Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 604

xt () xn ()

Bandpass filte r

ADC

f

Spectrum for x(n) for
f s = even integer x B Hz

s

(a)

UL UL L U L U L U

f

Spectrum for x(n) for
f s = odd integer x B Hz

(b)

UL UL L U L U L U

f

FIGURE 12.41

Spectrum of the undersampled signal.

where mðtÞ is the message signal with a bandwidth of 2 Hz. Using a sampling rate of 4 Hz by
substituting t ¼ nT , where T ¼ 1 =f s

into Equation (12.39) , we get the sampled signal as

t¼nT ¼ cos ð 2 p 20 n= 4 ÞmðnTÞ

xðnTÞ¼ cos ð 2 p 20 tÞmðtÞ (12.40)

Since 10 np ¼ 5 nð 2 pÞ is a multiple of 2 p ,

cos ð 2 p 20 n= 4 Þ¼ cos ð 10 pnÞ¼ 1 (12.41)

we obtain the undersampled signal as

xðnTÞ¼ cos ð 2 p 20 n= 4 ÞmðnTÞ¼mðnTÞ (12.42)

which is a perfect digital message signal. Figure 12.42 shows the bandpass signal and its sampled
signals when the message signal is 1 Hz, given as

mðtÞ¼ cos ð 2 ptÞ (12.43)

Case 2
If f c ¼ odd integer B and f c ¼ 2B , the sampled spectrum with all the replicas will be as
shown in Figure 12.41 (b), where the spectral portions L and U are reversed. Hence, frequency
reversal will occur. Then a further digital modulation in which the signal is multiplied by the
digital oscillator with a frequency of B Hzcanbeusedtoadjustthespectrumtobethesameas
that in Case 1.
As another illustrative example for Case 2, let us sample the following the bandpass signal with
a carrier frequency of 22 Hz, given by

xðtÞ¼ cos ð 2 p 22 tÞmðtÞ (12.44)