THE DISCRETE FOURIER TRANSFORM
Solution 9.1
(a)
X(k)
N-1
x(n) WNkn
=
=
RN(k)
n=0
=RN (k)
X (k)
= WNkn0 RN (k)
(c)
X(k)
N-1
an WNkn
=
RN(k)
n=0
1-aN W kN
1 N-ak
.- a
RN (k )
WN
1-a WN)
Solution 9.2
x( (n)) 4
0
T?
x((n-2)) 4
0
x((n-2)) 4 R4 (n)
0
x((-n) )
0
x((-n))
R (n)
0
Figure S9.2-1
S9.1
Solution 9.3
X1(0)
(k)
X
Since
=
-
=
( (N
-
RN(k)
((N))N
X
k) )N RN (k)
-
=
-
X
(0)
(0) = 0
Therefore X
Also, for N even,
--
))N
RN(k)
NN
=e -
(
X
)
=
Therefore X()
0.
Solution 9.4
x2(M)
x2
*1
()N
7
1
9
1
Figure 59.4-1
Solution 9.5
x 1(M)
0
1
2
3
4
5
x 2 ( (-M))
0
6
R 6 (M)
m
Ix 2 ( (2-m)M
0
m
x 1(n)
Figure S9.5-1
S9.2
x 2 (n)
Note that this corresponds to x1 (n) circularly shifted to the right by
two points.
Solution 9.6
We wish to compute X1 (k) given by
9
X1 (k) =a
x(n) zk-n
R1 0 (k)
n=0
j27k ­
where zk = 0.5 e 10 ej0
so that
9
j-n 1
j2Trk
X (k) =
x (n)
e
x(n)
e 10
10 e 10
R10 (k)
n=0
9
-n
jn Tk
e
10
R1 0 (k)
Thus X (k) is the 10-point DFT of the sequence
x (n)
=
x(n)[j e 10
Solution 9.7
In all of the following equations the DFT computed is valid only in the
range O<k<N-1 and is zero outside that range.
This permits us to keep
the equations somewhat cleaner by suppressing the use of the function
RN (k).
N-1
G1 (k) =
x(N -1-n)
WN
n=0
N-1
x(m) WNk(N-1-m)
=
m=0
N-1
=
j27k j2rr
x(m) e N e N
m=0
27rk
= ej N
-j27r k
X(e
)
= H7 (k)
N-1
G2 (k) =
N-1
(-1 )n x(n) WNkn
=
N
x(n) WN
kn
2
WN
n=0
S9.3
N-1
-j2
(k+ N)n
2=
N
e
x(n)
=
X (e
j
N
(k+
2
n=0
H8 (k)
=
N-1
2N-l
Z
G 3 (k)
x (n)
W2Nnk + Zx(n
n= 0
N-1
=
-
N)
W 2nk
n=N
x (n)
W 2 N (n+N) k
w7
2 Nnk +
x (n)
W2 N
n=0
N-1
=
[1 + W2 NNk
j2
n=0
1+(-1)
]X
r
(e
k
kN
=
H 3 (k)
N
--1
2
+ x(n + N ))WNnk
2
k 4 =
x (n)
N
f-
n=0
x (n)
=
n=0
N- 1
WNk
+
G = x (n)
n=0
N) k
wNWN(n- 2
N
n=f
2
N-i
x (n)
nk
= X(e
r
N
H 6 (k)
=
2N-1
G 5 (k)
x (n)
=
W2Nnk =X(e
N
H 2 (k)
=
n= 0
j 2rR
N-i
G 6 (k)
=
a x (n)
n= 0
N_
G 7 (k)
x (2n)
=
n= 0
w2nk
2N
N )
=
WNk
f
N-1
x2E
X(e
X
x()1+ (-1)n
nk/2
n=0
N-1
nk + W n(k+N/2)
2 n= x (n)WWN
NI
S9.4
Hi(k)
)
=
j sk
X (e3 2
j
+ Xe
2x
(k+N/
(k+N/2))]
(k
H5 (k)
All of the above properties can alternatively be obtained from the basic
DFT properties of sections 8.7 and 8.8, or the z-transform properties of
section 4.4. Many of the properties used in this problem have important
practical applications. g5 (n), for example, corresponds to augmenting a
finite length sequence with zeros so that a computation of the DFT for
this augmented sequence provides finer spectral sampling of the Fourier
transform.
S9.5
MIT OpenCourseWare
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Resource: Digital Signal Processing
Prof. Alan V. Oppenheim
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