Introduction to Analog Communication

Angle Modulation
FM is used over AM when better sound quality, noise resistance, and stable transmission are needed, while
AM is preferred for long-distance broadcasting and lower-cost transmissions

Phase Modulation
In phase modulation, the phase deviation ∅ 𝑡 is varied in such a way that at any instant of time, t, it is
proportional to the instantaneous amplitude of the modulating signal, x(t)

Kp unit of radians per volt

When speed v, is varying with time and is denoted by v(t), a function of time, we know that the distance s(t) covered
in say t seconds, is given by

𝜃 𝑡 = 𝜔𝑡 and 𝜔 = 2𝜋𝑓, if the frequency is varying with respect to time,

In frequency modulation, the instantaneous frequency of the modulated wave changes in such a way that at any
instant, the change from the unmodulated carrier frequency is directly proportional to the instantaneous amplitude of
the modulating signal,

the change in fi(t) from fc, the unmodulated carrier frequency, called the frequency deviation,
 Frequency modulation using a phase modulator

Phase modulation using a frequency modulator

FM PM
(ii) For FM:
Sketch FM and PM waves for the modulating signal m(t) shown in Fig. The constants kf and kp are
2𝝅 x 105 and 10𝝅 , respectively, and the carrier frequency fc is 100 MHz.
 BANDWIDTH OF ANGLE-MODULATED WAVES

Angle modulation is nonlinear and no properties of Fourier transform can be directly applied for its bandwidth analysis.
▪ The modulated wave consists of an unmodulated carrier plus various amplitude-modulated terms, such as a(t) sin 𝝎ct, a2 (t)
cos 𝝎ct, a3 (t) sin 𝝎ct, . . . .
▪ The signal a(t) is an integral of m(t). If M (f) is band-limited to B, A(f) is also band-limited to B.
▪ The spectrum of a2 (t) is simply A(f) * A(f) and is band-limited to 2B.
▪ Similarly, the spectrum of an(t) is band-limited to nB.
▪ Hence, the spectrum consists of an unmodulated carrier plus spectra of a(t) , a2 (t) , . . . , an(t) , . . . , centered at 𝝎c

Because n! increases much faster than |kfa(t)|n ,

▪ Most of the modulated-signal power resides in a finite bandwidth.

 Narrowband Angle Modulation Approximation

-------- Equation -1

▪ When kf is very small such that

▪ The FM signal follow this condition is called narrowband FM (NBFM)

▪ Then all higher order terms in Eq. 1 are negligible except for the first two

-------- Equation -2

▪ This expression similar to that of the AM signal with message signal a(t) .

▪ Because the bandwidth of a (t) is B Hz, the bandwidth of FM in Eq. 2 is 2B Hz

𝜋
▪ The sideband spectrum for FM has a phase shift of 2 with respect to the carrier, where as that of AM is in phase with the
carrier.
 Wideband FM (WBFM) Bandwidth Analysis

▪ Consider a low-pass m(t) with bandwidth B Hz. This signal is well approximated by a staircase signal m(t).
▪ The signal m(t) is now approximated by pulses of constant amplitude.
▪ For convenience, each of these pulses will be called a "cell."

▪ fs = 2B is called the Nyquist rate

▪ Sampling interval Ts = 1 /2B is called the
Nyquist interval

▪ Hence, the FM signal corresponding to this cell is a sinusoid of frequency 𝝎𝒄 + 𝒌𝒇 𝒎(𝒕𝒌 ) and duration T = 1 /2B,
▪ The FM spectrum for 𝒎(t)
ෝ consists of the sum of
the Fourier transforms of these sinusoidal pulses
corresponding to all the cells.

▪ Hence, the maximum and the minimum significant

frequencies in this spectrum are
𝝎𝒄 + 𝒌𝒇 𝒎(𝒕𝒑 ) + 𝟒𝝅𝑩 and 𝝎𝒄 - 𝒌𝒇 𝒎(𝒕𝒑 ) − 𝟒𝝅𝑩

▪ The FM spectrum bandwidth is approximately

▪ The peak frequency deviation in hertz by

▪ we observe that for the case of NBFM, 𝑘𝑓 is very small. Hence, given a fixed 𝑚𝑝 , ∆𝑓 is very small (in comparison to B)
for NBFM. In this case, we can ignore the small ∆𝑓 term

▪ Earlier that for narrowband, the FM bandwidth is approximately 2B Hz.

▪ This indicates that a better bandwidth estimate is

▪ This formula goes under the name Carson's rule in the literature
▪ Observe that for a truly wideband case, where ∆𝒇 ≫ 𝑩 , can be approximated as

▪ Carson's rule can be expressed in terms of the deviation

▪ We define a deviation ratio 𝜷 as ratio as

▪ 𝜷 is called the modulation index

 Spectral Analysis of Tone Frequency Modulation

▪ We use this special case to verify the FM bandwidth approximation.

▪ Tone modulation is a special case for which a precise spectral analysis is possible

▪ The bandwidth of m(t) is 𝟐𝝅𝑩 = 𝝎𝒎 rad/s

▪ The deviation ratio (or in this case, the modulation index) is

𝟐𝝅
▪ Note that 𝒆𝒋𝜷𝒔𝒊𝒏𝝎𝒎 𝒕 is a periodic signal with period and can be expanded by the exponential Fourier series, as
𝝎𝒎
usual,

▪ This integral has been extensively tabulated and is denoted by 𝑱𝒏 (𝜷)

▪ The Bessel function of the first kind and the nth order
▪ The strength of the nth sideband at 𝝎 = 𝝎𝒄 +n𝝎𝒎 is 𝑱𝒏 (𝜷)
From the plots of 𝑱𝒏 (𝜷) in Fig, it can be seen that for a
given 𝜷, 𝑱𝒏 (𝜷) decreases with n.

▪ It can be seen from that 𝑱𝒏 (𝜷) is negligible for 𝒏 > 𝜷 + 1

▪ The number of significant sideband impulses is 𝜷 + 1

▪ The bandwidth of the FM carrier is given by

An FM transmitter has a frequency deviation constant of 100
Hz/volt. To the modulator of this transmitter, a sinusoidal
modulating signal of rms value 2 volts and a frequency of 1 kHz,
is applied. Determine the peak frequency deviation and the
deviation ratio.
The message signal shown in the Fig. phase modulates a carrier signal Accoswct,
where fc =1MHz. If a maximum frequency deviation of 80 kHz is needed,
determine the value of the phase constant kp to be used by the modulator. With
this value of kp, what will be the range of variation of the carrier frequency?
An FM signal with single-tone modulation has a frequency deviation of 15 kHz
and a bandwidth of 50 kHz. Find the frequency of the modulating signal.
 GENERATING FM WAVES
▪ There are two ways of generating FM waves: indirect and direct.
▪ the narrowband FM generator that is utilized in the indirect FM generation of wideband angle modulation signals.

NBFM/NBPM GENERATION
▪ In this method, NBFM generated has some distortion because of the approximation.

▪ The output of this NBFM modulator also has some amplitude variations.

▪ A nonlinear device designed to limit the amplitude of a bandpass signal can remove most of
this distortion.
▪ The amplitude variations of an angle-modulated carrier can be eliminated by what is known
as a bandpass limiter,
 Hard limiter input output characteristic

Hard limiter input and the corresponding output.

▪ The output v0 (t) of the hard limiter is + 1 or - 1 , depending on whether vi(t) = A(t) cos 𝜃(t) is positive or negative

▪ Because A(t) ≥0, v0(t) can be expressed as a function of 𝜃

▪ v0 as a function of 𝜃 is a periodic square wave function with period 2𝜋, which can be expanded by a Fourier series

▪ We can pass the output of the hard limiter through a bandpass filter with a center frequency 𝝎𝐜 and a
bandwidth BFM
 Indirect Method of FM Generation (Armstrong )
▪ The NBFM is then converted to WBFM by using additional frequency multipliers.

▪ A frequency multiplier can be realized by a nonlinear device followed by a bandpass filter

▪ A bandpass filter centered at 2𝜔𝑐 would recover an FM signal with twice the original instantaneous frequency
▪ A nonlinear device may have the characteristic of
▪ A bandpass filter centering at 𝑛𝜔𝑐 can recover an FM signal whose instantaneous frequency has been
multiplied by a factor of n.
▪ These devices, consisting of nonlinearity and bandpass filters, are known as frequency multipliers.
▪ we want a twelfth-fold increase in the frequency deviation, we can use a twelfth-order nonlinear device or two
second-order and one third-order devices in cascade.
▪ A commercial FM transmitter using Armstrong's method used to find The final output is required to have a carrier frequency
of 91.2 MHz and frequency deviation ∆f = 75 kHz.

▪ Multiplication of 64 can be obtained by six doublers in cascade, and a multiplication of 48 can be obtained by four
doublers and a tripler in cascade.
▪ This scheme has an advantage of frequency stability, but it suffers from inherent noise caused by excessive
multiplication and distortion at lower modulating frequencies
In a WBFM generator of the Armstrong type shown in Figure, the initial low-
frequency carrier is of 200 kHz frequency. The maximum frequency deviation range
from 100 Hz to 15 kHz, and the final maximum frequency deviation and the carrier
frequency are to be 75 kHz and 102.4 MHz respectively. Choose an appropriate
multiplier and the mixer oscillator frequency.
 Direct Generation of FM
▪ Voltage-controlled oscillator (VCO), the frequency is controlled by an external voltage. The oscillation frequency varies
linearly with the control voltage.
▪ Another way of accomplishing the same goal is to vary one of the reactive parameters (C or L) of the resonant circuit of an
oscillator.
▪ A reverse-biased semiconductor diode acts as a capacitor whose capacitance varies with the bias voltage.

▪ The capacitance of these diodes, known under several trade names (e.g., Varicap, Varactor, Voltacap), can be approximated
as a linear function of the bias voltage m(t) over a limited range.

▪ In Hartley or Colpitt oscillators, for instance, the frequency of oscillation is given by

▪ If the capacitance C is varied by the modulating signal m(t), that is, if

 ∆𝑓
▪ In practice, is usually small, and, hence, ∆𝐶 is a small
𝑓𝑐
fraction of Co, which helps limit the harmonic distortion
that arises because of the approximation used in this
derivation.
▪ Direct FM generation generally produces sufficient
frequency deviation and requires little frequency
multiplication.
▪ But this method has poor frequency stability. In practice,
feedback is used to stabilize the frequency.
 DEMODULATION OF FM SIGNALS
▪ The information in an FM signal resides in the instantaneous frequency 𝜔𝑖 =𝜔𝑐 +𝑘𝑓 𝑚(𝑡)
▪ A frequency-selective network with a transfer function of the form ∣𝐻(𝑓)∣=2𝜋𝑓+𝑏 over the FM band would yield an
output proportional to the instantaneous frequency
▪ There are several possible circuits with such characteristics. The simplest among them is an ideal differentiator
with the transfer function 𝑗2𝜋𝑓.
If we apply 𝝋𝑭𝑴 (t) to an ideal differentiator, the output is

Both the amplitude and the frequency of the signal 𝝋ሶ 𝑭𝑴 (t) are modulated the envelope being A 𝝎𝒄 + 𝒌𝒇 𝒎(𝒕) .

Two conditions should be follow for envelop detector

 Single tuned discriminator/single slope detector

▪ It consists of a resonant circuit which is tuned to a frequency slightly above the carrier frequency.
▪ This circuit converts the FM signal into the corresponding AM signal.
▪ This AM signal is then fed to the diode detector for detection.
Advantages:
(i) It is simple.
(ii) It is economical and hence suitable, when the cost is more important.
Disadvantages:
(i) The characteristic of the circuit is non linear.
(ii) To reduce the distortions, the frequency deviation has to be less.
(iii) The amplitude variation may rise due to noise and other factors.
 Balance Slope Detector

▪ To overcome the problem of non-linearity encountered in the simple slope detector discussed earlier,
▪ Foster and Seeley proposed the dual-slope detector.
▪ This makes use of two resonant circuits with identical responses but with slightly different resonant
frequencies.
▪ The technique used in order to obtain a larger linear range