期末数字信号处理练习题

此题为考试题型提示及部分复习范围，全部考试范围为最后讲义及课堂所讲有关内容，一、填空题(本题共10小题，每小题4分，满分40分)1. The sample rate is fs, the analysis frequencies of m-th bin in N-point DFT is ( )Hz2. Using function of window can reduce the ripple level, but the ( ) is widened3. Frequency sampling filters can be consider as a ( ) and more complex ( ) in cascade.4. The terms FIR filter coefficients and ( ) are synonymous.5. Half-band FIR filter stop pass f f =( ).6. We hope to have more flexibility in trading off between, a window's main lobe width and （sidelobe levels ） in FIR filter designing7. The number of complex multiplications, for an N-point FFT, is approximately ( ).8. The spectrum of real signal is symmetrical about （zero ） Hz9. FIR filter design technique include (Window Design ) Method and (Optimum )Method.10. bandpass sampling is known as ( ) sampling, ( ) sampling, ( ) sampling, and ( ) sampling.11. A sudden change in the values of the coefficient sequence, causes ripples, or (sidelobes), in the frequency response.12. Decreasing the sampling rate is known as （decimation ）,increasing the sampling rate is known as （interpolation ）.13. To upsample xold(n) by a factor of four, we typically insert （three zeros ）between each sample.14. In quadrature processing, by convention, the real part of the spectrum is called the( ) component and the imaginary part of the spectrum is called the ( ) component.15. The window method of FIR filter design include ( ), ( ), etc.16. The magnitude of the N-point DFT results are directly proportional to ( ), and the DFT's frequency resolution is ( ).17. There are three major categories of finite-wordlength errors that plague IIR filter implementations: (coefficient quantization), (overflow errors) and (round off errors.)18. The impulse invariance method are susceptible to aliasing problems because practical analog filters cannot be perfectly (band-limited).19. The fourier transfer of a complex sinusoid t j e t x 0)(Ω=,)(Ωj X =( ).20. The fourier transfer of Time-domain impulse sequence∑∞-∞=-k nT t )(δ,)(Ωj X =( ).21. The major methods of designing IIR filter contains:( ),( ), ( ).22. The bilinear transform method has great utility because,it allows us simply to substitute a function of z for s in ( ) to get ( ), it maps the entire s-plane to the z-plane, enabling us to completely avoid the frequency-domain ( ).23. The DFT of finite sequence x(n) (0≤n ≤N-1) is X(k) (0≤k ≤N-1). ThenX(0)=( ).24. The definition of group delay is( ).二、选择答案1. Multiplying any number on the complex panel by j results in rotating ( ). A ．clockwise B ．anticlockwise2. twiddle factor 68W =( )A ．34WB ．14W3. Real signals includes ( ) spectral components.A ．positive frequencyB ．negative frequency C. positive and negative frequency4. In the following system, ( ) is linear system. (y(n)is the output sequence, x(n)is the input sequence) A ．y(n)=x 3(n) B ．y(n)=x(n)x(n+2) C ．y(n)=x(n)+2 D ．y(n)=x(n 2)5. The real and imaginary components of fourier transfer of real sequence are ( ).A ．even function and odd functionB ．odd function and even functionC ．odd function and odd functionD ．even function and even function6. The wrong description of the IIR filter characters is ( ) A ．The impulse response h(n) is infinite.B ．The structure of IIR filter must be recursiveC ．IIR filter is always stable filterD ．The system function H(z) of IIR filter have poles at finite Z-plane(0<|z|<∞)二、判断题(判断下列各题，正确的在题后括号内打“√”，错的打“×”。

每小题2分，共10分)1. The linear system must be shift-invariant system ( )2. The two sequences which have the same Z transfers are identical(相同的). ( )3. All discrete signal is digital signal. ( )4. All digital signal is discrete signal. ( )5. If discrete function X(k) (k ＝0，l ，2，…) is a periodic function, its frequency spectrum must be a continuous periodic function. ( )6. Increasing the numberof FIR filter tap can reduce the filter ripple. ( )三、简答题1. The sequence x(n) is assumed as⎪⎪⎩⎪⎪⎨⎧=====31211101)(n n n n n xy(n)=x(n)+x(n-1). Calculate the discrete sequence y(n). 2.四、计算与证明题（每小题10分，共50分）1. Plot the mapping the ωj axial in s-plane to z-plane in bilinear transform IIR filter designmethod and write out the mapping range.2. A system H(z) poles are as follow, please plot their discrete time-domain impulse responses.3. Show the relationship of Laplace transform, Fourier and Z transform in formula.4. If the original desired lowpass filter's passband width is fpass, its stopband begins at fstop, and the transition region width is ftrans = fstop–fpass, then the prototype subfilter's normalized frequency parameters are=-passpf=-transpf=-stopirf5. The transfer function of a general shaping FIR filter is∑-=-=1) () (pNkkM pshzkhzH6. Write out the expression of IDFT.7. Show the full decimation-in-time FFT implementation of an 8-point DFT8. The difference equation of a linear shift-invariance system isy(n)=x(n)+2x(n-1)-2x(n-2)-x(n-3)(1)Plot the system structure(2)Is its phase response of the filter linear?9. The difference equation of a linear shift-invariance system isy(n)-y(n-1)-6y(n-2)=x(n-1)please write out H(z) of the system10. The difference equation of a linear shift-invariance system is y(n)=x(n)-x(n-1)-0.5y(n-1) (1)Please derive H(Z)(2)Plot the zeros and poles in z plane and determine the system stability. (3)What kind of is the filter?11. The difference equation of a linear shift-invariance system is12. write out z transform of discrete signal below (1)y(n)=x(n+4)-x(n-4)(2)y(n)=3x(n+2)13. z transform can be derived by Laplace transform. Try to prove ∑∞-∞=-=n snT ssT ssenT x eX )()([])()()()()()()()(s ssT n snT ssts n sst s s st s e X snT x dt e nT t nT x dt e nT t nT x dt e nT x s X ==-=-==∑⎰∑⎰⎰∞-∞=-∞∞--∞-∞=∞∞--∞∞--δδ14. Please write out the Fourier transform of ∑∞-∞=-=n nT t t x )3()(δ,)(f X.对于抽样信号()sk t nT δ∞=-∞-∑的傅里叶变换为12()[()]k ssF j F j n T T πωω∞=-∞=-ΩΩ=∑所以 12()()[()]33k X f X j F j kTTπωω∞=-∞==-∑15. In bilinear transform IIR filter design method the transform relationship of z domain and s domain is:16. ⎩⎨⎧>-≤≤=N n N n n x 0101)(, write out its z transform and magnitude response of frequency.17. The relationship of a system ’ output and input is y(n)=nx(n). Try to determine linearity of system and a time-invariance of system.解：设111()[()]()y n T x n nx n == 222()[()]()y n T x n nx n ==令12()()()x n ax n bx n =+则121212()[()()][()][()]()()y n n ax n bx n a nx n b nx n ay n by n =+=+=+ 满足线性性的要求，所以系统是线性的。