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

4.2 Double-Sideband Amplitude Modulation 147
In this circuit there are two inputs: m(t) and cos wet. The output of the last summer, z(t),
no longer contains one of the inputs, the carrier signal cos wet. Consequently, the carrier signal
does not appear at the input of the final bandpass filter. The circuit acts as a balanced bridge
for one of the inputs (the carrier). Circuits that have this characteristic are called balanced
circuits. The nonlinear modulator in Fig. 4.3 is an example of a class of modulators known as
balanced modulators. This circuit is balanced with respect to only one input (the carrier); the
other input m(t) still appears at the final bandpass filter, which must reject it. For this reason, it
is called a single balanced modulator. A circuit balanced with respect to both inputs is called
a double balanced modulator, of which the ring modulator (see later: Fig. 4.6) is an example.
Switching Modulators: The multiplication operation required for modulation can be
replaced by a simpler switching operation if we realize that a modulated signal can be obtained
by multiplying m(t) not only by a pure sinusoid but by any periodic signal ¢ (t) of the fundamental
radian frequency We. Such a periodic signal can be expressed by a trigonometric
Fourier series as
00
<p (t) = L C n cos (ncvet + 0 n )
n=O
(4.4a)
Hence,
00
m(t)<p (t) = L C n m(t) cos (ncvct + 0 n )
n=O
(4.4b)
This shows that the spectrum of the product m(t)<p (t) is the spectrum M (cv) shifted to
±eve, ±2cvc, . .. , ±ncvc, .... If this signal is passed through a bandpass filter of bandwidth
2B Hz and tuned to We, then we get the desired modulated signal c1m(t) cos (wet + 01).*
The square pulse train w(t) in Fig. 4.4b is a periodic signal whose Fourier series was found
earlier (by rewriting the results of Example 2.4) as
1 2 1 1
w(t) = - + - ( cos Wet - - cos 3cvet + - cos 5cvct - · · · )
2 IT 3 5
(4.5)
The signal m(t)w(t) is given by
1 2 1 1
m(t)w(t) = -m(t) + - [ m(t) cos Wet - -m(t) cos 3cvct + -m(t) cos 5cvct - · · ·
2 IT 3 5
]
(4.6)
The signal m(t)w(t) consists not only of the component m(t) but also of an infinite
number of modulated signals with carrier frequencies eve, 3cvc, 5cvc, .... Therefore, the spectrum
of m(t)w(t) consists of multiple copies of the message spectrum M (f), shifted to
0, ±Jc, ±3fc , ±5fc, ... (with decreasing relative weights), as shown in Fig. 4.4c.
For modulation, we are interested in extracting the modulated component m(t) cos Wet
only. To separate this component from the rest of the crowd, we pass the signal m(t)w(t) through
a bandpass filter of bandwidth 2B Hz (or 4ITB rad/s), centered at the frequency ±Jc - Provided
the carrier frequency f c '.'.':: 2B (or We '.'.':: 4ITB), this will suppress all the spectral components
not centered at ±Jc to yield the desired modulated signal (2/IT )m(t) cos Wet (Fig. 4.4d).
We now see the real payoff of this method. Multiplication of a signal by a square pulse train
is in reality a switching operation in which the signal m(t) is switched on and off periodically; it
* The phase 01 is not important.