英文版数字信号处理去年考试题.doc

1.Consider an LTI system with input x[n] and output y[n] that satisfies the difference equation

a. Find the system function H(z). How many ROCs are associated with H(z)? For each case ,determine what type of the corresponding impulse response h[n]?
b. If this system is causal, then is it stable ? Justify your answer .And whether exists or not ? If it exists , determine .

2.Consider the finite-length sequence x[n]={1,0,2,1},0≤n≤3 , with an 4-point DFT given by x[k] .
a. If the 4-point DFT y[k] of length-4 sequence y[n] is given by y(k)=, determine y[n].
b.If the 4-point DFT w[k] of length-4 sequence w[n] is denoted by w(k)=x[k],determine w[n].
c. If N-point DFTs are used in the two-step procedure , how should we choose N so that w[n]=x[n]*x[n] for 0≦n≦N-1 ? Determine w[n] in the case also .
Note: using the DFT properties without computing x[k].

3.Given x[n]={0,1,2,3,4,5,6} be a length-7 sequence defined for 0≦n≦6 , with X denoting its DTFT
a) Evaluate the following function without computing the transform itself :
X() ; ;
b) Define Y[k] =X(e), 0≦n≦4, with y[n] denoting its 5-point IDFT.Determine y[n] without computing Y[k] and its IDFT .Can you recover X[n] from y[n].

4. An IIR filter is described by the following system function :
H(z)=
Determine and draw the following structures:
(a) Direct from II
(b) Cascade form.

5.Verify the identity equation

6.A continuous-time signal X=cos(2x300t)+ (2x500t)+ (2x1200t) is sampled at a 2kHz rate , and the sampled sequence is passed through an ideal lowpass filter with a cutoff frequency of 900Hz , generating a continuous-time signal y .
a. Determine the discrete-time signal x[n] generated by periodically sampling X at F=2khz
b.What are the frequency componts present in the reconstructed signal y.
c.If y is equal to the original continuous signal ,determine the sampling frequency in this case
d.If we want to pass the frequency components at 300Hz ,what type filter we should choose ? If the transition-width is assumed to as 100Hz, the minimum stopand attenuation , which window functions can we choose from Table1 ? Determine the length of the filter for the window you selectd.
Table1
Window Name
Transition Width
Min. stopband attenuation
Rectangular

20.9dB
Hanning

43.9dB
Hamming

54.5dB
Blackman

75.3dB
一、(20 points) 1. The input-output pair of a stable LTI system is shown in Fig.1(a).
(a) Determine the response to the input x1[n] in Fig.1(b).
(b) Determine the impulse response and frequency response of the system.
(c) Sketch the magnitude-frequency response of the system.

(a)

(b)
Fig.1
二、(20 points)Let h[n] be a Type-4 real-coefficient linear-phase FIR filter .
(1)If this filter has the following zeros:, please determine the locations of the remaining zeros.
(2)Please determine the FIR transfer function and realize it in cascade form and direct form I.
三、(20 points) The difference equation of a LTI discrete-time system is :
where x[n] and y[n] are the input and output sequences respectively.
(a) Please give the transform function H(z) as well as its poles and zeros;
(b) If the system is casual and stable, please give the ROC of the z-transform;
(c) Give the impulse response h[n] of this casual stable system.

四、(20 points) Design a DIGITAL low pass filter to meet the following requirements:

Select a window to design a linear-phase FIR filter, using the lowest order filter to meet the specifications.
五、(20 points) Fig.2(a) shows a 6-point discrete time sequence x[n]. Assume that x[n] = 0 outside the interval shown. The value of x[4] is not known and is represented as b. Let X(ejω) be the DTFT of x[n], and X1[k] be samples of X(ejω) with sampling interval π/2, i.e.,

The 4-point sequence x1[n] is the 4-point Inverse DFT of X1[k], and is shown in Fig.2(b).

(a) (b)
Fig.2
(a) Please determine the value of b.
(b) Let X2[k] be samples of X(ejω) with sampling interval π/3, i.e.,

Please determine and sketch the finite-length sequence y[n] whose 6-point DFT is Y[k] = W64kX2[k].
Determine and sketch the finite-length seque
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