(lecture Outline #2)
A (very) brief review of continuous
time signals and LTI sytems
time signals and LTI sytems
Examples of discrete time signals
Discrete time systems
time systems
Building blocks
Examples
How do we characterize these systems?
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Copyright 2005, S. K. Mitra/ 2006-2010 R.Leahy
Continuous time review
Continuous time signals
A real or complex function x(t) defined on the
real or complex function x(t)
on the
real line
Continuous time systems
A mapping from an input signal x(t) to an
output signal y(t)
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Continuous time systems
Linear time invariant system
Superposition: if y1 (t ) = H { x1 (t )} , y2 (t ) = H { x2 (t )}
and y (t ) = H { x1 (t ) + x2 (t )} then
th
y (t ) = y1 (t ) + y2 (t )
Time invariance
If y (t ) = H { x (t )} then
H { x(t t0 )} = y (t t0 ) t0
Characterized by impulse response
y (t ) =
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x(t )h( )d = x( )h(t )d
Example and the Dirac Delta
Dirac Delta and the sifting property
y (t ) =
x(t ) ( )d =
(t ) x( )d = x(t )
Simple Echo
y (t ) = x(t ) + ax(t t0 ) =
x(t ) { ( ) + a ( t )} d
0
y (t ) = x(t )* ( (t ) + a (t t0 ) )
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Eigenfunctions of LTI systems
What are the eigenfunctions of an LTI
system? i.e. signals for which output is a
system?
signals for which output is
scaled version of the input
x(t)
H(t)
y(t)=x(t)
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Eigenfunctions and the Fourier
Transform
Let x(t ) = exp { jt}
real.
Then: y (t ) = h( ) x(t )d
Then:
= h( ) exp{ j(t )}d
= exp{ jt} h( ) exp{ j }d
= x(t ) H ( j}
So x(t ) = exp { jt} is an Eigenfunction of an
LTI system. What are the eigenvalues?
What about x(t ) = exp { t} for complex?
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Fourier Transform
Definitions
X ( j ) =
x(t )e
j t
R
dt
1
x(t ) =
2
X ( j )e
j t
d
t R
Existence:
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Dirichlet conditions absolutely integrable,
finite number of discontinuties
More relaxed forms including generalized
functions
Examples
Dirac Delta (t )
Fourier transform (by sifting property):
D ( j ) =
(t )e
jt
dt = 1
Delayed delta function:
Dt0 ( j) =
(t t )e
0
jt
dt = e jt0
Note: you cannot find inverse of these
transforms through standard integrals as the
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examples of generalized functions.
Examples
Complex Sinusoid
x(t ) = exp { j0t} X ( j) = 2 ( 0 )
inverse Fourier transform:
1
x(t ) =
2
X ( j )e
j t
1
d =
2
( )e
0
jt
d = e j0 t
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Examples
Rect Function:
.5
X ( j ) =
e
.5
1
x(t ) =
0
j t
t 0.5
t > 0.5
e jt
dt =
j
.5
.5
e j /2 e j /2 sin( / 2)
=
=
j
/2
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sinc( / 2 )
Characterization of Continuous time
LTI systems
Impulse response: h(t)
Linear differential equations
Characterizes realizable LTI systems through LDE involving
inputs and outputs
Frequency response H(j)
- Fourier transform of impulse response
System function: H(S)
Laplace transform of impulse response
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Analog vs. Discrete Time Systems
Impulse response: h(t)
Impulse response: h(n)
differential equations
Linear differential equations
Characterizes realizable LTI
systems through LDE
involving inputs and outputs
Linear difference equations
Linear difference equations
Characterizes realizable LTI
systems through LDE
involving inputs and outputs
Frequency response H(jW)
- Fourier transform of
impulse response
Frequency response H (e j )
Discrete Time Fourier
transform of impulse
of impulse
response
System function: H(S)
Laplace transform of impulse
response
System function: H(Z)
Z-transform of impulse
response
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Signals represented as sequences of
numbers called samples
numbers, called samples
Sample value of a typical signal or sequence
denoted as x[n] with n being an integer in
the range n
x[n] defined only for integer values of n and
undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Discrete-time signal may also be written as
a sequence of numbers inside braces:
sequence of numbers inside braces:
{x[n]} = {, 0.2, 2.2,1.1, 0.2, 3.7, 2.9,}
In the above, x[1] = 0.2, x[0] = 2.2, x[1] = 1.1,
etc.
The arrow is placed under the sample at
time index n = 0
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Graphical representation of a discrete-time
signal with real
signal with real-valued samples is as shown
samples is as shown
below:
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
In some applications, a discrete-time
sequence {x[n]} may be generated by
periodically sampling a continuous-time
signal xa (t ) at uniform intervals of time
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Here, n-th sample is given by
x[n] = xa (t ) t = nT = xa (nT ), n = , 2, 1,0,1,
The spacing T between two consecutive
samples is called the sampling interval or
sampling period
Reciprocal of sampling interval T, denoted
as FT , is called the sampling frequency:
1
FT =
T
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Unit of sampling frequency is cycles per
second, or hertz (Hz), if T is in seconds
if
Whether or not the sequence {x[n]} has
been obtained by sampling, the quantity
x[n] is called the n-th sample of the
sequence
sequence
{x[n]} is a real sequence, if the n-th sample
x[n] is real for all values of n
Otherwise, {x[n]} is a complex sequence
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
A complex sequence {x[n]} can be written
as
as {x[n]} = {xre [n]} + j{xim [n]} where
where
xre [n] and xim [n] are the real and imaginary
parts of x[n]
The complex conjugate sequence of {x[n]}
is given by {x * [n]} = {xre [n]} j{xim [n]}
Often the braces are ignored to denote a
sequence if there is no ambiguity
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DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Example - {x[n]} = {cos 0.25n} is a real
sequence
sequence
{y[n]} = {e j 0.3n} is a complex sequence
can We write
{y[n]} = {cos 0.3n + j sin 0.3n}
where
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= {cos 0.3n} + j{sin 0.3n}
{yre [n]} = {cos 0.3n}
{yim [n]} = {sin 0.3n}
DiscreteDiscrete-Time Signals:
TimeTime-Domain Representation
Example {w[n]} = {cos 0.3n} j{sin 0.3n} = {e j 0.3n}
is the complex conjugate sequence of {y[n]}
That is, {w[n]} = {y * [n]}
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Basic Sequences
1, n = 0
Unit sample sequence - [n] =
0, n 0
1
n
4
3
2
1
0
1
2
3
4
5
6
1, n 0
0, n < 0
Unit step sequence -
[ n] =
1
22
4
3
2
1
0
1
2
3
4
5
6
n
Basic
Basic Sequences
Real sinusoidal sequence x[n] = A cos(o n + )
where A is the amplitude, o is the angular
frequency, and is the phase of x[n]
Example = 0.1
o
2
Amplitude
1
0
-1
-2
0
10
20
Time index n
30
40
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Basic Sequences
Exponential sequence x[n] = A n , < n <
where A and are real or complex numbers
j
If we write = e( o + jo ) , A = A e ,
then we can express
x[n] = A e je(o + jo ) n = xre [n] + j xim [ n],
where
xre [n] = A eon cos(o n + ),
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xim [ n] = A eon sin(o n + )
Basic
Basic Sequences
xre [ n] and xim [n] of a complex exponential
sequence are real sinusoidal sequences with
constant (o = 0 ), growing (o > 0 ), and
decaying (o < 0 ) amplitudes for n > 0
Imaginary part
Real part
0.5
0.5
Amplitude
1
Amplitude
1
0
-0.5
0
-0.5
-1
0
10
20
Time index n
30
-1
0
40
10
20
Time index n
1
x[n] = exp( 12 + j )n
6
25
30
40
Basic Sequences
Real exponential sequence x[n] = A n , < n <
where A and are real numbers
= 0.9
= 1.2
20
50
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Amplitude
Amplitude
40
30
20
5
10
0
0
26
10
5
10
15
20
Time index n
25
30
0
0
5
10
15
20
Time index n
25
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Basic
Basic Sequences: Relations
An arbitrary sequence can be represented in the
time-domain as a weighted sum of some basic
sequence and its delayed
sequence and its delayed (advanced) versions
versions
x(n)
x[n] = 0.5 [n + 2] + 1.5 [n 1] [n 2]
+ [n 4] + 0.75 [n 6]
x[n] = x(2) [n + 2] + x(1) [n 1] + x(2) [n 2]
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Basic Sequences: Relations
General relationship between sequences
and unit impulse
x[n] =
x(m) [n m]
m =
Unit step function and Unit Impulse
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m =
u ( n) =
m =0
u (m) (n m) = (n m)
Operations
Operations on Sequences
A single-input, single-output discrete-time
system operates on sequence called the
system operates on a sequence, called the
input sequence, according some prescribed
rules and develops another sequence, called
the output sequence, with more desirable
properties
x[n]
Discrete-time
system
y[n]
Output sequence
Input sequence
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Basic Operations
Product (modulation) operation:
x[n]
Modulator
y[n]
y[n] = x[n] w[n]
w[n]
An application is in forming a finite-length
sequence from an infinite
sequence from an infinite-length sequence
sequence
by multiplying the latter with a finite-length
sequence called a window sequence
Process called windowing
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Basic
Basic Operations
Addition operation:
Adder
x[n]
+
y[n]
y[n] = x[n] + w[n]
w[n]
Multiplication operation
Multiplier
A
y[n] = A x[n]
y[n]
x[n]
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Basic Operations
Time-shifting operation: y[n] = x[n N ]
where N is an integer
If N > 0, it is delaying operation
Unit delay
x[n]
z 1
y[n]
y[n] = x[n 1]
If N < 0, it is an advance operation
x[n]
Unit advance
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z
y[n]
y[n] = x[n + 1]
Basic
Basic Operations
Time-reversal (folding) operation:
y[n] = x[n]
Branching operation: Used to provide
multiple copies of a sequence
x[n]
x[n]
x[n]
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Basic Operations
Example - Consider the two following
sequences of length defined for
sequences of length 5 defined for 0 n 4:
{a[n]} = {3 4 6 9 0}
{b[n]} = {2 1 4 5 3}
New sequences generated from the above
two sequences by applying the basic
two sequences by applying the basic
operations are as follows:
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Basic
Basic Operations
{c[n]} = {a[n] b[n]} = {6 4 24 45 0}
{d [n]} = {a[n] + b[n]} = {5 3 10 4 3}
{e[n]} = 3 {a[n]} = {4.5 6 9 13.5 0}
2
As pointed out by the above example,
operations on two or more sequences can be
carried out if all sequences involved are of
carried out if all sequences involved are of
same length and defined for the same range
of the time index n
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Combinations of Basic
Operations
Example -
y[n] = 1x[n] + 2 x[n 1] + 3 x[n 2] + 4 x[n 3]
(Youll see this many times in this course its an FIR filter)
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A Useful Example
Useful
M-point moving-average system M 1
1
y[ n] = M x[ n k ]
k =0
1 = 2 = ... = M
M=3
= 1/ M
An application: Consider
application: Consider
x[n] = s[n] + d[n],
where s[n] is the signal corrupted by a noise
d[n]
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Simple moving average filter
s[n] = 2[ n(0.9) n ], d[n] - random signal
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d[n]
s[n]
x[n]
6
Amplitude
x(n)=s(n)+d(n)
4
2
0
-2
0
10
20
30
Time index n
40
50
7
s[n]
y[n]
6
5
Amplitude
y(n) =
filtered/smoothed
version of x(n)
4
3
2
1
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0
0
10
20
30
Time index n
40
50
Another
Another Example
n
Accumulator - y[n] = x[ ]
n 1
=
= x[ ] + x[n] = y[n 1] + x[n]
=
The output y[n] at time instant n is the sum
of the input sample x[n] at time instant n
and the previous output y[n 1] at time
instant
instant n 1, which is the sum of all
which is the sum of all
previous input sample values from to n 1
The system cumulatively adds, i.e., it
accumulates all input sample values
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Accumulator as recursive filter
We can implement the accumulator as
x(n)
+
1
y(n)=x(n)+y(n-1)
y(n)
Z-1
This is a discrete-time approximation to an integrator. Its
also an example of a recursive (feedback) system
Can modify using a lossy integrator
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y(n)=x(n)+y(n-1)
So.
Weve seen a couple of practically useful
systems but
does it make sense to use them?
Are they stable (and what do we mean by
stable)
What are their properties and how do we
define these properties?
Can we find better ways of doing these
operations (smoothing and accumulating) .. and
others
Well answer these in the next few weeks
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