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.
OPTIMIZED RATE ALLOCATION OF HYPERSPECTRAL IMAGES IN COMPRESSED DOMAIN USING ...Pioneer Natural Resources
This paper studies the application of bit allocation using JPEG2000 for compressing multi-dimensional remote sensing data. Past experiments have shown that the Karhunen- Lo
`
e
ve transform (KLT) along with rate distortion optimal(RDO) bit allocation produces good compression perfor-mance. However, this model has the unavoidable disadvan-tage of paying a price in terms of implementation complex-ity. In this research we address this complexity problem byusing the discrete wavelet transform (DWT) instead of theKLT as the decorrelator. Further, we have incorporated amixed model (MM) to find the rate distortion curves instead of the prior method of using experimental rate distortioncurves for RDO bit allocation. We compared our results tothe traditional high bit rate quantizer bit allocation modelbased on the logarithm of variances among the bands. Our comparisons show that by using the MM-RDO bit rate al-location method result in lower mean squared error (MSE)compared to the traditional bit allocation scheme. Our ap- proach also has an additional advantage of using DWT asa computationally efficient decorrelator when compared tothe KLT
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.
OPTIMIZED RATE ALLOCATION OF HYPERSPECTRAL IMAGES IN COMPRESSED DOMAIN USING ...Pioneer Natural Resources
This paper studies the application of bit allocation using JPEG2000 for compressing multi-dimensional remote sensing data. Past experiments have shown that the Karhunen- Lo
`
e
ve transform (KLT) along with rate distortion optimal(RDO) bit allocation produces good compression perfor-mance. However, this model has the unavoidable disadvan-tage of paying a price in terms of implementation complex-ity. In this research we address this complexity problem byusing the discrete wavelet transform (DWT) instead of theKLT as the decorrelator. Further, we have incorporated amixed model (MM) to find the rate distortion curves instead of the prior method of using experimental rate distortioncurves for RDO bit allocation. We compared our results tothe traditional high bit rate quantizer bit allocation modelbased on the logarithm of variances among the bands. Our comparisons show that by using the MM-RDO bit rate al-location method result in lower mean squared error (MSE)compared to the traditional bit allocation scheme. Our ap- proach also has an additional advantage of using DWT asa computationally efficient decorrelator when compared tothe KLT
Signal, Sampling and signal quantizationSamS270368
Signal sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing its amplitude at regularly spaced intervals of time. This is typically done using an analog-to-digital converter (ADC). The rate at which samples are taken is called the sampling frequency, often denoted as Fs, and is measured in hertz (Hz). The Nyquist-Shannon sampling theorem states that to accurately reconstruct a signal from its samples, the sampling frequency must be at least twice the highest frequency component present in the signal (the Nyquist frequency). Sampling at a frequency below the Nyquist frequency can result in aliasing, where higher frequency components are incorrectly interpreted as lower frequency ones.
Signal, Sampling and signal quantizationSamS270368
Signal sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing its amplitude at regularly spaced intervals of time. This is typically done using an analog-to-digital converter (ADC). The rate at which samples are taken is called the sampling frequency, often denoted as Fs, and is measured in hertz (Hz). The Nyquist-Shannon sampling theorem states that to accurately reconstruct a signal from its samples, the sampling frequency must be at least twice the highest frequency component present in the signal (the Nyquist frequency). Sampling at a frequency below the Nyquist frequency can result in aliasing, where higher frequency components are incorrectly interpreted as lower frequency ones.
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
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
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.
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.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
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.
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.
3. Introduction to Waveform Coding
• Waveform coding is some kind of approximately
lossless coding, as it deals with speech signal as
any kind of ordinary data.
• The resulting signal is close as possible as the
original one.
• Codecs using this techniques have generally low
complexity and give high quality at rates 16 Kbps.
• The simplest form of waveform coding is Pulse
Code Modulation (PCM).
4. Pulse Code Modulation (PCM)
• It involves sampling and quantizing the input
waveform.
• PCM consists of three steps to digitize an
analog signal:
1. Sampling
2. Quantization
3. Binary encoding
5.
6. Prediction Filtering
• Linear prediction is a mathematical operation
where future values of a discrete-time signal
are estimated as a linear function of previous
samples.
• In digital signal processing, linear prediction is
often called linear predictive coding (LPC).
• linear prediction can be viewed as a part of
mathematical modelling or optimization.
7. The Prediction Model
• The most common representation is
• Where is the predicted signal value, x(n-i)
the previous observed values, and the
predictor coefficients.
• The error generated by this estimate is
• Where x(n) is the true value.
8. Differential pulse-code modulation
(DPCM)
• Differential pulse-code modulation (DPCM) is
a signal encoder that uses the baseline of
pulse-code modulation (PCM) but adds some
functionalities based on the prediction of the
samples of the signal.
• The input can be an analog signal or a digital
signal.
10. • DPCM code words represent differences between samples
unlike PCM where code words represented a sample value.
• Basic concept of DPCM - coding a difference, is based on
the fact that most source signals show significant
correlation between successive samples so encoding uses
redundancy in sample values which implies lower bit rate.
• Realization of basic concept (described above) is based on a
technique in which we have to predict current sample value
based upon previous samples (or sample) and we have to
encode the difference between actual value of sample and
predicted value.
12. Delta Modulation
• A Delta modulation (DM or Δ-modulation) is an analog-to-digital
and digital-to-analog signal conversion technique used for
transmission of voice information where quality is not of primary
importance.
• To achieve high signal-to-noise ratio, delta modulation must use
oversampling techniques, that is, the analog signal is sampled at a
rate several times higher than the Nyquist rate.
• Derived forms of delta modulation are continuously variable slope
delta modulation, delta-sigma modulation, and differential
modulation.
• Differential pulse-code modulation is the super-set of DM.
13. Features
• the analog signal is approximated with a series of
segments
• each segment of the approximated signal is compared to
the original analog wave to determine the increase or
decrease in relative amplitude
• the decision process for establishing the state of
successive bits is determined by this comparison
• only the change of information is sent, that is, only an
increase or decrease of the signal amplitude from the
previous sample is sent whereas a no-change condition
causes the modulated signal to remain at the same 0 or 1
state of the previous sample.
15. Differential Pulse Code Modulation
(DPCM)
• What if we look at sample differences, not the
samples themselves?
– dt = xt-xt-1
– Differences tend to be smaller
• Use 4 bits instead of 12, maybe?
16. Differential Pulse Code Modulation
(DPCM)
• Changes between adjacent samples small
• Send value, then relative changes
– value uses full bits, changes use fewer bits
– E.g., 220, 218, 221, 219, 220, 221, 222, 218,.. (all values between 218
and 222)
– Difference sequence sent: 220, +2, -3, 2, -1, -1, -1, +4....
– Result: originally for encoding sequence 0..255 numbers need 8 bits;
– Difference coding: need only 3 bits
17. Adaptive Differential Pulse Code
Modulation (ADPCM)
• Adaptive similar to DPCM, but adjusts the width of the
quantization steps
• Encode difference in 4 bits, but vary the mapping of bits to
difference dynamically
– If rapid change, use large differences
– If slow change, use small differences
19. A large step size is required when sampling those parts
of the input waveform of steep slope. But a large
step size worsens the granularity of the sampled
signal when the waveform being sampled is changing
slowly.
• A small step size is preferred in regions where the
message has a small slope. This suggests the
need for a controllable step size – the control
being sensitive to the slope of the sampled signal
• Hence ADM is prefered.
23. Basic Concepts of LPC
• It is a parametric de-convolution algorithm
• x(n) is generated by an unknown sequence e(n)
exciting a unknown system V(Z) which is supposed to
be a linear non time-variant system.
• V(Z) = G(Z)/A(Z), E(Z)V(Z) = X(Z)
• G(Z) = Σj=0 gjZ , A(Z) = Σi=0 aiZ
Q -j P -i
• Where ai and gj are parameters, real and a0 = 1
• If an algorithm could estimate all these parameters,
then V(Z) could be found, and E(Z) could be found
also. This finishes de-convolution.
24. • There are some limitations for the model
• (1) G(Z) = 1 then V(Z) = 1/A(Z) this is so called “Full
Poles εodels” and the parametric de-convolution
became coefficients(ai) estimation problem.
• (2) e(n) sequence is of form Ge(n), where e(n) is a
periodic pulse or a Gaussian white noise sequence.
For the first case e(n) = Σ6(n-rNp) and for the second
case R(k) = E[e(n)e(n+k)] = 6(k) and the value of
e(n) satisfied with Normal distribution. G is a non-
negative real number controlling the amplitude.
• The way is x(n)->V(Z)(P,ai)->e(n),G->type of e(n)
25. • Suppose x(n) and type of e(n) are known, what is
the optimized estimation of P and ai, e(n) and G? It is
the LMS algorithm.
• Suppose x(n) is the predicted value of x(n), it is the
linear sum of previous P’ known values of x:
• x(n) = Σi=1
P’ ai x(n-i)
• The predicted error
• s(n) = x(n)-x(n) = x(n) - Σi=1
P’ ai x(n-i)
• It is a stochastic sequence. The variance of it could
be used to evaluate the quality of prediction.
26. • σ2 = Σnε2(n) (time average replaced means)
• It could be proved that if x(n) is generated by “full
poles” model : x(n) = -Σi=1
P ai x(n-i) + Ge(n) and
optimized P’ = P
, optimized ai = ai, σ2 is minimal.
• σ2 = Σn [x(n) -Σi=1
P ai x(n-i)]2
• ={Σn x2(n)}-2Σi=1
P ak{Σn x(n-k)x(n)}+
• Σk=1
PΣi=1
P akai{Σn x(n-k)x(n-i)}
• By setting ð(σ2 )/ ðak = 0 we can get
• -2 {Σn x(n-k)x(n)}+2Σi=1
P ai{Σn x(n-k)x(n-i)}=0
• Or Σi=1
P aiφ(k,i) = φ(k,0)
• if φ(k,i) =Σn x(n-k)x(n-i) 1<=i<=P and 1<=k<=P
27. i=1 i
• Σ P a φ(k,i) = φ(k,0), k=1~P is called δPC
canonical equations. There are some different
algorithms to deal with the solution.
k=0 k 0
• [σ2]min = Σ P a φ(k,0) with a = 1
• So if we have x(n), φ(k,i) could be calculated,
and equations could be solved to get ai and
[σ2]min also could be obtained. For short-time
speech signal according to different lower and
upper limitation of the summary we could
have different types of equations. We will
discuss these different algorithms later.
28. Auto-Correlated Solution of LPC
• Suppose windowed signal is xw(n)
• φ(k,i) = Σn xw(n-k)xw(n-i)
• If window length is N-1 then the summation
range will be 0~N+P-1
• φ(k,i) = Σm xw(m+(i-k))xw(m) = R(i-k) if n-i
= m
• φ(k,i) = R(i-k) = R(k-i) = R(|i-k|) <= R(0)
i=1 i
• The equations became Σ P a R(|i-k|)= - R(k)
1<=k<=P
• These are Toplitz equations and have high
efficient solution.
29. • |R(0) R(1) …… R(P-1)| |a1| | R(1) |
• |R(1) R(0) …… R(P-2)| |a2| | R(2) |
• |………………………………….| |...| = …...
• |R(P-1) R(P-2) … R(0) | |ap| | R(P) |
• 6.2.1 Durbin Algorithm
• 1. E(0) = R(0)
• 2. Ki = [ R(i) - Σ aj R(i-j)]/E
(i-1) (i-1)
• 3. ai = Ki
(i)
• 4. aj = aj – Kiai-j
(i) (i-1) (i-j)
• 5. E(i) = (1-Ki )E
2 (i-1)
• Final solution is aj = aj(p)
1<=i<=p
1<=j<=i-1
1<=j<=p
30. • For iteration i we got a set of
coefficients for the predictor of i-th
order and the minimal predicted error
energy E(i). We also can get it by {R(k)}
:
• E(i) = R(0) –Σk=1 akR(k), 1<=i<=p
i
• Ki is the reflect coefficient : -1<=Ki<=1
It is a sufficient and necessary condition
for stable H(z) during iteration.
31. • 6.2.2 Schur algorithm
• At first an auxilary sequence is defined. Its properties
are :
• (1) qi(j) = R(j) when i = 0
• (2) qi(j) = 0 when i > 0, j=1~p
• (3) qp(0) = E(p) is the predicted error energy.
• (4) |qi(j)| <= R(0), it is equal only if i=j=0
• The algorithm is as following:
• 1. r(j) = R(j)/R(0), r(-j) = r(j), j=0~p
• 2. a0 = 1, E(0) = 1
• 3. q0(j) = r(j) -p<j<p
32. • 4. i = 1, k1 = r(1)
• 5. For i-p<=j<=p
qi(j) = qi-1(j) + ki *qi-1 (i-j)
ki = qi-1(j)/qi(0)
aj = qi-1(i-j)
(i)
E(i) = E(i-1)(1-ki
• 6. If i<p, back to step 5
• 7. Stop
• If we only calculate ki, then only first two expressions
in step 5 are enough. It is suitable for fix-point
calculation (r<=1) or hardware implementation.
33. Covariance Solution of LPC
k=1~p, i=0~p
let n-i=m, m=-i~N-i-1
|φ(1,0)|
|φ(2,0)|
• If not using windowing, but limiting the range of
summation, we could get :
• σ2 = Σn=0
N-1ε2(n) n=0~N-1
• φ(k,i) = Σn=0
N-1 x(n-k)x(n-i)
• = Σm=-
N-i-1 x(m+(i-k))x(m)
i
• The equations will be like following :
• |φ(1,1) φ(1,2) …… φ(1,p)| |a1|
• |φ(2,1) φ(2,2) …… φ(2,p)| |a2|
• .………………………………………………=…………
• |φ(p,1) φ(p,2) …… φ(p,p)| |ap| |φ(p,0)|
34. Covariance Solution of LPC
• The matrix is a covariance matrix and it is positive
determined, but not Toplitz. There is no high efficient
algorithm to solve. Only common used LU algorithm
could be applied. Its advantage is not having big
predicted error on the two ends of window. So when
N~P the estimated parameters have more accuracy
than auto-correlated method. But in speech
processing very often N>>P, so the advantage is not
obvious.
35. LPC parameters and their relationships
1<=j<=p
1<=j<=i-1
at last aj= aj 1<=j<=p
(p)
• (1) Reflect Coefficients
• Also known as PARCOR coefficients
• If {aj} are known, ki could be found as following :
• aj = aj
(p)
• ki = ai(i)
• aj = (aj + aj ai-j )/(1-ki )
(i-1) (i) (i) (i) 2
• The inverse process :
• aj = ki
(i)
• aj = aj - kj ai-j
(i) (I-1) (i-1)
• -1<=ki<=1 is the sufficient and necessary condition
for stable system function
36. • (2) Coefficients of Logarithm Area Ratio
• gi = log(Ai+1/Ai) = log[(1-ki)/1+ki]) i=1~p
• Where A is the intersection area of i-th
segment of the lossless tube.
• ki = (1-exp(gi))/(1+exp(gi)) i=1~p
• (3) Cepstrum Coefficients
• cn = an + Σk=1 kckan-k/n, 1<=n<=p+1
n
•
n-1 kc a /n, n>p+1
= an + Σk=n-p k n-k
37. • (4) The Roots of Predictor
• A(z) = 1 – Σk=1 akz = Πk=1 (1-zk z ) = 0
p -k p -1
• Transfer to S-plane: zi = exp(siT)
• Suppose si = σi + jΩi , zi = zir + jzii , then
• Ωi = tan(zii/zir)/T andσi= log(zii
2 + zir )/(2T)
2
• (5)The impulse response of full poles system
• h(n) = Σk=1 akh(n-k)+σ(n) n>=0
p
• = 0 n<0
Σ
38. • (6) Auto-correlated Coefficients of impulse
response of the full poles system
• H(z) = S(z)/U(z) = G/(1- Σk=1 k
p a z-k)
• The auto-correlated coefficients of h(n) is :
• R(i) = Σn=0 h(n)h(n-i) = R(-i)
• It could be proved that :
k=1 k
• R(i) = Σ p a R(|i-k|) 1<=i<=p
• And R(0) = Σk=0 k
p a R(k) + G2
• {ak} -> {R(i)} and {R(i)} -> {ak} are
equivalent
39. • (7) Auto-correlated coefficients of impulse
response of the predicted error filter (inverse
filter)
• A(z) = 1 - Σ akz-k
• The impulse response is :
• a(n) = 6(n) - Σk=1
p ak6(n-k)
• = 1, n = 0; an , 0<n<=p; 0, otherwise
• Its auto-correlated function is :
• R(i) = Σk=1
p a(k)a(k+i) 0<=i<=p
40. • (8)Line Spectrum Pair (LSP) or Line
Spectrum Frequency (LSF)
• A(p)(z)=1-Σk=1 akz
p -k ( p is even)
• Define P(z) = A(p)(z)+z-(p+1)A(p)(z-1)
Q(z) = A(p)(z)- z-(p+1)A(p)(z-1)
• It could be proved that : All roots of P(z) and
Q(z) are on the unit circle and alternatively
arranged on it provided the roots of A(z) are
inside the unit circle.
41. • Replace z with expjω:
• P(expjω)=|A(P)(expjω)|expjφ(ω)[1+exp[-j((p+1)ω+
2φ(ω))]
• Q(expjω)=|A(P)(expjω)|expjφ(ω)[1+exp[-j((p+1)ω+
2φ(ω)+π)]
• If the roots of A(P)(z) are inside the unit circle, whenωis
0~π, φ(ω) changes from 0 and returns to 0, the amount
[(p+1)ω+2φ(ω)] will be 0~(p+1)π
• P(expjω)=0 : [(p+1)ω+2φ(ω)]=kπ, k=1,3,…P+1
• Q(expjω)=0 : [(p+1)ω+2φ(ω)]=kπ, k=0,2,…P
• The roots of P and Q : Zk = expjωk [(p+1)ω+2φ(ω)]=kπ,
k=0,1,2,…P+1
• And ω0 < ω1 < ω2 < … < ωP < ωP+1
42. • If a1 ~ap are known, LSP could be found by
(p) (p)
A(z) -> P(z) -> p(w) -> f1, f2, … fp
• If f1~fp are known, ai
(p) could be found by P(z)
1 p
and Q(z) -> A(z) = P(z) + Q(z) -> a (p)~a (p)