13005810.ppt

ELEN 4304/5346 Digital Signal Processing Fall 2008
1
Lecture 01: Introduction
Based on ECE 4624 by Dr. A.A. (Louis) Beex,
Virginia Tech
Instructor:
Dr. Gleb V. Tcheslavski
Contact: gleb@ee.lamar.edu
Office Hours: Room 2030
Class web site:
http://ee.lamar.edu/gleb/dsp/ind
ex.htm

2
Syllabus overview
• Pre-req.: ELEN 3313 Signals and Systems or their equivalent
• Required book: Sanjit K. Mitra, Digital-Signal Processing: A Computer-Based
Approach, McGraw-Hill Co., Third edition, 2004, ISBN: 0-07-286546-6.
• Required software: The Mathworks, The Student Edition of MATLAB, Release
2006a or later.
• Structure: Two 75-minute lectures per week. One homework, three projects,
midterm exam, and the final examination.
• Tests: The Midterm and Final exams will be closed book/notes. Your
performance on the Projects will account for the bulk of your grade.
• Honor System: Discussions on lecture subject material, to clarify your
understanding, are highly encouraged. However, it is your personal understanding
only that should be reflected in all work that you turn in. Any copyright violations
(including copying articles and/or web pages to your reports) will be prosecuted!

3
Styles, notations, legends…
1. Colors: Normal text and formulas
Something more important (imho)
Important formulas and results
Very Important Formulas
Miscellaneous
2. Equations notations: (2.17.3)
Lecture # Slide # Formula #

4
Linear algebra overview
It is convenient in many applications to represent signals and system coefficients by
vectors and matrices. Therefore, a brief overview of linear algebra is considered.
1. Vectors
A vector (denoted by a lowercase bold letter) is an array of real-valued or complex-
valued numbers or functions. We will assume column vectors. The N-dimensional
vector is:
1
2
N
x
x
x
 
 
 

 
 
 
x
The transpose of a vector is a row vector:
 
1 2
T
N
x x x

x
(1.4.1)
(1.4.2)

5
The Hermitian transpose is the complex conjugate of the transpose of x:
 
* * * *
1 2
H T
N
x x x
 
   
x x
It might be convenient in some cases to consider a set of values xn containing the
signal values in the certain range:
1
1
n
n
n
n N
x
x
x

 
 
 
 

 
 
 
x
(1.5.1)
(1.5.2)

6
The measure of the “magnitude” of the vector is the norm.
The Euclidean or L2 norm:
2
2
1
N
i
i
x

 
x
The L1 norm:
1
1
N
i
i
x

 
x
The L norm: max i
i
x


x
We will be using the second norm unless stated otherwise.
If the vector has a non-zero norm, it can be normalized as follows:
x 
x
v
x
(1.6.1)
(1.6.2)
(1.6.3)
(1.6.4)
The vector vx is a unit norm vector that lies in the same direction as x.

7
Lastly, the norm can be used to measure the distance between two vectors:
 
2
1
,
N
i i
i
d x y

   

x y x y (1.7.2)
For two complex vectors a and b of the same length, the inner product is a scalar
defined as
*
1
,
N
H
i i
i
a b

  
a b a b
For two real vectors, the inner product becomes
1
,
N
T
i i
i
a b

  
a b a b
(1.7.3)
(1.7.4)
The squared norm represents the energy in the signal:
1
2 2
0
N
n
n
x


 
x (1.7.1)

8
where  is the angle between two vectors. Therefore, two nonzero vectors a and b
are orthogonal if their inner product is zero:
The inner product defines the geometrical relationship between two vectors:
, cos

a b a b (1.8.1)
, 0

a b
Two vectors that are orthogonal and have unit norms are called orthonormal.
Since |cos|  1, the inner product is bounded:
, 
a b a b
(1.8.3) is known as the Cauchy-Schwarz inequality. The equality holds iff a and b
are colinear, i.e.
,
 
  
a b const
(1.8.2)
(1.8.3)
(1.8.4)

9
Another useful inequality is
2 2
2 ,  
a b a b (1.9.1)
The output of an LTI FIR filter can be represented using the inner product. If the
filter output is the convolution of its unit pulse response and the input
1
0
N
n k n k
k
y h x



 
then, expressing x as in (1.5.2) and representing hn as
 
0 1 1
T
N
h h h 

h
the filter output may be written as the inner product:
T
n n
y  h x
(1.9.2)
(1.9.3)
(1.9.4)

10
A set of n vectors v1, v2, … vn is linearly independent if
1 1 2 2 0
n n
  
   
v v v
implies that i = 0 for all i. If a set of nonzero i can be found satisfying (1.10.1), the
vectors are linearly dependent and at least one of the vectors, say v1 can be
expressed as a linear combination of the remaining vectors:
1 2 2 3 3 n n
  
   
v v v v
for some set of scalars i.
For vectors of dimension N, no more than N vectors may be linearly independent.
Therefore, any set of N-dimensional vectors containing more than N vectors will
always be linearly dependent.
(1.10.1)
(1.10.2)

11
For a set of N vectors v1, v2, … vn
 
1 2 n

V v v v
Consider the set of all vectors V that may be formed from a linear combination of
the vectors vi
1
N
i i
i


 
v v
This set forms a vector space, and the vectors vi are said to span the space V.
Furthermore, if the vectors vi are linearly independent, they form a basis for the
space V, and the number of vectors in the basis N is the dimension of the space.
For instance, the set of all real vectors of the form
 
1 2
T
N
x x x

x
forms an N-dimensional vector space, denoted by RN that is spanned by the basis
vectors
(1.11.1)
(1.11.2)
(1.11.3)

12
 
 
 
1
2
1 0 0 0
0 1 0 0
0 0 0 1
T
T
T
N



u
u
u
In terms of basis, any vector
 
1 2
T
N
v v v

v
may be uniquely decomposed as follows:
1
N
i i
i
v

 
v u
It should be pointed out, however, that the basis for a vector space is not unique.
(1.12.1)
(1.12.2)
(1.12.3)

13
2. Matrices
An n x m matrix is an array of real or complex numbers or functions having n rows
and m columns.
11 12 13 1
21 22 23 2
1 2 3
m
m
n n n nm
a a a a
a a a a
a a a a
 
 
 

 
 
 
A
An n x m matrix of
numbers
An n x m matrix of
functions
11 12 13 1
21 22 23 2
1 2 3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
m
m
n n n nm
a z a z a z a z
a z a z a z a z
z
a z a z a z a z
 
 
 

 
 
 
A
If n = m, A is a square matrix.
(1.13.1)
(1.13.2)

14
The output of an LTI FIR filter can be written in vector form as follows:
T T
n n n
y  
h x x h
If xn = 0 for n < 0, the filter output may be expressed for n  0 as
0

y X h
where X0 is the convolution matrix
0
1 0
2 1 0
0
1 2 3 0
0 0 0
0 0
0
N N N
x
x x
x x x
x x x x
  
 
 
 
 
  
 
 
 
 
X
and  
0 1 2
T
y y y

y
(1.14.1)
(1.14.2)
(1.14.3)
(1.14.4)
We observe that, in addition to
its structure of having equal
values along each of the
diagonals, X0 has N-1 columns
and an infinite number of rows.

15
An n x m matrix can be represented as a set of m column vectors
 
1 2 m

A c c c
or as a set of n row vectors
1
2
H
H
H
n
 
 
 

 
 
 
 
r
r
A
r
(1.15.1)
(1.15.2)
An n x m matrix may be partitioned into submatrices as follows
11 12
21 22
 
  
 
A A
A
A A
where A11 is p x q, A12 is p x (m-q), A21 is (n-p) x q, and A22 is (n-p) x (m-q).
(1.15.3)

16
If A is an n x m matrix, then the transpose AT is the m x n matrix that is formed by
interchanging the rows and columns of A. Therefore, the (i, j)th element becomes
the (j, i)th element and vice versa. If the matrix is square, the transpose is formed
by reflecting its elements with respect to the diagonal.
A square matrix A is symmetric if
T

A A
For complex matrices, the Hermitian transpose is defined as
   
*
* T
H T
 
A A A
A square complex matrix A is Hermitian if
H

A A
(1.16.1)
(1.16.2)
(1.16.3)

17
Useful properties of Hermitian transpose are
 
 
 
1.
2.
3.
H H H
H
H
H H H
   
 
 
A B A B
A A
AB B A
Replacing the Hermitian transpose with an ordinary transpose, we obtain
 
 
 
1.
2.
3.
T T T
T
T
T T T
   
 
 
A B A B
A A
AB B A
(1.17.1)
(1.17.2)

18
Let A be an n x m matrix partitioned in a set of m column vectors
 
1 2 m

A c c c (1.18.1)
The rank of A, rank(A) is defined as the number of linearly independent columns in
A, i.e., the number of linearly independent vectors in the set {c1, c2, … , cm}. One of
the properties of the rank is that
( ) ( )
H
rank rank
 
A A
Therefore, if A is partitioned in a set of n row vectors
1
2
H
H
H
n
 
 
 

 
 
 
 
r
r
A
r
Then, the rank of A is also equal to the number of linearly independent row vectors.
(1.18.2)
(1.18.3)

19
The important property of the rank is
( ) ( ) ( )
H H
rank rank rank
 
A AA A A (1.19.1)
Since the rank of a matrix is equal to the number of linearly independent rows and
the number of linearly independent columns, then, if A is an m x n matrix:
( ) min( , )
rank m n

A
If A is an m x n matrix and rank(A) = min(n,m), then A is of full rank.
If A is a square matrix of full rank, then there exists a unique matrix A-1 called the
inverse of A such that
1 1
 
 
A A AA I
(1.19.2)
(1.19.3)

20
where 1 0 0
0 1 0
0 0 1
 
 
 

 
 
 
I (1.20.1)
is the identity matrix with ones along the main diagonal and zeros everywhere else.
In this case, A is said to be invertible or nonsingular. If A is not of full rank, then it is
noninvertible or singular and does not have an inverse.
If A and B are invertible, then
 
   
   
1 1 1
1 1
1
1 1 1 1 1 1
1.
2.
3.
H
H
  
 

     
 
 
    
AB B A
A A
A BCD A A B C DA B DA
(1.20.2)
(1.20.3)
(1.20.4)

21
(1.20.4) is called the matrix inversion lemma. It is assumed that A is n x n, B is n
x m, C is m x m, and D is m x n with A and C nonsingular matrices.
A special case of this lemma occurs when C = 1, B = u, and D = vH where u and v
are n-dimensional vectors. In this case
 
1 1
1 1
1
1
H
H
H
 
 

  

A uv A
A uv A
v A u
Which is sometimes referred to as a Woodbury’s identity. As a special case, if A =
I, (1.21.1) becomes
 
1
1
H
H
H

  

uv
I uv I
v u
(1.21.1)
(1.21.2)

22
If A = a11 is a 1 x 1 matrix, the determinant is defined as det(A) = a11. For an n x n
matrix, the determinant is defined recursively as
   
1
det( ) 1 det
n
i j
ij ij
i
a


 

A A
where Aij is the (n-1) x (n-1) matrix formed by deleting the ith row and the jth column
of A.
Property: An n x n matrix A is invertible iff
det( )
A 0
(1.22.1)
(1.22.2)

23
For A and B being n x n matrices, if A is invertible, and a constant , the following
properties hold:
     
   
   
   
1
1. det det det
2. det det
3. det det
1
4. det
det
T
n
 

 
 
 
 
AB A B
A A
A A
A
A
For an n x n matrix A, the trace function is defined as
 
1
n
ii
i
tr a

 
A
(1.23.1)
(1.23.2)
(1.23.3)
(1.23.4)
(1.23.5)

24
3. Linear equations
Consider the following set of n linear equations in the m unknowns xi, i = 1,2,…,m
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
...
...
...
m m
m m
n n nm m n
a x a x a x b
a x a x a x b
a x a x a x b
   
   
   
These equations may be written in matrix form as follows

Ax b
(1.24.1)
(1.24.2)
Where A is an n x m matrix with entries aij, x is an m-dimensional vector of
unknowns xi, and b is an n-dimensional vector with elements bi.

25
A convenient way to view (1.24.2) is as an expansion of vector b in terms of a
linear combination of the column vectors ai of the matrix A:
1
m
i i
i
x

 
b a
Solving (1.25.1) depends on a number of factors including the relative size of m
and n, the rank of A, and the elements of b.
(1.25.1)
3.1. Square matrix: m = n.
The solution to (1.25.1) depends upon whether or not A is singular. If A is
nonsingular, then the inverse A-1 exists and the solution is uniquely defined by
1


x A b (1.25.2)
However, if A is singular, then there may either be no solution (the equations are
inconsistent) or many solutions.

26
For the case in which A is singular, the columns of A are linearly dependent and
there exist nonzero solutions to the homogeneous equation

Az 0
In fact, there will be k = n – rank(A) linearly independent solutions to the
homogeneous equations. Therefore, if there is at least one vector x0 that solves
(1.24.2), then any vector of the form
0 1 1 ... k k
 
   
x x z z
will also be a solution where zi, i = 1,2,…,k are linearly independent solutions to
(1.24.2).
(1.26.1)
(1.26.2)

27
3.2. Rectangular matrix: n < m.
In this situation, there are fewer equations then unknowns. Therefore, if the
equations are not inconsistent, there are many vectors satisfying the equations: the
solution is undetermined or incompletely specified. A common approach to define a
unique solution is to find the vector that satisfies the equations and has a minimum
norm; i.e.
min suchthat
   
x Ax b
If rank(A) = n (the rows of A are linearly independent), then the n x n matrix AAH is
invertible and the minimum norm solution is
 
1
0
H H 

x A AA b
 
1
H H 


A A AA
The matrix
is called the pseudo-inverse of the matrix A for the undetermined problem.
(1.27.1)
(1.27.2)
(1.27.3)

28
3.3. Rectangular matrix: n > m.
In this situation, there are more equations then unknowns and, in general, no
solution exists. The equations are inconsistent and the solution is overdetermined.
For the case of 3 equations in 2 unknowns, this
problem is illustrated:
Since an arbitrary vector b cannot be
represented as a linear combination of the
columns of A, the goal is to find the coefficients
xi producing the best approximation to b:
1
ˆ
m
i i
i
x a

 
b (1.28.1)

29
(1.29.1)
The approach commonly used in this situation is to find the least squares solution;
i.e., the vector x minimizing the norm of the error
2 2
 
e b Ax
The least squares solution has the property that the error
 
e b Ax
is orthogonal to the column vectors of A. This orthogonality implies that
0
H

A e
or
H H

A AX A b
which are known as the normal equations.
(1.29.2)
(1.29.3)
(1.29.4)

30
If the columns of A are linearly independent (A has full rank), the matrix AHA is
invertible and the least square solution is
 
1
0
H H


x A A A b
or
0


x A b
where  
1
H H



A A A A
is the pseudo-inverse of the matrix A for the overdetermined problem. Furthermore,
the best approximation of b is given by the projection of the vector b onto the
subspace spanned by the vectors ai
 
1
0
ˆ H H

 
b Ax A A A A b
(1.30.1)
(1.30.2)
(1.30.3)
(1.30.4)

31
or
where  
1
H H
A


P A A A A
ˆ
A

b P b (1.31.1)
(1.31.2)
is called the projection matrix. Finally, using the orthogonality condition, the
minimum mean square error is
2
0
min H H H
  
e b e b b b Ax (1.31.3)

32
4. Special matrix forms
4.1. Diagonal matrix – a square matrix having all entries equal to zero except,
possibly, for those along the main diagonal:
11
22
0 0
0 0
0 0 nn
a
a
a
 
 
 

 
 
 
A
The diagonal matrix may be written as
 
11 22
diag nn
a a a

A
(1.32.1)
(1.32.2)

33
4.2. Identity matrix – a diagonal matrix with ones along the diagonal:
 
1 0 0
0 1 0
diag 1 1 1
0 0 1
 
 
 
 
 
 
 
I (1.33.1)
4.3. Block diagonal matrix – a diagonal matrix whose entries along the diagonal are
replaced with matrices:
11
22
0 0
0 0
0 0 nn
 
 
 

 
 
 
A
A
A
A
(1.33.2)

34
(1.34.1)
4.4. Exchange matrix – a symmetric matrix with ones along the cross-diagonal and
zeros everywhere else. Since J2 = I, then J is its own inverse.
0 0 1
0 1 0
1 0 0
 
 
 

 
 
 
J
The effect of multiplying a vector v by the exchange matrix is a vector with
reversed order of the entries; i.e.,
1
2 1
1
n
n
n
v v
v v
v v

   
   
   

   
   
 
 
J (1.34.2)

35
Similarly, if a matrix A is multiplied on the left by the exchange matrix, the effect is
to reverse the order of each column. For example:
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
 
 
  
 
 
A
then 31 32 33
21 22 23
11 12 13
T
a a a
a a a
a a a
 
 
  
 
 
J A
(1.35.1)
(1.35.2)

36
Similarly, if a matrix A is multiplied on the right by the exchange matrix, the effect is
to reverse the order of each row:
13 12 11
23 22 21
33 32 31
a a a
a a a
a a a
 
 
  
 
 
AJ
Finally, the effect of forming the product JTAJ is to reverse the order of each row
and column
33 32 31
23 22 21
13 12 11
T
a a a
a a a
a a a
 
 
  
 
 
J AJ
thereby reflecting each element of A about the central element.
(1.36.1)
(1.36.2)

37
4.5. Upper triangular matrix – a square matrix, in which all entries below the
diagonal are zero:
0
ij
a for i j
   
For example, for n = 4
11 12 13 14
22 23 24
33 34
44
0
0 0
0 0 0
a a a a
a a a
a a
a
 
 
 

 
 
 
A
4.6. Lower triangular matrix – a square matrix, in which all entries above the
diagonal are zero:
0
ij
a for i j
   
(1.37.1)
(1.37.2)
(1.37.3)

38
Some properties of lower and upper triangular matrices are:
1. The transpose of a lower (upper) triangular matrix is an upper (lower) triangular
matrix;
2. The determinant of a lower or upper triangular matrix is
3. The inverse of an upper (lower) triangular matrix is upper (lower) triangular;
4. The product of two upper (lower) triangular matrices is upper (lower) triangular.
1
det( )
n
ii
i
a

 
A (1.38.1)

39
4.7. Toeplitz matrix – a square n x n matrix, in which all entries along each of the
diagonals have the same value
1, 1 ,
ij i j
a a i n j n
 
    
1 3 5 7
2 1 3 5
4 2 1 3
6 4 2 1
 
 
 

 
 
 
A
(1.39.1)
(1.39.2)

40
4.8. Hankel matrix – a square n x n matrix, in which all entries along each of the
cross-diagonals have the same value
1, 1 ,
ij i j
a a i n j n
 
    
1 3 5 7
3 5 7 4
5 7 4 2
7 4 2 1
 
 
 

 
 
 
A
(1.40.1)
(1.40.2)
Another example of Hankel matrices is the exchange matrix J.

41
Toeplitz matrices are a special case of matrices known as persymmetric.
4.9. Persymmetric matrix – a matrix symmetric about the cross-diagonal:
1, 1
ij n j n i
a a    

For example, for n = 4, a persymmetric but non-Toeplitz matrix:
1 3 5 7
2 2 4 5
4 4 2 3
6 4 2 1
 
 
 

 
 
 
A
(1.41.1)
(1.41.2)
If a Toeplitz matrix is symmetric (or Hermitian in the case of a complex matrix), we
only need to specify the first row and the first column to completely determine a
Toeplitz matrix.

42
Symmetric Toeplitz matrices are a special case of matrices known as
centrosymmetric.
4.10. Centrosymmetric matrix – a matrix that is both symmetric and persymmetric.
For example, for n = 4, a centrosymmetric but non-Toeplitz matrix:
1 3 5 6
3 2 4 5
5 4 2 3
6 5 3 1
 
 
 

 
 
 
A (1.42.1)
If A is a symmetric Toeplitz matrix, then
T

J AJ A
If A is a Hermitian Toeplitz matrix, then
*
T

J AJ A
(1.42.2)
(1.42.3)

43
Some important properties are…
1. The inverse of a symmetric matrix is symmetric;
2. The inverse of a persymmetric matrix is persymmetric;
3. The inverse of a Toeplitz matrix is not, in general, Toeplitz. However, since a
Toeplitz matrix is persymmetric, the inverse will always be persymmetric.
Furthermore, the inverse of a symmetric Toeplitz matrix will be centrosymmetric.
The relationship between the
symmetries of a matrix and
its inverse

44
4.11. Orthogonal matrix – a real n x n matrix with orthogonal columns (and rows):
 
1 2 n

A a a a
and
If
1
0
T
i j
for i j
for i j
  

 
  

a a
then A is orthogonal.
We observe that if A is orthogonal, then
T

A A I
Therefore, the inverse of an orthogonal matrix is equal to its transpose:
1 T


A A
(1.44.1)
(1.44.2)
(1.44.3)
(1.44.4)

45
4.12. Unitary matrix – a complex n x n matrix with orthogonal columns (rows):
1
0
H
i j
for i j
for i j
  

 
  

a a
H

A A I
1 H


A A
(1.45.2)
(1.45.3)
then
(1.45.1)
The inverse of a unitary matrix equals to its Hermitian transpose

46
The quadratic form of a real symmetric n x n matrix A is the scalar defined by
 
1 1
n n
T
A i ij j
i j
Q x a x
 
  
x x Ax
where x is a vector of n real variables. Observe that the quadratic form is a
quadratic function in the n variables x1, x2, …, xn. For example, the quadratic form
of
3 1
1 2
 
  
 
A
is
  2 2
1 1 2 2
3 2 2
T
A
Q x x x x
   
x x Ax
(1.46.1)
(1.46.2)
(1.46.3)

47
Similarly, for a Hermitian matrix, the Hermitian form is
  *
1 1
n n
H
A i ij j
i j
Q x a x
 
  
x x Ax (1.47.1)
If the quadratic form of a matrix A is positive for all nonzero vectors x,
  0
A
Q 
x
then A is said to be positive definite and we write A > 0. For example, if
(1.47.2)
2 0
0 3
 
  
 
A
which has the quadratic form
    1 2 2
1 2 1 2
2
2 0
2 3
0 3
A
x
Q x x x x
x
 
 
  
 
 
   
x
(1.47.3)
(1.47.4)
is positive definite since QA(x) > 0 for all x  0.

48
If the quadratic form of a matrix A is non negative for all nonzero vectors x,
  0
A
Q 
x
then A is said to be positive semidefinite.
(1.48.1)
If the quadratic form of a matrix A is negative for all nonzero vectors x,
  0
A
Q 
x
then A is said to be negative definite.
(1.48.2)
If the quadratic form of a matrix A is non positive for all nonzero vectors x,
  0
A
Q 
x
then A is said to be negative semidefinite.
(1.48.3)
A matrix A that is none of the above is called indefinite.

49
5. Eigenvalues and Eigenvectors
Let A be an n x n matrix and consider the following set of linear equations:


Av v
where  is a constant. Equivalently:
  0

 
A I v
(1.49.1)
(1.49.2)
In order for a nonzero vector v to be a solution to this equation, it is necessary for
the matrix A - I to be singular. Therefore, the determinant of A - I must be zero:
   
det 0
p  
  
A I
where p() is an nth order polynomial in . This polynomial is called the
characteristic polynomial of the matrix A and its n roots i for i = 1, 2,…, n are
called the eigenvalues of A.
(1.49.3)

50
For each eigenvalue i for i = 1, 2,…, n the matrix A - I will be singular and there
will be at least one nonzero vector vi that solves (1.49.1); i.e.
i i i


Av v
These vectors vi are called the eigenvectors of A. For any eigenvector vi, vi will
also be an eigenvector for any constant . Therefore, eigenvectors are often
normalized to have unit norm.
Property 1: The nonzero eigenvectors v1, v2, … vn corresponding to the distinct
eigenvalues 1, 2, …, n are linearly independent.
If A is an n x n singular matrix, then there are nonzero solutions to the
homogeneous equation
0
i 
Av
and it follows that  = 0 is an eigenvalue of A.
There are n – rank(A) linearly independent solutions to (1.50.2). Therefore, A will
have rank(A) nonzero eigenvalues and n – rank(A) eigenvalues that are equal zero.
(1.50.1)
(1.50.2)

51
Property 2: The eigenvalues of a Hermitian matrix are real.
Property 3: A Hermitian matrix is positive definite A > 0 iff eigenvalues of A are
positive: k > 0.
Property 4: The eigenvectors of a Hermitian matrix corresponding to distinct
eigenvalues are orthogonal; i.e.
The determinant of a matrix is related to its eigenvalues as
 
1
det
n
i
i


 
A (1.51.1)
, 0
i j i j
if then
 
     
v v (1.51.2)
Spectral Theorem: Any Hermitian matrix A can be decomposed as
1 1 1 2 2 2 ...
H H H H
n n n
  
    
A VΛV v v v v v v (1.51.3)
where i are the eigenvalues of A and vi are a set of orthonormal eigenvectors.

52
Property 5: Let B be an n x n matrix with eigenvalues i and let A be a matrix that is
related to B as follows

 
A B I
Then A and B have the same eigenvectors and the eigenvalues of A are i + .
(1.52.1)
Property 6: For a symmetric positive definite matrix A, the equation
1
T

x Ax (1.52.2)
defines an ellipse in n dimensions whose axes are in the
directions of the eigenvectors vj of A with the half-length
of these axes equal to 1 j


53
Concepts of signals and systems
Signals play an important role in our daily life. Examples of signals are speech,
music, pictures, video… a signal is a function of independent variables such as
time, distance, temperature, etc.
For instance, a speech signal represents
air pressure (or a voltage output from a
microphone) as a function of time;
a black and white picture is a
representation of light intensity as a
function of two spatial coordinates;
Video is a function of two spatial coordinates and
time.

54
Most signals are generated naturally; however, a signal can
also be generated synthetically by a computer simulation.
A signal carries information, and one of the objective of signal
processing is to extract useful information from the signal.
The method of information extraction depends on the type of
signal and the nature of information.

55
The term system usually indicates a physical device used to
generate or manipulate signals. Thus, we may consider a
system as a mathematical function applied to a signal.
Alternatively, a system can be viewed as any process that
results in the transformation of signals. Therefore, a system
may have an input signal and an output signal that is related
to the input through the system transformation.
For instance, an audio amplifier takes a recorded music
signal, amplifies it and outputs to the speakers usually
allowing us to control the amplification (i.e. “loudness”) and
tone (amounts of “bass” and “treble”).

56
Why Digital?? Music recording and
reproduction: some history
Past…

57
Music recording and reproduction
Still past… And (almost) present

58
Music recording and reproduction: some
physics behind
 Mechanical wave (sound) generates electrical signal in a microphone.
 Signal is amplified
 And recorded (stored), which usually involves some conversion…
1. Recording
Music recording techniques can be divided into analog and digital
according to the signal’s representation at the storage stage.
2. Reproduction
 A recorded signal is converted back to electrical current.
 Amplified
 A mechanical wave (sound) is generated by a speaker (electrical oscillation is
converted into mechanical one).

59
Music recording and reproduction: The
media
Magnetic tape – Analog recording:
Magnetization of the media changes
according to an analog electrical signal
CD/DVD – digital recording:
Optical properties are modified to
encode a digital signal

60
So, why DSP???
Most of the portable
(and not only
portable!) electronic
items we are dealing
with in our everyday
life are digital…
Digitally recorded
data needs to be
manipulated…

61
Sampling
Let us consider a sinusoid x(t) = A cos(t + ), t is a continuous time variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Volts
Time, sec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Volts
Time, sec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Volts
Time, sec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Volts
Time, sec
By sampling, we can reduce storage requirements considerably!
Ts

62
Sampling: Ideal C/D converter
Ideal C/D
converter
x(t)
Ts
xn
Therefore, a sampled signal: ( )
n s
x x nT

s
t nT

(1.62.1)
for a sinusoid: ( ) cos( ) cos( )
n s s
x x nT A nT A n
  
     
1
s
s
f
T

The sampling frequency: (1.62.2)
(1.62.3)
Normalized radial frequency:
Fractional frequency:
s s
T f
   
n s
f f f

(1.62.4)
(1.62.5)

63
Aliasing
Let us consider two discrete-time sinusoids:
1, cos(0.4 )
n
x n

 2, cos(2.4 )
n
x n


2, 1,
cos(2.4 ) cos(2 0.4 ) cos(0.4 )
n n
x n n n n x
   
    
They are obviously at two different frequencies BUT trigonometry shows:
Result: we may reconstruct more than one sinusoid from samples!
Aliasing (more than one signal represented by the same
frequency samples) is due to the 2n periodicity of sin/cos…
0 100 200 300 400 500 600 700 800 900 1000
-1
-0.5
0
0.5
1
Amplitude
x1
(t)
x2
(t)
x1n

64
Aliasing
Intuitively, we would lower the storage requirements if the sampling frequency
can be reduced… Then we would collect less samples…
How would we mitigate aliasing?
Restrict ourselves to the signals at frequencies [- ]…
…which leads us to an appropriate selection of sampling
rate (sampling frequency)!

65
How many samples do we need to
reconstruct the initial sinusoid?
Let us consider two periods of a sinusoid and a uniform sampling…
As many
as:
4
uniform
samples/
period
Can be
reconstructed!
Can fit >1 sin?
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts

66
66
How many signal’s samples do we need to reconstruct the initial sinusoid?
Let us consider two periods of a sinusoid and uniform sampling…
As many
as:
2
uniform
samples/
period.
Still can be
reconstructed!
Can fit >1 sin?
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts

67
Let us consider two periods of a sinusoid and a uniform sampling…
As many
as:
1
sample/
period.
Can draw
MORE than
one sinusoid!
Initial signal is
lost.
Can fit >1 sin?
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time, sec
Volts

68
The Sampling Theorem
We can conclude that at least two uniform signal samples per period are necessary
to recover the initial sinusoidal signal.
Harry Nyquist Claude Shannon Kotelnikov
Nyquist-Shannon Theorem: a sinusoidal signal of frequency f0 can be
perfectly reconstructed from it’s uniformly sampled version if samples
were taken at the frequency at least 2 f0 (twice the signal’s frequency).

69
The Sampling Theorem
More accurately:
A band-limited signal can be perfectly reconstructed from
its uniformly sampled version iff samples were taken at a
frequency at least twice higher than the highest frequency
in the signal’s spectrum.
One half of the sampling frequency – the maximum
frequency component that can be represented – is called the
Nyquist frequency.
What’s about systems? Do they become discrete too?

70
Classification of sequences
1 2
1 2
;
N n N
N N
 
    
A discrete-time signal can be classified into several types based on its
specific characteristics…
1. Duration: a discrete-time signal may be a finite-length or an infinite-
length sequence.
A finite-length (finite-duration; finite-extend) sequence is defined for a
finite time interval n where
Therefore, the length (duration) N of the above finite-length digital
sequence may be computed as
2 1 1
N N N
  
A length-N discrete-time sequence consists of N samples.
(1.70.1)
(1.70.2)

71
1
0
n
x for n N
   
There are three types of infinite-length sequences:
A right-sided sequence xn has zero-valued samples for n < N1:
where N1 is a finite integer that can be positive or negative. If N1 is
positive, a right-sided sequence is usually called a causal sequence.
A left-sided sequence xn has zero-valued samples for n > N2:
2
0
n
x for n N
   
where N2 is a finite integer that can be positive or negative. If N2 is
negative, a left-sided sequence is usually called an anticausal sequence.
A general two-sided sequence xn is defined for both positive and
negative values of n.
(1.71.1)
(1.71.2)

72
*
n n
x x
 
2. Symmetry with respect to the time-index n = 0.
A sequence xn is called a conjugate-symmetric sequence if
A real conjugate-symmetric sequence is called an even sequence.
A sequence xn is called a conjugate-antisymmetric sequence if
*
n n
x x

A real conjugate-antisymmetric sequence is called an odd sequence.
For a conjugate-antisymmetric sequence xn, the sample value at n = 0
must be purely imaginary.
For an odd sequence x0 = 0.
(1.72.1)
(1.72.2)

73
cs ca
n n n
x x x
 
Any complex sequence xn can be expressed as a sum of its conjugate-
symmetric part xcs
n and its conjugate-antisymmetric part xca
n:
where
 
 
*
*
1
2
1
2
cs
n n n
ca
n n n
x x x
x x x


 
 
Therefore, the computation of conjugate-symmetric and conjugate-
antisymmetric parts of a sequence involves conjugation, time-reversal,
addition, and multiplication operations.
(1.73.1)
(1.73.2)
(1.73.3)

74
Similarly, any real sequence xn can be expressed as a sum of its even
part xev
n and its odd part xod
n:
ev od
n n n
x x x
 
where
 
 
1
2
1
2
ev
n n n
od
n n n
x x x
x x x


 
 
The symmetry properties of sequences often simplify their respective
frequency-domain representation.
(1.74.1)
(1.74.2)
(1.74.3)

75
3. Periodicity.
A sequence xn satisfying
n n kN
x x n

  (1.75.1)
is called a periodic sequence with a period N, where N is a positive
integer and k is any integer. Otherwise, a sequence is called an
aperiodic sequence. The fundamental period Nf of a periodic signal is the
smallest value of N for which (1.75.1) holds.
Sum or product of two or more periodic sequences is also a periodic
sequence. For instance, a sum of two periodic sequences xa
n and xb
n
with fundamental periods Na and Nb is a periodic sequence with a
fundamental period N
 
,
a b
a b
N N
N
GCD N N
 (1.75.2)
The greatest common divisor

76
2
x n
n
E x


 
4. Energy and Power signals.
The total energy of a sequence xn is defined as
An infinite-length sequence with finite sample values may or may not
have finite energy.
The average power of an aperiodic sequence xn is defined as
2
1
lim
2 1
K
x n
K
n K
P x
K





(1.76.1)
(1.76.2)

77
2
,
K
x K n
n K
E x

 
,
1
lim
2 1
x x K
K
P E
K



The average power of a sequence can be related to its energy by
defining its energy over a finite interval -K  n  K as
Then
The average power of a periodic sequence xn with a period N is
1
2
0
1 N
x n
n
P x
N


 
The average power of an infinite-length sequence may be finite or
infinite.
(1.77.1)
(1.77.2)
(1.77.3)

78
An infinite energy signal with finite average power is called a
power signal.
A finite energy signal with zero average power is called an
energy signal.
An example of a power signal is a periodic sequence that
has a finite average power but infinite energy.
An example of an energy signal is a finite-length sequence
which has finite energy but zero average power.

79
n x
x B
  
n
n
x


 

5. Other classifications.
1) A sequence xn is called bounded if each of its samples is of finite
magnitude:
2) A sequence xn is called absolutely summable if
3) A sequence xn is called square-summable if
2
n
n
x


 

(1.79.1)
(1.79.2)
(1.79.3)

80
sin
,
a c
n
n
x n
n


     
Therefore, a square-summable sequence has finite energy and is an
energy signal if it also has zero power.
An example of a sequence that is square-summable but not absolutely
summable is
Examples of sequences that are neither absolutely summable nor
square-summable are
sin ,
,
b
n c
c
n
x n n
x K n

    
    
where K is a constant.
(1.80.1)
(1.80.2)
(1.80.3)

81
Important quantities in discrete time
Kronecker delta (unit sample, unit
impulse) function:
1, 0
0, 0
n
n
n

 

 
 

(1.81.1)
Shifted by k samples unit sample
function:
1,
0,
n k
n k
n k
 
 

 
 

(1.81.2)
k = 2

82
The unit step function: 1, 0
0, 0
n
n
n

 

 
 

(1.82.1)
Shifted by k samples unit step
function:
1,
0,
n
n k
n k

 

 
 

(1.82.2)
k = -2

83
0
1
n
n n m k
m k
n n n
  
  


 

 
 
 
The unit sample and unit step sequences are related as follows:
(1.83.1)
(1.83.2)
The real sinusoidal sequence with constant amplitude:
 
0
cos ,
n
x A n n
 
     
where A, 0,  are real numbers: the amplitude, the angular frequency,
and the phase of the sinusoidal sequence xn.
(1.83.3)

84
i q
n n n
x x x
 
The real sinusoidal sequence can also be written as
where xi
n and xq
n are the in-phase and quadrature of xn that are given by
   
0 0
cos cos ; sin sin
i q
n n
x A n x A n
   
   
(1.84.1)
(1.84.2)

85
,
n
n
x A n

     
 
0 0
,
j j
e A A e
  
 
  
   
   
0 0 0
0
0 0
0 0
cos sin
j n j n
n
n
n n
x Ae A e e
A e n j A e n
   

 
   
 
 
   
The exponential sequence is
where A and  are real or complex numbers computed as
Therefore:
If
re im
n n n
x x x
 
then
   
0 0
0 0
cos ; sin
n n
re im
n n
x A e n x A e n
 
   
    
(1.85.1)
(1.85.2)
(1.85.3)
(1.85.4)
(1.85.5)

86
With both A and  real, (1.85.1) reduces to a real exponential sequence.
For n  0, such a
sequence with | |
< 1 decays
exponentially as n
increases and with
| | > 1 grows
exponentially as n
increases.

87
System’s impulse response:
n k n k
k
h b  
 
Discrete convolution:
n n n n n l n l
l
y x h h x x h



     
All LTI systems satisfy the convolution equation where:
xn is an input to the system, yn is its output, and hn is the
system’s impulse response.
(1.87.1)
(1.87.2)
Convolution with an impulse:
0 0
n n n n n
x x
  
  (1.87.3)

88
Question time!
Remark: strictly speaking, to make a signal
digital, in addition to discretization in time
(sampling), we need to quantize the samples
(make them discrete in amplitude), which will
be discussed later.

89
Introduction to Matlab
Matlab is an interactive, matrix-based software complex for
scientific and engineering computations.
Interface: the main window

90
Interface: the editor window

91
Interface: figure windows
Whenever you plot results, they will be plotted in one of the figure windows.

92
We may say that there are two ways to use Matlab:
• We can enter the data (manually or load from files or
external sources), manipulate the data (filter in some
manner) and plot (or save) the results through the main
window.
• Or we can write a code (actually, the same sequence of
commands we would enter one by one if we used the main
window) and save it as an *.m file to be able to come back
to it later. This is what the editor window is used for.

93
We will concentrate on the “command (main) window (line)”
approach.
For help in Matlab:
1. Type help commandname where “commandname” stands for a name of a
command.
Ex: help plot
You will get all necessary information about the command: its syntactic,
arguments, etc. Pay attention to similar (or related) commands listed at the end.
2. If you don’t really know the name of a command you need,
type lookfor keyword where “keyword” is a word corresponding to what you are
looking for.
Ex: lookfor matrix
You will get a list of names and brief descriptions of commands containing the
“keyword”.

94
• There is no need to describe (pre-allocate) any variables,
constants etc. before you use them.
• Matlab normally treats all the variables as matrices.
• For our purposes, we will usually work with 1D matrices –
vectors.
• Matlab is case sensitive! A and a are two different variables.
• Matlab remembers your entries: use the up-arrow to call
previous commands.
• Matlab is an expression language: variable = expression. If
variable and = are omitted, a variable ans is automatically
created, to which the result is assigned.

95
To manually enter a matrix from the keyboard, type
A = [1,2,3;4,5,6;7,8,9];
or A = [1 2 3;4 5 6;7 8 9];
Commas (or spaces) divide entries within the same row; semicolons divide rows.
All rows MUST have the same number of elements!
We can call (and modify if needed) matrix elements by their indexes: A(2,3)
would bring 6 on the screen; A(2,3) = 21 will replace an entry 6 by 21.
The semicolon “;” at the end of the expression instructs not to show the result on
the screen.
So, if you need to check what A equals to, just type A and press “Enter”!
1 2 3
4 5 6
7 8 9
A
 
 
  
 
 

96
You can merge together two or more matrices
Ex: A = [1 2 3]; B = [4 5 6 7 8]; C = [A B A];
Will create a vector C = [1 2 3 4 5 6 7 8 1 2 3]
Ex: A = [1 2 3]; A = [A;A.^2];
Will modify a vector A as follows:
1 2 3
1 4 9
A
 
 
 
Btw, we can call particular rows or columns of a matrix (a
vector) or their combinations:
A(2,:) will return [1 4 9] – all entries of the second row
A(:,2) will return - all entries of the second column
2
4
 
 
 
C = A(1:end,2:end)
is 2 3
4 9
C
 
 
 

97
There are numerous functions that generate some common matrices:
zeros(n,m) – n by m array of zeros
ones(n,m) – n by m array of ones
eye(n) – n by n identity matrix
rand(n,m) – n by m array of UNIFORMLY distributed pseudo-random numbers
randn(n,m) – n by m array of NORMALLY distributed pseudo-random numbers
Trick: If you need to create a vector A of linearly increasing (decreasing) numbers,
Type something like: A = first:step:end;
where “first” is a value of the first element, “step” is the increment (decrement),
“end” is the value of the last element.
Ex: n = 1:100;
Ex: a = 21:0.14:531.2;

98
Matrix operations
symbol description
+ Addition
- Subtraction
‘ Transpose
^ Power
* Multiplication
/ Right division
Left division
symbol description
.^ Power
.* Multiplication
./ Right division
. Left division
Array operations
(entry by entry)
Ex:
1 2 5 6 19 22 5 12
; ; .*
3 4 7 8 43 50 21 32
A B A B A B
       
       
       
       
Note: “*” denotes multiplication here only! In this class, “*” implies convolution!

99
All data you enter during the current session is stored in the
memory usually referred to as a workspace.
You may save your workspace by a command save.
You may erase (clear) any variable from the workspace by a
command clear variablename, where “variablename” stands
for a name of the variable you wish to erase. Command
clear all erases all nonpermanent variables!
To see what is in the workspace, use the command whos.

100
FOR, WHILE, IF
FOR variable = expression, statement, ..., statement END
1. FOR – repeat the statement(s) a number of times
Ex: A = []; b = 4.35;
for ind = 1:5
A = [A ind^3];
B = b*A;
end
A = []; b = 4.35;
for ind = 1:5
A = [A ind^3];
end
B = b*A;
Better
style:
A = []; b = 4.35; for ind = 1:5, A = [A ind^3] end; B = b*A;
Equivalent:

101
2. WHILE – repeat the statement(s) while the expression is true
WHILE expression, statement, ..., statement END
Ex: A = []; b = 4.35; ind = 1;
while ind <= 5
A = [A ind^3];
ind = ind + 1;
end
B = b*A;

102
3. IF – execute the statement(s) if the expression is true
IF expression, statement, ..., statement END
Ex: A = randn(1);
B = 0.5;
if A < B
A = A+B;
end

103
Relational operators
Operator Description
< Less than
> Greater than
<= Less or equal
>= Greater or equal
== Equal to
~= Non equal to
Operator Description
& And
| Or
~ Not
Logical operators
We may compare entire matrices (vectors).

104
Example: generate 100 samples of a sinusoid at the frequency 0.04
1. “Traditional approach”:
for n = 1:100
x(n) = sin(0.04*pi*n);
end
2. “Matlab approach”:
x = sin(0.04*pi*[1:100]);
Much faster! Whenever data (or operations) can be vectorized, it speeds up
your code.

105
Some build in Matlab functions…
sin(x) – “x” must be in radians
cos(x)
tan(x)
asin(x)
acos(x)
atan(x)
exp(x)
log(x) – natural log
rem(x) – remainder
abs(x) – absolute value
sqrt(x) – square root
sign(x) – sign of x
round(x) – rounds towards nearest integer
floor(x) – rounds towards minus infinity (down)
ceil(x) – rounds towards plus infinity (up)
abs(x) – where “x” is complex - modulus
angle(x) – phase angle in radians
real(x) – real part of complex “x”
imag(x) – imaginary part of “x”

106
Some build in Matlab vector functions…
max
min
sort
sum
prod
median
mean
std
var
any
all
And some build in constants
pi
eps – machine precision
j, i
size(x) shows the size of “x” (number of
rows and columns)

107
Figures and plotting stuff
figure – creates new figure
plot(x) – a planar plot of vector x on the current axes
polar(x) – a polar plot of vector x on the current axes
hold on; - tells Matlab to keep (hold) the existing plot such that the new plot
will be added
hold off; - “unhold” the existing plot will be replaced by the new plot
xlabel(string) – puts the “string” as a label on x-axis
ylabel(string) – puts the “string” as a label on y-axis
title(string) – puts the string as a figure title
legend(string1,string2,…) – ads a legend to the current axes
grid – manipulates gridlines
subplot – allows placing multiple axes on a single figure
xlim(v) – sets the limits (given in the vector v) for the x-axis
ylim(v) – sets the limits (given in the vector v) for the y-axis

108
x = sin(0.04*pi*[1:100]);
plot(x,’-’,’linewigth’,2)
grid;
ylim([-1.1,1.1])
xlabel(‘Time, sample’)
ylabel(‘Amplitude’)
Example: the following code generates this
figure:
To copy your figure to another application (such as MS Word), go to the Edit
menu in the Figure window and select “Copy Figure”. Now, your figure is in the
computer memory and can be pasted into another application.

109
Matlab summary
Remember:
1. Use help! No point to memorize everything.
2. Think in “matrix/vector” terms.
3. If Matlab stops responding and says it’s Busy,
“Ctrl”+”c” breaks the operation
4. If “Ctrl”+”c” did not help, “Ctrl”+”q” quits Matlab.

110
Considerations and Definitions
We need to make distinctions between a software model (or
our paper/pencil work) and its hardware implementation.
System:
Filter = System
Continuous time Discrete time
We may use: time-, frequency-, and z-domain descriptions of a system:
hn - impulse response;
H() - frequency response (BIBO!);
H(z) – system function.
These descriptions (if they exist) are equivalent.
Additionally may be used:
(xn, yn) - Input/output pairs; {a, b, c,…} - state matrices; difference equations…

111
Continuous vs. discrete time: sampling
from Mitra’s book:
Sampling frequency:
1
s
s
f
T

In general, sampling may lead to information been lost…
(1.111.1)

112
System’s characteristics
0 0 0
n n n n n n n
x y x y x n
 
    
( k n) - no output before input
n n n k
x y y fct x
        
1. Linearity:
2. Time-invariance:
3. Causality:
4. Stability (BIBO):
1 1 2 2 1 2 1 2 1 2
, , ,
n n n n n n n n n n
x y and x y ax bx ay by x x a b
           
for every bounded input,
the output must be bounded
n x n y
x B y B
          
   
Important: use random input signals to test your system!
Important: we don’t really know whether the system is stable: it is stable only
for the particular (bounded) input!
(1.112)
Important: these properties are specified for relaxed systems only!

113
Description of systems…
{ }
n k n k n m m k n k n n
k m k
y x h x h h x h
m n k linear convolutio
x n
  
      
   
  
' : Kroneker delta function
n k n k n
k
Let the system s input x x where is
 

         

If the system is linear:
' : ( ) { }
n k n k
k
k k n k
k
the system s output y x h
and I x
T
   
      

since ,
; ( ). :
n n n k n k n k n k
h h in general For TI h

    
       

Therefore:
hn is a complete I/O (input/output) description of an LTI system
n
0
n-k
n = k
1
(1.113.1)
(1.113.2)

114
Notes on stability (BIBO) of LTI systems
{ : }
n k n k k n k n k x x k
k k k
Let y h x h x bounded input x B B h
  
       
  
for BIBO it is sufficient that: k
k
h absolute summability
  

0 0 0
Maximum ( ) ( )
( ) (
out
)
put:
n x n k n k n k x k
k k
k k x k x
k
k
k
Let x B and y h x max y h B si
h n
gn h
h sign h B sign h B is als ecessary
o

        
 
 
 


 
 
( 0
,
, 0 )
k h
k
n n n n
LTI BIBO
LTI Cau
h B
h h u
s l h n
a
  
 
 
 


 
 


(1.114.1)
(1.114.2)
(1.114.3)
(1.114.4)
(1.114.5)

115
Causality test (for LTI)
0 0
0 0
0 0
1, 1, 1,
2, 2, 2,
n k n k k n k k n k
k k n n k n n
n k n k k n k
k n n k n n
n k n k k n k
k n n k n n
y h x h x h x n
y h x h x
y h x h x
  
   
 
   
 
   
   
  


 


  
 
 
1, 2, 0
0
n n
Since x x n n
    
0
1, 2, 1, 2, 1, 2, 0
0 0
0 ( ) {( ) 0} 0
0 0
n n k n k n k n k n k
k n n
k
y y h x x x x n n
h k n n and n n k
   
 
         
         

, 0 0
n
LTI causal h n
    
(1.115.1)
(1.115.2)

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