Lecture 2 LTI systems: convolutions, impulse …...EITF75 Systems and Signals system LTI systems A system is LTI if-and-only if: • It is linear • It is time-invariant Linear system

EITF75 Systems and Signals

Lecture 2 LTI systems: convolutions, impulse responses (and more)

Fredrik Rusek


system

LTI systems

A system is LTI if-and-only if: • It is linear • It is time-invariant

Linear system Time invariant system


system

LTI systems


Linear system Time invariant system

LTI systems have compact mathematical representation We next provide two ways the reach the representation


LTI system


Method I (not in book, but I find it illuminating)

Input x(n): A sequence of numbers Output y(n): A sequence of numbers

In the Linear algebra course, how did we represent a sequence of numbers?


LTI system




In the Linear algebra course, how did we represent a sequence of numbers? With a vector

IN OUT


LTI system





IN OUT Why is the linear algebra course dealing so much with matrices?


LTI system





IN OUT Why is the linear algebra course dealing so much with matrices? Because every linear function can be represented by a matrix


LTI system



Summary so far: A linear system can be represented as

where


LTI system



A linear system can be represented as

where

But, our system is LTI, not only linear, so this imposes restrictions on i.e., must have a special structure


LTI system




Let us understand this special structure


LTI system





Assume


LTI system





Assume

The output must be the first column of


LTI system





Assume


LTI system





Assume

The output must be the second column of


LTI system



Now recall that system is time-invariant Implication?


LTI system




The outputs should be The same, but one step delayed


LTI system




The outputs should be The same, but one step delayed i.e.


LTI system




The outputs should be The same, but one step delayed i.e. equal values along all diagonals


LTI system



Summary. An LTI system is any discrete-time system that can be described by

Som

e v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or


LTI system



Summary. An LTI system is any discrete-time system that can be described by

Som

e v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Alternative formulation of course-goal: Understand properties of a matrix of the form

Som

e v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or


LTI system

Som

e v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

Same v

ect

or

The LTI system is FULLY characterized by one vector/sequence of numbers/discrete signal


LTI system


Method II (to reach the same conclusion: an LTI system can be described by a signal)

Input signal Output signal

Impulse response (definition)


LTI system





Time-invariant


LTI system





Time-invariant

Linearity (scaling part)


LTI system





Time-invariant

Linearity (scaling part)

Linearity (summation part)


LTI system





LTI system




Final result: The system is described by h(n) The formula is named convolution. Super-important


Summary LTI

Input/Output relation (Convolution)

Short-hand notation


Agenda

Get familiar with through some examples For what do we have BIBO stability? See relationship between and Some notes on correlation functions

Today

In the long run


Agenda


Today

In the long run (Loosely speaking)

Study in detail via z-transform, and 2 types of Fourier transforms The sampling-reconstruction issues


Agenda


Today

In the long run (Loosely speaking)

Study in detail via z-transform, and 2 types of Fourier transforms The sampling-reconstruction issues


Example

Given: Input signal and impulse response

Find: Output signal


Example


Find: Output signal

Let us start with computation of

1 2 3 4 -1


Example


Find: Output signal


4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal


1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal


1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal


1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal


1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1

? ?


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1

Stays the same

?


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1

Stays the same Moves


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1



Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


”Home work 1” Verify that causal x(n) and causal h(n) yields causal y(n)

”Home work 2” If x(n) starts at -3, and h(n) at -4. When does y(n) start?

”Home work 3” etc


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1

?


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

computation of

1 2 3

4

4

2

6

-3 -2 -1 0

1 2 3 4 -1


Example


Find: Output signal

Repeating gives


Example


Find: Output signal


Example


Find: Output signal

Three more methods


Example


Find: Output signal

Three more methods Method 1


Example


Find: Output signal



Example


Find: Output signal



Example


Find: Output signal



Example



Put numbers in a table and multiply


Example



Sum the diagonals


Example



Result


Example



Result


Make sure that you understand why a convolution of a length K signal with a length L signal has length K+L-1


Example Three more methods Method 3: Analytical solution





Minor trick. Not really needed, but slightly simpler. Try without the trick at home













Thus


Standard Properties

Commutativity

Associativity

Distributivity


Some consequences

Commutativity

Associativity

Distributivity


Some consequences

Commutativity

Associativity

Distributivity


Some consequences

Commutativity

Associativity

Distributivity


BIBO stability

A system is BIBO stable if


BIBO stability


For an LTI system


BIBO stability


For an LTI system


BIBO stability


For an LTI system


BIBO stability


For an LTI system

An LTI system is stable if


Relation to difference equations

We have seen that an LTI system is fully described by an impulse response h(n)




We have also mentioned that difference equations are important for LTI systems




We have also mentioned that difference equations are important for LTI systems This means that every impulse response h(n) is equivalent to a difference equation We now investigate this



Suppose



Suppose

We then get



Suppose

We then get This is a convolution, with a finite length impulse response b(n)



Suppose

We then get This is a convolution, with a finite length impulse response b(n)

The class of systems described by difference equations encompasses LTI systems with finite length impulse responses



Consider now



Consider now

We then get



Consider now

We then get



Consider now

We then get



Consider now

We then get



Consider now

We then get

Pattern recognition, suitably done at home, gives



Consider now

We then get


Convolution



Consider now

We then get


Infinite Impulse response (IIR)



Consider now

We then get


What is this?




Consider now

We then get



What is this?

It does not depend on x(n)



Consider now

We then get



What is this?

It does not depend on x(n) If y(-1)≠0, we have output without any input



Consider now

We then get



What is this?

It does not depend on x(n) If y(-1)≠0, we have output without any input Not Linear system Not time-invariant



Consider now

We then get



If y(-1)=0, we say that the system is at rest. System is LTI



Consider now

We then get




If y(-1) ≠ 0, we say that the system is not at rest/has initial conditions. Strictly speaking: Not LTI. However, so common, so still treated within a study of LTI systems



Consider now

We then get


Note: Only makes sense to assume So, transient will fade out and ”after a while it is LTI”





Consider now

We then get





The class of systems described by difference equations encompasses LTI systems with infinite length impulse responses



Consider now

We then get





Every impulse response corresponds to one difference equation.


Brief info on correlation

Not focal point of course, but highly important in signal processing

Correlation measures similarity between two signals




Auto correlation


Cross correlation

Measures similarity between time shifted versions of the same signal

Measures similarity between time shifted versions of different signals




Auto correlation


Cross correlation



Example: 5G communication system

UE1

UE2

When a user (UE) wants to connect, it sends a known signal, x1(n) or x2(n) x1(n)

x2(n)




Auto correlation


Cross correlation




UE1

UE2


x2(n)

Cross correlation between x1(n) and x2(n) should be small (to know who is connecting)




Auto correlation


Cross correlation




UE1

UE2


x2(n)

Auto correlation of x1(n) (and x2(n)) should be delta (to know when a user is connecting)



Cross correlation for input and output signals

Lecture 2 LTI systems: convolutions, impulse …...EITF75 Systems and Signals system LTI systems A system is LTI if-and-only if: • It is linear • It is time-invariant Linear system

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