Electrical Engineering Department DSP for 4th Year electronic and
communication
Minia University Faculty of Engineering
Digital Signal Processing
Tel: 01284240601 E-mail:
Lecture 6
Dr / Gerges Mansour
3
Convolution is a mathematical way of combining two signals to form a third signal.
What does convolution mean?
INPUT X(N)
OUTPUT Y(N)
X(N) Y(N)
# In image processing
In digital image processing convolutional filtering
plays an important role in many important algorithms in edge detection and related processes.
What are the applications of convolution?
Gaussian blur can be used to obtain a smooth grayscale
digital image of a halftone print
# Convolutional neural networks
apply multiple cascaded convolution kernels with applications
in machine vision and artificial intelligence
Given an input signal x(n)
as shown in fig(1)and system response h(n)
as shown in fig(2)find the result of convolution
fig(1) fig(2)
Ring of input k=1:2 when k=1
Y(n)= x(1)h(n-1) + x(2)h(n-2)
Summation Rules
X(n) =u(n) & h(n)=(3/4)^n
ANS : Y(n) = 4 . (4/3)^n
find the result of convolution
U(k) , k
>=0
find the result of convolution
X(n) =u(n+3) & h(n)=u(n-3)
ANS : Y(n) = n+1
u(n+3) . u(n-k-3)
k
>=-3 , k<=n-3 1
y(n)=1 . (n-3+3+1)
Convolve the following two rectangular sequences using the table method :
X(n)
h(n)
Y(0) =0
Y(1) = 0+1 = 1 Y(2) = 0+1+1 =2 Y(3) = 1+1 = 2 Y(4) = 1
Y(n) = [ Y(0) , Y(1) , Y(2) , Y(3), Y(4)]
Y(n) = [ 0 , 1 , 2 , 2 , 1 ]
X(n) ={1,1,1}
h(n) ={0,1,1}
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• Stability (in the sense of bounded-input bounded-output)[ BIBO]
• A system is stable if and only if every bounded input produces a bounded output.
• A stable system satisfies the BIBO bounded input for bounded output condition .
• Here , bounded means finite in amplitude
b n
y
a n
x Input
] [
Output
] [
• Output should be bounded or finite at every instant of time
• Some examples of bounded inputs are functions
of sine , cosine , signum and unit step .
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• Example 10 :
• Square:
[ ] 2
]
[ n x n
y
]
2[ by
bounded is
Output
] [
by bounded
is Input
a n
y
a n
x
Stable system
• Example 11 :
• Determine which the system is stable or not : y (t) = sine[ x(t)]
• Stable system
•
sine function has a definite boundary of values ,
which lies between -1 to +1
• Unstable system do not satisfy the BIBO condition , therefore , for a bounded input , we can not expect a bounded output .
• Example 3 :
• y (t) = ( X(t) / sine(t) ).
Unstable system
• Sine function has a definite raneg from -1 to +1 ; but
here it is present in the denominator , so for t =0 , sine
function becomes 0 , then the whole system tends to
infinity , therefore the system is unstable
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• Example 12 :
• Log
[ ]
log ]
[ n 10 x n
y
n x
a n
x
log 10
0 y
0 n
for x bounded
not is
output
] [
by bounded
is
input
Questions !
1- What are two uses of digital sequences ? 2- Given the linear system
y(n)=2x(3n),
• determine whether each of the following systems is time invariant.
3-define the impulse response
4- Determine which the system is stable or not : y (t) = sine[ x(t)]
5-
X(n) =u(n) & h(n)=(3/4)^n
find the result of convolution
#What is the use of digital sequences
?
• Calculate the encoded sample
amplitude for a given sample number n;
• Generate the sampled sequence for
simulation.
# Given the linear system
y(n)=2x(3n),
determine whether each of the following systems is time invariant.
• . Let the input and output be x1(n) and y1(n)
then the system output is y1(n)=2x1(3n)
Again, x2(n) and y2(n), where x2(n)=x1(n-n0),
y2(n)=2x2(3n). Since x2(n)=x1(n-n0),
replacing n by 3n leads to x2(3n)=x1(3n-n0)
y2(n) # y1(n-n0).
The system is not time invariant
#define the impulse response
• A linear time-invariant system can be completely described by its unit-impulse response, which is defined as the system response due to the
impulse inputδ(n) with zero-initial conditions
Determine which the system is stable or not :
y (t) = sine[ x(t)]
• Stable system
•
sine function has a definite boundary of values , which lies between
-1 to +1
#
solution X(n) =u(n) & h(n)=(3/4)^n
find the result of convolution
ANS : Y(n) = 4 . (4/3)^n U(k) , k
