basic signals linearity, stationarity, causality · introduction to signal theory basic...

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

Basic signalsLinearity, stationarity, causalityModeling Systems and Processes (11MSP)

Bohumil Kovar, Jan Prikryl, Miroslav Vlcek

Department of Applied MahematicsCTU in Prague, Faculty of Transportation Sciences

2nd lecture 11MSP2019

verze: 2019-03-04 14:27


Table of contents

1 Introduction to Signal Theory

2 Basic continuous-time signals

3 Basic discrete-time signals

4 System Response


Table of contents



Basic continuous-time signals

Dirac delta function

Unit step function

Exponential function

Periodic and harmonic functions


4 System Response


Dirac delta functionApproximation

This function is defined at a time interval for all t, and its nonzerovalue is assumed only around t = 0. The area of these functions isequal to 1 for each ε > 0.

δε(t)

t−ε +ε

12ε

δε(t)

t0 +ε

1ε

δε(t)

t−ε +ε

1ε

We define δ(t) as δ(t) = limε→0 δε(t).


Diracuv impulsDefinition

The function δ(t) is called Dirac’s impulse, Dirac’s δ function orunit impulse. The value of δ(t) for t 6= 0 is δ(t) = 0. Its value int = 0 is not defined as a function, an integral definition is used∫ ∞

−∞δ(t) dt =

∫ ε

−εδ(t) dt =

∫ 0+

0−δ(t) dt = 1

pro each ε > 0.


Unit step function

The unit step function or Heaviside step function, usuallydenoted by 1(t) is defined as

1(t) =

{1 for t ≥ 00 for t < 0.

1(t)

t

1


Unit step functionδ(t) and 1(t) relation

It is true that

δ(t) =d

dt1(t).

1(t)

t

1

ε


Exponential functionReal variant

Consider an exponential function

f (t) = eαt ,

where α is a real constant, as shown in the following figure.

eαt

t

eαt

t


Exponential functionComplex variant

Exponential functionf (t) = A eαt ,

where α ∈ C is especially interesting when α = iω,

f (t) = A e iωt = A (cosωt + i sinωt) .


Periodic function

The continuous-time signal f (t) is said to be periodic with theperiod 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 asT0.


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, angularfrequency and phase shift. Sinus function is periodic with baseperiod T = 2π/ω.


Table of contents




Discrete unit impulse and step function

Discrete sinusoidal sequence

4 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 thetemperature 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.


Discrete unit impulse

The discrete unit impulse δ[n] is defined as

δ[n] =

{1 for n = 00 for n 6= 0 .

δ[n]

n

δ[n − 2]

n


Discrete step function

Discrete step function 1[n] is defined as

1[n] =

{1 for n ≥ 00 for n < 0

1[n]

n

1[n − 1]

n


Discrete sinusoidal sequence

Take the sine wave signal f (t) = A sin(ωt + Φ) with periodT = 2π/ω. If we sample this signal with the period Ts > 0, weobtain 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


Periodic signal

Discrete signal f [n] is periodic if there exists a positive integer Nfor 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 calleddiscrete signal period.

The smallest possible N is called fundamental period and wedenote it as N0.


Periodic signalDiscrete sinusoidal sequence may not be periodic!

Discrete sine signal not necessarily periodic, depending on thechoice of sampling period Ts. For a periodic discrete sinusoidalsignal with period N it must be valid

N = m · 2π

Ts,

where m ∈ N. We also have N ∈ N, so 2π/Ts must be rationalnumber.

Example (Non-periodic sine signal)

Signaly [n] = sin n

is not periodic for Ts = 0.1 s, because 2π/Ts is not rationalnumber.


Table of contents




4 System Response

Discrete-time systems

Linearnı a nelinearnı

Time invariant (stationary) systems

Causal systems

Continuous-time systems

Autonomnı a neautonomnı system


Discrete-time systemsLTI system response to general input

u[n] LTI y [n]

u[n]

n

y [n]

n


Discrete-time systemsImpulse response

Definition (Impulse response)

The response of the system to the unit impulse δ[n] will be calledimpulse response and denoted as h[n],

h[n] = S{δ[n]}h[n,m] = S{δ[n −m]} .


Discrete-time systemsStep response

Definition (Step response)

The response of the system to 1[n] will be called step responseand denoted as s[n],

s[n] = S{1[n]} = S

{n∑

m=0

δ[n −m]

}.


Linear systems

Definition (Linearity)

In mathematics, we denote the function f (x) as linear in the casethat 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]} .


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]}


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]


Example

Example (Non linear systems)

The numerical calculation of the square root can be written by therecursive 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] y2[n]1 3 92 3,165 10,0172253 3,162278 10,000002149284 3,162277660 9,999999999568...

......


Linear systemsResponse 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 byits responses to the variously shifted unit pulses h[n,m].


Linear systemsStep response

The step response s[n] of the discrete linear system is given by asimple 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].


Time invariant systems

The system is called term time invariant if all events aredependent 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 toa simple vector form

h[n,m]→ h[n −m] = S{δ[n −m]} .


Time invariant systemsSuperposition 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



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



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


Convolution

Due to the time invariance, we get system response to the generalinput 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] 6= y [n]

u[n]


Example

Example (Time invariant systems)

Let us consider the microeconomic system of price variationsdescribed 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 isnot a function of n, this equation remains the same when changingn rightarrown −m. The impulse response is then

h[n] = (−a)n1[n].


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 thisequation does not remain the shape when changingn rightarrown −m. The impulse response can be written in theform

h[n] = (−1)n n! 1[n].


Causal systems

The system is causal if its output depends only on current andpast 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]︸︷︷︸

0

+∞∑k=0

h[k] u[n − k]

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


Causal systems

The convolutional sum for the linear, time-invariant and causalsystem is

y [n] =∞∑k=0

h[k] · u[n − k] =0∑

k=−∞u[k] · h[n − k].

If we additionally require that the input and output signals have awell-defined start, ie that ∀n < 0 : u[n] = 0, y [n] = 0 (both signalscan 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].


Continuous-time systemsImpulse and step response

Definition (Impulsnı odezva)

Response to Dirac’s impulse δ(t) will be called impulse responseand 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 responseand denote s(t),

s(t) = S{1(t)} = S{∫ t

0δ(t − τ) dt

}.


Continuous-time systemsConvolution

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!


Continuous-time systemsConvolution Example


Continuous-time systems

For u(t) = δ(t)

y(t) = S{u(t)} =

∫ ∞−∞

h(τ) · δ(t − τ) dτ = h(t).


Continuous-time systemsCausal system

The output signal y(t) of the continuous causal system dependsonly on input values for previous time points. The convolutionalintegral is then

y(t) =

∫ ∞−∞

h(τ) u(t − τ) dτ

=

∫ 0

−∞h(τ) u(t − τ) dτ︸︷︷︸

0

+

∫ ∞0

h(τ) u(t − τ) dτ

and the impulse response value for t < 0, we again considerh(t) = 0.


Continuous-time systemsKonvoluce pro kauzalnı LTI system

The convolutional integral for the linear, time-invariant and causalsystem is

y(t) =

∫ ∞0

h(τ) · u(t − τ) dτ =

∫ 0

−∞u(τ) · h(t − τ) dτ.

If we additionally require that the input and output signals have awell-defined origin, ie that ∀t < 0 : u(t) = 0, y(t) = 0 (bothsignals can be nonzero members only for t ≥ 0) then

y(t) =

∫ t

0h(τ) · u(t − τ) dτ =

∫ t

0u(τ) · h(t − τ) dτ.


System characteristicsAutonomous and non-autonomous systems

Definition (Autonomnı system)

For an autonomous system we consider one that does not haveinput.

y [n + 1] + a y [n] = 0.

The output of the autonomous system is a response to the initialconditions.

If the system has input u[n], that is

y [n] + a y [n − 1] = u[n],

the system is considered non-autonomous.

basic signals linearity, stationarity, causality · introduction to signal theory basic...

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