Nachrichtentechnik I WS 2005/2006

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1 Nachrichtentechnik I WS 2005/ Signals & Systems wt 10/2005 1

2 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation signal space representation Stochastic signals properties white Gaussian noise Systems definition & classification linear time-invariant systems wt 10/2005 2

3 Motivation Signal: Physical means to represent information Necessary for information transmission Here: mathematical concept required to model communications System: Mathematical model for the transmission medium which transports information Examples: wire, optical fiber, radio transmission, filter Results in a transformation of the input signal Input Signal s(t) System g(t) Output Signal Transformation wt 10/2005 3

4 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation signal space representation Stochastic signals properties white Gaussian noise Systems definition & classification linear time-invariant systems wt 10/2005 4

5 Signals Definition (cf Lüke: Signalübertragung ) Change in a physical quantity in order to attract attention and to transfer meaning Examples: Pressure fluctuations Ink distribution on a white paper Brightness distribution on a screen Voltage / current variations Mathematical description (time signal): s(t) range (Wertebereich) domain (Definitionsbereich) wt 10/2005 5

6 Signal Classification (1) range domain real-valued real-valued / complex-valued continuous-time & continuous-range integers continuous-time & discrete-range analog signal integers discrete-time & continuous-range discrete-time & discrete-range digital signal wt 10/2005 6

7 k s(t) k+1 s(t) k+2 s(t) Signal Classification (2) Deterministic signals 6 Described by a closed mathematical expression, eg s( t) = cos(2πf 0t) - t Stochastic signals 6 Characterized by a stochastic process: - Probability distribution function / cumulative distribution function - Power density spectrum Represented by sample functions: Examples: speech signal, noise, 6 - t - t wt 10/2005 7

8 Signal Properties (1) Scalar product: Orthogonality: Cross-correlation function (CCF): (s(t),g(t)) = Z s(t) g (t) dt (s(t),g(t)) = 0 ϕ E sg(τ) =(s(t),g(t + τ)) Z = s (t) g (t + τ) dt Autocorrelation function (ACF): ϕ E ss(τ) = Z s(t) s (t + τ) ) dt wt 10/2005 8

9 Signal Properties (2) Distance between signals (norm): d = s(t) g(t) = p (s(t) g(t),s(t) g(t)) Signal energy: E s = s(t) 2 =(s(t),s(t)) = ϕ E ss(0) = Z s(t) 2 dt Mean signal power: P s = lim T 1 2T T T s( t) 2 dt Energy signals: Es < and Ps = 0 Power signals: Es = and Ps < wt 10/2005 9

10 Basic signals (1) Rectangular impulse: rect(t) = ½ 1 0 for otherwise t 1 2 Triangular impulse: Λ(t) = ½ 1 t for t 1 0 otherwise Sine x over x: si(πt) = sin(πt) πt Gaussian impulse: 1 t e 2π 2 2 wt 10/

11 Basic signals (2) Step function: ε(t) = ½ 1 0 for t 0 otherwise Dirac delta function δ(t): Z δ(τ)dτ =1 Z s(t τ)δ(τ)dτ = s(t) Sha function: III(t) = X i δ(t i) wt 10/

12 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation signal space representation Stochastic signals properties white Gaussian noise Systems definition & classification linear time-invariant systems wt 10/

13 Signal Transformations Time reflection: even signal: odd signal: Scaling: Time shift (delay element): Fourier transform: s( t) s ( t) = s( t) s( t) = s( t) s(at) s( t t0) S(f)= Z s(t)= Z s(t)e j2πft dt S(f)e j2πft df wt 10/

14 Properties of the Fourier Transform Linearity Convolution Multiplication Modulation Time Shift Similarity Similarity & Shift Differentiation t-multiplication Complex Conjugate Parseval s Theorem Time Domain Frequency Domain c 1s1( t) + c2s2( t) c 1S1( f ) + c2s2( f ) s1( t) s2( t) S1( f ) S2( f ) S f ) S ( s1( t) s2( t) 1( 2 f ) j2πf t S( f f0) 0 s( t) e s ( t t ) j2πft0 0 S( f ) e 1 f s (at) S( a ) a s a ( t t 1 f j2πft0 )) S( a ) e ( 0 d n n s(t) ( j2πf ) n S( f ) dt n n j2πt ) s( ) d S( f ) ( t s * ( ± t) S *( f ) x ( t), s( t) = X ( f ), S( f ) a df n wt 10/

15 Symmetry Properties of the Fourier Transform Time Domain s(t) Frequency Domain S(f) real even real odd imaginary even imaginary odd real even imaginary odd imaginary even real odd wt 10/

16 Basic Signals and their Spectra (1) Basic waveform Spectrum Informationstechnik Universität Ulm wt 10/

17 Basic Signals and their Spectra (2) Basic waveform Spectrum wt 10/

18 Signal Space Representation Coordinates in signal space: a i =(s(t),b i (t)) i =0, 1,,M 1 = Z s(t) b i(t) dt Linear combination: s(t) = M 1 X i=0 a i b i (t) Signal vectors: a s =(a 0,a 1,,a M 1 ) wt 10/

19 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation signal space representation Stochastic signals properties white Gaussian noise Systems definition & classification linear time-invariant systems wt 10/

20 Stochastic Signals: Properties (1) Linear ensemble mean: m s (t 0 )= 1 s(t 0 )= lim n n nx k= 1 k s(t 0 ) Ensemble mean square: P = s 2 (t 0 )= n 1 lim n nx k= 1 k s 2 (t 0 ) Variance: σs(t 2 0 )= s(t0 ) m s (t 0 ) 2 ensemble average s 2 (t 0 )=σ 2 s(t 0 )+m 2 s (t 0 ) wt 10/

21 Stochastic Signals: Properties (2) time average Linear time mean: k m = m t (k 0 ) = k s(t) = lim T Z 1 T 2 T T k s (t) dt Time mean square: Z T k 2 1 s (t) = lim k 2 s (t) T 2 T T dt Stationarity: ensemble averages are time invariant Ergodicity: time average and ensemble average coincide ergodicity stationarity wt 10/

22 Stochastic Signals: Properties (3) Autocorrelation function: ϕ ss (t 0,τ)= s(t 0 ) s(t 0 + τ) ϕ ss (t 0, 0) = s(t 0 ) s(t 0 )= s 2 (t 0 ) Autocovariance function: μ ss (t 0,τ)= s(t0 ) m s (t 0 ) s(t 0 + τ) m s (t 0 +τ ) Power density spectrum: Φ ss (t 0,f)= Z ϕ ss (t 0,τ) e j2πfτ dτ Φ ss ( 0 t0, f ) ϕss( t, τ ) wt 10/

23 Stochastic Signals: Properties (4) Crosscorrelation function: ϕ sg (t 0,τ)= s(t 0 ) g (t 0 + τ) Crosscovariance function: μ sg (t 0,τ)= s(t0 ) m s (t 0 ) g(t 0 + τ) m g (t 0 + τ) wt 10/

24 Stochastic Signals: Properties (5) Cumulative distribution function (cdf) (Verteilungsfunktion): P s (x, t 0 )= k s(t 0 ) x Prob 1 P s (x 1,t 0 ) P s (x 2,t 0 ) 1 P s (,t 0 ) = 1 P s (,t 0 ) = 0 for x 1 x 2 Probability density function (pdf) (Verteilungsdichtefunktion): p s (x, t 0 )= d dx P s(x, t 0 ) wt 10/

25 p e Gaussian Distribution (1) Probability density function (pdf): p s (x) = 1 (x m) 2 e 2 πσ 2 2σ 2 p s (x) 6 1 p x - m ; m m + Informationstechnik Universität Ulm wt 10/ P s (x)

26 1 e 2 2 Gaussian Distribution (2) p Cumulative distribution function (cdf): m ; m m + P s (x) = Z x p s (ϑ) dϑ - x P s (x) mean value - x Mittelwert wt 10/

27 Gaussian Distribution (3) Cdf of Gaussian distribution: P s (x) = Z x = 1 2 p s (ϑ) dϑ m x erfc 2 σ 2 Error function: erf (x) = Z x 2 e ϑ 2 π dϑ 0 Error function complement: erfc(x) =1 erf (x) wt 10/

28 White Gaussian Noise Process Probability distribution function: Gaussian Ergodic Process Mean value: m s = 0 Variance: 2 σ s = ϕ ss (0) Power density spectrum: Φ ss (t 0,f)=Φ ss (f) =N 0 white Autocorrelation function: ϕ ss (τ) =N 0 δ(τ) wt 10/

29 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation signal space representation Stochastic signals properties white Gaussian noise Systems definition & classification linear time-invariant systems wt 10/

30 Systems Definition (cf Oppenheim Signals and Systems ) A system can be viewed as any process that results in the transformation of signals Input Signal s(t) System g(t) $ $ Output Signal Examples: filter wire Transformation/ Mapping wt 10/

31 A System Can Be Continuous-time or discrete-time Deterministic or stochastic Memory-less or with memory Causal or non-causal Stable or unstable Linear or nonlinear Time-invariant or time-variant wt 10/

32 Block-diagram Representation of Systems s(t) = s 1 (t) g 1 (t) = s 2 (t) g 2 (t) system 1 system 2 s(t) g(t) = g 2 (t) + g 3 (t) system 3 s(t) = s 3 (t) g 3 (t) wt 10/

33 Basic Systems (1): n(t) + g(t) = s(t) + n(t) (addition of two signals) s(t) g(t) a g(t) = a s(t) (multiplication by a s(t) g(t) constant coefficient) T s(t) g(t) g(t) = s(t-t) (delay element) n(t) x g(t) = s(t) n(t) (multiplication of two s(t) g(t) signals) wt 10/

34 Basic Systems (2): SIN g(t) = sin [s(t)] (nonlinear operation) s(t) g(t) h(t) s(t) g(t) g(t) = s(t) * h(t) (LTI-system) s(t) h 1 (t) g(t) (feedback system) h 2 (t) wt 10/

35 Linear Time-Invariant (LTI) Systems (1) Linear: s i(t) $ g i(t) $ X s i (t) $ X g i (t) $ i i Superposition principle Time-Invariant: Impulse response: Z s(t) $ g(t) $ s(t τ) $ g(t τ) $ δ(t) s(t τ) δ(τ) dτ h(t) Z s(t τ) h(τ) dτ Convolution integral: Z s(t τ) h(τ) dτ = s(t) h(t) wt 10/

36 Linear Time-Invariant (LTI) Systems (2) Input Signal s(t) $ h(t) Output Signal g(t) =s(t) h(t) $ impulse response Properties of the convolution integral: commutative law: s t) h( t ( ) = h( t) s( t) associative law: distributive law: s s ( 2 t) [ h1 ( t) h2 ( t)] = [ s( t) h1 ( t)] h ( t) ( 2 t) [ h1 ( t) + h2 ( t)] = [ s( t) h1 ( t)] + [ s( t) h ( t)] wt 10/

37 Some Properties of LTI Systems Transfer function: h(t) H ( f ) g( t) = s( t) h( t) G( f ) = S( f ) H ( f ) without memory: h( t) = K δ ( t) causality: h( t) = 0 t < 0 for wt 10/

38 Special LTI Systems (1) s(t) δ(t) g(t) =s(t) Ideal system: h ideal (t)=δ(t) Output signal of an ideal system: g(t) =δ(t) s(t) = s(t) wt 10/

39 Special LTI Systems (2) Ideal lowpass filter: h TP (t)=2f g si(π2f g t) H TP (f)= rect( f ) 2f g f g : cutoff frequency Ideal bandpass filter: h BP (t)=f si(πf t) 2cos(2πf 0 t) H BP (f)= ( ( f f 0 )+ ( f + f 0 rect rect )) f f f : f 0 : bandwidth center frequency wt 10/

40 Special LTI Systems (3) Short time integration: Integration: h KI ( t) g( t) g( t) = t h Int (t)=ε(t) t T = rect T t T = rect T T T s(τ ) dτ 2 2 * s( t) integration time g(t) =ε(t) s(t) = Z t s(τ) dτ wt 10/

41 LTI Systems with Stochastic Input Signals (1) Signals, sample functions: k s(t) h(t) k g(t)=h(t) k s(t) sample function of input signal Autocorrelation functions (Wiener Lee relation) Power density spectrum (Wiener-Khintchine theorem) sample function of output signal ϕ gg (τ)=ϕ ss (τ) ϕ E hh(τ) Φ gg (f)=φ ss (f) H(f) 2 wt 10/

42 LTI Systems with Stochastic Input Signals (2) Example: WGN at the input Power density spectrum: Autocorrelation function: h(t) n (t) n e (t) Φ ne n e (f) =N 0 H(f) 2 ϕ ne n e (τ) =N 0 ϕ E hh(τ) Variance at the output: σ 2 n e = N 0 ϕ E hh(0) = N 0 E h LTI system: ideal lowpass filter Φ ne n (f) =N e 0 rect ( f ) 2f g ϕ ne n e (τ) =N 0 2f g si(π2f g t) 2 σ ne = N 0 2f g wt 10/

43 LTI Systems with Stochastic Input Signals (3) Generalized Wiener Lee relation k s(t) h ( t 1 ) k g ( t 1 ) h ( t 2 ) k g ( t 2 ) Crosscorrelation function: WGN at the input: ϕ g1 g 2 (τ)=ϕ ss (τ) ϕ E h 1 h 2 (τ) ϕ ne1 n e2 (τ)=n 0 ϕ E h 1 h 2 (τ) h ( ) ( ) and orthogonal: 1 t h ϕ (τ)= 2 t n e1 n e2 0 wt 10/

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