d spy 2003 final
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7/27/2019 d Spy 2003 Final
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Final Examination of Digital Signal ProcessingJan. 2003
PART-I. Conceptual Problems: (40%) (Close Book)
1-1. Explain the following four terminologies: (20%)
(a) Decimation in Frequency FFT (8-point); (b) Chirp transform algorithm;
(c) Goertzel algorithm; (d) FIR Filter Design by Windowing.
1-2. There are eight filters, whose transfer functions are characterized by the following poles and zeros
H1(z) : Poles: 0.5, (1± j)/2 ................ Zeros: 2, ( 2 ± 2 j) , 0, 0H2(z) : Poles: 0.4, 3 .......................... Zeros:0, 0, ±1, ± j
H3(z) : Poles: 0.5, −0.3± j0.6............ Zeros: 0, −0.2, 5
H4(z) : Poles: 0, 0.5 .......................... Zeros: 0.4, +0.4± j0.3, −0.4± j0.3
H5(z) : Poles: None .......................... Zeros: (−1± j)/2, 2/3 ± 2/2 j
H6(z) : Poles: 0, 0 ............................. Zeros: 1± j, 0.5± j0.5
H7(z) : Poles: −0.3, 0.2..................... Zeros: 0, 1, −1, 2
H8(z) : Poles: 0.5± j0.5, 0................. Zeros: 0, 2 ± 2 j
Based on pole-zero characterization, please give your decision reasons first and identify which filters are
(a) Linear phase filter (5%) (b) Minimum phase filter (5%)
(c) All-pass filter (5%) (d) Stable filter (5%)
PART II. DFT Properties and Computation (50%)
2-1. An 8-point sequence is given as x 8[n] = [3 4 -2 7 -4 1 -5 2], its DFT is expressed by X 8[k ]. By 16-point
DFT program, we can compute X 16[k ] = DFT16(x [n]), where we pad 8 extra zeros in the end). Please
compute the following results, (20%)
(a) ∑=
−15
0
16 ][)1(k
k k X ; (b) DFT(DFT(x 8[n])); (c) X 16[8]; (d) X 8[8]
2-2. Cosider the real finite-length sequence x[n] shown in below:
(a) Sketch the finite-length sequence y[n], whose 8-point DFT is ][][ 38 k X W k Y k = . (5%)
(b) Sketch the finite-length sequence w[n], whose 8-point DFT is ]}[Re{][ k X k W = . (5%)(c) Sketch the finite-length sequence q[n], whose 8-point DFT is 3,2,1,0],2[][ == k k X k Q . (5%)
2-3. Followed by Problem 2-1, in terms of X 16[k ] or X 8[k ], please find the 8-point DFTs of the following
sequences. (You don’t need to compute X 16[k ] or X 8[k ], you only express, for example, Y 5[k] = 2 X 8[k] if
y5[n] = 2 x[n].)
(a) Time reverse sequence: y1[n] = x[ N −1−n]; (8-point DFT) (5%)
(b) Alternate sign change sequence: y2[n] = (−1)n x[n]; (8-point DFT) (5%)
(c) Alternate zero insertion sequence: y4[n] =odd
even
,0
],2/[
n
nn x
. (16-point DFT) (5%)
0 1 2 3 4 5
n
2
3
4
5
-
1
- 76
6
7/27/2019 d Spy 2003 Final
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PART III. Filter Properties and Filter Design (60%)
3-1. Consider two systems defined by Figures 3-1(a) and 3-1(b).
(a) If these two systems are equal, please find values of a, b, and c. (5%)
(b) Are these two systems all-pass filters? Please explain it. (5%)
3-2. If H lp( Z ) is a given lowpass filter is transferred by Z −1=1
1
5.01 −
−
−
+−
z
z α
. (a) Determine α such that the Z-plane
to z-plane transform satisfies the transformation constraints. (5%) (b) What is the frequency response
(lowpass, bandpass or highpass) of the new transformed filter? Plot the frequency transferring curve and
explain (10%)
3-3. Consider the discrete-time linear causal system defined by system function H ( z )=421
321
362
5231−−−
−−−
+−+
+−+
z z z
z z z ,
(a) Give difference equations, which characterize direct form II; (5%) (b) Draw its direct form II.(5%)
3-5. (a) Please describe the design procedures of FIR filters by using the windowing technique (5%). (b) What
are selection criteria for choosing the window function, which is the most important? Why? (5%)
3-6. Give an interesting legend for DSP course, which can make students pleasant! (10%)
3-7. Give other suggestions with more than 20 Chinese characters to the DSP course. (5%)
Happy Chinese New Year and Have a Good Winter Vacation!
Useful Formular:
1. z-transform: ∑∞
−∞=
−=n
n z n x z X ][)(
2. Fourier transform: ∑∞
−∞=
ω−ω =
n
jn j en xe X ][)(
3. Convolution: ∑∑∞
−∞=
∞
−∞=
−=−==k k
k xk nhk hk n xnhn xn y ][][][][][*][)(
4. 2-D Convolution: ∑ ∑∞
−∞=
∞
−∞=
−−==k l
l k hl mk n xmnhmn xmn y ],[],[),(*),(),(
5. DFT: ∑∑−
=
π−−
=
==1
0
/21
0
][][][ N
n
N nk j N
n
nk N en xW n xk X
6. Two dimensional DFT: ∑ ∑−
=
−
=
=1
0
1
0
],[],[ N
n
ml M
nk N
M
m
W W mn xl k X
z -1 z -1 0.357
−1
1 1 [n]
z -1 1 1[n]
a b
c
Figure 3-1(a) Figure 3-1(b)
[n][n]
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