中南大学课件.ppt

中南大学课件.pptDiscrete LTI systems: the convolution sum
Consider signal x[n]:
An arbitrary sequence can be representedas a bination of shifted unit implulses [n-k], where
the weights in this bination are x[k].
Write as :
…
…
-4
-3
-2
-1
0
1
2
3
4
n
x[n]
Discrete LTI systems: the convolution sum
( sifting property —筛选性)
Discrete LTI systems: the convolution sum
Let h[n] denote the response of linear system to [n].
. h[n] — the unit impulse response.
then, each [n] of x[n]response:
….
….
Discrete LTI systems: the convolution sum
….
….
This result is referred to as the convolution sum, and the operation on the right-hand side of EQ. is known as the convolution sum of x[n] and h[n].
Discrete LTI systems: the convolution sum
We represent the operation as
y[n] =x[n]  h[n] ()
The same,
is referred to as the convolution sum
of x[n] and [n].
Some notes:
An LTI system pletely characterized by its h[n].
Discrete LTI systems: the convolution sum
The graph of convolution sum.
Transform independent variable:
x[n], h[n]x[k], h[k]
and h[k]  h[-k]
Shift h[-k] n steps h[n-k].
For any n, x[k] multiplied by h[n-k]
and x[k]·h[n-k] .
Discrete LTI systems: the convolution sum
Example determine y[n]=x[n]h[n].
Answer:
(a)
Discrete LTI systems: the convolution sum
and h[k]h[-k]
(b) Shift h[-k] to the
right(n>0) or to the
left (n<0).
n<0, y[n]=0
n=0, y[0]=x[0]h[0] =1=
n=1, y[1]=x[0]h[1]+x[1]h[0]
=+2=
Discrete LTI systems: the convolution sum
(c) For any particular
value of n, we multiply
these two signals and
sum over all values of k .
n=2, y[2]=x[0]h[2]+x[1]h[1]
=+2=
n=3, y[3]=x[1]h[2]=2
n4, y[n]=0
Discrete LTI systems: the convolution sum
or y[n]={, , , 2} n=0, 1, 2, 3,
The main steps of convolution sum graph:
reversal shift multiply sum
(3) Calculation of convolution sum can be done by upright multiplication.