数字信号处理期末考试A卷答案

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电子科技大学二零零八至二零零九学年第一学期期末考试

数字信号处理课程考试题半期考卷（120 分钟）考试形式：开卷考试日期2008年12月30 日

课程成绩构成：平时20 分，期中20 分，实验20 分，期末40 分

一二三四五六七八九十合计

1.Given a causal system with the transfer function being1

12

3

()

252

z

H z

z z

-

--

-

=

-+

.

(1)Try to calculate it’s differen ce equation.

(2)Try to give the zero-pole plot of the system, then determine whether the system is stable.

(3)Calculate it’s impulse response.

(4)Give a canonic direct realization of the system and plot it.

Solution:（共20分）

(1)Difference equation is: 2y[n]-5y[n-1]+2y[n-2]=-3x[n-1]（3分）

(2)zero-pole plot:（5分）

The ROC is |z|>2, so the system is not stable.（2分）

(3)111

12211

33120.52

()

25225221210.512

z z z z

H z

z z z z z z z z

---

----

--

===-=-

-+-+----

11

[]0.5(0.5)[]2(2)[]{(0.5)(2)}[]

n n n n

h n n n n

μμμ

--

∴=-=-（5分）

(4)（5分）

z-1

z-1

2.5

x[n]

-1

-1.5

y[n]

x[n]

-1.5

y[n]

2.5

z-1

z-1

-1

2.Consider a length-9 sequence x [n ] = {2, -3, -1, 0, -4, 3, 1, 2, 4}, -2 ≤ n ≤ 6. The z -transform X (z ) of x [n ] is sampled at N points ωk = 2πk /N, 0 ≤ k ≤ N -1, on the unit circle yielding the frequency samples:

2/[]()|,0,1,,1j k N z e X k X z k N π===-%L

Determine the periodic sequence []x

n % whose discrete discrete Fourier series coefficients are given by []X k %when N = 4 and N = 11, respectively ( without evaluating []X k %).

Solution:（共15分）

According to the definition of z-transform,

6

2

()[]n

n X z x n z

-=-=

∑ (0.1)（5分）

Then, []X

k % can be written as

26

2

[][],0,1,,1kn

j

N

n X k x n e

k N π-=-=

=-∑%L (0.2)（5分）

According to the definition of DFS

21

0[][],,0,1,,1kn

N j

N

n X k x n e

n k N π--===-∑%%L (0.3)（5分）

when N = 4, [][4]x n x n r =+%%, and its principle period is {0, 2, 2, 0}, 0 ≤ n ≤ 3

when N = 11, [][11]x n x n r =+%%, and its principle period is {-1, 0, -4, 3, 1, 2, 4, 0, 0, 2, -3}, 0 ≤ n ≤ 10

3．Knowing the transfer function of a causal system as 1

11()10.9z H z z --+=-, we get a new transfer

function 41()()H z H z =. Please:

(1) Sketch the magnitude response of 1()j H e ω

.

(2) Determine the ω values of peaks and dips in magnitude of 1()j H e ω

.

(3) If the transfer function H(z) is cascaded with a system 11

()(1)(10.9)G z z z --=+- to get a new system

transfer function H 2(z), try to compute the phase function and the group delay of 2()H z .

Solution:（共20分）

(1) 4141()10.9z H z z

--+=-→4141()10.9j j j e H e e ωω

ω--+=-→4141()10.9j j j e H e e ωωω--+=- magnitude response:

（共9分）

(2) ω values of peaks in magnitude of 1()j H e ω:-π/2,0,π/2, π, （2分）

ω values of dips in magnitude of 1()j H e ω:,－3π/4, －π/4，π/4, 3π/4（2分）

（3）1212

2()()()(1)12H z H z G z z z z ---==+=++g

22()122(cos 1)

j j j j H e e e e ωωωωω---∴=++=+

The phase function ()θωω=-，The group delay ()

1g d d θωτω

=-=（共7分）

https://www.360docs.net/doc/ca15639097.html,ing windowed Fourier series method, design a causal low-pass FIR digital filter with the following

specifications:

Pass band edges f p : 6kHz Stop band edges f s : 8kHz Pass band ripple αp : 1dB Stop band attenuation αs : 40dB

Sampling frequency FT: 20kHz

Solution:（共15分）

(1) Perform some specifications transform.

22347,,,555210

p p s s p s s p c T

T f f F F πωωππππ

πωωωωωω+=

=

==?=-===

（每个1分，共4分）

(2) The ideal impulse response is

sin [],c LP n

h n n n

ωπ=

-∞<<∞（1分） (3) According the requirement of stop band attenuation, the Hann （or Hamming,Blackman ） window should be chose. The transition bandwidth of hann window is 3.11π/M, i.e.,

3.1115.555

M M ππ

≤?≥（2分） So, M = 16 and the Hann window function is

2[]0.50.5cos ,21n w n M n M M π??

=+-≤≤ ?+??

（2分）

(4) Windowing: h t [n ] = h LP [n ]w [n ], -16 ≤ n ≤ 16（2分） Casual system: h [n ] = h t [n -16], 0 ≤ n ≤ 33（2分）

(5) Test ：Compute H(e jw )=DTFT(h[n]) to verify.（1分）

5. The normalized transfer function of an order-2 low-pass Butterworth analog filter is as follows:

1

21)(2++=

s s s H an .

With bilinear transform, an order-2 low-pass IIR Butterworth digital filter with -3dB cutoff frequency at 1000Hz and sampling frequency at 4000Hz has been designed. The structure of the filter is sketched in figure 1. Determine the value of constant K and M and expression of prototype analog filter H a (s).

Solution:（共20分）