Download Introduction To Neural Networks

Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Sequences
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Exponential Sequences
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s is a complex number
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If s is a real number and a=e then the sequence is
called real exponential
Slide 1
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Sequences
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Geometric Sequence: real exponential sequence
defined as:
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Sequences
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Sinusoidal sequence
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Properties of Sequences
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Sum of two signals:
w=x+y
w(k) = x(k) + y(k)
Multiplication of two signals:
w=x.y
w(k) = x(k) y(k)
Multiplication of signal by a scalar:
w=cx
w(k) = c x(k)
Energy of signal:
If a signal is delayed by m time units then x(k)
becomes x(k-m)
Sequence: sum of scaled, delayed unit samples
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Properties of Sequences
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Example:
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Signal Measures
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The signal norm is defined:
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Some properties of the signal norm are:
Norm 1: Sum of the magnitudes of each signal sample
It is used to determine system stability
Slide 6
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Signal Measures
Norm 1: Sum of the magnitudes of each signal sample.
It is used to determine system stability.
Norm 2: Provides a measure of the signal power.
It is the most frequent used measure.
Norm infinity: Gives the peak magnitude of the signal.
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
Discrete–time system: Converting input sequence
x=x(n) into output sequence y=y(n) through
transformation φ[.]
y(n) = φ[x(n)]
Α linear system is defined by the principle of
superposition. If,
Then a system is linear if and only if,
Slide 8
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
Example 1: Is the following system linear?
y(n) = 10x(n) - 5y(n-1)
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____________________________________________
Yes, the system is linear
Slide 9
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
Example 2: Is the following system linear?
_________________
______________________________________
No, the system is not linear
Slide 10
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
A system is time–invariant or shift- invariant if,
y(n) is response to x(n) then
y(n-k) is response to x(n-k)
: a signal delay of k samples
Example 1: Is the following system shift-invariant?
y(n) = 10x(n) - 5y(n-1)
________________________
_________________________
Yes, the system is shift-invariant
Slide 11
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
Example 2: Is the following system shift-invariant?
y(n) = n x(n)
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No, the system is not shift-invariant
We said that we can express:
The system output response is:
Slide 12
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear, Shift-Invariant Systems
If the system is linear,
the response of the system to a sum of inputs is the
same as the sum of the system’s responses to each of
the individual inputs:
By definition:
φ[δ(k)] = h(k)
If the system is shift-invariant: φ[δ(n-k)] = h(n-k)
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If a system is linear and shift-invariant,
the convolution sum applies y(n) = x(n) * h(n)
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear Convolution
The graph method of computing the Convolution sum
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Folding one of the sequences x(n) or h(n) over the
horizontal axis and getting x(-k) or h(-k)
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Shifting the folded sequence creating x(n-k) or
h(n-k)
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The addition of the product of the two sequences at
time n yields the output y(n)
Example: What is the response y(n) if h(n)={1,2,3}
and x(n) = {3,1,2,1}
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Linear Convolution
As a result, y(n)={3,7,13,8,8,3}
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If x(n): N samples, h(n): M samples,
y(n): N+M-1 samples
Slide 15
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Stability and Causality
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A system is stable if a bounded input produces a
bounded output.
Necessary and sufficient condition
This is the norm as defined in a previous session
Example: is the following system stable:
y(n) = x(n) + b y (n-1)
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Calculating the norm we have:
The system is stable
Slide 16
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Stability and Causality
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A causal system is a system that at time m
produces a system output that is depended only on
current and past inputs, that is: n<m
This is always true for a unit impulse response
it is zero for n<0
A discrete-time, linear, shift-invariant system is
causal if and only if h(n)=0 for n<0
Example: is the following system causal:
y(n) = x(n) + b y (n-1)
Since the unit-sample response is zero for n<0,
the system is causal.
Slide 17
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Digital Filters
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A broad class of digital filters are discribed by
linear, constant coefficient, difference equations
{ai} {bi} characterize the system
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Given: initial conditions x(i), y(i) i=-1,-2,…,-M
input sequence: x(n)
output sequence: y(n)
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The system is causal.
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The system is Mth-order.
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Two main classes of digital filters:
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Infinite Impulse Response (IIR)
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Finite Impulse Response (FIR)
Slide 18
Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Digital Filters
 Infinite Impulse Response (IIR): current and past
input samples and past output samples.
Example: Determine impulse response for the firstorder IIR filter. y(n) = x(n) + b y (n-1)
Assume: x(n)=0, y(n)=0 for n<0
x(n)=δ(n)
h(n)=δ(n) + b h(n-1)
h(n)=0
h(0)=1
h(1)=0
h(2)=0
h(3)=0
……..
+
+
+
+
n<0
b0 =1
b1 =b
b b = b2
b b2 = b3
h(n)= bn u(n)
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Technological Educational Institute Of Crete
Department Of Applied Informatics and Multimedia
Neural Networks Laboratory
DISCRETE SIGNALS AND
SYSTEMS
Digital Filters
 Finite Impulse Response (FIR): current and past
input samples.
 The coefficients of the FIR filter are equivalent to
the filter’s impulse response. Why?
Remember convolution?
h(k) = bk
h(k) = 0
k=0,1,2,…,M
otherwise
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