TSDT14 Signal Theory Lecture 1 - Communication Systems

TSDT14 Signal TheoryLecture 1Introduction and Repetition of Signals & SystemsMikael OlofssonDepartment of EE (ISY)Div. of Communication Systems

TSDT14 Signal Theory - FormaliaInformation & course material:Lecturer & examiner:Tutorials and laborations:Examination:www.commsys.isy.liu.se/TSDT14Mikael Olofsson, mikael@isy.liu.seMirsad Čirkić, mirsad@isy.liu.seReza Moosavi, reza@isy.liu.seJohannes Lindblom, lindblom@isy.liu.seLaborations (2hp):Study 1, 2, 3 (3x2 hours)Sign-up on the webReport before end of exam periodWritten exam (4hp):1 simple task – Demand: 2/3 OK5 tasks (5 points each), max 25Pass: 10 points2013-09-02TSDT14 Signal Theory - Lecture 1 2

Course Aims 1(2)After passing the course, the student shouldbe able to clearly define central concepts regarding stochasticprocesses, using own words. (task 1)be able to reliably perform standard calculations regarding stochasticprocesses, e.g. LTI filtering (both time continuous and time discrete),sampling and pulse amplitude modulation. (task 1)be able to reliably perform standard calculations regarding stochasticprocesses being exposed to certain momentary non-linearities thatare common in telecommunication, especially uniform quantizationand monomial non-linearities of low degrees. (task 1)with some reliability be able to solve problems that demandintegration of knowledge from different parts of the course.(tasks 2-6)2013-09-02TSDT14 Signal Theory - Lecture 1 3

Course Aims 2(2)After passing the course, the student shouldbe able to account for the connection between different concepts inthe course in a structured way using adequate terminology.(lab report)be able to estimate the auto correlation function and power spectraldensity of a stochastic process based on a realization of the process.Also, clearly and logically account for those estimations andconclusions that can be drawn from them. (lab report)2013-09-02TSDT14 Signal Theory - Lecture 1 4

Languages in Tutorial Sessions (Lessons)Three tutorial tracks:• Group A: Johannes Lindblom, in Swedish• Group B: Reza Moosavi, in English• Group C: Mirsad Cirkic, in SwedishYou are free to follow any tutorial group you wish.You cannot demand language changes for the group teaching.Individual teaching can be done in any language that works.2013-09-02TSDT14 Signal Theory - Lecture 1 5

Standard SituationNoiseSourceEncodingModulationChannelDecodingDemodulationDestinationPartly or completely unknown ⇒Probabilistic modelWe can haveLinear and non-linear filteringSampling and reconstructionUp- and down-samplingModulationAlso: Error Correction, Packing, Cryptology,…2013-09-02TSDT14 Signal Theory - Lecture 1 6

What are Signals & Systems?Signals: Measurable physical quantities.Voltages, currents, pressures, fluxes, temperatures, …Systems: Something that is affected by signals and that generatesignals as a response.Electrical networks, mechanical systems, hydraulic systems,economical systems,…x(t)Systemy(t)Signals & Systems: A theory for analyzing signals and systems.Differential equations, convolutions, transforms2013-09-02TSDT14 Signal Theory - Lecture 1 7

What’s the Use of Signals & Systems?TelecomFiltering, modulation, estimation, equalizing, …Automatic controlModeling, control, feedback,…Signal processingSpectral analysis, detection, …Image processingMultidimensional filtering, detection of objects, …ElectronicsFilters, implementation of all of the above, …2013-09-02TSDT14 Signal Theory - Lecture 1 8

Fundamental Signalsj 2πf tComplex exponential: e = cos(2πf t) + j sin(2πf t)00 0Unit step: u(t) =⎧0, t0Unit impulse:δ(t):∞∫ x(t)δ(t) dt = x(0)−∞Property:tu(t) = ∫ δ(τ ) dτ−∞2013-09-02TSDT14 Signal Theory - Lecture 1 9

Classification of SignalsDeterministic–StochasticOnedimensional–Multidimensionalx( t)=e−t2Unknown signalsnoisex ( t)x ( a,b,c)Signal TheoryImage ProcessingPeriodic–Non-periodicTime-continuous–Time-discrete⋯⋯( t + T ) x( t)x =Amplitude continuous–Amplitude discreteDigital Filters2013-09-02TSDT14 Signal Theory - Lecture 1 10

Special OutputsImpulse response:δ(t)Energy-freesystemh(t)Step response:u(t)Energy-freesystemg(t)General case:x(t)Energy-freesystemy(t)2013-09-02TSDT14 Signal Theory - Lecture 1 11

Linear systems– preserve linear combinationsA system is linear if a linear combination of inputs result in thecorresponding linear combination of outputs.( t) y ( t)x1 →1( t) y ( t)x2 →2⇒( t) + bx ( t) → ay ( t) by ( t)ax1 21+2Example linear:yy( t) = x( t − 3)( ) (2t x t )y =( t) = x( − t)yddt( t) x( t)y =( t) = x( t) cos( ωt)( t) t x( t)y =Example non-linear:y2( t) = x ( t)( t) = x( t)y 12013-09-02TSDT14 Signal Theory - Lecture 1 12

Time-invariant systems– always behave the same wayA system is time-invariant if a time-shift of its input results in thecorresponding time-shift of its output. This must hold for all inputs and alltime-shifts. (The opposite is called time-variable.)x( t) → y( t)⇒ x( t −τ ) → y( t −τ )Example time-invariant:Example time-variable:y ( t) = x( t − 3)y ( t) = x( t)y( t) = x( t) cos( ωt)y( t) = x( − t)y2( t) = x ( t)ddty t 1 x( ) ( t)( ) ( )= ( ) (2y t = t x t y t = x t )2013-09-02TSDT14 Signal Theory - Lecture 1 13

LTI– Linear and time-invariantA system is called LTI if it is both linear (L) and time-invariant (TI).1( t) → y ( t)a1x1( t −τ1) + a2x2( t −τ2)⇒( t) y ( t)a1 y1( t −τ1) + a2y2( t −τ2)x1x2 →2Example LTI:y( t) = x( t − 3)ddt( t) x( t)y =Example non-LTI:y2( t) = x ( t)( ) (2t x t )y =( t) t x( t)y =yy( t) = x( t)y 1( t) = x( − t)( t) = x( t) cos( ωt)2013-09-02TSDT14 Signal Theory - Lecture 1 14

ConvolutionThe convolution of the signals a(t) and b(t):∞∫−∞( a *b)( t) a( τ ) b( t −τ)= dτProperties:Bilinear:Commutative:Fix one of them, linear w.r.t the other.( a * b)( t) = ( b*a)( t)Associative: (( a * b)* c)( t) = ( a*( b*c))( t)∞∫−∞( )Convergence: If a t dt is convergent and b(t) finite.…or the other way around.2013-09-02TSDT14 Signal Theory - Lecture 1 15

Output of an LTI systemx(t)H{·}y(t)( t) H{ x( t)}y =Linearity:Time-invariance:∞⎧= H ⎨ ∫ x⎩−∞∞∫−∞∞∫−∞( τ ) δ ( t −τ)( τ ) H{ δ ( t −τ)}= xdτ( τ ) h( t −τ)= x dτ⎫dτ⎬⎭=( x*h)( t)Step responce:( t) = ( u h)( t)g *∞∫−∞( τ ) u( t −τ)= h dτ=t∫−∞h( τ )dτ2013-09-02TSDT14 Signal Theory - Lecture 1 16

Cascaded LTI systemsh(t)x(t)h 1 (t)y(t)h 2 (t)z(t)A system consisting of LTI systems is an LTI system.( t) = ( x*h )( t)( t) = ( y * h )( t)= (( * h ) h )( t)*(h h ))(t)y1z2x * 1 2associativityx * 1 2Identify in( t) = ( x h)( t)⇒ ( t) = ( h h )( t)z *h * 1 22013-09-02TSDT14 Signal Theory - Lecture 1 17

Causal Systems– Do Not Know the FutureA system is causal if its output does not depend on future input values.This should hold for all inputs and all time instances.Example of causal systems:yyyyy( t) = x( t) cos( ωt)2( t) = x ( t) cos( ωt)( t) = x( t − 3)2( t) = x ( t − 3)( t) = x( τ )t∫t−1dτ– also momentary– also dynamic2013-09-02TSDT14 Signal Theory - Lecture 1 18

Anti-Causal Systems– Do Not Know the HistoryA system is anti-causal if its output does not depend on historical inputvalues. This should hold for all inputs and all time instances.Example of anti-causal systems:yyyyy( t) = x( t) cos( ωt)2( t) = x ( t) cos( ωt)( t) = x( t + 3)2( t) = x ( t + 3)t+1( t) = x( τ )∫tdτ– Also momentary– also dynamic2013-09-02TSDT14 Signal Theory - Lecture 1 19

Generally Non-Causal Systems– Know Both the History and the FutureA system that is neither causalt nor anti-causal is called generallynon-causal.Generally non-causal systems are dynamic.Example of generally non-causal systems:yyy( t) = x( t + 3) + x( t − 3)2( t) = x ( t + 3) − x( t − 3)+ 11 t2∫t−1( t) = x( τ )dτ2013-09-02TSDT14 Signal Theory - Lecture 1 20

Stable systems– behave in a controlled mannerA system is called stable (BIBO-stable) if a limited input results in alimited output. This must hold for all limited inputs and all time instances.x ( t) < M⇒y ( t) < NExample stable:yy( t) = x( t − 3)y( t) = x( − t)2( t) = x ( t)( ) (2y t = x t )y( t) = x( t) cos( ωt)Exampel unstable:ddty ( t) = x( t)y( t) = 1 x( t)( t) t x( t)y =2013-09-02TSDT14 Signal Theory - Lecture 1 21

Marginally stable– a special case of unstableUnstable systems can be divided into two types – Strictly unstable andmarginally stable. Marginally stable systems behave as if it was stablefor some limited inputs, but not for others.Example marginally stable systems:t∫−∞x( τ )dτddt( t) x( t)y =2013-09-02TSDT14 Signal Theory - Lecture 1 22

Causality and Stability of LTI SystemsAn LTI system is completely described by its impulse response.Causality⇔h ( t) = 0 , t < 0 ⇔ g( t) = 0 , t < 0Anti-causality h ( t) = 0 , t > 0g( t) = 0 , t > 0⇔⇔Stability∞∫−∞⇔ h( t)dtconvergent (absolute integrable)Marginal stability ⇔ h ( t) < M (limited)and∞∫−∞h( t)dtnon-convergent2013-09-02TSDT14 Signal Theory - Lecture 1 23

The Principle sine in – sine outFor LTI systems, we have:Sine in – Sine out (the same frequency)Compare to particular solution of a differential equation.Also: the jω method.Linearity implies:Σ Sine in – Σ Sine outDesireable:Describe signals in terms of sines.Thus: Fourier series and Fourier transforms2013-09-02TSDT14 Signal Theory - Lecture 1 24

Fourier Series ExpansionDemands on the signal x(t):1. Periodic, period T.2. Absolute integrable:( t) dt < ∞3. A finite number of local min & max in a period.4. A finite number of discontinuities in a period.T∫0xJean Baptiste Joseph Fourier1768 – 1830Then X , ˆ , ϕ exist for k ∈ 1, 2,...,∞ such thatX k0{ }xk( t) = X + Xˆsin( k πft + ϕ )∑ ∞ 0 k2k = 1holds with f = 11 T .Tone number k.1k2013-09-02TSDT14 Signal Theory - Lecture 1 25

Complex Fourier Series ExpansionEuler:sin( k2πft + ϕ )1k=ej( k 2πft+ϕ ) − j( k πft+ϕ )− ej21 k21kResult:x( t) = X + Xˆsin( k πft + ϕ )∑ ∞ 0 k2k = 11k=X0+∑ ∞k = 1⎛⎜⎝Xˆjej2kϕkejk 2πft1−Xˆkej2− jϕke− jk 2πft1⎞⎟⎠=∑ ∞k = −∞C 1e jk 2πftkC 0CkC − kRelations: X0= C0X ˆ = 2 C , ∀ k >kk0C= C*− k k,( ),∀ > 0πϕ k= + arg C kk2∀ k2013-09-02TSDT14 Signal Theory - Lecture 1 26

Determining C kConsider:T∫0x( t)e− jkπft2 1dt=T∞∫ ∑0 m=−∞Cmejmeπf1 t − jk 2πft2 1dt=∞∑m=−∞CmT∫0ej( m−k)πft2 1dt= TC kThus:Ck=T1 − jk πftT∫0x( t)e2 1dt⎧0,= ⎨⎩T,m≠km=kBut also:Ck=t0+ T1 − jk 2π f tx ( t + T ) = x( t)1x( t)e dtT∫t0Reason:e− jk 2πf1( t+T )= e( f t )− jk 2π1 +1=e1− jk 2πft2013-09-02TSDT14 Signal Theory - Lecture 1 27

Properties of Fourier Series ExpansionsLet x(t) and y(t) be periodic signals with period T and Fourier seriescoefficients C k and D k , respectively.Signal( t) by( t)x( t −τ )x( at)ax + a C k+ bDkddtx( t)Fourier series coefficient no kCkC e −kjk 2πf1 τ(period T/a)jk2πf1C k2013-09-02TSDT14 Signal Theory - Lecture 1 28

Fourier Transform( )Demands on the signal x t :Absolute integrable:∞∫−∞( t) dt < ∞Finite number of discontinuities.∞∫−∞ Limited variation: x' ( ) < ∞xtdtTransform:Xx∞∫−∞( f ) F { x( t)} = x( t)− j t= e2 πfdtInverse transform:∞- j2πft1( t) F { X ( f )} = X ( f )= e df∫−∞Spectrum of x(t):Amplitude spectrum:Phase spectrum:XXarg( f )( f ){ X ( f )}2013-09-02 TSDT14 Signal Theory - Lecture 1 29

Important Property: DerivationNotation: X ( f ) F { x( t)}= We have:F⎧⎨⎩ddtx⎫⎬⎭( t) = j2πfX ( f )Example:i( t)R+x ( t)C+y( t)−−Capacitance:ddt( t) C y( t)i =Energy freey(t)initially 0.Resistance: x ( t) − y( t) = Ri( t)(1)(2)ddtdRC y t + y t =dt(1) in (2) ⇒ x ( t) − y( t) = RC y( t)⇒ ( ) ( ) x( t)Transform ⇒j 2πfRCY ( f ) + Y ( f ) = X ( f )⇒ ( j 2πfRC +1) Y ( f ) = X ( f )⇒Y1j2πfRC( f ) =X ( f )+ 1Transform and inverse transform usually using a table.2013-09-02 TSDT14 Signal Theory - Lecture 1 30

Signal Power and Signal Energy – ParsevalSignal power:( t) 2x Signal energy: x( t)∞∫−∞2dtParseval’s relation (special case):∞∫−∞x2*( t) dt x( t) x ( t)∞∫−∞Energy spectrum: X ( f ) 2∞∫= dt−∞∞*( f ) x ( t)∫j= X e2 πf tdt df−∞∞∞= ∫ ∫−∞ −∞Xj 2πft *( f ) e df x ( t) dt =∞∫−∞*( f ) X ( f )= X df∞∫−∞( f )2= X df**Parseval’s relation (generally): a( t) b ( t) dt A( f ) B ( f )∞∫−∞∞∫= df−∞2013-09-02 TSDT14 Signal Theory - Lecture 1 31

Periodic signalsObservation:Thus:FF∞-1j2πft{ δ ( f − f0)} = ∫δ( f − f0) e−∞j 2πf{ 0 te } = δ ( f − f )Euler: F { cos( 2πf t)}F0{ sin( 2πf t)}0= F= F0j 2πft⎧ e + e⎨⎩ 2⎧e⎨⎩j 2πftx( t)periodic with period T:( ) ∑ ∞ t =k = −∞jk 1x C ke− j0 2− ej2− j0 2dtπft00πft⎫⎬⎭⎫⎬⎭fe0j2πft1212 j= 1∑ ∞ Xk1k = −∞2π f t⇒ ( ω) = C δ ( f − kf )=1T( δ ( f − f ) + ( f + f ))= δ0( δ ( f − f ) − ( f + f ))= δ0002013-09-02 TSDT14 Signal Theory - Lecture 1 32

Output from an LTI SystemNotation: A( f ) = F { a( t)}B( f ) = F { b( t)}Property: F∞∞= ∫ ∫−∞ −∞∞a∞∞ ∞− j2πf t= dτe dt− j2πf t{( a * b)( t)} ( a*b)( t) e dt = a( τ ) b( t −τ)∫−∞− j2πf t( τ ) b( t −τ) e dτdt =∞λ = t −τdλ= dt− j2π f τ− j2πf λ( τ ) e dτb( λ) e dλ= A( f ) B( f )= ∫ a ∫−∞LTI System:−∞h(t)H(f )∫ ∫−∞ −∞∞ ∞∫ ∫−∞ −∞x(t)y(t) = (x∗h)(t)X(f ) Y(f ) = X(f )H(f )− j2πf ( τ + λ )( τ ) b( λ) e= adτdλ2013-09-02 TSDT14 Signal Theory - Lecture 1 33

The Principle sine in – sine outFor LTI systems, we have:Sine in – Sine out (the same frequency)More precisely:Input: x( t) = Xˆsin( 2π f0t+ϕ)y ( t) = XˆH ( f ) sin( 2πft + ϕ + { H ( f )})Output:0 0arg0Amplitude characteristic:Phase characteristic:Harg( f ){ H ( f )}This is the jω method in condensed form.2013-09-02TSDT14 Signal Theory - Lecture 1 34

Classification of Frequency Selective FiltersA (frequency selective) filter is an LTI system. Usually it letssome frequency band through or stops it.Usually an electrical network – either passive or active.Notation Ideal amplitude characteristics Real amplitude char.Lowpass filter (LP filter):Highpass filter (HP filter):Bandpass filter (BP filter):Bandstop filter (BS filter):Allpass filter (AP filter):2013-09-02TSDT14 Signal Theory - Lecture 1 35

Time-Discrete Signals and Systems2013-09-02TSDT14 Signal Theory - Lecture 1 36

TransformationsSamplingPulse-Amplitude Modulation (PAM)2013-09-02TSDT14 Signal Theory - Lecture 1 37

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