Digital Signal Processing Reference
In-Depth Information
1
1
0.5
0
0.5
−0.5
−1
0
−1
−0.5
0
0.5
1
0
10
20
30
40
(b) Real
(a) n
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
10
20
30
40
0
10
20
30
40
(c) n
(d) n
Figure 2.12:
(a) Magnitude of
W
= (0.99*exp(j
π
/18))
n
where n = 0:1:40; (b) Plot of entire power sequence
in complex plane; (c) Real part of entire power sequence of
W
for powers 0 to 40; (d) Imaginary part of
entire power sequence of
W
for powers 0 to 40.
Let's compute powers 0:1:3 for the complex number [0 + j] and describe or characterize the
resultant real and imaginary parts. The power sequence is [j
0
,j
1
,j
2
,j
3
], which reduces to [1, j, -1, -j],
with the real parts being [1,0,-1,0] and the imaginary parts being [0,1,0,-1]. These may be described as
four-sample, single-cycle cosine and sine waves. Another way to write this would be
y = cos(2*pi*(0:1:3)/4) + j*sin(2*pi*(0:1:3)/4)
which returns the following:
y = [1,(0 +1i),(-1 + 0i),(-0 - 1i)]
Let's compute the complex power sequence
W
n
where
n
= 0:1:4 and
W
=
(
√
2
/
2
)(
1
+
j)
. Note
45. Then
W
0
:
1
:
4
= [1, 1
initially that
W
=1
45, 1
90, 1
135, 1
180], which reduces to
[1, 0.707(1+j), j, 0.707(-1+j), -1]
To compute the expression using m-code, make the call
n = 0:1:4; W = (sqrt(2)/2)*(1+j); y = W.ˆn
