ltipropertiesexample solved
TRANSCRIPT
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ECE 2610 Example Page1
LTI System Properties Example
Determine if the system
is (1) linear (2) time invariant
To check both linearity and time invariance we follow the proof templates in the text/notes
Linearity:
Form
Form with
The system is linear since
Time Invariance
Form (delayed input)
Form
We see that does not equal , so the system is not time invariant
Two system are connected in cascade, that is the output ofS1 is
connected into the input ofS2
Find the impulse response, , of the cascade
y n x n 0.2n cos=
w n w n x1 n 0.2n cos x2 n 0.2n cos +=
y n x n x1 n x2 n +=y n x1 n x2 n + 0.2n cos=
x1 n 0.2n cos x2 n 0.2n cos+=
w n y n =
w n
w n x n n n n0
0.2n cos x n n0 0.2n cos= =
y n n0 n n0 x n 0.2n cos n n n0
x n n0 0.2 n n0 cos= =
w n y n n0
S1: y1 n x1 n x1 n 2 x1 n 3 +=
S2: y2 n x2 n 2x2 n 1 +=
h n
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ECE 2610 Example Page2
Draw the direct form block diagram for the first system S1
To find the impulse response of a two subsystem cascade, we need to convolve the individ-ual impulse responses, i.e., form
By inspection the impulse response ofs1 is
By inspection the impulse response ofs2 is
We can perform the convolution using a table
The direct form block diagram ofs1 is follows from the text/notes
h n h1 n h2 n +=
h1 n n n 2 n 3 + 1 0 1 1 = =
n 0=
h2 n n 2 n 2 + 1 2 = =
n 0=
n = 4
n = 4
n = 3
n = 2
n = 0
n = 1
0 0 010 00000 020 0 00
0
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01 0
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0 0-10 00 010 0 1 0 00 00
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h1[k] h[n]
Outputs forn < 0 and
n > 4 are all 0
sum of products formed between h[k]and x[n-k] inside red box.
Flipped andshifted: h2[n-k]
Expected output range
[0+0, 3+1] = [0,4]
>> filter([1 0 1 -1],1,[1 2 0 0 0 0])
ans = 1 2 1 1 -2 0
n=0 n=4
MATLAB
Check
Unit
Delay
Unit
Delay
Unit
Delay
x[n]
y[n]
1
-1
1x[n- 1]
x[n- 2]
x[n- 3]