The demodulation of an FM signal requires a circuit that yields an output proportional to the frequency deviation of the input. Such circuits are known asfrequency discriminators.1If the input to an ideal discriminator is the angle-modulated signal
𝑥𝑟(𝑡) = 𝐴𝑐cos[2𝜋𝑓𝑐𝑡 + 𝜙(𝑡)] (4.76) the output of the ideal discriminator is
𝑦𝐷(𝑡) = 1 2𝜋𝐾𝐷𝑑𝜙
𝑑𝑡 (4.77)
1The termsfrequency demodulator and frequency discriminator are equivalent.
Output voltage
f fc
KD 1
Input frequency
Figure 4.13 Ideal discriminator.
For FM,𝜙(𝑡)is given by
𝜙(𝑡) = 2𝜋𝑓𝑑∫
𝑡𝑚(𝛼)𝑑𝛼 (4.78)
so that (4.77) becomes
𝑦𝐷(𝑡) − 𝐾𝐷𝑓𝑑𝑚(𝑡) (4.79)
The constant𝐾𝐷is known as thediscriminator constant and has units of volts per Hz. Since an ideal discriminator yields an output signal proportional to the frequency deviation of a carrier, it has a linear frequency-to-voltage transfer function, which passes through zero at𝑓 = 𝑓𝑐. This is illustrated in Figure 4.13.
The system characterized by Figure 4.13 can also be used to demodulate PM signals. Since 𝜙(𝑡)is proportional to𝑚(𝑡)for PM,𝑦𝐷(𝑡)given by (4.77) is proportional to the time derivative of𝑚(𝑡)for PM inputs. Integration of the discriminator output yields a signal proportional to 𝑚(𝑡). Thus, a demodulator for PM can be implemented as an FM discriminator followed by an integrator. We define the output of a PM discriminator as
𝑦𝐷(𝑡) = 𝐾𝐷𝑘𝑝𝑚(𝑡) (4.80)
It will be clear from the context whether𝑦𝐷(𝑡)and𝐾𝐷refer to an FM or a PM system.
An approximation to the characteristic illustrated in Figure 4.13 can be obtained by the use of a differentiator followed by an envelope detector, as shown in Figure 4.14. If the input to the differentiator is
𝑥𝑟(𝑡) = 𝐴𝑐cos[2𝜋𝑓𝑐𝑡 + 𝜙(𝑡)] (4.81) the output of the differentiator is
𝑒(𝑡) = −𝐴𝑐 [
2𝜋𝑓𝑐+ 𝑑𝜙 𝑑𝑡
]
sin[2𝜋𝑓𝑐𝑡 + 𝜙(𝑡)] (4.82) This is exactly the same form as an AM signal, except for the phase deviation𝜙(𝑡). Thus, after differentiation, envelope detection can be used to recover the message signal. The envelope of𝑒(𝑡)is
𝑦(𝑡) = 𝐴𝑐 (
2𝜋𝑓𝑐+ 𝑑𝜙 𝑑𝑡
)
(4.83) and is always positive if
𝑓𝑐> − 1 2𝜋
𝑑𝜙
𝑑𝑡 for all𝑡
Envelope detector Differentiator
yD(t) Bandpass
f ilter Bandpass limiter Limiter
xr(t)
Figure 4.14
FM discriminator implementation.
which is usually satisfied since𝑓𝑐is typically significantly greater than the bandwidth of the message signal. Thus, the output of the envelope detector is
𝑦𝐷(𝑡) = 𝐴𝑐𝑑𝜙
𝑑𝑡 = 2𝜋𝐴𝑐𝑓𝑑𝑚(𝑡) (4.84)
assuming that the DC term,2𝜋𝐴𝑐𝑓𝑐, is removed. Comparing (4.84) and (4.79) shows that the discriminator constant for this discriminator is
𝐾𝐷= 2𝜋𝐴𝑐 (4.85)
We will see later that interference and channel noise perturb the amplitude𝐴𝑐 of𝑥𝑟(𝑡). In order to ensure that the amplitude at the input to the differentiator is constant, a limiter is placed before the differentiator. The output of the limiter is a signal of square-wave type, which is𝐾sgn [𝑥𝑟(𝑡)]. A bandpass filter having center frequency𝑓𝑐 is then placed after the limiter to convert the signal back to the sinusoidal form required by the differentiator to yield the response defined by (4.82). The cascade combination of a limiter and a bandpass filter is known as abandpass limiter. The complete discriminator is illustrated in Figure 4.14.
The process of differentiation can often be realized using a time-delay implementation, as shown in Figure 4.15. The signal 𝑒(𝑡), which is the input to the envelope detector, is given by
𝑒(𝑡) = 𝑥𝑟(𝑡) − 𝑥𝑟(𝑡 − 𝜏) (4.86) which can be written
𝑒(𝑡)
𝜏 = 𝑥𝑟(𝑡) − 𝑥𝑟(𝑡 − 𝜏)
𝜏 (4.87)
Since, by definition,
𝜏→lim0
𝑒(𝑡) 𝜏 = lim
𝜏→0
𝑥𝑟(𝑡) − 𝑥𝑟(𝑡 − 𝜏)
𝜏 = 𝑑𝑥𝑟(𝑡)
𝑑𝑡 (4.88)
it follows that for small𝜏,
𝑒(𝑡) ≅ 𝜏𝑑𝑥𝑟(𝑡)
𝑑𝑡 (4.89)
This is, except for the constant factor𝜏, identical to the envelope detector input shown in Figure 4.15 and defined by (4.82). The resulting discriminator constant𝐾𝐷is2𝜋𝐴𝑐𝜏. There are many other techniques that can be used to implement a discriminator. Later in this chapter we will examine the phase-locked loop, which is an especially attractive, and common, implementation.
Envelope detector Time delay
Approximation to differentiator τ
+ Σ
−
e(t) yD(t)
xr(t)
Figure 4.15
Discriminator implementation using a time delay and envelope detection.
EXAMPLE 4.5
Consider the simple RC network shown in Figure 4.16(a). The transfer function is
𝐻(𝑓) = 𝑅
𝑅 + 1∕𝑗2𝜋𝑓𝐶 = 𝑗2𝜋𝑓𝑅𝐶
1 + 𝑗2𝜋𝑓𝑅𝐶 (4.90)
The amplitude response is shown in Figure 4.16(b). If all frequencies present in the input are low, so that
𝑓 ≪ 1
2𝜋𝑅𝐶
C
R
) b ( )
a (
(c) H(f)
fc f 1
0.707
2π1RC
Filter Envelope detector
Figure 4.16
Implementation of a simple frequency discriminator based on a high-pass filter. (a) RC network. (b) Transfer function. (c) Discriminator.
the transfer function can be approximated by
𝐻(𝑓) = 𝑗2𝜋𝑓𝑅𝐶 (4.91)
Thus, for small𝑓, the RC network has the linear amplitude--frequency characteristic required of an ideal discriminator. Equation (4.91) illustrates that for small𝑓, the RC filter acts as a differentiator with gain RC. Thus, the RC network can be used in place of the differentiator in Figure 4.14 to yield a discriminator with
𝐾𝐷= 2𝜋𝐴𝑐𝑅𝐶 (4.92)
■ This example again illustrates the essential components of a frequency discriminator, a circuit that has an amplitude response linear with frequency and an envelope detector.
However, a highpass filter does not in general yield a practical implementation. This can be seen from the expression for 𝐾𝐷. Clearly the 3-dB frequency of the filter,1∕2𝜋𝑅𝐶, must exceed the carrier frequency 𝑓𝑐. In commercial FM broadcasting, the carrier frequency at the discriminator input, i.e., the IF frequency, is on the order of 10 MHz. As a result, the discriminator constant𝐾𝐷is very small indeed.
A solution to the problem of a very small𝐾𝐷 is to use a bandpass filter, as illustrated in Figure 4.17. However, as shown in Figure 4.17(a), the region of linear operation is often unacceptably small. In addition, use of a bandpass filter results in a DC bias on the discriminator output. This DC bias could of course be removed by a blocking capacitor, but the blocking capacitor would negate an inherent advantage of FM---namely, that FM has DC response. One can solve these problems by using two filters with staggered center frequencies 𝑓1and𝑓2, as shown in Figure 4.17(b). The magnitudes of the envelope detector outputs following the two filters are proportional to|𝐻1(𝑓)|and|𝐻2(𝑓)|. Subtracting these two outputs yields the overall characteristic
𝐻(𝑓) =|𝐻1(𝑓)|−|𝐻2(𝑓)| (4.93)
as shown in Figure 4.17(c). The combination, known as abalanced discriminator, is linear over a wider frequency range than would be the case for either filter used alone, and it is clearly possible to make𝐻(𝑓𝑐) = 0.
In Figure 4.17(d), a center-tapped transformer supplies the input signal𝑥𝑐(𝑡)to the inputs of the two bandpass filters. The center frequencies of the two bandpass filters are given by
𝑓𝑖 = 1 2𝜋√
𝐿𝑖𝐶𝑖 (4.94)
for 𝑖 = 1, 2. The envelope detectors are formed by the diodes and the resistor--capacitor combinations𝑅𝑒𝐶𝑒. The output of the upper envelope detector is proportional to|𝐻1(𝑓)|, and the output of the lower envelope detector is proportional to|𝐻2(𝑓)|. The output of the upper envelope detector is the positive portion of its input envelope, and the output of the lower envelope detector is the negative portion of its input envelope. Thus,𝑦𝐷(𝑡)is proportional to|𝐻1(𝑓)|−|𝐻2(𝑓)|. The termbalanced discriminator is used because the response to the undeviated carrier is balanced so that the net response is zero.
H1(f)
H2(f)
f1
f2 f1 f
f f
–H2(f)
H1(f)
H1(f) Linear region
Linear region (a)
(b)
(c)
(d)
Amplitude responseAmplitude responseAmplitude response
H(f) =H1(f) – H2(f)
R
R
D
D L1
C1
C2
C3 Re
Re
C4 L2
xc(t) yD(t)
Bandpass Envelope detectors
Figure 4.17
Derivation of a balanced discriminator. (a) Bandpass filter.
(b) Stagger-tuned bandpass filters.
(c) Amplitude response of a balanced discriminator. (d) Typical
implementation of a balanced discriminator.