Study: Spectral analysis of a Musical Signal, FFT
Symmetric
Note Conjugate symmetry
as data are real:
Symmetric
Anti-Symmetric
90
38. The Effect of data length N
N=32
FFT
N=128
FFT
N=1024
FFT
Low
resolution
High
resolution
92
39. The DFT approximation to the DTFT
DTFT at frequency
:
DFT:
• Ideally the DFT should be a `good’ approximation to the DTFT
• Intuitively the approximation gets better as the number of data
points N increases
• This is illustrated in the previous slide – resolution gets better as
N increases (more, narrower, peaks in spectrum).
•
How to evaluate this analytically?
View the truncation in the summation as a multiplication by a
rectangle window function
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46. Now, imagine what happens when the sum of two frequency components is DFT-ed:
The DTFT is given by a train of delta functions:
Hence the windowed spectrum is just the convolution of the window spectrum with
the delta functions:
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47. DFT for the data
Both components
separately
Both components
Together
ωΤ
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48. Brief summary
•
•
•
•
The rectangular window introduces broadening of any frequency
components (`smearing’) and sidelobes that may overlap with other
frequency components (`leakage’).
The effect improves as N increases
However, the rectangle window has poor properties and better choices
of wn can lead to better spectral properties (less leakage, in particular)
– i.e. instead of just truncating the summation, we can pre-multiply by a
suitable window function wn that has better frequency domain
properties.
More on window design in the filter design section of the course – see
later
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49. Summary so far …
• The Fourier Transform (FT) is a way of transforming
a continuous signal into the frequency domain
• The Discrete Time Fourier Transform (DTFT) is a
Fourier Transform of a sampled signal
• The Discrete Fourier Transform (DFT) is a discrete
numerical equivalent using sums instead of integrals
that can be computed on a digital computer
• The FFT is a highly elegant and efficient algorithm,
which is still one of the most used algorithms in
speech processing, communications, frequency
estimation, etc – one of the most highly developed
area of DSP
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