This document discusses different types of digital modulation techniques. It begins by defining modulation as the process of modifying characteristics of a high frequency carrier signal using a low frequency information signal. The key types of digital modulation covered are:
1) Amplitude Shift Keying (ASK) which modifies the amplitude of the carrier based on the binary input signal.
2) Frequency Shift Keying (FSK) which shifts the frequency of the carrier between two frequencies (mark and space) depending on the binary input.
3) Phase Shift Keying (PSK) which shifts the phase of the carrier by 180 degrees depending on the binary input state for Binary PSK, and by multiple phase shifts for techniques like Quadrature P
The document discusses the anatomy and course of the facial nerve. It is divided into 3 parts - intracranial, intratemporal, and extracranial. The intratemporal part passes through the internal auditory meatus, labyrinthine segment, tympanic segment, and mastoid segment. The facial nerve has motor, sensory and parasympathetic components. It is vulnerable in certain segments like the labyrinthine portion. Understanding its relationships to surrounding structures is important for temporal bone surgery to avoid injury.
Materi tugas saluran transmisi dan matching impedanceEmyu Rahmawan
Saluran transmisi adalah media yang menghubungkan sumber sinyal dan beban untuk mentransmisikan gelombang elektromagnetik. Terdapat beberapa jenis saluran transmisi seperti two-wire, koaksial, mikrostrip, dan waveguide. Karakteristik utama saluran transmisi adalah impedansi karakteristiknya yang merupakan rasio antara tegangan dan arus pada ujung saluran yang panjangnya tak terhingga.
Temporal bone & Mastoid anatomy - Arjun Antony GraisonArjun Graison
The temporal bone is formed from the fusion of four bones during development. It houses important structures of the ear. The mastoid air cells begin developing in utero and continue growing after birth, providing increasing protection to the facial nerve. Key landmarks include the mastoid tip, digastric groove, sigmoid sinus, and Trautman's triangle, which is important for neurosurgical approaches. The temporal bone has complex pneumatization patterns that can vary between individuals.
This document discusses various methods for evaluating facial nerve function, including physical examination, quantitative tests, imaging, and electrodiagnostic testing. It describes the House-Brackmann grading system for assessing facial paralysis and limitations of this system for evaluating acute facial paralysis. Various topognostic tests are described but are noted to provide limited information. Imaging techniques like MRI are useful for tumor detection but show enhancement in Bell's palsy cases. Electrodiagnostic tests can help determine prognosis and predict recovery.
This document discusses different types of digital modulation techniques. It begins by defining modulation as the process of modifying characteristics of a high frequency carrier signal using a low frequency information signal. The key types of digital modulation covered are:
1) Amplitude Shift Keying (ASK) which modifies the amplitude of the carrier based on the binary input signal.
2) Frequency Shift Keying (FSK) which shifts the frequency of the carrier between two frequencies (mark and space) depending on the binary input.
3) Phase Shift Keying (PSK) which shifts the phase of the carrier by 180 degrees depending on the binary input state for Binary PSK, and by multiple phase shifts for techniques like Quadrature P
The document discusses the anatomy and course of the facial nerve. It is divided into 3 parts - intracranial, intratemporal, and extracranial. The intratemporal part passes through the internal auditory meatus, labyrinthine segment, tympanic segment, and mastoid segment. The facial nerve has motor, sensory and parasympathetic components. It is vulnerable in certain segments like the labyrinthine portion. Understanding its relationships to surrounding structures is important for temporal bone surgery to avoid injury.
Materi tugas saluran transmisi dan matching impedanceEmyu Rahmawan
Saluran transmisi adalah media yang menghubungkan sumber sinyal dan beban untuk mentransmisikan gelombang elektromagnetik. Terdapat beberapa jenis saluran transmisi seperti two-wire, koaksial, mikrostrip, dan waveguide. Karakteristik utama saluran transmisi adalah impedansi karakteristiknya yang merupakan rasio antara tegangan dan arus pada ujung saluran yang panjangnya tak terhingga.
Temporal bone & Mastoid anatomy - Arjun Antony GraisonArjun Graison
The temporal bone is formed from the fusion of four bones during development. It houses important structures of the ear. The mastoid air cells begin developing in utero and continue growing after birth, providing increasing protection to the facial nerve. Key landmarks include the mastoid tip, digastric groove, sigmoid sinus, and Trautman's triangle, which is important for neurosurgical approaches. The temporal bone has complex pneumatization patterns that can vary between individuals.
This document discusses various methods for evaluating facial nerve function, including physical examination, quantitative tests, imaging, and electrodiagnostic testing. It describes the House-Brackmann grading system for assessing facial paralysis and limitations of this system for evaluating acute facial paralysis. Various topognostic tests are described but are noted to provide limited information. Imaging techniques like MRI are useful for tumor detection but show enhancement in Bell's palsy cases. Electrodiagnostic tests can help determine prognosis and predict recovery.
This chapter discusses small-scale fading and multipath propagation effects in mobile radio channels. It explains that multipath waves traveling along paths of different lengths interfere at the receiver, causing rapid fluctuations in signal strength over short distances. The key points are:
1) Small-scale fading is caused by multipath interference and depends on factors like surrounding objects, signal bandwidth, and mobile speed.
2) Multipath propagation can be modeled using the time-varying impulse response of the channel.
3) Important parameters used to characterize fading include coherence bandwidth, Doppler spread, coherence time, delay spread, and Ricean/Rayleigh distributions.
This document is the preface to the 2005 edition of the Teletraffic Engineering Handbook published jointly by the ITU and ITC. The handbook covers the basic theory of teletraffic engineering using elementary probability theory. It includes 15 chapters covering topics like introduction, mathematical background, telecommunication loss models, data communication delay models, and measurements. The purpose is to serve as both a handbook and textbook to help engineers understand ITU recommendations on traffic engineering, evaluate tools and methods, and stay up-to-date. The handbook was initiated by the ITC and adopted by the ITU-D Study Group 2 to benefit both developing and developed countries.
This document describes the design and working of a wireless FM transmitter circuit that can transmit frequency modulated signals within the range of 88 MHz to 108 MHz using low power over a short range of 20 to 30 meters. The circuit uses components like transistors, capacitors, resistors, and wire to build an inductor and antenna. When an audio signal is input, it is mixed with the FM frequency for transmission. The circuit allows for wireless audio signal transmission that can be used in various places and institutions.
Konsepsi mengenai sinyal ini terdiri atas pengertian dari sinyal, klasifikasinya, operasi yang dapat dilakukan pada sinyal dan bagaimana cara merepresentasikan sinyal. Untuk memahami konsep sinyal, pada uraian berikut akan dibahas pengertian sinyal dan contoh-contoh sinyal.
Dalam banyak kasus, hal yang penting untuk memahami makna suatu operasi pada variabel bebas. Transformasi variabel bebas sangat berguna untuk menguji sifat-sifat dari suatu sinyal. Pada uraian ini dibahas beberapa operasi pada sinyal yang banyak dijumpai dalam praktik.
Metode yang dapat digunakan untuk memproses suatu sinyal bergantung kepada karakteristik sinyal. Beberapa metode/teknik hanya dapat digunakan pada kelompok sinyal tertentu. Karena itu pemahaman akan klasifikasi sinyal sangat diperlukan sebelum melakukan analisis sinyal. Pada bagian ini akan ditelaah macam-macam klasifikasi sinyal.
Sinyal dasar adalah sinyal yang dapat digunakan untuk menyusun atau merepresentasikan sinyal-sinyal yang lain. Ada beberapa sinyal dasar/elementer yang sering digunakan dalam praktik. Dengan merepresentasikan suatu sinyal dalam bentuk sinyal elementer, pemahaman tentang sifat-sifat sinyal dan sistem menjadi lebih mudah. Beberapa diantara sinyal-sinyal dasar tersebut memiliki karakteristik yang menjadikan penyelesaian persoalan teknik atau rekayasa menjadi lebih mudah. Pada bagian ini akan dibahas macam-macam sinyal dasar, baik sinyal waktu kontinu maupun sinyal waktu diskret.
The document provides an overview of SONET (Synchronous Optical Networking) and SDH (Synchronous Digital Hierarchy) architectures and technologies. It describes the four layers of SONET (path, line, section, photonic), SONET frame structure including overhead bytes, and how lower-rate STS frames are multiplexed into higher-rate frames. It also discusses different types of SONET networks including linear, ring and mesh, as well as the use of virtual tributaries to transport digital signals of different rates over SONET.
This document discusses Chirp Spread Spectrum (CSS) communication. It defines key terms like spectrum, spread spectrum, and chirps. It explains that chirps are signals where frequency increases or decreases over time. CSS uses chirp signals that are spread over a wide bandwidth for communication. It provides several benefits - high robustness against interference due to its wide bandwidth, resistance to multipath fading and Doppler shift, low power consumption, and low latency.
- A cochlear loss typically results in acoustic reflexes present at normal hearing levels (below 100 dB HL), but at reduced sensation levels (less than 65 dB above the hearing threshold). Significant reflex decay is not expected.
- A conductive loss usually results in absent ipsilateral acoustic reflexes in the ear with the loss. A contralateral reflex may be present if the loss is unilateral and not severe. Any reflex found would be at a normal sensation level but a higher hearing level due to the elevated threshold.
- A retrocochlear loss may result in absent reflexes or ones present at elevated hearing and sensation levels. Early on a reflex may be present but reflex decay would be found.
The facial nerve originates from the pons and has both motor and sensory components. It passes through the internal acoustic meatus, facial canal within the temporal bone, and exits at the stylomastoid foramen. Within the facial canal it has several segments separated by genu. Important surgical landmarks along the facial canal include the geniculate ganglion, pyramidal eminence, and digastric ridge. The facial nerve gives off branches like the chorda tympani and greater petrosal nerve. It terminates in five branches that innervate muscles of facial expression. Knowledge of the facial nerve anatomy is important for otologic and parotid surgeries to avoid injury.
This document discusses the anatomy and pathways of the facial nerve (cranial nerve VII). It notes that the facial nerve is composed of approximately 10,000 neurons that innervate the muscles of facial expression. It describes the various segments and branches of the facial nerve from the brainstem to the muscles of the face. It also discusses the embryological development of the facial nerve and its central connections in the brainstem and cortex.
The document provides an overview of spread spectrum techniques, including:
- A brief history noting its invention in the 1940s and military applications since the 1950s.
- Three main types of spread spectrum are described: direct sequence, frequency hopping, and time hopping.
- Direct sequence spread spectrum is explained in more detail, showing how the information signal is modulated by a spreading sequence.
- Advantages of spread spectrum techniques include resistance to jamming, ability to handle multipath interference, privacy, and allowing multiple access through different spreading codes.
middle ear spaces an important topic otorhinolaryngologyDrsiyaMedfriend
This document discusses the detailed anatomy of the various spaces within the middle ear. It begins with the embryology of the middle ear and then describes each individual space in detail, including their boundaries, contents, and surgical importance. The main spaces discussed are the epitympanum/attic, hypotympanum, mesotympanum, protympanum, retrotympanum, and their subdivisions. For each space, the document outlines their anatomical boundaries, relationships to surrounding structures, and clinical relevance for ear surgeries.
Dokumen tersebut berisi soal ujian akhir semester mata kuliah Sistem Komunikasi Serat Optik. Terdapat empat soal yang membahas karakteristik laser diode, LED, photodiode, dan sistem transmisi serat optik.
This document discusses various parameters for auditory brainstem response (ABR) testing, including stimulus type, intensity, and presentation rate. It describes advantages and disadvantages of different transducer types (insert earphones, supra-aural earphones, bone vibrators) and stimulus types (clicks, tone bursts, chirps, speech). Factors that influence ABR waveforms like latency and amplitude are discussed in relation to stimulus intensity, frequency, duration, and envelope. Guidelines are provided for optimal application of different stimulus parameters in clinical ABR testing.
This chapter discusses small-scale fading and multipath propagation effects in mobile radio channels. It explains that multipath waves traveling along paths of different lengths interfere at the receiver, causing rapid fluctuations in signal strength over short distances. The key points are:
1) Small-scale fading is caused by multipath interference and depends on factors like surrounding objects, signal bandwidth, and mobile speed.
2) Multipath propagation can be modeled using the time-varying impulse response of the channel.
3) Important parameters used to characterize fading include coherence bandwidth, Doppler spread, coherence time, delay spread, and Ricean/Rayleigh distributions.
This document is the preface to the 2005 edition of the Teletraffic Engineering Handbook published jointly by the ITU and ITC. The handbook covers the basic theory of teletraffic engineering using elementary probability theory. It includes 15 chapters covering topics like introduction, mathematical background, telecommunication loss models, data communication delay models, and measurements. The purpose is to serve as both a handbook and textbook to help engineers understand ITU recommendations on traffic engineering, evaluate tools and methods, and stay up-to-date. The handbook was initiated by the ITC and adopted by the ITU-D Study Group 2 to benefit both developing and developed countries.
This document describes the design and working of a wireless FM transmitter circuit that can transmit frequency modulated signals within the range of 88 MHz to 108 MHz using low power over a short range of 20 to 30 meters. The circuit uses components like transistors, capacitors, resistors, and wire to build an inductor and antenna. When an audio signal is input, it is mixed with the FM frequency for transmission. The circuit allows for wireless audio signal transmission that can be used in various places and institutions.
Konsepsi mengenai sinyal ini terdiri atas pengertian dari sinyal, klasifikasinya, operasi yang dapat dilakukan pada sinyal dan bagaimana cara merepresentasikan sinyal. Untuk memahami konsep sinyal, pada uraian berikut akan dibahas pengertian sinyal dan contoh-contoh sinyal.
Dalam banyak kasus, hal yang penting untuk memahami makna suatu operasi pada variabel bebas. Transformasi variabel bebas sangat berguna untuk menguji sifat-sifat dari suatu sinyal. Pada uraian ini dibahas beberapa operasi pada sinyal yang banyak dijumpai dalam praktik.
Metode yang dapat digunakan untuk memproses suatu sinyal bergantung kepada karakteristik sinyal. Beberapa metode/teknik hanya dapat digunakan pada kelompok sinyal tertentu. Karena itu pemahaman akan klasifikasi sinyal sangat diperlukan sebelum melakukan analisis sinyal. Pada bagian ini akan ditelaah macam-macam klasifikasi sinyal.
Sinyal dasar adalah sinyal yang dapat digunakan untuk menyusun atau merepresentasikan sinyal-sinyal yang lain. Ada beberapa sinyal dasar/elementer yang sering digunakan dalam praktik. Dengan merepresentasikan suatu sinyal dalam bentuk sinyal elementer, pemahaman tentang sifat-sifat sinyal dan sistem menjadi lebih mudah. Beberapa diantara sinyal-sinyal dasar tersebut memiliki karakteristik yang menjadikan penyelesaian persoalan teknik atau rekayasa menjadi lebih mudah. Pada bagian ini akan dibahas macam-macam sinyal dasar, baik sinyal waktu kontinu maupun sinyal waktu diskret.
The document provides an overview of SONET (Synchronous Optical Networking) and SDH (Synchronous Digital Hierarchy) architectures and technologies. It describes the four layers of SONET (path, line, section, photonic), SONET frame structure including overhead bytes, and how lower-rate STS frames are multiplexed into higher-rate frames. It also discusses different types of SONET networks including linear, ring and mesh, as well as the use of virtual tributaries to transport digital signals of different rates over SONET.
This document discusses Chirp Spread Spectrum (CSS) communication. It defines key terms like spectrum, spread spectrum, and chirps. It explains that chirps are signals where frequency increases or decreases over time. CSS uses chirp signals that are spread over a wide bandwidth for communication. It provides several benefits - high robustness against interference due to its wide bandwidth, resistance to multipath fading and Doppler shift, low power consumption, and low latency.
- A cochlear loss typically results in acoustic reflexes present at normal hearing levels (below 100 dB HL), but at reduced sensation levels (less than 65 dB above the hearing threshold). Significant reflex decay is not expected.
- A conductive loss usually results in absent ipsilateral acoustic reflexes in the ear with the loss. A contralateral reflex may be present if the loss is unilateral and not severe. Any reflex found would be at a normal sensation level but a higher hearing level due to the elevated threshold.
- A retrocochlear loss may result in absent reflexes or ones present at elevated hearing and sensation levels. Early on a reflex may be present but reflex decay would be found.
The facial nerve originates from the pons and has both motor and sensory components. It passes through the internal acoustic meatus, facial canal within the temporal bone, and exits at the stylomastoid foramen. Within the facial canal it has several segments separated by genu. Important surgical landmarks along the facial canal include the geniculate ganglion, pyramidal eminence, and digastric ridge. The facial nerve gives off branches like the chorda tympani and greater petrosal nerve. It terminates in five branches that innervate muscles of facial expression. Knowledge of the facial nerve anatomy is important for otologic and parotid surgeries to avoid injury.
This document discusses the anatomy and pathways of the facial nerve (cranial nerve VII). It notes that the facial nerve is composed of approximately 10,000 neurons that innervate the muscles of facial expression. It describes the various segments and branches of the facial nerve from the brainstem to the muscles of the face. It also discusses the embryological development of the facial nerve and its central connections in the brainstem and cortex.
The document provides an overview of spread spectrum techniques, including:
- A brief history noting its invention in the 1940s and military applications since the 1950s.
- Three main types of spread spectrum are described: direct sequence, frequency hopping, and time hopping.
- Direct sequence spread spectrum is explained in more detail, showing how the information signal is modulated by a spreading sequence.
- Advantages of spread spectrum techniques include resistance to jamming, ability to handle multipath interference, privacy, and allowing multiple access through different spreading codes.
middle ear spaces an important topic otorhinolaryngologyDrsiyaMedfriend
This document discusses the detailed anatomy of the various spaces within the middle ear. It begins with the embryology of the middle ear and then describes each individual space in detail, including their boundaries, contents, and surgical importance. The main spaces discussed are the epitympanum/attic, hypotympanum, mesotympanum, protympanum, retrotympanum, and their subdivisions. For each space, the document outlines their anatomical boundaries, relationships to surrounding structures, and clinical relevance for ear surgeries.
Dokumen tersebut berisi soal ujian akhir semester mata kuliah Sistem Komunikasi Serat Optik. Terdapat empat soal yang membahas karakteristik laser diode, LED, photodiode, dan sistem transmisi serat optik.
This document discusses various parameters for auditory brainstem response (ABR) testing, including stimulus type, intensity, and presentation rate. It describes advantages and disadvantages of different transducer types (insert earphones, supra-aural earphones, bone vibrators) and stimulus types (clicks, tone bursts, chirps, speech). Factors that influence ABR waveforms like latency and amplitude are discussed in relation to stimulus intensity, frequency, duration, and envelope. Guidelines are provided for optimal application of different stimulus parameters in clinical ABR testing.
Prezentare electiva 1 ing.vlad marius DINAMICA STRUCTURILOR MECANICE COMPLEXEMarius Vlad
Prezentare Materie Electiva 1 . in cadrul Scolii doctorale din Universitatea Tehnica de Constructii Bucuresti, Facultatea de Utilaj Tehnologic.
METODE NUMERICE ȘI APROXIMATIVE ÎN STUDIUL VIBRAȚIILOR SISTEMELOR DISCRETE
Metoda iterației matriceale (METODA STODOLA)
ANALIZA NUMERICĂ A VIBRAȚIILOR
Transformata Fourier Discretă (DFT)
Transformata Fourier Rapidă (FFT)
STABILITATEA MIȘCĂRII
Sisteme dinamice neliniare. Stabilitatea punctelor de echilibru. Portret de stare.
2. Analiştii de semnal dispun de un impresionant arsenal de
instrumente. Poate cel mai cunoscut dintre ele este analiza Fourier,
care descompune semnalul într-o sumă de sinusoide de diferite
frecvenţe, amplitudini sau faze.
Analiza Fourier poate fi interpretată ca o tehnică matematică
pentru transformarea percepţiei semnalului din domeniul timp (ca
funcţie de timp) în domeniul frecvenţă (ca o funcţie de frecvenţă).
21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
1. Ferestre de timp şi de frecvenţă
3. Analiza Fourier este folositoare deoarece frecvenţele conţinute
de semnal prezintă o mare importanţă, oferind o serie de informaţii
despre semnal.
Analiza Fourier are totuşi şi un dezavantaj. Prin transformarea
semnalului de timp într-unul de frecvenţă informaţia despre timp este
pierdută. De asemenea, dacă se cunoaşte transformata Fourier a
semnalului în domeniul timp, este imposibil să spunem când anume a
avut loc un eveniment de timp.
Considerăm un semnal oarecare . . Pentru acest semnal
definim:
)(tf
−=
−=
∫
∫
∞
∞−
∞
∞−
−
inversăFouriertatransformadeFtf
directăFouriertatransformadtetfF
tj
tj
ωω
π
ω
ω
ω
|
)(
2
1
)(
)()(
Prelucrarea matematică a semnalelor (PMS)21-Jan-13
4. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Se observă că dacă dispunem de putem spune orice
despre conţinutul de frecvenţă al semnalului dar nu ştim nimic
despre . Pentru a avea o informaţie despre la o anumită
frecvenţă ω este nevoie să integrăm pe un interval de timp infinit,
pierzându-se astfel informaţia temporală punctuală. Dual, transformata
Fourier inversă oferă informaţie completă în domeniul timp integrand
pe un domeniu infinit în frecvenţă. Se constată
că există o dualitate timp/frecvenţă: informaţie completă într-un
domeniu (timp sau frecvenţă) conduce la lipsă de informaţie totală în
domeniul dual (frecvenţă sau respectiv timp). în practică majoritatea
semnalelor conţin numeroase caracteristici nestaţionare sau tranzitorii:
schimbări abrupte de nivel, formă, fază sau declanşarea sau terminarea
unor evenimente. Cum analiza Fourier nu a fost concepută pentru a
detecta sau surprinde aceste caracteristici (care se produc pe intervale
scurte sau foarte scurte de timp), a fost nevoie de tehnici noi pentru
corectarea deficienţei prezentate anterior. Astfel a apărut:
Analiza Fourier pe timp scurt (Short-Time Fourier Analysis)
)(ωF
)(tf
)(tf f
)(tf
)(ωF
5. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
În 1946 Dennis Gabor a adaptat transformarea Fourier pentru
a analiza semnalul pe porţiuni mici de timp, tehnică numită ferestruirea
(windowing) semnalului. Această adaptare a lui Gabor, numită
transformarea Fourier pe timp scurt, Short-Time Fourier Transform
(STFT), mapează semnalul de timp într-o funcţie bidimensională, de
timp şi de frecvenţă.
Short-Time Fourier Analysis
6. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
În practica analizei Fourier, un semnal continuu în timp este
ferestruit cu o fereastră de o anumită dimensiune . Spectrul discret
de frecvenţe se obţine utilizând un algoritm discret numit - Discret
Fourier Transform.
Ferestruirea în timp are, evident, un corespondent în domeniul
frecvenţă. Astfel, este binecunoscut faptul că spectrul de frecvenţe al
unui semnal eşantionat este unul periodic determinat prin repetarea
spectrului semnalului neeşantionat centrat pe multipli ai frecvenţei de
eşantionare.
Ferestruirea în timp sau în frecvenţă creează însă nişte efecte
secundare, unul dintre acestea şi poate cel mai important fiind
fenomenul Gibbs.
T∆
7. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
1.1. Fenomenul Gibbs
Fenomenul Gibbs :
•Apare la reconstituirea semnalelor discontinue pe baza spectrului dat
de integrala Fourier .
•Constă în apariţia unor oscilaţii în jurul unor tranziţii abrupte în
frecvenţă (frecvenţă de tăiere).
Atât în proiectarea filtrelor numerice cât şi la estimarea
densităţii de putere spectrală, alegerea unei funcţii de ferestruire
(ferestre) joacă un rol important în determinarea calităţii rezultatelor
obţinute.
Rolul principal al ferestrei constă în diminuarea efectelor
fenomenului Gibbs provocat de trunchierea unei serii de timp infinite.
8. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Exemplu:
Pentru a proiecta un filtru - Finite Impulse Response prin
ferestruire, se porneşte de la răspunsul său ideal în frecvenţă
(caracteristica de frecvenţă dorită) , se calculează răspunsul la
impuls (ce reprezintă - Infinite Impulse Response) şi apoi se
trunchiază (ferestruieşte) răspunsul la impuls pentru a obţine un număr
finit de coeficienţi.
Trunchierea răspunsului ideal la impuls conduce la un
efect cunoscut ca fenomenul Gibbs şi constă într-un comportament
oscilant în jurul unor tranziţii abrupte în frecvenţă (frecvenţa de tăiere).
Dacă vom calcula răspunsul în frecvenţă al filtrului FIR obţinut
vom constata oscilaţii în frecvenţă în jurul frecvenţei de tăiere.
Se observă că:
•ferestruirea se face în timp (se trunchiază semnalul în domeniul timp)
•fenomenul Gibbs se manifestă în domeniul frecvenţă.
)(ωH
)(th
)(th
9. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
•Dacă ferestruirea se face în frecvenţă, atunci fenomenul Gibbs se va
observa în domeniul timp.
•Ferestruirea în frecvenţă apare inerent la sinteza unui semnal periodic
pe baza coeficienţilor Fourier.
•Sumarea efectivă nu se poate face practice pentru un număr infinit de
frecvenţe.
•Semnalul de timp aproximat se va sintetiza prin sumare finită, deci
practic spectrul semnalului periodic aproximat se obţine prin
ferestruirea în frecvenţă a spectrului semnalului original.
Concluzie:
Fenomenul Gibbs este reprezentat de oscilaţii ale semnalelor
sintetizate în jurul punctelor de discontinuitate (în timp sau în
frecvenţă), oscilaţii datorate ferestruirii semnalelor în domeniul
complementar (în frecvenţă, respectiv timp).
10. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Tipuri de ferestre:
•1. Fereastra dreptunghiulară de frecvenţă
•2. Fereastra triunghiulară de frecvenţă
•3. Fereastra Hamming
•4. Ferestre Cosinus Generalizate
1.2. Ferestre de semnal în domeniul frecvenţă
11. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
1. Fereastra dreptunghiulară de frecvenţă
Fie un semnal în domeniul timp şi transformata sa Fourier.
Considerăm o trunchiere de bandă finită a acesteia, astfel:
care rescrisă sub forma unde este un
impuls dreptunghiular unitar de frecvenţă de lăţime , centrat în
(figura a)
a) Reprezentare intrare-ieşire b) Caracteristica de frecvenţă
)(tu )(ω
∧
u
>
≤
=
∧
∧
B
Bu
uB
ω
ωω
ω
,0
),(
)(
)()()( ωωω BB puu
∧∧
= )(ωBp
B2
0=ω
12. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Semnalul de ieşire din acest sistem va fi o aproximaţie a
semnalului iniţial de intrare. Conform relaţiei generale de convoluţie
obţinem:
(2.1)
(2.2)
(2.3)
Funcţia pondere astfel determinată este cunoscută: forma ei
exprimă faptul că sistemul astfel definit este necauzal.
∫ ∫
∞
∞−
∞
∞−
−=−= ττττττ dtuwdutwtu DDB )()()()()(
)(sinsin
1
2
1
2
1
)( Btc
B
Bt
t
e
jt
detw B
B
tj
B
B
tj
D
πππ
ω
π
ωω
==== −
−
∫
( ) ( )∫ ∫
∞
∞−
∞
∞−
−=−= τττ
π
τττ
π
dtuBc
B
dutBc
B
tuB )(sin)()(sin)(
13. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Forma sa de undă sugerează ca dacă în (2.3) este un punct
de discontinuitate a lui , atunci pe măsură ce “alunecă”
spre discontinuitate creşte frecvenţa oscilaţiilor, dar amplitudinea
maximă a oscilaţiilor rămâne constantă.
Fie, de exemplu, funcţia treaptă, adică . Deducem:
Considerăm funcţia specială (neexprimabilă prin funcţii elementare)
sinus integral:
(2.4)
Deducem că:
t
)(tu )(⋅Dw
)()( ttu 1=
∫∫∫∫ ∞−∞−∞−
∞
∞−
===−=
BtBtt
DB dcdd
B
dwttu λλ
π
λ
λ
λ
π
τ
τ
τ
π
τττ )(sin
1sin1sin1
)()(1)(
λλ dcxSi
x
∫=
0
)(sin)(
)(
2
1
)(sin
1
)(sin
1
)(sin
1
)(
0
0
BtSidcdcdctu
Bt Bt
B +=+== ∫ ∫ ∫∞− ∞−
λλ
π
λλ
π
λλ
π
14. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Expresia (2.4) este reprezentată în figura de mai jos:
Mărirea lui B (lăţimii ferestrei, a benzii de frecvenţă) conduce la
apropierea momentului maximului de t = 0 , dar nu-I afectează
amplitudinea de 1.094.
Creşterea lărgimii de bandă a sistemului definit de fereastra
dreptunghiulară nu face decât să comprime oscilaţiile către punctul de
discontinuitate; în punctul de discontinuitate se va manifesta o eroare
sistematică de 9% (fenomenul Gibbs).
15. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
2. Fereastra triunghiulară de frecvenţă
(2.5)
Caracteristică de frecvenţă este impulsul triunghiular unitar din figura
de mai jos:
b) Caracteristica de frecvenţă
Semnalul de ieşire din acest sistem va fi o aproximaţie a
semnalului de intrare iniţial. Conform relaţiei generale de
convoluţie obţinem:
(2.6)
>
<−
=Λ
B
B
BB
ω
ω
ω
ω
,0
,1
)(
τττ dutwtu DB )()()( ∫
∞
∞−
−=
)(tu
16. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
unde este funcţia pondere a ferestrei triunghiulare de
frecvenţă. Aceasta se obţine din transformata Fourier inversă a lui
:
(2.7)
Fie, funcţia treaptă, . Din relaţia (2.5) deducem:
(2.8)
de
B
detw tj
B
B
B
B
tj
BD
ωω
ω
π
ωω
π ∫ ∫− −
−=Λ= 1
2
1
)(
2
1
)(
)()( ttu 1=
τλλλτττ −==−= ∫∫ ∞−
∞
tdwdtwtu
t
DDB ,)()(1)()(
0
)(twD
)(ωBΛ
17. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
3. Fereastra Hamming
Caracteristica de frecvenţă a acestei ferestre este:
Pe măsură ce lăţimea ferestrei Hamming de frecvenţă creşte
(odată cu creşterea lui ), lăţimea lobului central de timp a ferestrei
corespondente de timp se îngustează, iar frecvenţa lobilor laterali
creşte.
Aproximarea semnalului treaptă unitară este mai bună.
>
<+
=
B
B
BwHM
ω
ω
πω
ω
,0
,cos46.054.0
)(
18. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
4. Ferestre Cosinus Generalizate
•ferestrele Blackman, Flat Top, Hamming, Hann şi rectangulare sunt
toate cazuri speciale de ferestre cosinus generalizate.
•sunt o combinaţie de secvenţe sinusoidale cu frecvenţele 0, şi
2 .
Ferestrele pot fi descrise în general sub forma:
unde A, B şi C sunt constante pe care putem să le definim.
B
πω
B
πω
⋅+
⋅+=
B
C
B
BAW
πωπω
ω 2coscos)(
19. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Conceptul din spatele acestor ferestre este că prin sumarea
termenilor individuali din definiţia ferestrelor, vârfurile frecvenţelor joase
se combină în frecvenţă de o asemenea manieră încât amplitudinea
lobilor laterali să scadă. Efectul secundar este însă creşterea lăţimii
lobului principal.
•ferestrele Hamming şi Hann sunt ferestre cosinus generalizate cu doi
termeni
•fereastra Blackman este o fereastră cu trei termeni
•Fereastra cu vârf plat (Flat Top window) este o fereastră cu 5 termeni
şi este folosită pentru calibrare.
Fereastra Blackman are mai multe variante. Forma originală
este:
în care coeficienţii au următoarele valori:
+
+
+=
B
a
B
a
B
aaW
πωπωπω
ω 3cos2coscos)( 3210
20. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Fereastra dreptunghiulară în domeniul timp
(3.1)
Egalitatea:
(3.2)
sugerează faptul că aplicarea ferestrei de timp se traduce într-o
convoluţie în domeniul frecvenţă. Relaţia (3.2) înseamnă de fapt
trecerea semnalului printr-un sistem a cărui caracteristică de
frecvenţă este (vezi figura de mai jos)
a) Reprezentare intrare-ieşire b) Caracteristica de frecvenţă
1.3. Ferestre de semnal în domeniul timp
>
≤
=
Bt
Bttu
tuB
,0
),(
)(
λλλω
π
λλωλ
π
ω dupdupu BBB )()(
2
1
)()(
2
1
)(
∧∞
∞−
∧∧∞
∞−
∧∧
∫∫ −=−=
)(ωu
)()( ωω BpjH
∧
=
21. 21-Jan-13 Prelucrarea matematică a semnalelor (PMS)
Caracteristica de frecvenţă a ferestrei dreptunghiulare din
figura b) va fi:
(3.3)
Cu (3.3), relaţia (3.2) se scrie sub forma:
Toate rezultatele care s-au obţinut pentru ferestrele de
frecvenţă vor fi valabile şi în domeniul timp.
)(sin2sin
21
)( ωω
ωω
ω ωω
BcBBe
j
dtep B
B
tj
B
B
tj
B ⋅==−== −
−
−
−
∧
∫
λλλω
π
λλωλ
π
ω duBc
B
duBc
B
uB )())((sin)()(sin)(
∧∞
∞−
∧∞
∞−
∧
⋅−⋅=−⋅⋅= ∫∫
22. Va multumim pentru atentie !
21-Jan-13 Prelucrarea matematică a semnalelor (PMS)