CE105 X L TN HIU S thi cui k - 25/12/2012

Thi gian: 120 pht

H tn:

M s sinh vin:

Ch : thi c tng cng 5 cu.

Cu 1. (2)

Mt h thng LTI c m t nh sau:

y[n] = x[n] + 2x[n-1] + x[n-2]

a) Xc nh p ng xung h[n] ca h thng.

b) H thng c n nh khng? Ti sao?

c) Xc nh p ng tn s (frequency response) )( ieH ca h thng.

Cu 2. (2)

Cho mt h thng LTI nhn qu (causal), nu tn hiu u vo ca h thng l

]1[2][)2

1(

4

1][ 2 nununx nn

th h thng s sinh ra tn hiu u ra y[n] c bin i z tng ng l:

)31)(21)(3

11(

)1)(1()(

111

11

zzz

zzzY

a) Xc nh X(z) v H(z)

b) V ROC cho X(z), Y(z) v H(z)

c) Tm p ng xung (impulse response) h[n] ca h thng v cho bit h thng c

n nh (stable) khng?

Cu 3. (2)

a) Cho x(n) = {2, 5, 6, 7}, s dng FFT (decimation in time) tm X(K)?

a) Cho DTFT ca tn hiu x1(n) = {1, 0, -1, 0} l )(1 X , biu din DTFT )(2 X ca tn

hiu x2(n) = {1, 0, -1, 0, 1, 0, -1, 0} thng qua )(1 X .

b) Xc nh DTFT )(X ca tn hiu )1(3.0)( 2 nunx n

Cu 4. (2)

a) Tnh p ng xung ca b lc thng thp (lowpass) c chiu di 3 c tn s ct b

(cutoff frequency) l 800 Hz v tn s ly mu l 8000 Hz s dng ca s

Hamming.

b) Xc nh hm truyn t (transfer function) v phng trnh sai phn (difference

equation) ca h thng.

Cu 5. (2)

Mt b lc Butterworth thng cao (highpass) c tng cng ti a l 5 dB, tn s dy

thng tn (passband frequency) l 1500 HZ m ti tng cng ti thiu l 3 dB,

tn s dy chn tn (stopband frequency) l 214.3 Hz, v suy gim dy chn tn

(stopband attenuation) l 40 dB. Cho thi gian ly mu 0.1 s.

a) Tm ?,,, spsp AA

b) Tm bc N ca b lc v h s gn (ripple factor) .

c) Tm cc cc (pole) ca b lc Nsss ,,, 21 .

d) Tm hm truyn t (transfer function) )(sH ca b lc.

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Fourier transform properties

Sequences x[n], y[n] Transforms X(ej), Y (ej) Propertyax[n] + by[n] aX(ej) + bY (ej) Linearityx[n nd] e

jndX(ej) Time shiftej0nx[n] X(ej(0)) Frequency shiftx[n] X(ej) Time reversal

nx[n] jdX(ej)

d Frequency diff.x[n] y[n] X(ej)Y (ej) Convolutionx[n]y[n] 12pi

R pipi

X(ej)Y (ej())d Modulation

Common Fourier transform pairsSequence Fourier transform[n] 1

[n n0] ejn0

1 ( < n 1

u[n 1] 11z1

|z| < 1

[nm] zm All z except 0 or anu[n] 1

1az1|z| > |a|

anu[n 1] 11az1

|z| < |a|

nanu[n] az1

(1az1)2|z| > |a|

nanu[n 1] az1

(1az1)2|z| < |a|(

an 0 n N 1,

0 otherwise1aNzN

1az1|z| > 0

cos(0n)u[n]1cos(0)z

1

12 cos(0)z1+z2

|z| > 1

rn cos(0n)u[n]1r cos(0)z

1

12r cos(0)z1+r2z2

|z| > r

CE105_Dec2012_Final1.pdfUntitled1.pdf