0195136047.pdf

642 SIGNAL PROCESSING

Am cos 2πfmt

t

Signal

cos 2πfct

t

Carrier wave

Am cos 2πfmt × cos 2πfct

Product

0

f

Am

fm

1

0

f

fm fc

0

f

fm fc − fm fc fc + fm

Am

2

Am

2

Figure 14.2.2Frequency translation waveforms and line spectra.

0 W

f

fc − W fc fc + W

Amplitude

(a)

0

2 W

f

Amplitude

Lower

sideband

Upper

sideband

(b)

Figure 14.2.3Amplitude spectra in double-sideband modulation (DSB).(a)Amplitude spectrum of low-

pass modulation signal.(b)Amplitude spectrum of bandpass modulated signal.

Now, in order to recoverx(t) fromxc(t), theproduct demodulatorshown in Figure 14.2.4(a),

which has a local oscillator synchronized in frequency and phase with the carrier wave, can be

used. The inputy(t) to the low-pass filter is given by

x(t)cos 2πfct=x(t)cos^22 πfct

=

1

2

x(t)+

1

2

x(t)cos 2π( 2 fc)t (14.2.4)

indicating that the multiplication has produced both upward and downward frequency translation.

In Equation (14.2.4), the first term is proportional tox(t), while the second looks like DSB at

carrier frequency 2fc. Then, if the low-pass filter in Figure 14.2.4(a) rejects the high-frequency

components and passesf≤W, the filtered outputz(t) will have the desired formz(t)=Kx(t).