Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 606

C a se 2: f c = odd n umber x B=22 Hz, fs = 2B=4 H z

1

Phase
distortion

0.5

0

-0.5

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s ec . )

Case 2: fc= odd number x B=22 Hz, fs=2B=4 Hz, with B=2 Hz oscillator

1

Message
signals

0.5

0

-0.5

-1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s ec . )

FIGURE 12.43

Plots of the bandpass signals and sampled signals for Case 2.

Since

cos ð 11 pnÞ cos ðpnÞ¼ 1 (12.49)

it follows that

xðnTÞ cos ð 2 p 2 n= 4 Þ¼ cos ðp 11 nÞmðnTÞ cos ðp nÞ¼mðnTÞ (12.50)

which is the recovered message signal. Figure 12.43 shows the sampled bandpass signals with the
reversed message spectrum and the corrected message spectrum, respectively, for a message signal
having a frequency of 0.5 Hz; that is,

mðtÞ¼ cos ð 2 p 0 : 5 tÞ (12.51)

Case 3
If f c ¼ noninterger B , we can extend the bandwidth B to B such that

f c ¼ integer B and f s ¼ 2 B (12.52)

Then we can apply Case 1 or Case 2. An illustration of Case 3 is included in the following
example.