This digital method is built using chirp z-transform(CZT) and provides 100% alias-free bandwidth such as using ideal LPF. This noble method is efficient for economic and practical considerations.
- Obtained the Fast Fourier Transform of signals.
- Designed and Validated Low Pass, High Pass, and Band Pass filters in compliance with the specifications.
- Produced and compared graphs of the results upon processing.
These slides cover the fundamentals of data communication & networking. It covers Channel Capacity It is useful for engineering students & also for the candidates who want to master data communication & computer networking.
Suppression of Chirp Interferers in GPS Using the Fractional Fourier TransformCSCJournals
In this paper we apply the Fractional Fourier Transform (FrFT) to remove chirp interferers that corrupt Global Positioning System (GPS) signals. The concept is based on the fact that in the time-frequency plane, known as the Wigner Distribution (WD), chirps are represented as lines. Using an FrFT with some rotational parameter ‘a’, we rotate to a new time axis ta that transforms the chirp to a tone, in which the energy of the tone is contained in usually just one or two samples. The best `a', and the correct time sample along the ta axis, may be found without a priori knowledge by searching for the peak in the FrFT, since compression to one or two time samples results in an energy spike. Once the peak is found, we zero out the tone, and hence the underlying chirp. Rotation back to the original time domain via an inverse FrFT produces an improved GPS signal. This method can apply to multiple chirp interferers, and we describe how to easily determine the number of interferers, K, by finding peaks in the FrFT space over the parameter `a'. We also describe how to easily notch the interferers once converted to tones by computing a threshold based on the power of the coarse acquisition (C/A) code and noise. We show that for signal-to-noise ratios (SNRs) greater than at least 10 dB, interferers can be notched regardless of the ratio of the C/A code power to the combined interferer power, denoted as carrier-to-interference ratio (CIR).
METHOD FOR REDUCING OF NOISE BY IMPROVING SIGNAL-TO-NOISE-RATIO IN WIRELESS LANIJNSA Journal
The signal to noise ratio (SNR) is one of the important measures for reducing the noise.A technique that uses a linear prediction error filter (LPEF) and an adaptive digital filter (ADF) to achieve noise reduction in a speech and image degraded by additive background noise is proposed. Since a speech signal can be represented as the stationary signal over a short interval of time, most of speech signal can be predicted by the LPEF. This estimation is performed by the ADF which is used as system identification. Noise reduction is achieved by subtracting the reconstructed noise from the speech degraded by additive background noise. Most of the MR image accelerating methods suffers from degradation of acquired images, which is often correlated with the degree of acceleration. However, Wideband MRI is a novel technique that transcends such flaws.In this paper we proposed LPEF and ADF for reducing the noise in speech and also we demonstrate that Wideband MRI is capable of obtaining images with identical quality as conventional MR images in terms of SNR in wireless LAN.
- Obtained the Fast Fourier Transform of signals.
- Designed and Validated Low Pass, High Pass, and Band Pass filters in compliance with the specifications.
- Produced and compared graphs of the results upon processing.
These slides cover the fundamentals of data communication & networking. It covers Channel Capacity It is useful for engineering students & also for the candidates who want to master data communication & computer networking.
Suppression of Chirp Interferers in GPS Using the Fractional Fourier TransformCSCJournals
In this paper we apply the Fractional Fourier Transform (FrFT) to remove chirp interferers that corrupt Global Positioning System (GPS) signals. The concept is based on the fact that in the time-frequency plane, known as the Wigner Distribution (WD), chirps are represented as lines. Using an FrFT with some rotational parameter ‘a’, we rotate to a new time axis ta that transforms the chirp to a tone, in which the energy of the tone is contained in usually just one or two samples. The best `a', and the correct time sample along the ta axis, may be found without a priori knowledge by searching for the peak in the FrFT, since compression to one or two time samples results in an energy spike. Once the peak is found, we zero out the tone, and hence the underlying chirp. Rotation back to the original time domain via an inverse FrFT produces an improved GPS signal. This method can apply to multiple chirp interferers, and we describe how to easily determine the number of interferers, K, by finding peaks in the FrFT space over the parameter `a'. We also describe how to easily notch the interferers once converted to tones by computing a threshold based on the power of the coarse acquisition (C/A) code and noise. We show that for signal-to-noise ratios (SNRs) greater than at least 10 dB, interferers can be notched regardless of the ratio of the C/A code power to the combined interferer power, denoted as carrier-to-interference ratio (CIR).
METHOD FOR REDUCING OF NOISE BY IMPROVING SIGNAL-TO-NOISE-RATIO IN WIRELESS LANIJNSA Journal
The signal to noise ratio (SNR) is one of the important measures for reducing the noise.A technique that uses a linear prediction error filter (LPEF) and an adaptive digital filter (ADF) to achieve noise reduction in a speech and image degraded by additive background noise is proposed. Since a speech signal can be represented as the stationary signal over a short interval of time, most of speech signal can be predicted by the LPEF. This estimation is performed by the ADF which is used as system identification. Noise reduction is achieved by subtracting the reconstructed noise from the speech degraded by additive background noise. Most of the MR image accelerating methods suffers from degradation of acquired images, which is often correlated with the degree of acceleration. However, Wideband MRI is a novel technique that transcends such flaws.In this paper we proposed LPEF and ADF for reducing the noise in speech and also we demonstrate that Wideband MRI is capable of obtaining images with identical quality as conventional MR images in terms of SNR in wireless LAN.
Wireless Communication Networks and Systems 1st Edition Beard Solutions Manualpuriryrap
Full download : http://alibabadownload.com/product/wireless-communication-networks-and-systems-1st-edition-beard-solutions-manual/
Wireless Communication Networks and Systems 1st Edition Beard Solutions Manual
The Presentation includes Basics of Non - Uniform Quantization, Companding and different Pulse Code Modulation Techniques. Comparison of Various PCM techniques is done considering various Parameters in Communication Systems.
Speech Processing in Stressing Co-Channel Interference Using the Wigner Distr...CSCJournals
The Fractional Fourier Transform (FrFT) can provide significant interference suppression (IS) over other techniques in real-life non-stationary environments because it can operate with very few samples. However, the optimum rotational parameter ‘a’ must first be estimated. Recently, a new method to estimate ‘a’ based on the value that minimizes the projection of the product of the Wigner Distributions (WDs) of the signal-of-interest (SOI) and interference was proposed. This is more easily calculated by recognizing its equivalency to choosing ‘a’ for which the product of the energies of the SOI and interference in the FrFT domain is minimized, termed the WD-FrFT algorithm. The algorithm was shown to estimate ‘a’ more accurately than minimum mean square error FrFT (MMSE-FrFT) methods and perform far better than MMSE Fast Fourier Transform (MMSE-FFT) methods, which only operate in the frequency domain. The WD-FrFT algorithm significantly improves interference suppression (IS) capability, even at low signal-to-noise ratio (SNR). In this paper, we apply the proposed WD-FrFT technique to recovering a speech signal in non-stationary co-channel interference. Using mean-square error (MSE) between the SOI and its estimate as the performance metric, we show that the technique greatly outperforms the conventional methods, MMSE-FrFT and MMSE-FFT, which fail with just one non-stationary interferer, and continues to perform well in the presence of severe co-channel interference (CCI) consisting of multiple, equal power, non-stationary interferers. This method therefore has great potential for separating co-channel signals in harsh, noisy, non-stationary environments.
source coding systems is the very first concept in digital communications.In this slide, i have mentioned about the basics of digital communications and sampling theorem.
It consists of PCM,DPCM,ADPCM,delta modulation.
Wireless Communication Networks and Systems 1st Edition Beard Solutions Manualpuriryrap
Full download : http://alibabadownload.com/product/wireless-communication-networks-and-systems-1st-edition-beard-solutions-manual/
Wireless Communication Networks and Systems 1st Edition Beard Solutions Manual
The Presentation includes Basics of Non - Uniform Quantization, Companding and different Pulse Code Modulation Techniques. Comparison of Various PCM techniques is done considering various Parameters in Communication Systems.
Speech Processing in Stressing Co-Channel Interference Using the Wigner Distr...CSCJournals
The Fractional Fourier Transform (FrFT) can provide significant interference suppression (IS) over other techniques in real-life non-stationary environments because it can operate with very few samples. However, the optimum rotational parameter ‘a’ must first be estimated. Recently, a new method to estimate ‘a’ based on the value that minimizes the projection of the product of the Wigner Distributions (WDs) of the signal-of-interest (SOI) and interference was proposed. This is more easily calculated by recognizing its equivalency to choosing ‘a’ for which the product of the energies of the SOI and interference in the FrFT domain is minimized, termed the WD-FrFT algorithm. The algorithm was shown to estimate ‘a’ more accurately than minimum mean square error FrFT (MMSE-FrFT) methods and perform far better than MMSE Fast Fourier Transform (MMSE-FFT) methods, which only operate in the frequency domain. The WD-FrFT algorithm significantly improves interference suppression (IS) capability, even at low signal-to-noise ratio (SNR). In this paper, we apply the proposed WD-FrFT technique to recovering a speech signal in non-stationary co-channel interference. Using mean-square error (MSE) between the SOI and its estimate as the performance metric, we show that the technique greatly outperforms the conventional methods, MMSE-FrFT and MMSE-FFT, which fail with just one non-stationary interferer, and continues to perform well in the presence of severe co-channel interference (CCI) consisting of multiple, equal power, non-stationary interferers. This method therefore has great potential for separating co-channel signals in harsh, noisy, non-stationary environments.
source coding systems is the very first concept in digital communications.In this slide, i have mentioned about the basics of digital communications and sampling theorem.
It consists of PCM,DPCM,ADPCM,delta modulation.
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introduction to pulse shaping and equalization in advanced digital communication, it's characterisation, signal design of band limited signal, design of bandlimited signal for no ISI and design of bandlimited signal with controlled ISI-partial response, linear equalization,
AREA EFFICIENT & COST EFFECTIVE PULSE SHAPING FILTER FOR SOFTWARE RADIOS ijasuc
In this paper area efficient and cost effective techniques for design of pulse shaping filter have been
presented to improve the computational and implementation complexity. Pulse shaping filters have been
designed and implemented by using Raised cosine filter, Nyquist filter and optimized half band filters for
software defined radio (SDR) based wireless applications. The performance of different filters is compared
in terms of BER and hardware requirements. The results show that the BER performance of the optimized
designs is almost identical to the Raised cosine filter with significant reduction in hardware requirements.
The hardware saving of 60% to 90% can be achieved by replacing the Raised cosine filter with proposed
filters to provide cost effective solution for wireless communication applications.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
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Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
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.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
De-mystifying Zero to One: Design Informed Techniques for Greenfield Innovati...
Digital anti aliasing filter
1. 1
Abstract Anti-aliasing filter is indispensable to digital signal processing, especially analog to digital converter. An ideal
anti-aliasing filter is an ideal low pass filter(LPF), which passes all input frequencies below cutsoff frequency and stop
the undesired frequencies above . Such an ideal LPF is not physically realizable. In practice, feasible filter is analog LPF,
which was well developed. However due to the transition band of analog LPF, which contains a gradual attenuation of the
input frequencies, analog LPF attenuates all frequencies 20% beyond the bandwidth to zero. Therefore alias-free range is
80% of bandwidth. In practice, the sampling frequency should be greater than 2.5 times the maximum frequency
of input signal. For some conservative design, can be chosen 10 times higher than the filter cut-off frequency . For
this reason, analog filter is not efficient in terms of economic consideration. Now we introduce the new digital approach
for anti-aliasing filter. This digital method is built using chirp z-transform(CZT) and provides 100% alias-free bandwidth
such as using ideal LPF. This noble method is efficient for economic and practical considerations.
I. INTRODUCTION
liasing can occur when a signal is sampled at rate,
which is less than half of the Nyquist frequency .
Anti-aliasing filter cuts off all the undesired higher
frequencies greater than and passes all the
appropriate input frequencies below . It ensure that
the signal is bandlimited, | | or at least nearly so.
Consider rectangular spectral filter for ideal LPF,
| | (1)
its Fourier transform pair is sinc functuon,
↔ (2)
Suppose is the input signal for sampling, and is
output after passing LPF.
(3)
where is a truncated function of with
interval . Since sinc function is time
unlimited function, therefore it should be applied within
certain interval.
(4)
where .
(5)
From above equation, possibly implementable LPF is
, not . Set , is
Figure 1. feasible LPF with
The ideal low pass filter has sharp cutoff frequency
, which the reconstructed signal spectrum is
defined on the interval | | . However is
not good as ideal LPF due to many overshoots and
transition band. An ideal anti-aliasing filter is ideal low
pass filter, which passes all the input frequencies, less
than and cuts off all the frequencies, greater than .
In practice, the transition band between and ,
attenuate the input frequencies. Even we want to pass
only signals with frequencies less than , those signals
in the transition band could still cause aliasing.
For example, we have an input signal whose
maximum frequency , and sampling it with
sampling frequency , which is equal to
Nyquist rate . Bandwidth of anti-aliasing filter is
. However even , the actual usable
bandwidth can be less than 100 Hz. The anti-aliasing
filter reduces the bandwidth to 80 Hz and below. In
many data acquisition systems, there is an anti-aliasing
filter which starts reducing the amplitude of the signal
starting at 80% of the bandwidth.[1] Therefore in
practice, the sampling frequency should be at least
greater than 2.5 times the maximum frequency of
input signal. As an example, the maximum frequency
of an audio signal is up to 20 KHz. The Nyquist sampling
rate is theoretically 40 KHz. However it is commonly
sampled with from 44.1 KHz to 96 KHz.
Well known classical analog anti-aliasing filters are
Elliptical, Butterworth, Chebyshev, Bessel, Cauer and as
well as many others. Among these filters, Butterworth
filter the most commonly used.
Analog anti-aliasing filer is not efficient in bandwidth
usage. This is the main reason why we introduce the
digital approach.
Digital Anti-Aliasing Filter
Deahyun Kim
A
March 15, 2020
2. 2
II. DIGITAL ANTI-ALIASING FILTER
A. Introduction of Digital Filter
Instead of removing extraneous frequencies prior to
aliasing, the digital approach uses a much higher
oversampling frequency and utilize CZT and iFFT to
obtain the sampled signal with undersampling rate .
This process does not require analog filter that provide
insufficient bandwidth efficiency due to transition band.
Suppose a signal is given, with maximum frequency
and we are going to take undersampling with
. is the sampled discrete signal after
digital anti-aliasing filter. The following is diagram of
digital process to avoid aliasing without analog filter.
Figure 2. oversampling for digital anti-aliasing filter
Oversampling is first step for digital process.
Figure 3. diagram of digital anti-aliasing filter
where CZT is ‘Chirp z-transform’ and iFFT is ‘inverse
FFT’. CZT perform as ideal LPF function with the sharp
cutoff frequency , which is necessary to maximize use
of the available bandwidth for undersampling without
aliasing. In the above diagram, the signal data is digitally
processed and undersampled with the desired sampling
frequency .
B. Key Statement
Suppose is defined on , band unlimited
function, and its Fourier transform exist.
Statement 1. In Fourier transform domain, as time
sampling period is getting small, frequency period of
is longer and bandwidth of | |
increase. Therefore become close to bandlimited
function, and can be recovered from with
sufficiently small sampling period .
Statement 2. Transform maximum frequency in
bandwidth of | | , is proportional to
sampling frequency .
(6)
where is the maximum frequency represented by
calculation of Fourier transform. Statement 2 means
that according to sampling frequency , transform
maximum frequency is decided, and vice versa.
Statement 3. If we select cutoff frequency of in
figure 3 as,
(7)
and denote otherwize =0,
then samped signal with undersampling frequency
can be determined from .
The above eq(7) is basic concept for our digital filter
model.
C. Procedure of Digital Filter
Suppose that is a bandlimited function with and
we want to undersampling with .
sampling rate
maximum frequency of
Nyquist frequency,
oversampling frequency,
undersampling frequency,
cutoff frequency of LPF
Table 1 frequency symbols
[Step 1] Sampling with oversampling frequency
and obtain .
This process is oversampling to obtain . In
other words, by sufficiently oversampling, it is close to
without provocative aliasing.(statement 1)
Oversampling frequency should be integer multiple
of to perform later Fourier transform.
(8)
[Step 2] Take CZT of to calculate ,
where , and is desired undersampling rate
(statement 3). The reason why to use CZT instead of
DFT is because there is no need to calculate DFT beyond
. Using CZT, we calculate .
[Setp 3] Take iFFT of , then the result
is identical to by undersampling frequency
without aliasing through ideal LPF.
This digital process is simple and provides 100%
frequency bandwidth efficiency without aliasing. Also it
doesn’t require analog filter, and will reduce the cost of
implementation.
D. Introduction of CZT
Suppose is N-point sequence. Chirp z-transform
along spiral arcs in the z-plane is defined as[2]
{ } ∑ ∑ (9)
for , where
(10)
(11)
A is the complex starting point, are respectively
starting angle and end angle.
W is the complex ratio between points, and M is the
number of points to calculate, then eq(9) become
3. 3
{ } ∑ [ ] [ ] (12)
Set ,
then CZT calculate the z-transform of on this
contour.
Figure 4. chirp z-transform
If set in eq(12), spiral set is M evenly
spaced samples around the unit circle. Suppose N is the
length of the input sequence. As known, FFT provides a
frequency resolution of . When we improve the
resolution of only certain narrowband, it is inefficient to
improve entire bandwidth by FFT with sufficiently large
N. In this case, CZT can improve only certain frequency
band such as data zoom in. CZT is also another method
to calculate DFT fast as FFT. CZT is more efficient than
the DFT algorithm for the computation of arbitrary size,
including the prime size.[3]
Set for eq(12),
{ } ∑ (13)
for
Apply concept of CZT into LPF with cutoff frequency ,
set
(14)
Simply using eq(13),
{ } ∑ (15)
for .
E. LPF Kernel,
Recall figure 3 again, is rearranged to make
to perform iFFT of .
{ (16)
is consisted of iFFT of and ,
(17)
where
(18)
(19)
∑ (20)
Using eq(15), the are respectively,
∑ [∑ ] (21)
∑ [∑ ] (22)
∑ (23)
where is LPF kernel,
∑ (24)
where
{ (25)
Set , is
Figure 5. with
is similar with Dirichlet kernel. Its properties are
is complex number
periodic function with period
maximum value,
integral, ∫ for
where is non-zero positive integer.
zeros at
F. Concept of Digital Filter
Continue to derive , similar with eq(21),
∑ [∑ ] (26)
∑ (27)
∑ (28)
{
( )
(29)
4. 4
Combined with eq(25) and eq(29), figure 3 is described
by
(30)
Figure 6. concept of digital anti-aliasing filter
III. SIMULATION
A. Introduction
A speech voice signal is given, maximum frequency
KHz. The speech is “Hi I am emma. One of the
available high quality text to speech voices.”
We want sampling with = Hz using digital
method in figure 3.
Figure 7. given speech voice signal
Recall figure 3 again. is the sampled signal, digital
audio with oversampling frequency Hz,
which is a common sampling frequency.[2] Recall eq(8),
, (31)
Since KHz, if is less than Nyquist frequency
KHz, then aliasing occurs in and voice of
will be distorted without anti-aliasing process.
Compare with of sampling frequency
Hz and Hz without anti-aliasing filter.
Figure 8. Comparison of with Hz and
Hz without anti-aliasing filter.
The magnitude of is normalized to compare each
other. As shown in the above figure, we can recognize
the aliasing effect in by undersampling
Hz without anti-aliasing process.
B. Digital Filter
Now is given, its data sample size
and time length is 5.6689 sec. In real process, data
samples acquired per frame. Frame size is amount of
time data collected to perform a transform. And block is
data samples acquired during one frame.
For example, is sampled with Hz and
divided by 5 frames, then its frame size is 1.1338 sec
and block size is 50,000. However main purpose of this
section is to understand digital anti-aliasing filter
process, then we process without frame segment,
therefore is processed as one frame.
[Step 1] Using eq(15), take chirp z-transform of to
calculate , , where Hz.
For eq(15), and .
Figure 9. ,
[Step 2] Make , using .
Figure 10. ,
The data sample size of is .
[Step 3] Take iFFT of with point.
The result is a sampled signal with Hz
using digital anti-aliasing filter. Data length of is
from eq(31).
C. Results
The original voice signal is now sampled by
undersampling frequency Hz with digital
anti-aliasing filter.
Figure 11. by Hz with digital AA process
The two voices will be easily distinguishable. Refer to
appendix for code of each figure.
5. 5
IV. CONCLUSION
As mentioned, digital filter is implemented by CZT of
and iFFT of and provide 100% bandwidth
usage. Recall figure 3 and 6, we have the knowledge that
output is related to the partial sum of Fourier
series. Also this is represented by interpolation of iFFT.
Using these relationship, we will introduce another new
digital method, which use only FFT.
APPENDIX
Figure 1
fs=5; fc=fs/2;a=1;ts=0.01;N=10000;
t=-a:ts:a-ts;wt=2*fc.*sinc(2.*fc*t);
xf=fft(wt,N);
f0=1/ts/N;f=0:f0:fs-f0;fL=length(f);
wla=abs(xf(1:fL)).*ts; plot(f,wla)
Figure 4
a0=0.7;w0=0.99;w1=pi/3;w2=5*pi;
M=80;dw=(w2-w1)/(M-1);k=0:M-1;
a=a0*exp(-1i*w1);
w=w0*exp(-1i*dw);
z=a*(w.^-(k')); zplane(z)
Figure 5
M=20;fc=5;Ts=0.001;t=-6:Ts:6;
sk=zeros(size(t));
for k=0:M-1
sk=sk+exp(2i.*pi.*fc.*t.*k./M);
end;
sk=sk./2/M;plot(t,abs(sk))
Figure 7
[s,fos] = audioread('emma.wav');N=250000;so=s(1:N,1);
t=0:1/fos:N/fos-1/fos;plot(t,so);figure;
sf=fft(so); f0=1/N;f=0:f0:.3-f0;f=f.*fos;fL=length(f);
plot(f,abs(sf(1:fL)))
sound(so,fos)
Figure 8
clear all
[s,fos] = audioread('emma.wav');N=250000;so=s(1:N,1);
fus=4410;fc=fus/2;fr=fos/fus;
sf=fft(so)/fr; f0=1/N;f=0:f0*fos:fc-f0*fos;fL=length(f);
plot(f,abs(sf(1:fL)));figure
s4=so(1:fr:end);s4L=length(s4);%sampling fs=4.4k
sf4=fft(s4); f0=1/s4L;f=0:f0*fus:fc-f0*fus;fL=length(f);
plot(f,abs(sf4(1:fL)))
Figure 9
clear all
[s,fos] = audioread('emma.wav');N=250000;so=s(1:N,1);
fus=4410;fc=fus/2;fr=fos/fus;M=N/fr/2;
f0=1/N;f=0:f0*fos:fc-f0*fos;
w=exp(-1i*2*pi*fc/(M*fos));a=1;
Scf=czt(so,M,w,a)/fr;plot(f,abs(Scf))
Figure 10&11
clear all
[s,fos] = audioread('emma.wav');N=250000;
fus=4410;fc=fus/2;fr=fos/fus;M=N/fr/2;
f0=1/N;f=0:f0*fos:fc-f0*fos;
%==so(n)==
so=s(1:N,1)';
%==CZT of so(n)==
w=exp(-1i*2*pi*fc/(M*fos));b=1;
Scf=czt(so,M,w,b)/fr;
%==Sus(k)==
sa=so(1:2:end);sb=-so(2:2:end);a=sum([sa sb]);
Sus=[Scf a flip(conj((Scf(2:M))))];
f=0:f0*fos:fus- f0*fos;plot(f,abs(Sus));
%==s(n) with fus with digital AA=
sn=real(ifft(Sus));
t=0:1/fus:2*M/fus-1/fus;figure;plot(t,sn);figure;
%==s(n) with fus without AA==
s4=so(1:fr:end);plot(t,s4);
%== sound s(n) with fus with digital AA=
sound(sn,fus)
REFERENCES
[1] Siemens Digital Industries Software-Digital Signal
Processing: Sampling Rates, Bandwidth, Spectral
Lines, and more
[2] Rabiner, Schafer, Rader, “The Chirp z-Transform
Algorithm”. The Bell System Technical Journal, May
1969, page 84.
[3] Ronald Allen, Duncan Mills, “Signal Analysis: Time,
Frequency, Scale, and Structure”, Wiley-IEEE Press,
page 574
Deahyun Kim was born in Andong, South Korea. He
received the Bachelor's Degree in Electronic
Engineering from Chung-Ang University, Seoul in 1998,
the Master's Degree in Electrical Engineering in New
York University, Tandon School of Engineering in 1993
and his Ph.D. Degree in Digital Signal Processing, New
York University, Tandon School of Engineering in 1995.
In 1994, he joined the SK telecom R&D center located at
Daedeok Science Town in Daejeon, South Korea. After
1995, he had worked at SK telecom R&D center in New
Jersey. He was involved in the project of pager chipset,
CDMA chipset development, and participated in CDMA
standard organization e.g.CDG, 3GPP2, TIA. He currently
work in research on Fourier space such as remove Gibbs
phenomenon, digital anti-aliasing filter, etc.