LTI ht gt gt ht Example gn un u3n hn n n1 LTI hn gn gn hn Convolution methods Method 1 running sum Plot ID: 248685
Download Presentation The PPT/PDF document "Convolution" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Convolution
LTI:
h(t)
g(t)
g(t)
h(t)
Example: g[n] = u[n] – u[3-n]
h[n] = [n] + [n-1]
LTI: h[n]
g[n]
g[n]
h[n]Slide2
Convolution methods:
Method 1: “running sum”Plot x and h
vs. mFlip h over vertical axis to get h[-m]Shift h[-m]
to obtain h[n-m]Multiply to obtain x[m]h[n-m]Sum on m,
m x[m] h[n-m]Increment n and repeat steps 3~6Slide3
Convolution methods:
Method 2: Superposition methodWe know
[n] h[n] and the system is LTI. Therefore,
a0[n-n0
] a0h[n-n
0]a0
[n-n0] + a
1[n-n1] + … a
0h[n-n0] + a1h[n-n1]
…Since the input can be expressed as a sum of shifted unit samples:x[n] = … x[-2]
[n+2] + x[-1][n+1] + x[0]
[n] + x[1][n-1] + …
The out is then a sum of shifted unit sample responses:x[n0
]
[n-n
0
] + x[n
1
]
[n-n
1
] + … x[n
0
]h[n-n
0
] + x[n
1
]h[n-n
1
] …
1
k=-
1
x[k]h[n-k]
(The Convolution Sum!)