Nachrichtentechnik I WS 2005/2006 1 Signals & Systems wt 10/2005 1
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
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
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
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
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
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
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
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
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 10/2005 10
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 10/2005 11
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 12
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 10/2005 13
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 10/2005 14
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 10/2005 15
Basic Signals and their Spectra (1) Basic waveform Spectrum Informationstechnik Universität Ulm wt 10/2005 16
Basic Signals and their Spectra (2) Basic waveform Spectrum wt 10/2005 17
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 10/2005 18
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 19
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 10/2005 20
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 10/2005 21
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 10/2005 22
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 10/2005 23
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 10/2005 24
p e 2 2 6 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 2 2 1 x - m ; m m + Informationstechnik Universität Ulm wt 10/2005 25 P s (x)
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) 6 1 1 2 mean value - x Mittelwert wt 10/2005 26
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 10/2005 27
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 10/2005 28
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 29
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 10/2005 30
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 10/2005 31
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 10/2005 32
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 10/2005 33
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 10/2005 34
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 10/2005 35
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 10/2005 36
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 10/2005 37
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 10/2005 38
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 10/2005 39
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 10/2005 40
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 10/2005 41
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 10/2005 42
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 10/2005 43