B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

150 AMPLITUDE MODULATIONS AND DEMODULATIONS
Hence, we have
4 1 1
Vi (t) = m(t)wo(t) = - [ m(t) cos W e t - -m(t) cos 3w e t + -m(t) cos 5w e t - · · · ]
]'( 3 5
(4.7b)
The signal m(t)wo(t) is shown in Fig. 4.6d. When this waveform is passed through a bandpass
filter tuned to W e (Fig. 4.6a), the filter output will be the desired signal (4/]'()m(t) cos W e t,
In this circuit there are two inputs: m(t) and cos W e t, The input to the final bandpass filter
does not contain either of these inputs. Consequently, this circuit is an example of a double
balanced modulator.
Example 4.2
Frequency Mixer or Converter
We shall analyze a frequency mixer, or frequency converter, used to change the carrier
frequency of a modulated signal m(t) cos W e t from W e to another frequency WJ .
This can be done by multiplying m(t) cos W e t by 2cos Wmixt, where Wmix = W e + WJ or
W e - WJ, and then bandpass-filtering the product, as shown in Fig. 4.7a.
Fi g
ure 4.7
Frequency mixer
or converter.
x(t)
Bandpass
filter
tuned to wI
2 cos (w e ± w 1 )t
(a )
0
I
I
I
I
,--,
I \
\
/-,
\ I \
\ I \
WI 2w c
- WI 2w c
(b )
/-'
/ \ W-►
2w c +w 1
The product x(t) is
x(t) = 2m(t) cos W e t cos Wmixt
If we select Wmix = W e - w1 , then
If we select Wmix = W e + WJ , then
= m(t)[cos (w e - wmix)t + cos (w e + Wmix)t]
x(t) = m(t)[cos wi t + cos (2we - w1 )t]
x(t) = m(t)[cos wit + cos (2we + WJ )t]
In either case, as long as W e - w1 2: 27( B and w1 2: 2rr B, the various spectra in Fig. 4. 7b
will not overlap. Consequently, a bandpass filter at the output, tuned to w1 , will pass the
term m(t) cos wi t and suppress the other term, yielding the output m(t) cos wit. Thus,
the carrier frequency has been translated to wr from W e .