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Unformatted text preview: if you ' ". .3 "'{).1 0 '" 0.2 '~'. t  Fig. P 5.17 F lat t op s ampling.
5 .17 I n E xample 5.3, t he s ampling of a signal j (t) was accomplished by multiplying t he
signal by a pulse t rain PT(t), r esulting in t he s ampled signal depicted in Fig. 5.8d.
T his p rocedure is known as t he n atural s ampling. F igure P 5.17 shows t he so called
f lat t op s ampling o f t he s ame signal j (t) = sinc 2(51l"t).
( a) Show t hat t he signal j (t) c an b e recovered from fiat t op s amples if t he s ampling
r ate is no less t han t he N yquist rate.
( b) E xplain how you would recover j et) from t he fiat t op samples.
( c) F ind t he expression for t he s ampled signal s pectrum F(w) a nd sketch i t roughly.
Hint: F irst show t hat t he fiat t op s ampled signal c an b e generated by passing t he
signal j(t)OT(t) t hrough a filter whose impulse response is h (t) = p T(t). For signal
recovery from t he samples, follow t he reverse procedure.
5 .18 A c ompact disc ( CD) r ecords audio signals digitally by using P CM. Assume t he a udio
signal b andwidth t o b e 15 kHz.
( a) W hat is t he N yquist rate?
( b) I f t he N yquist samples are quantized into 65536 levels ( L = 65536) a nd t hen
b inarycoded, w hat n umber o f b inary digits is r equired to encode a sample.
( c) D etermine t he n umber o f binary d igits/s ( bits/s) r equired t o e ncode t he a udio
signal.
( d) For practical reasons discussed in t he t ext, signals are sampled a t a r ate well
above t he N yquist rate. P ractical CDs use 44100 samples/so I f L = 65536, determine
t he n umber of pulses/ s required to encode t he signal.
5.19 A T V signal (video a nd audio) h as a b andwidth o f 4.5 MHz. T his signal is s ampled,
quantized a nd b inarycoded t o o btain a P CM (pulse code modulated) signal.
( a) D etermine t he s ampling r ate if t he signal is t o b e sampled a t a r ate 20% above
t he N yquist rate.
( b) I f t he s amples a re q uantized into 1024 levels, w hat n umber of binary pulses is
required to encode each sample.
( c) D etermine t he b inary pulse r ate ( bits/s) o f t he b inary coded signal. 3 60 5 S ampling  10 Prove t hat a s ignal cannot be simultaneously timelimited and bandlimited.
Hint: Show t hat contrary assumption leads to contradiction. Assume a signal simultaneously timelimited and bandlimited so t hat F(w) = 0 for Iwl > 27rB. I n this
case F(w) = F (w)rect (4,,"'B1) for B ' > B. This fact means t hat J(t) is equal t o
J(t) * 2 B'sinc(27rB't). The latter cannot be timelimited because the sine function
tail extends to infinity.
~1 For a signal Jet) t hat is timelimited to 10 ms and has an essential bandwidth of 10
kHz, determine No, the number of signal samples necessary t o compute a power of
2FFT with a frequency resolution Fa of a t least 50 Hz. Explain if any zero padding
is necessary. ContinuousTime S ystem Analysis
U sing t he Laplace T ransform
I~ 2 2 To compute t he D FT of signal J(t) in Fig. P5.22, write the sequence !k (for k = 0 to
No  1) if the frequency resolution Fa mu...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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