Lt i Systems

8/12/2019 Lt i Systems

http://slidepdf.com/reader/full/lt-i-systems 1/71



A system is homogeneous

if it multiplies all input

amplitudes of a certain

frequency by a constant:if:

inp(t)

outp(t)then:

k x inp(t) k x outp(t)

Homogeneity



Time (s)

0 0.02-1

1

0

Time (s)

0 0.02-1

1

0

Time (s)

0 0.02-1

1

0

Time (s)

0 0.02-1

1

0

SYSTEMamplifier

Time (s)

0 0.02-1

1

0

input amplitude (mV)

output amplitude (mV)

25

Time (s)

0 0.02-1

1

0

300 Hz 300 Hz

37.5

50 75

75

112,5

25/50/75 mV 37.5/75/112,5 mV

linear relationship

amplification factorof 1.5 for each

amplitude value

line crosses 0

Homogeneity



A system is additive if the sum of individual

output signals eaquals the sum of the sameindividual input signals after having passed

through the system, i.e:

if:

inp1(t) outp1(t)

inp2(t) outp2(t)

then:

[inp1(t) + inp2(t)] [outp1(t) + outp2(t)]

Additivity



Time (s)

0 0.02-0.5116

0.5075

0

Time (s)

0 0.5.05411

0.1753

0

Time (s)

0 0.02-0.5

0.5

0

Time (s)

0 0.02-0.5

0.5

0+

=

Time (s)

0 0.02-0.8801

0.8801

0

SYSTEMlow-pass filter

cutoff 450 Hz

Time (s)

0 0.02-0.51

0.5146

0

+=

sinewave

300 Hz

sinewave

600 Hz

no effect

amplitudeset to 0

Additivity



A system is time-invariant if it does notchange the output over time, i.e.:

if:inp(t) outp(t)

then:

inp(t) + d seconds outp(t) + d seconds

Time-Invariance



Time (s)

0 0.00907029-2

2

0

Time (s)

0 0.00907029-2

2

0

SYSTEM

amplifier

(by factor 2)

Time (s)

0 0.00907029-2

2

0

Time (s)

0 0.00907029-2

2

0

s i g n a l 1 : p u l s e

s i g n a l 2 : p u l s e

delayed by 3 msIMPORTANT: delayed of 3 ms stays

constant

Time-Invariance



SYSTEM

Frequency

R e s p o n s e

output amplitudeinput amplitude

Input signal Output signal

Frequency Frequency

A m p l i t u d e

A m p l i t u d e

transfer function or

frequency response

Frequency Domain

Characterisation of LTIs



SYSTEMInput signal

(impulse)

Output signal

time

phase spectrum

amplitude spectrum

0

k

phase spectrum

amplitude spectrum

phase response

amplitude response

Time-Domain

Characterisation of LTIs



transfer functionor

frequency response

SYSTEM

Frequency

R

e s p o n s e

Charaterisation of signals and systems

Frequency

A m

p l i t u d e

SIGNALS

Time (s)

0 0.01-1.167

1.083

0

frequencydomain

timedomain

frequencydomain



Question:

Can systems becharacterised by a time

domain?

Answer:YES!



Time-domain characterisation of LTI systems

3

0

LTI-SYSTEM

3

0

3

1

0

3

1

0

* 1/3

system ishomogenious so

output will be *1/3 as well

* 1/3

1/3 ms




3

0

LTI-SYSTEM

3

0

3

0

3

0

system is timeinvariant so

delay will remain3 ms

delay: 3 msdelay: 3 ms




3

0LTI-

SYSTEM

3

0

3

0

3

0

additivity

+ +




LTI-SYSTEM

3

0

3

0

What we can do already:

If we know the system response to one single pulse,we can predict the output for a 'complicated' chain of pulses.




pulses of 1/3 ms




pulses of 1/7 ms




relationship between pulse duration and output amplitude:

the shorter the puls the less energy it contains,the smaller will be the amplitude of the output signal




In order not to loose energy of the output signal we

increase the amplitude of the input pulse ...

... untill at some pointthe puls has no

duration and itsamplitude isinfinitely high




relationship between puls duration and output amplitude:

An infinitesimally narrowand infinitely small pulse

of finite energy is knownas an impulse. True impulses areonly a mathematicalconcept and do notoccur in real life.




IMPORTANT:

• With an impulse response ANY output of an LTI systemcan be predicted by the input.

• Therefor: In the time-domain LTI systems are completelycharacterised by their impulse response.




Concept of CONVOLUTION

• True impulses cannot be used in signal processing.

• Convolution is used to calculate the time response of an LTI

system on a given pulse response.

Convolution requires thorough knowledge of calculus.




The relationship between theimpulse response and the frequency response

impulse response and frequency response are two differentpoints of view to look at the same thing: the response of a system

We should actually be able e.g. to determine the impulse responsefrom the frequency domain.

This is what we will do in the following.




Suppose we know the frequency response to a system and want to knowwhat its output is to an impulse. Remember how to do that?

Output amplitude (f) = Input amplitude (f) X Amplitude response (f)

And for the phase spectrum?

Output phase (f) = Input phase (f) + Phase response (f)




Output amplitude (f) = Input amplitude (f) X Amplitude response (f)

Output phase (f) = Input phase (f) + Phase response (f)

What is the phase spectrum of an impulse?

Answer: 0

What is the amplitude spectrum of an impules?

Answer: constant (k)

0

k




conclusion

Output phase (f) = Phase response (f)

The phase spectrum of the impulse response of asystem is simply the phase response of the system

Output amplitude (f) = k X Amplitude response (f)

The spectrum of a system impulse response is simply

the amplitude response of the system.




SYSTEMInput signal Output signal

time

phase spectrum

amplitude spectrum

0

k

phase spectrum

amplitude spectrum

phase response

amplitude response



EE-2027 SaS, L3: 4/20

Examples of Simple Systems

To get some idea of typical systems (and their properties), considerthe electrical circuit example:

which is a first order , CT differential equation.

Examples of first order , DT difference equations:

where y is the monthly bank balance, and x is monthly net deposit

which represents a discretised version of the electrical circuit

Example of second order system includes:

System described by order and parameters (a, b, c)



EE-2027 SaS, L3: 5/20

First Order Step Responses

People tend to visualise systems in terms of their responsesto simple input signals (see Lecture 4…)

The dynamics of the output signal are determined by thedynamics of the system, if the input signal has no

dynamics

Consider when the input signal is a step at t, n = 1, y(0) = 0

First order CT differential system First order DT difference system

t

u(t)

y(t)



EE-2027 SaS, L3: 6/20

System Linearity

Specifically, a linear system must satisfy the two properties:1 Additive: the response to x1(t)+x2(t) is y1(t) + y2(t)

2 Scaling: the response to ax1(t) is ay1(t) where aC

Combined: ax1(t)+bx2(t) ay1(t) + by2(t)

E.g. Linear y(t) = 3*x(t) why?

Non-linear y(t) = 3*x(t)+2, y(t) = 3*x2(t) why?

(equivalent definition for DT systems)

x

y

The most important property that a system

possesses is linearity

It means allows any system response to be

analysed as the sum of simpler responses(convolution)

Simplistically, it can be imagined as a line



EE-2027 SaS, L3: 7/20

Bias and Zero Initial Conditions

Intuitively, a system such as:

y(t) = 3*x(t)+2

is regarded as being linear. However, it does not satisfy the

scaling condition.

There are several (similar) ways to transform it to anequivalent linear system

Perturbations around operating value x*, y*

Linear System Derivative

Locally, these ideas can also be used to linearise a non-

linear system in a small range



EE-2027 SaS, L3: 8/20

Linearity and Superposition

Suppose an input signal x[n] is made of a linear sum of

other (basis/simpler) signals xk[n]:

then the (linear) system response is:

The basic idea is that if we understand how simple signals

get affected by the system, we can work out how complex

signals are affected, by expanding them as a linear sum

This is known as the superposition property which is true for

linear systems in both CT & DT

Important for understanding convolution (next lecture)



EE-2027 SaS, L3: 9/20

Definition of Time Invariance

A system is time invariant if its behaviour and characteristics arefixed over time

We would expect to get the same results from an input-outputexperiment, if the same input signal was fed in at a different time

E.g. The following CT system is time-invariant

because it is invariant to a time shift, i.e. x2(t) = x1(t-t0)

E.g. The following DT system is time-varying

Because the system parameter that multiplies the input signal istime varying, this can be verified by substitution



EE-2027 SaS, L3: 10/20

System with and without Memory

A system is said to be memoryless if its output for each value ofthe independent variable at a given time is dependent on the

output at only that same time (no system dynamics)

e.g. a resistor is a memoryless CT system where x(t) is current

and y(t) is the voltage

A DT system with memory is an accumulator (integrator)

and a delay

Roughly speaking, a memory corresponds to a mechanism in thesystem that retains information about input values other than

the current time.





EE-2027 SaS, L3: 12/20

System Stability

Informally, a stable system is one in which small input signals leadto responses that do not diverge

If an input signal is bounded, then the output signal must also bebounded, if the system is stable

To show a system is stable we have to do it for all input signals.To show instability, we just have to find one counterexample

E.g. Consider the DT system of the bank account

when x[n] = [n], y[0] = 0

This grows without bound, due to 1.01 multiplier. This system isunstable.

E.g. Consider the CT electrical circuit, is stable if RC>0, because itdissipates energy



EE-2027 SaS, L3: 13/20

Invertible and Inverse Systems

A system is said to be invertible if distinct inputs lead to distinctoutputs (similar to matrix invertibility)

If a system is invertible, an inverse system exists which, whencascaded with the original system, yields an output equal to

the input of the first signal

E.g. the CT system is invertible:

y(t) = 2x(t)

because w(t) = 0.5*y(t) recovers the original signal x(t)

E.g. the CT system is not-invertible

y(t) = x2(t)

because distinct input signals lead to the same output signalWidely used as a design principle:

– Encryption, decryption

– System control, where the reference signal is input



EE-2027 SaS, L3: 14/20

Systems are generally composed of components (sub-systems).

We can use our understanding of the components and their

interconnection to understand the operation and behaviour of

the overall system

Series/cascade

Parallel

Feedback

System Structures

System 1 System 2x y

System 1

System 2

x y

+

System 2

System 1x y

+



EE-2027 SaS, L3: 15/20

Systems In Matlab

A system transforms a signal into another signal.

In Matlab a discrete signal is represented as an indexedvector.

Therefore, a matrix or a for loop can be used totransform one vector into another

Example (DT first order system)

>> n = 0: 10;>> x = ones( si ze( n) ) ;

>> x( 1) = 0;

>> y( 1) = 0;

>> f or i =2: 11y( i ) = ( y( i - 1) + x( i ) ) / 2;

end

>> pl ot ( n, x, ‘ x’ , n, y, ‘ . ’ )

S t Lib i i Si li k



EE-2027 SaS, L3: 16/20

System Libraries in Simulink





Discrete-time LTI systems

x[n] - a discrete-time signal

We want to rewrite x[n] using unit impulse functions

More specifically, utilizing the sampling property

Define

We can rewrite

Superposition of scaled versions of time shifted unit impulses




Why do we want to rewrite x[n] in such a cumbersome way?!?

Let

output is a sum of time-shifted

versions of h with weights x[k]




A discrete-time LTI system is completely characterized by

Called (unit) impulse response(unit) impulse response

is known as convolution sumconvolution sum or

superposition sumsuperposition sum Operation is called the convolutionconvolution of sequences

Notation:



Evaluation of convolution sum:




Method #1

2

2

1

5

32

32

E l ti f l ti



Take

Shift it to the rightright by n

Take the product

Take summation

Example: y[-1]

1

22

1 12


Method #2

1

E l i f l i




Method #2 Example: y[2]

122

12

1



Continuous-time LTI systems

We can rewrite using the unit impulse function

Continuous-time counterpart (after similar exercise)

where (called (unit) impulse response(unit) impulse response)

Integration called convolution integralconvolution integral or superposition integralsuperposition integral

Notation:

CT LTI system characterized by its impulse response

Procedure for evaluating




convolution

Take Shift it to the rightright by t

Take the product

Integrate over the “intersectionintersection”

Examples:

0

P d f l ti




convolution

-1

0

1

••

0

0

0

0

0

0

00

0





convolutionExample: For t > 0,

Answer:



P ti f LTI t



Properties of LTI systems

3. Associative property

Impulse response of the cascade of two LTI systems is the convolution

of their individual impulse responses

Order of the systems does not matter!Order of the systems does not matter!

Can be generalized to the cascade of an arbitrary number of LTIsystems

Require both linearity and time invarianceboth linearity and time invariance




4. Invertibility of LTI systems

When S is LTI and invertible, there exists an inverse system which is LTI

5. Causality for LTI system

Recall that in order for S to be causal, should depend only on

DT:

CT:

P ti f LTI t




6. LTI systems with and without memory

A system is memorylessmemoryless if, for any two pairs of input-outputsand given , implies

DT:

CT:





7. Stability for LTI systems

BIBO stability: every bounded input leads to a bounded output

DT LTI system S (with impulse response h) is stable iff (if and only if)

Called “absolutelyabsolutely summablesummable”

CT LTI system S (with impulse response h) is stable iff (if and only if)

Called “absolutelyabsolutely integrableintegrable”

Exploiting Superposition and Time-Invariance



Question: Are there sets of “basic” signals so that:

a) We can represent rich classes of signals as linear combinations of

these building block signals.

b) The response of LTI Systems to these basic signals are both simpleand insightful.

Fact: For LTI Systems (CT or DT) there are two natural choices for

these building blocks

Focus for now: DT Shifted unit samples

CT Shifted unit impulses

Representation of DT Signals Using Unit Samples



That is ...



That is ...

Coefficients Basic Signals

The Sifting Property of the Unit Sample



DT System x[n] y[n]

• Suppose the system is linear, and define hk [n] as the

response to δ [n - k ]:

From superposition:



DT System x[n] y[n]

• Now suppose the system is LTI, and define the unit

sample response h[n]:

From LTI:

From TI:

Convolution Sum Representation of

Response of LTI Systems



Response of LTI Systems

Interpretation

n n

n n

Visualizing the calculation of



y[0] = prod of

overlap for

n = 0

y[1] = prod of

overlap for

n = 1

Choose value of n and consider it fixed

View as functions of k with n fixed

Calculating Successive Values: Shift, Multiply, Sum



g , p y,

-1 1 × 1 = 1

(-1) × 2 + 0 × (-1) + 1 × (-1) = -3

(-1) × (-1) + 0 × (-1) = 1

(-1) × (-1) = 1

4

0 × 1 + 1 × 2 = 2

(-1) × 1 + 0 × 2 + 1 × (-1) = -2

Properties of Convolution and DT LTI Systems



1) A DT LTI System is completely characterized by its unit sample

response



Unit Sample response

Th C t ti P t



The Commutative Property

Ex: Step response s[n] of an LTI system

“input” Unit Sample response

of accumulator

step

input

The Distributive Property



Interpretation

The Associative Property



Implication (Very special to LTI Systems)

Properties of LTI Systems



1) Causality ⇔

2) Stability ⇔

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Lt i Systems