南京理工大学数字信号处理（英）考试试卷

南京理工大学课程考试试卷( 学生考试用 )

课程教学 大纲编号： 04027704 课程名称： 数字信号处理（英） 学分： 3.5 试卷编号： 001 考试方式： 闭卷笔试 考试时间： 120分钟 满分分值： 100分 组卷年月： 组卷教师： 审定教师： Notice! 1. All the answers should be written in the answer sheet! 2. The following may be used for your answers: Properties of common window functions: Width of mainlobe Peak approximation error (dB) Rectangular Bartlett 4p/(M+1) -21 8p/M -25 Hanning 8p/M -44 Hamming 8p/M -53 Blackman 12p/M -74 3. Some useful equations:

4. Some useful signs:

---- modular operation;

----- N-point circular convolution

* ------ convolution sum I. Basic concepts: ( 40’ ) 1. 8. Consider the following systems (a) to (d), where y[n] are the output sequences, x[n] are the input sequences. The linear systems include ; the shift-invariant systems include ; the causal systems include ; the stables include . (a) ； (b)；(c)； (d)

For each of the following systems, where y[n] are the output sequence, x[n] are the input sequence. The linear systems are ; the shift-invariant systems are ; the causal systems are ; the stables are . (a) ； (b)；(c)； (d)

2. Sequence

conjugate symmetric sequence antisymmetric sequence

, then the

；and the conjugate 。 3. A signal has a continuous and period frequency spectrum. 4. Let

, then

, , , , 3. The inverse z transform of

. is sequence

4. Let sequence

DFT is X[k], 0￡ k ￡7, then （a）

；（b） ； , its 8-point

（c） ；（d） 。 5. A 2nd-order causal and stable allpass system with real impulse response has one

pole at, then its transfer function is , its ROC is .

6. To compute the N-point DFT of a sequence x[n] (0￡ n ￡ N-1)directly by the DFT analysis equation, complex multiplications and complex additions are needed. But using a radix-2 FFT algorithm, only complex multiplications and complex additions are needed.

II. Computation: ( 30’ )

1. (6’) Compute the 4-ponit DFT X[k] of sequence x[n] = {3, 2, 0, 2 }.

2. (8’) Let x[n] = {1, -2, 0, 2},0 ￡ n￡ 3, h[n]= {2, -1, 1, -2},0 ￡ n￡ 3, Determine: (1) y[n]=x[n] * h[n];

(2) w[n]=x[n] h[n].

3.(8’) A causal LTI system is described by the difference equation

(1) Determine the impulse response h[n] of this system; (2) If the input sequence is x[n] =

, determine the output sequence y[n].

4.(8’) Consider a length- N sequence x[n], 0 ￡ n￡ N -1, with an N-point DFT X[k], 0 ￡ k￡ N -1, where N is an even number..

(1) Let sequence , i.e.,

y[n]= x[<-n> N], 0 ￡ n￡ N -1. Determine the N-point DFT Y[k], 0 ￡ k￡ N -1;

(2) Let sequence

, determine the N-point DFT Z[k], 0 ￡ k￡ N -1;

(3) Let sequence , 0 ￡ n￡ N -1, determine the N-point

DFT W[k], 0 ￡ k￡ N -1;

III. Analysis and design: ( 40’ ) 1.（8）A causal IIR LTI discrete-time system is characterized by the input-output relation:

(1) Determine the system function H(z) ;

(2) Sketch the magnitude response |H(e jw)|and phase responsej (w).

2. (9’) A causal LTI discrete-time system is characterized by system function:

(1) Determine the impulse response h[n] of this system; (2) Prove that the system has a linear phase response;

(3) Plot the linear phase structure flowgraph to realize this system.

3. (10’) (1) A causal analog system function is , Design a

corresponding digital filter by bilinear transform method ( assume T=2), determine the digital system function H(z);

(2) Design a highpass FIR filter with the smallest length using the window-based approach and meeting the specifications:

Select a suitable window function w[n], determine the filter length N and impulse response h[n].

4. (13’) Use DIT-FFT algorithm to compute the N-point DFT X[k], 0 ￡ k￡ N -1, of a length-N sequence x[n], 0 ￡ n￡ N -1,