Basic signals Linearity, stationarity, causality Modeling Systems - PowerPoint PPT Presentation

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Basic signals Linearity, stationarity, causality Modeling Systems and Processes (11MSP) Bohumil Kov´ aˇ r, Jan Pˇ rikryl, Miroslav Vlˇ cek Department of Applied Mahematics CTU in Prague, Faculty of Transportation Sciences 2nd lecture 11MSP 2019 verze: 2019-03-04 14:27

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Table of contents 1 Introduction to Signal Theory 2 Basic continuous-time signals 3 Basic discrete-time signals 4 System Response

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Table of contents 1 Introduction to Signal Theory 2 Basic continuous-time signals Basic continuous-time signals Dirac delta function Unit step function Exponential function Periodic and harmonic functions 3 Basic discrete-time signals 4 System Response

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Dirac delta function Approximation This function is defined at a time interval for all t , and its nonzero value is assumed only around t = 0. The area of these functions is equal to 1 for each ε > 0. δ ǫ ( t ) δ ǫ ( t ) δ ǫ ( t ) 1 1 1 2 ǫ ǫ ǫ t t t − ǫ + ǫ + ǫ − ǫ + ǫ 0 We define δ ( t ) as δ ( t ) = lim ǫ → 0 δ ǫ ( t ).

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Dirac˚ uv impuls Definition The function δ ( t ) is called Dirac’s impulse , Dirac’s δ function or unit impulse. The value of δ ( t ) for t � = 0 is δ ( t ) = 0. Its value in t = 0 is not defined as a function, an integral definition is used � 0+ � ∞ � ǫ δ ( t ) d t = δ ( t ) d t = δ ( t ) d t = 1 0 − −∞ − ǫ pro each ǫ > 0.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Unit step function The unit step function or Heaviside step function , usually denoted by 1 ( t ) is defined as � 1 for t ≥ 0 1 ( t ) = 0 for t < 0 . 1 ( t ) 1 t

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Unit step function δ ( t ) and 1 ( t ) relation It is true that δ ( t ) = d d t 1 ( t ) . 1 ( t ) 1 t ǫ

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Exponential function Real variant Consider an exponential function f ( t ) = e α t , where α is a real constant, as shown in the following figure. e α t e α t t t

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Exponential function Complex variant Exponential function f ( t ) = A e α t , where α ∈ C is especially interesting when α = i ω , f ( t ) = A e i ω t = A (cos ω t + i sin ω t ) .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Periodic function The continuous-time signal f ( t ) is said to be periodic with the period T , if ∀ t : f ( t + T ) = f ( t ) and therefore also for any one k ∈ Z f ( t ) = f ( t + T ) = f ( t + 2 T ) = · · · = f ( t + k · T ) The smallest possible T is called fundamental period , denoted as T 0 .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Sinus function f ( t ) = A sin ( ω t + Φ) , A sin( ω t + Φ) T = 2 π/ω A A sin(Φ) t The constants A , ω and Φ are called amplitude , angular frequency and phase shift . Sinus function is periodic with base period T = 2 π/ω .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Table of contents 1 Introduction to Signal Theory 2 Basic continuous-time signals 3 Basic discrete-time signals Discrete unit impulse and step function Discrete sinusoidal sequence 4 System Response

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Formation of discrete signals How do discrete signals arise? • naturally (average daily temperatures, daily conversion rates, student numbers) • by sampling continuous-time signals (measuring the temperature every hour, measuring the flow every 15 minutes) Discrete signals to be dealt with in the subject are discrete in time, but continuous in function value.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete unit impulse The discrete unit impulse δ [ n ] is defined as � 1 for n = 0 δ [ n ] = 0 for n � = 0 . δ [ n ] δ [ n − 2] n n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete step function Discrete step function 1 [ n ] is defined as � 1 for n ≥ 0 1 [ n ] = 0 for n < 0 1 [ n ] 1 [ n − 1] n n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete sinusoidal sequence Take the sine wave signal f ( t ) = A sin( ω t + Φ) with period T = 2 π/ω . If we sample this signal with the period T s > 0, we obtain a discrete sine signal f [ n ] = f ( nT ) = A sin( ω nT s + Φ) = A sin( ξ n + Φ) , where n = 0 , ± 1 , ± 2 , . . . a ξ = ω T s . A sin( ξ n + Φ) t

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Periodic signal Discrete signal f [ n ] is periodic if there exists a positive integer N for that f [ n ] = f [ n + N ] = f [ n + 2 N ] = · · · = f [ n + k · N ] for all n ∈ Z (from ( −∞ , ∞ )) and for any k ∈ Z . N is called discrete signal period . The smallest possible N is called fundamental period and we denote it as N 0 .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Periodic signal Discrete sinusoidal sequence may not be periodic! Discrete sine signal not necessarily periodic, depending on the choice of sampling period T s . For a periodic discrete sinusoidal signal with period N it must be valid N = m · 2 π , T s where m ∈ N . We also have N ∈ N , so 2 π/ T s must be rational number. Example (Non-periodic sine signal) Signal y [ n ] = sin n is not periodic for T s = 0 . 1 s, because 2 π/ T s is not rational number.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Table of contents 1 Introduction to Signal Theory 2 Basic continuous-time signals 3 Basic discrete-time signals 4 System Response Discrete-time systems Line´ arn´ ı a neline´ arn´ ı Time invariant (stationary) systems Causal systems Continuous-time systems Autonomn´ ı a neautonomn´ ı syst´ em

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete-time systems LTI system response to general input u [ n ] y [ n ] u [ n ] y [ n ] LTI n n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete-time systems Impulse response Definition (Impulse response) The response of the system to the unit impulse δ [ n ] will be called impulse response and denoted as h [ n ], h [ n ] = S{ δ [ n ] } h [ n , m ] = S{ δ [ n − m ] } .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Discrete-time systems Step response Definition (Step response) The response of the system to 1 [ n ] will be called step response and denoted as s [ n ], � � n � s [ n ] = S{ 1 [ n ] } = S δ [ n − m ] . m =0

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Linear systems Definition (Linearity) In mathematics, we denote the function f ( x ) as linear in the case that it is 1 additive f ( x 1 + x 2 ) = f ( x 1 ) + f ( x 2 ) a 2 homogeneous, f ( α x ) = α f ( x ) . Similarly, this applies to linear systems. Definition (Linear system) The system is linear if for two different input signals u 1 [ n ] and u 2 [ n ] S{ u 1 [ n ] + u 2 [ n ] } = S{ u 1 [ n ] } + S{ u 2 [ n ] } , S{ α u [ n ] } = α S{ u [ n ] } .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Superposition principle Definition (Superposition principle) For two different input signals u 1 [ n ] and u 2 [ n ] with outputs y 1 [ n ] and y 2 [ n ] y 1 [ n ] = S{ u 1 [ n ] } y 2 [ n ] = S{ u 2 [ n ] } and for u [ n ] = α u 1 [ n ] + β u 2 [ n ] also α y 1 [ n ] + β y 2 [ n ] = y [ n ] = S{ u [ n ] } = S{ α u 1 [ n ] + β u 2 [ n ] } Generally � � � u [ n ] = a i u i [ n ] → y [ n ] = a i y i [ n ] = a i S{ u i [ n ] } i i i

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response Example Example (Linear systems) Consider the system y [ n ] + a y [ n − 1] = u [ n ] . If there is a linear combination of two different input signals u [ n ] = b 1 u 1 [ n ] + b 2 u 2 [ n ] then on output is y [ n ] = b 1 ( y 1 [ n ] + a y 1 [ n − 1]) + b 2 ( y 2 [ n ] + a y 2 [ n − 1]) where y 1 [ n ] + a y 1 [ n − 1] = u 1 [ n ] y 2 [ n ] + a y 2 [ n − 1] = u 2 [ n ]