This document discusses the transmission of frequency modulated (FM) waves. It explains that the bandwidth of an FM wave is theoretically infinite but is effectively limited in practice. Carson's rule provides an estimate of the bandwidth but is not always accurate. The document also describes Armstrong's method for generating FM waves using narrowband phase modulation and frequency multiplication, and demodulating FM waves using envelope detection after taking the derivative of the received signal.

1. TELE3113 Analogue and Digital
Communications
Transmission of FM Waves
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. Wideband FM Signal
In the last lecture, we studied:
The FM wave of the single-tone message signal is given by
∞
s(t) = Ac Jn (β) cos[2π(fc + nfm )t].
n=−∞
The spectrum of s(t) is given by
∞
Ac
S(f ) = Jn (β)[δ(f − fc − nfm ) + δ(f + fc + nfm )].
2 n=−∞
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.1/

3. Transmission BW of FM (1)
In theory, an FM wave contains an infinite number of
side-frequencies, so the BW is infinite.
In practice, the FM wave is effectively limited to a finite
number of significant side-frequencies.
Specifically, by observing the spectrum,
for large β, the BW approaches 2∆f = 2βfm .
for small β, the BW approaches 2fm .
Carson’s rule:
1
BT ≈ 2∆f + 2fm = 2∆f 1+ .
β
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.2/

4. Transmission BW of FM (2)
Carson’s rule is simple, but not an accurate estimate.
An accurate estimate of the BW can be defined as the
separation between the two frequencies beyond which
none of the side frequencies is greater than 1% of the
unmodulated carrier amplitude.
Mathematically, the BW is given by (universal rule)
nmax
BT = 2nmax fm = 2∆f
β
where nmax is the largest value of the integer n that satisfies
|Jn (β)| > 0.01.
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.3/

5. Transmission BW of FM (3)
β 0.1 0.3 0.5 1 2 5 10 20 30
2nmax 2 4 4 6 8 16 28 50 70
2
10
BT 2nmax
∆f = β
BT/∆ f in term of β
Normalized bandwidth, B /∆ f
T
1
10
0
10
−1 0 1
10 10 10
Modulation index, β
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.4/

6. Transmission BW of FM (4)
Relative error of BW estimate with Carson’s rule compared to
the universal rule is defined by
Carson
BT 1+β
= 1 − Universal = 1− × 100%.
BT nmax
Usually, Carson’s rule underestimates the BW by 25%.
40
35
Relative error of BW estimate (%)
30
25
20
15
10 Relative error of Carson’s rule in term of β
5
0
−5
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.5/
−10 −1 0 1

7. Generation of FM Waves (1)
Armstrong’s method:
The message signal is first used to produce a narrow-band
FM.
Then, a frequency multiplication is used to produce a
wide-band FM.
Message Narrow-
signal band phase Frequency
Integrator multiplier
m(t) Modulator Wide-
band
FM wave
Narrow-band phase
modulator
Crystal
controlled
oscillator
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.6/

8. Generation of FM Waves (2)
Frequency multiplier
FM wave
FM wave s(t) s’(t) with
with carrier Bandpass
Memoryless v(t) filter with carrier
frequency fc nonlinear frequency nfc
and mid-band
device frequency nfc and
modulation modulation
n
The input-output relation of the nonlinear device is
v(t) = a1 s(t) + a2 s2 (t) + · · · + an sn (t),
where a1 , a2 , · · · , an are coefficients determined by the device
and n is the highest order of nonlinearity.
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.7/

9. Generation of FM Waves (3)
Consider the input s(t) is an FM wave defined by
t
s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ .
0
Suppose that bandpass filter is designed to have a BW
equal to n times the BW of s(t).
After bandpass filtering of the output v(t), we have
t
s (t) = Ac cos 2πfc t + 2πkf m(τ )dτ ,
0
where fc = nfc and kf = nkf .
TELE3113 - Transmission of FM Waves. August 25, 2009. – p.8/

10. Demodulation of FM Waves
Recall that the FM signal is given by
t
s(t) = Ac cos 2πfc t + 2πkf m(τ )dτ .
0
After taking the derivative of s(t) with respect to t, we get
t
ds(t)
= −2πAc [fc + kf m(t)] sin 2πfc t + 2πkf m(τ )dτ .
dt 0
The derivative is indeed a band-pass signal with amplitude
modulation and the amplitude is 2πAc fc [1 + kf m(t)/fc ].
If fc is large enough such that the carrier is not
overmodulated, then we can recover the message signal
m(t) with an envelop detection. TELE3113 - Transmission of FM Waves. August 25, 2009. – p.9/