数字信号处理 期末试题A

《数字信号处理》课程试卷（A卷）

课程代码：131300111

本试卷用于信息工程系2011级通信工程/电子科学与技术/电子信息专业本科学生

（时量：120分钟 总分100分）

注意：1、答案必须填写在答题纸上，填写在试卷上无效

2、答卷必须写明题目序号，并按题号顺序答题

3、请保持行距，保持卷面整洁

1. (15 points) The impulse response h n and the input sequence x n of and LTI discrete system are denoted as,

x n R8 n , h n R4 n

(a) Determine the zero-state response y n .

(b) Let H k and X k be the 16-point DFT of h n and x n and

Y1 k H k X k k 0,1,2,3, ,15

y1 n IDTFT Y1 k n 0,1,2,3, ,15

Determine the sequence of y1 n .

(c) Plot the DFT-based implementation of determining y n and determine the point L of the DFT used in the DFT-based implementation.

2. (15 points) Consider a sequence:

x n {1,0,1,0,1,0,1},0 n 6

(a) Determine DTFT X(ej ) of x n .

(b) Determine the 8-point DFT X8(k) of x n using DIT-FFT approach.

3. (10 points) Prove the Parseval’s Theorem: The total energy of a

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length-N sequence g[n] can be computed by summing the square of the absolute values of the DFT samples G[k] divided by N, that is

1N 1

|g[n]| |G[k]|2 Nk 0n 02N 1

4. (20 points) Using bilinear transformation method, design a high-pass IIR digital filter based on Butterworth or Elliptic analog filter. The specifications are,

Pass-band edge frequency: 900Hz

Maximum pass-band attenuation: 0.1dB

Stop-band edge frequency: 600Hz

Minimum stop-band attenuation: 60dB p pSpectral transformation: s , where, sdenotes the Laplace s

transform variable of the prototype analog low-pass filter HLP(s) and s

). the variable of the desired analog filter HD(s

(a) Give the main design steps.

(b) Give the corresponding MATLAB program statements for each step.

5. (20 points)Consider the following three IIR transfer functions of a causal digital filter,

6 5z 1 z 2

i. H1 z 1 21 0.1z 0.56z

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3 7z 1 2z 2

ii. H2 z 1 21 0.1z 0.56z

1 5z 1 6z 2

iii. H3 z 1 21 0.1z 0.56z

(a) Are the transfer functions BIBO stable?

(b) What is the relation between their magnitude function?

(c) What is the relation between their phase function?

(d) Develop a canonic direct-form II realization for H3 z .

6. (20 points)Suppose a FIR digital filter shown in Figure 1.

x(n 1

1 1

y(n)

Figure 1

(a) Determine the transfer function H(z).

(b) Is it a linear-phase FIR filter and why?

(c) Determine the magnitude function and the phase function of the FIR digital filter.

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