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(ELEC211)02fallfinalsol.pdf
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1. We have two discrete-time systems defined as follows:
N . 1 N
System1: y[n] = Mean .x[n + k]
.= . x[n + k]
k =-N 2N +1 k =-N
System2: y[n] = Median .x[n + k] N .. = Median { x[n -N ],L, x[n],L, x[n + N ] }.
k =-N
Here, we defined that Median {x1,L, x2N +1}= x(N 1) , where x() i s are the rank-ordered version of xi s and
+
x 1 x 2 L x .
() () (2N +1)
(a) Whether is each of these two systems linear ? Justify your answers. (6 pts) System 1 : Linear
N . 1 N y1[n] = Mean .x1[n + k]
.= . x1[n + k]k =-N . 2N + 1 k =-N
N . 1 N y [n] = Mean x [n + k]
.= . x2[n + k]
2 . 2
k =-N . 2N + 1 k =-N
N . 1 N y [n] = Mean .( ax [n + k] + bx [n + k]) .= . ( ax1[n + k] + bx2[n + k])
3 12
k =-N (2N + 1) k =-N
1 . N 1 . N
= ax [n + k] + bx [n + k]
2N + 1 k =-N 12N + 1 k =-N 2
. 1 N .. 1 N .
= a .. . x1[n + k].. + b .. . x2[n + k]..
. 2N + 1 k =-N .. 2N + 1 k =-N .
= ay1[n] + by2[n]
System 2 : Non-linear
For example : Assume N = 2
y1[3] = Median {x1[n]} = 3
y2[3] = Median {x2[n]} = 3
1 2 3 4 5
If x3[n] = x1[n] + x2[n] = [ 7, 9, 7, 6,3 ]
y3[3] = Median {x3[n]} = 7
. y1[n] + y2[n] = 6
n
1 2 3 4 5 (b) Whether is each of these two systems time-invariant ? Justify your answers. (6 pts) System 1 : Time C invariant
N
N
1
[n]
Mean
[
n k]
[
k]
+
+
y
=
x
1
=
x
1
n
1
kN
2N
1
+
=-
k =-N
N
N
1
[n]
Mean
[
n k]
[
k]
+
+
y
=
x
=
x n
2 2 2
kN
2N
1
+
=-
k =-N
If x2[n]
=
x
1
[
1]
n
-
N
N
1
y
2
[n]
Mean
[
n
1
+
k]
[
n
1
+
k]
=
x
1
-
=
x
1
-
kN
2N
+
1
=-
k =-N
y
1
[
n
-
1]
=
System 2 : Time - invariant
N
[n]
Median
[
n k]
+
y
=
x
11
k =-N N
[n]
Median
[
n k]
+
y
=
x
2 2
k =-N
If x2[n]
=
x
1
[
n
1]
N
y
2
[n]
Median
[
n
1
+
k]
=
x
1
-
k =-N
y
1
[
n
-
1]
=
2. (12 pts) Consider the signal depicted in Figure P-2. Let the Fourier transform of this signal be written in rectangular form as X (ej W ) = A(W) + jB(W).
j W j W
Sketch the function of time corresponding to the transform Y(e ) = [B(W) + A(W) e ].
x[n]
Solution :
j W j W j W j W j W
Y(e ) = [B(W) + A(W) e ] =-j [ jJm{X (e )} ] + e e{X (e )}
j W j W. 1 . j W-j W
=-j [ jJm{X (e )} ] + e .. [ X (e ) + X (e )]. 2 .
11
y[n] =-j Od {x[n]} + x [n +1] + x [-n -1]
22
n
11
x [n + 1] + x [-n -1]
22
3. Let x(t) be a real signal whose spectrum X( jw) is drawn in Fig. P-3 where W = w -w < w . According
10 0
to the Nyquist sampling theorem, we need to set the sampling rate ws. 2w1 to avoid aliasing (as w1 is the highest frequency in x(t) ). However, we practically can use a lower sampling rate to completely avoid aliasing also.
(a) Determine the lowest sampling rate that can be used to do the sampling on x(t) so as to completely
wmin
avoid aliasing. (4 pts