Signals and Systems - Electrical Engineering

448 CHAPTER 7: Sampling Theory

where

1(t)=









∞∫

−∞

t^2 |x(t)|^2 dt

Ex









0.5

measures the duration of the signal for which the signal is significant in time, and

1()=









∞∫

−∞

^2 |X()|^2 d

Ex









0.5

measures the frequency support for which the Fourier representation is significant. The energy of

the signal is represented byEx. Compute1(t)and1()for the given signalx(t)and verify that the

uncertainty principle is satisfied.

7.5. Nyquist sampling rate condition and aliasing

Consider the signal

x(t)=

sin(0.5t)

0.5t

(a) Find the Fourier transformX()ofx(t).

(b) Isx(t)band limited? If so, find its maximum frequencymax.

(c) Suppose thatTs= 2 π. How doessrelate to the Nyquist frequency 2 max? Explain.

(d) What is the sampled signalx(nTs)equal to? Carefully plot it and explain ifx(t)can be reconstructed.

7.6. Anti-aliasing

Suppose you want to find a reasonable sampling periodTsfor the noncausal exponential

x(t)=e−|t|

(a) Find the Fourier transform ofx(t), and plot|X()|. Isx(t)band limited?

(b) Find a frequency 0 so that99%of the energy of the signal is in−o≤≤o.

(c) If we lets= 2 π/Ts= 5  0 , what would beTs?

(d) Determine the magnitude and bandwidth of an anti-aliasing filter that would change the original

signal into the band-limited signal with99%of the signal energy.

7.7. Sampling of modulated signals

Assume you wish to sample an amplitude modulated signal

x(t)=m(t)cos(ct)

wherem(t)is the message signal andc= 2 π 104 rad/sec is the carrier frequency.

(a) If the message is an acoustic signal with frequencies in a band of[0, 22] KHz, what would be the

maximum frequency present inx(t)?

(b) Determine the range of possible values of the sampling periodTsthat would allow us to samplex(t)

satisfying the Nyquist sampling rate condition.

(c) Given thatx(t)is a band-pass signal, compare the above sampling period with the one that can be

used to sample band-pass signals.