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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EITF75 Systems and Signals
Lecture 2 LTI systems: convolutions, impulse responses (and more)
Fredrik Rusek
EITF75 Systems and Signals
system
LTI systems
A system is LTI if-and-only if: • It is linear • It is time-invariant
Linear system Time invariant system
EITF75 Systems and Signals
system
LTI systems
A system is LTI if-and-only if: • It is linear • It is time-invariant
Linear system Time invariant system
LTI systems have compact mathematical representation We next provide two ways the reach the representation
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
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?
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
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? With a vector
IN OUT
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
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? With a vector
IN OUT Why is the linear algebra course dealing so much with matrices?
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
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? With a vector
IN OUT Why is the linear algebra course dealing so much with matrices? Because every linear function can be represented by a matrix
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
Summary so far: A linear system can be represented as
where
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
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
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
A linear system can be represented as
Let us understand this special structure
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
A linear system can be represented as
Let us understand this special structure
Assume
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
A linear system can be represented as
Let us understand this special structure
Assume
The output must be the first column of
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
A linear system can be represented as
Let us understand this special structure
Assume
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
A linear system can be represented as
Let us understand this special structure
Assume
The output must be the second column of
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
Now recall that system is time-invariant Implication?
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
Now recall that system is time-invariant Implication?
The outputs should be The same, but one step delayed
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
Now recall that system is time-invariant Implication?
The outputs should be The same, but one step delayed i.e.
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
Now recall that system is time-invariant Implication?
The outputs should be The same, but one step delayed i.e. equal values along all diagonals
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
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
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method I (not in book, but I find it illuminating)
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
EITF75 Systems and Signals
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
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
Impulse response (definition)
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
Impulse response (definition)
Time-invariant
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
Impulse response (definition)
Time-invariant
Linearity (scaling part)
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
Impulse response (definition)
Time-invariant
Linearity (scaling part)
Linearity (summation part)
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
EITF75 Systems and Signals
LTI system
A system is LTI if-and-only if: • It is linear • It is time-invariant
Method II (to reach the same conclusion: an LTI system can be described by a signal)
Input signal Output signal
Final result: The system is described by h(n) The formula is named convolution. Super-important
EITF75 Systems and Signals
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
EITF75 Systems and Signals
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 (Loosely speaking)
Study in detail via z-transform, and 2 types of Fourier transforms The sampling-reconstruction issues
EITF75 Systems and Signals
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 (Loosely speaking)
Study in detail via z-transform, and 2 types of Fourier transforms The sampling-reconstruction issues
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Let us start with computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
? ?
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
Stays the same
?
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
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
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
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
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
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
”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
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
?
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
computation of
1 2 3
4
4
2
6
-3 -2 -1 0
1 2 3 4 -1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Repeating gives
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Three more methods
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Three more methods Method 1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Three more methods Method 1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Three more methods Method 1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Find: Output signal
Three more methods Method 1
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Three more methods Method 2
Put numbers in a table and multiply
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Three more methods Method 2
Sum the diagonals
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Three more methods Method 2
Result
EITF75 Systems and Signals
Example
Given: Input signal and impulse response
Three more methods Method 2
Result
EITF75 Systems and Signals
Make sure that you understand why a convolution of a length K signal with a length L signal has length K+L-1
EITF75 Systems and Signals
Example Three more methods Method 3: Analytical solution
Minor trick. Not really needed, but slightly simpler. Try without the trick at home
EITF75 Systems and Signals
BIBO stability
A system is BIBO stable if
For an LTI system
An LTI system is stable if
EITF75 Systems and Signals
Relation to difference equations
We have seen that an LTI system is fully described by an impulse response h(n)
EITF75 Systems and Signals
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
EITF75 Systems and Signals
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 This means that every impulse response h(n) is equivalent to a difference equation We now investigate this
EITF75 Systems and Signals
Relation to difference equations
Suppose
We then get This is a convolution, with a finite length impulse response b(n)
EITF75 Systems and Signals
Relation to difference equations
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
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
Convolution
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
Infinite Impulse response (IIR)
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
What is this?
Infinite Impulse response (IIR)
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
Infinite Impulse response (IIR)
What is this?
It does not depend on x(n)
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
Infinite Impulse response (IIR)
What is this?
It does not depend on x(n) If y(-1)≠0, we have output without any input
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Pattern recognition, suitably done at home, gives
Infinite Impulse response (IIR)
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
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Infinite Impulse response (IIR)
It does not depend on x(n) If y(-1)≠0, we have output without any input Not Linear system Not time-invariant
If y(-1)=0, we say that the system is at rest. System is LTI
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Infinite Impulse response (IIR)
It does not depend on x(n) If y(-1)≠0, we have output without any input Not Linear system Not time-invariant
If y(-1)=0, we say that the system is at rest. System is LTI
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
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Infinite Impulse response (IIR)
Note: Only makes sense to assume So, transient will fade out and ”after a while it is LTI”
If y(-1)=0, we say that the system is at rest. System is LTI
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
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Infinite Impulse response (IIR)
Note: Only makes sense to assume So, transient will fade out and ”after a while it is LTI”
If y(-1)=0, we say that the system is at rest. System is LTI
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
The class of systems described by difference equations encompasses LTI systems with infinite length impulse responses
EITF75 Systems and Signals
Relation to difference equations
Consider now
We then get
Infinite Impulse response (IIR)
Note: Only makes sense to assume So, transient will fade out and ”after a while it is LTI”
If y(-1)=0, we say that the system is at rest. System is LTI
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
Every impulse response corresponds to one difference equation.
EITF75 Systems and Signals
Brief info on correlation
Not focal point of course, but highly important in signal processing
Correlation measures similarity between two signals
EITF75 Systems and Signals
Brief info on correlation
Not focal point of course, but highly important in signal processing
Auto correlation
Correlation measures similarity between two signals
Cross correlation
Measures similarity between time shifted versions of the same signal
Measures similarity between time shifted versions of different signals
EITF75 Systems and Signals
Brief info on correlation
Not focal point of course, but highly important in signal processing
Auto correlation
Correlation measures similarity between two signals
Cross correlation
Measures similarity between time shifted versions of the same signal
Measures similarity between time shifted versions of different signals
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)
EITF75 Systems and Signals
Brief info on correlation
Not focal point of course, but highly important in signal processing
Auto correlation
Correlation measures similarity between two signals
Cross correlation
Measures similarity between time shifted versions of the same signal
Measures similarity between time shifted versions of different signals
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)
Cross correlation between x1(n) and x2(n) should be small (to know who is connecting)
EITF75 Systems and Signals
Brief info on correlation
Not focal point of course, but highly important in signal processing
Auto correlation
Correlation measures similarity between two signals
Cross correlation
Measures similarity between time shifted versions of the same signal
Measures similarity between time shifted versions of different signals
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 of x1(n) (and x2(n)) should be delta (to know when a user is connecting)
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