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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

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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]