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. δǫ(t) t −ǫ +ǫ

1 2ǫ

δǫ(t) t +ǫ

1 ǫ

δǫ(t) t −ǫ +ǫ

1 ǫ

We define δ(t) as δ(t) = limǫ→0 δǫ(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 ∞

−∞

δ(t) dt = ǫ

−ǫ

δ(t) dt = 0+

0−

δ(t) dt = 1 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(t) = 1 for t ≥ 0 for t < 0. 1(t) t 1

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 dt 1(t). 1(t) t 1 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) = eαt, where α is a real constant, as shown in the following figure. eαt t eαt t

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Exponential function

Complex variant

Exponential function f (t) = A eαt, where α ∈ C is especially interesting when α = iω, f (t) = A eiωt = A (cos ωt + i sin ωt) .

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 + 2T) = · · · = f (t + k · T) The smallest possible T is called fundamental period, denoted as T0.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Sinus function

f (t) = A sin (ωt + Φ), A sin(ωt + Φ) t A A sin(Φ) T = 2π/ω The constants A, ω and Φ are called amplitude, angular frequency and phase shift. Sinus function is periodic with base period T = 2π/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 δ[n] = 1 for n = 0 for n = 0 . δ[n] n δ[n − 2] 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[n] = 1 for n ≥ 0 for n < 0 1[n] n 1[n − 1] 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(ωt + Φ) with period T = 2π/ω. If we sample this signal with the period Ts > 0, we

• btain a discrete sine signal

f [n] = f (nT) = A sin(ωnTs + Φ) = A sin(ξn + Φ), where n = 0, ±1, ±2, . . . a ξ = ωTs. A sin(ξn + 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 + 2N] = · · · = 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 N0.

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 Ts. For a periodic discrete sinusoidal signal with period N it must be valid N = m · 2π Ts , where m ∈ N. We also have N ∈ N, so 2π/Ts must be rational number. Example (Non-periodic sine signal) Signal y[n] = sin n is not periodic for Ts = 0.1 s, because 2π/Ts 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] LTI y[n] u[n] n y[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], s[n] = S{1[n]} = S

• n
• m=0

δ[n − m]

• .

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 (x1 + x2) = f (x1) + f (x2) a 2 homogeneous, f (αx) = αf (x).

Similarly, this applies to linear systems. Definition (Linear system) The system is linear if for two different input signals u1[n] and u2[n] S{u1[n] + u2[n]} = S{u1[n]} + S{u2[n]} , S{αu[n]} = αS{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 u1[n] and u2[n] with outputs y1[n] and y2[n] y1[n] = S{u1[n]} y2[n] = S{u2[n]} and for u[n] = αu1[n] + βu2[n] also αy1[n] + βy2[n] = y[n] = S{u[n]} = S{αu1[n] + βu2[n]} Generally u[n] =

• i

aiui[n] → y[n] =

• i

aiyi[n] =

• i

aiS{ui[n]}

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] = b1u1[n] + b2u2[n] then on output is y[n] = b1 (y1[n] + a y1[n − 1]) + b2 (y2[n] + a y2[n − 1]) where y1[n] + a y1[n − 1] = u1[n] y2[n] + a y2[n − 1] = u2[n]

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Example

Example (Non linear systems) The numerical calculation of the square root can be written by the recursive relation y[n + 1] = 1 2

• y[n] + u[n]

y[n]

• .

The square root of 10 is equal to (with 10 decimal precision) √ 10 = 3,16227766017. For u[n] = u[0] = 10 we get

n y[n] y 2[n] 1 3 9 2 3,165 10,017225 3 3,162278 10,00000214928 4 3,162277660 9,999999999568 . . . . . . . . .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Linear systems

Response to general input signal

The system response to the general input signal u[n] is then y[n] = S{u[n]} = S

• ∞
• m=−∞

u[m] δ[n − m]

• =

∞

• m=−∞

u[m] S{δ[n − m]} =

∞

• m=−∞

u[m] h[n, m] We see that the behavior of the system is entirely determined by its responses to the variously shifted unit pulses h[n, m].

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Linear systems

Step response

The step response s[n] of the discrete linear system is given by a simple sum of impulse responses for 0 ≤ m ≤ n. s[n] = S{1[n]} = S

• n
• m=0

δ[n − m]

• =

n

• m=0

S{δ[n − m]} =

n

• m=0

h[n, m].

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Time invariant systems

The system is called term time invariant if all events are dependent only on the time interval (difference of time events) n − m and not at each time point n and m separately. today . . . y[n] = S [u[n]] tomorrow . . . y[n − 1] = S [u[n − 1]] . . . Then, the impulse response equation also switches from matrix to a simple vector form h[n, m] → h[n − m] = S{δ[n − m]} .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Time invariant systems

Superposition of response y[n] from h[n − k]

h[n] n u[n] LTI y[n] y[n] = u[0] · h[n] u[n] n u[0] · h[n] n y[n] n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Time invariant systems

Superposition of response y[n] from h[n − k]

h[n] n u[n] LTI y[n] y[n] = u[0] · h[n] + u[1] · h[n − 1] u[n] n u[1] · h[n] n y[n] n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Time invariant systems

Superposition of response y[n] from h[n − k]

h[n] n u[n] LTI y[n] y[n] = u[0] · h[n] + u[1] · h[n − 1] + u[2] · h[n − 2] u[n] n u[2] · h[n] n y[n] n

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Convolution

Due to the time invariance, we get system response to the general input as convolution sum. y[n] =

∞

• m=−∞

h[n − m] · u[m] =

∞

• k=−∞

h[k] · u[n − k], which (to save space) we denote as y[n] = h[n] ∗ u[n]. Caution: it is not a multiplication! h[n] = y[n] u[n]

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Example

Example (Time invariant systems) Let us consider the microeconomic system of price variations described by the differential equation y[n] + a y[n − 1] = u[n]. Since its coefficients do not depend on time, ie a is constant and is not a function of n, this equation remains the same when changing n rightarrown − m. The impulse response is then h[n] = (−a)n1[n].

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Example

Example (Time variable systems) Let us now consider the slightly different differential equation y[n] + n · y[n − 1] = u[n]. The multiplication factor of y[n − 1] depends on time, and this equation does not remain the shape when changing n rightarrown − m. The impulse response can be written in the form h[n] = (−1)n n! 1[n].

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Causal systems

The system is causal if its output depends only on current and past input values. Thus, the output signal y[n] of the causal system depends only on {u[n], u[n − 1], u[n − 2], . . . }. In the convolution sum y[n] =

∞

• k=−∞

h[k] u[n − k] =

−1

• k=−∞

h[k] u[n − k]

• +

∞

• k=0

h[k] u[n − k] we have to put all the elements of the impulse response h[n] = 0 for n < 0.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Causal systems

The convolutional sum for the linear, time-invariant and causal system is y[n] =

∞

• k=0

h[k] · u[n − k] =

• k=−∞

u[k] · h[n − k]. If we additionally require that the input and output signals have a well-defined start, ie that ∀n < 0 : u[n] = 0, y[n] = 0 (both signals can have nonzero members only for n ≥ 0) then y[n] =

n

• k=0

h[k] · u[n − k] =

n

• k=0

h[n − k] · u[k].

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

Impulse and step response

Definition (Impulsn´ ı odezva) Response to Dirac’s impulse δ(t) will be called impulse response and denote h(t), h(t) = S{δ(t)} h(t, τ) = S{δ(t − τ)} . Definition (Step response) System response to step function 1(t) we will call step response and denote s(t), s(t) = S{1(t)} = S t δ(t − τ) dt

• .

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

Convolution

The system response to the general is convolutional integral y(t) = ∞

−∞

h(τ)u(t − τ) dτ = ∞

−∞

h(t − τ) · u(τ) dτ. We often write the operation in a simplified form y(t) = h(t) ∗ u(t). Again, we remind that this is not a multiplication!

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

Convolution Example

1 2 3 4

1 2

1 τ 1 2 3 4

1 2

1 t u(t) ∗ h(t) =

∞

• −∞

u(τ) h(t − τ) dτ (t − 0) u(τ)h(0 − τ) u(t)

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

For u(t) = δ(t) y(t) = S{u(t)} = ∞

−∞

h(τ) · δ(t − τ) dτ = h(t).

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

Causal system

The output signal y(t) of the continuous causal system depends

• nly on input values for previous time points. The convolutional

integral is then y(t) = ∞

−∞

h(τ) u(t − τ) dτ =

−∞

h(τ) u(t − τ) dτ

• +

∞ h(τ) u(t − τ) dτ and the impulse response value for t < 0, we again consider h(t) = 0.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

Continuous-time systems

Konvoluce pro kauz´ aln´ ı LTI syst´ em

The convolutional integral for the linear, time-invariant and causal system is y(t) = ∞ h(τ) · u(t − τ) dτ =

−∞

u(τ) · h(t − τ) dτ. If we additionally require that the input and output signals have a well-defined origin, ie that ∀t < 0 : u(t) = 0, y(t) = 0 (both signals can be nonzero members only for t ≥ 0) then y(t) = t h(τ) · u(t − τ) dτ = t u(τ) · h(t − τ) dτ.

Introduction to Signal Theory Basic continuous-time signals Basic discrete-time signals System Response

System characteristics

Autonomous and non-autonomous systems

Definition (Autonomn´ ı syst´ em) For an autonomous system we consider one that does not have input. y[n + 1] + a y[n] = 0. The output of the autonomous system is a response to the initial conditions. If the system has input u[n], that is y[n] + a y[n − 1] = u[n], the system is considered non-autonomous.