数字信号处理试卷及参考答案_数字信号处理试卷答案

数字信号处理试卷及参考答案由刀豆文库小编整理，希望给你工作、学习、生活带来方便，猜你可能喜欢“数字信号处理试卷答案”。

《数字信号处理》

参考答案

1.Determine the period of the sinusoidal sequence x(n)5sin(0.04n).Determine another distinct sinusoidal sequence having the same period as x(n).Solution: 4N

0.0

2rNselect50;r

1502r;r310.12

2.Without computing the DTFT, Determine which of the following sequences have real-valued DTFT and have imaginary-valued DTFT:

n,NnN

(a)

x1(n)

0,otherwise

2n,NnN

(b)

x2(n)

0,otherwiseSolution:(a): odd symmetry----imaginary-valued DTFT(b):even symmetry----real-valued DTFT

3.A continuous-time signal

xa(t)is composed of a linear combination of sinusoidal signals of frequencies 300 Hz, 500 Hz, 1.2kHz and 2.15k Hz.The signal xa(t)is sampled at an 2k Hz rate, and the sampled sequence is then paed through an ideal lowpa filter with a cutoff frequency of 900 Hz, generating a continuous-time signal

ya(t).What are the frequency

ya(t)? components present in the reconstructed signal Solution: kFTFi

300 Hz----300, 1700, 2300Hz...500 Hz----500, 1500, 2500Hz...1.2k Hz----1200, 800, 3200Hz...2.15kHz----2150, 150, 4150Hz...So

4.Consider a length-N sequence ya(t)are 150 Hz, 300 Hz, 500 Hz, 800 Hz.x(n),0nN1, which N even.Define 2 subsequences of length-N/2 each: x0(n)x(2n)and x1(n)x(2n1), 0nN2.Let X(k),0kN1, denote the N-point DFT of x(n), and X0(k)and X1(k), 0kN21, denote the N/2-point DFTs of x0(n)and x1(n), respectively.Expre X(k)as a function of X0(k)

and X1(k).nkX(k)x(n)WNn0N12r0N12r0N1Solution:

2rk(2r1)kx(2r)WNx(2r1)WNN12r0N12r0rkkrkx(2r)WNWNx(2r1)WN22N12r0N12r0

rkkrkx0(r)WNWNx(r)W1N22X0kWkXkNN12,N20kN1

5.Using z-transform methods, determine the explicit expreion for the impulse response h(n)of a causal LTI discrete-time system that develops an output y(n)2(0.3)nu(n)for an inputx(n)4(0.6)nu(n).Solution:

2,10.3z-14X（z）=,-11-0.6zY（z）=z0.3 z0.2

Y（z）0.5(10.6z-1)H（z）==,-1X（z）10.3z1.5110.3z-1z0.3

y(n)(n)1.5(0.3)nu(n).6.The transfer function of a causal stable filter is:

K(1z-1)（z）=, H-11-z

Where

01

 is real and K is a real constant.Determine the shape of the filter.Solution:

lowpa filter.7.Consider the third-order IIR block form:

Then give the transfer function.Solution:

20.44z0.362z0.02 H(z)2(z0.8z0.5)(z0.4)

0.440.362z10.02z2z1 121(10.8z0.5z)(10.4z)

8.Determine the transfer function of 1-order digital Butterworth lowpa filter with a cutoff frequency at 500Hz ,and sampling rate is 4 kHz.With bilinear transformation method.Solution: The cutoff frequency 500Hz is converted to a digital frequency

p

ppT0.25rad fs This frequency is pre-warped to the analog frequency

2ptanp2fstanp3313.7rad/s T22which makes the transfer function for the analog filter

p3313.7 H(s)sps3313.7

After the bilinear transformation, the digital transfer function is obtained:

3313.7 H(z)z1 160003313.7z1

3313.73313.7z10.2930.293z1

1 11313.74686.3z10.414z1

9.The lowpa filter specifications given below, design a FIR filter with the smallest length meeting the specification using the window-based approach: ωp= 0.4π, ωs= 0.5π , αp =1dB, αs= 40dB

Solution:

c(ps)/20.454

So, the ideal LP impulse response is: sincnh[n],n dnSelect a window based on table and specifications.Because of αs=40dB, select window Hann.Calculate order N: 3.110.1M31.132 spM

N=2M+1=65 The impulse response is: sin0.45(n32)h[n]wHann[n32],0n64(n32)