Angle Modulation

In frequency modulation, the instantaneous frequency of the carrier varies linearly with the message. For a single tone, the FM wave can be expanded with Bessel functions into a carrier and an infinite set of sidebands.

Instantaneous frequency and phase

Let $m(t) = A_m\cos\omega_mt$ and $c(t) = A_c\cos\omega_ct$. The instantaneous frequency is

$$f_i(t) = f_c + k_fm(t) = f_c + k_fA_m\cos\omega_mt = f_c + \Delta f\cos\omega_mt$$

where $k_f$ is the frequency sensitivity (Hz/V) and $\Delta f = k_fA_m$ is the peak frequency deviation. The phase is

$$\theta_i(t) = 2\pi\int_0^t f_i(\tau)\,d\tau = \omega_ct + \frac{\Delta f}{f_m}\sin\omega_mt = \omega_ct + \beta\sin\omega_mt$$

with modulation index $\beta = \Delta f/f_m$. So

$$s(t) = A_c\cos(\omega_ct + \beta\sin\omega_mt)$$

Complex envelope and Fourier series

$$s(t) = \text{Re}\left[A_c\,e^{j\beta\sin\omega_mt}\,e^{j\omega_ct}\right]$$

The term $\tilde s(t) = A_ce^{j\beta\sin\omega_mt}$ is periodic with period $T_m = 1/f_m$, so it has a Fourier series:

$$\tilde s(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_mt}$$ $$c_n = \frac{1}{T_m}\int_{-T_m/2}^{T_m/2} A_c e^{j\beta\sin\omega_mt}e^{-jn\omega_mt}dt$$

Put $x = \omega_mt$ ($dt = dx/\omega_m$, limits $-\pi$ to $\pi$):

$$c_n = \frac{A_c}{2\pi}\int_{-\pi}^{\pi} e^{j(\beta\sin x - nx)}dx = A_cJ_n(\beta)$$

where $J_n(\beta)$ is the Bessel function of the first kind of order n:

$$J_n(\beta) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx$$

Final expression

$$\tilde s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)e^{jn\omega_mt}$$ $$s(t) = \text{Re}\left[A_c\sum_n J_n(\beta)e^{j(\omega_c + n\omega_m)t}\right] = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[(\omega_c + n\omega_m)t]$$

Expanded, using $J_{-n}(\beta) = (-1)^nJ_n(\beta)$:

$$\begin{aligned} s(t) &= A_cJ_0(\beta)\cos\omega_ct \\ &\quad + A_cJ_1(\beta)[\cos(\omega_c+\omega_m)t - \cos(\omega_c-\omega_m)t] \\ &\quad + A_cJ_2(\beta)[\cos(\omega_c+2\omega_m)t + \cos(\omega_c-2\omega_m)t] + \dots \end{aligned}$$

Conclusions

1. The spectrum has a carrier at $f_c$ (amplitude $A_cJ_0(\beta)$) and infinite sidebands at $f_c \pm nf_m$ with amplitudes $A_c|J_n(\beta)|$.
2. Odd-order lower sidebands have reversed sign.
3. The carrier amplitude depends on $\beta$ and becomes zero when $J_0(\beta) = 0$ (β = 2.405, 5.52, ...).
4. Since $\sum J_n^2(\beta) = 1$, total power is constant: $P = A_c^2/2$.
5. Only about $\beta + 1$ sidebands on each side are significant, giving Carson's rule $B_T \approx 2(\beta+1)f_m = 2(\Delta f + f_m)$.

Pre-emphasis is the boosting of the high-frequency part of the message at the FM transmitter before modulation; de-emphasis is the matching attenuation of those frequencies at the receiver after demodulation.

Why they are needed

• In FM, the output noise power spectral density rises with the square of frequency (parabolic noise spectrum): $S_N(f) \propto f^2$.
• Audio signals have most of their energy at low frequencies; the high-frequency parts are weak.
• So the high audio frequencies have the worst SNR. Pre-emphasis raises them above the noise before transmission; de-emphasis restores the original balance and, at the same time, cuts the high-frequency noise.

Circuits

 Pre-emphasis (high-pass)   De-emphasis (low-pass)
   +--[ R ]--+                o--[ R ]--+----o
 o-+         +--+--o                    |
   +--||-----+  |                      === C
       C        > r                     |
 o--------------+--o          o---------+----o

• Pre-emphasis: $H_{pe}(f) = 1 + j\dfrac{f}{f_1}$, where $f_1 = \dfrac{1}{2\pi RC}$.
• De-emphasis: $H_{de}(f) = \dfrac{1}{1 + jf/f_1}$, so $H_{pe}H_{de} = 1$ and the message is undistorted.
• Standard time constant $RC = 75\ \mu$s ($f_1 \approx 2.1$ kHz) in USA FM broadcasting; 50 µs in Europe and many other countries.

 Gain
  ^      pre-emphasis
  |          ___/
  |      ___/
  |-----/---------- 0 dB
  |      \___
  |          \___ de-emphasis
  +-----+------------> f
        f1=2.1 kHz

Benefits

• Improves output SNR of FM by about 10–13 dB for audio.
• No extra bandwidth needed (for normal audio content).
• Also used in phonograph recording, tape recording and TV sound.

A stereo encoder (stereo multiplexer) combines the left (L) and right (R) audio channels into one composite baseband signal for FM stereo broadcasting, in a way that mono receivers can still use.

Block diagram

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

Working

1. A matrix circuit (adder and subtractor) forms $L+R$ (mono-compatible sum) and $L-R$ (difference).
2. Both are low-pass filtered to 15 kHz and pre-emphasized.
3. $L-R$ is DSB-SC modulated in a balanced modulator onto a 38 kHz sub-carrier, giving sidebands from 23 to 53 kHz.
4. The 38 kHz sub-carrier is made by doubling a 19 kHz crystal oscillator. A small 19 kHz pilot (about 10% deviation) is added so the receiver can regenerate the 38 kHz carrier in phase.
5. The composite signal is

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38000\,t) + K\cos(2\pi\,19000\,t)$$

Spectrum of composite signal

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

The total baseband bandwidth is 53 kHz, which then frequency-modulates the carrier. A mono receiver uses only 0–15 kHz ($L+R$); a stereo receiver recovers $L-R$ as well and forms $L$ and $R$ by adding and subtracting.

An FM superheterodyne receiver converts the received FM station (88–108 MHz) to a fixed intermediate frequency of 10.7 MHz, amplifies and limits it, and then recovers the audio with a frequency discriminator.

Block diagram

Ant
 |
[RF amp]->[Mixer]->[IF amp]->[Limiter]->[Discri-]
88-108M      ^     10.7 MHz             [ minator]
 |           |     BW 200k                  |
 |      [Local osc]<------AFC---------------+
 |       fLO=fs+10.7M                       |
 +--ganged--+               [De-emph]<------+
                               |
                            [AF amp]-->Spk

Working of each block

1. RF amplifier: tuned to the wanted station $f_s$. It gives gain at a low noise figure (important at VHF), improves sensitivity and rejects the image frequency $f_s + 2(10.7)$ MHz.
2. Local oscillator and mixer: the oscillator is ganged with the RF tuning so that $f_{LO} = f_s + 10.7$ MHz. The mixer produces sum and difference frequencies; the difference, 10.7 MHz, is selected. The frequency deviation (up to ±75 kHz) is unchanged by mixing.
3. IF amplifier: several fixed-tuned stages at 10.7 MHz with a bandwidth of about 200 kHz (Carson bandwidth for $\Delta f = 75$ kHz, $W = 15$ kHz is 180 kHz). Provides most of the gain and the selectivity.
4. Amplitude limiter: clips the IF signal to a constant amplitude, removing amplitude variations caused by noise, interference and fading. The information is in the frequency, so nothing useful is lost. This is the main reason FM is less noisy than AM.
5. Discriminator (FM detector): converts frequency variations into voltage variations, e.g. a Foster–Seeley discriminator, ratio detector, slope detector or PLL. Output is proportional to $k_fm(t)$.
6. De-emphasis network: an RC low-pass filter with 75 µs (or 50 µs) time constant. It undoes the transmitter's pre-emphasis and reduces high-frequency noise.
7. AF amplifier and speaker: amplifies the audio to drive the loudspeaker. In stereo sets a stereo decoder sits between discriminator and AF stage.
8. AFC (automatic frequency control): the DC part of the discriminator output shows any mistuning; it is fed back to a varactor in the local oscillator to keep the IF centred at 10.7 MHz.

Differences from AM superheterodyne

Example: for a station at 100 MHz, $f_{LO} = 110.7$ MHz and the image is at 121.4 MHz, outside the FM band.

Stereo FM sends two audio channels, left (L) and right (R), on one FM carrier using a composite baseband signal. It stays compatible with mono receivers because the sum $L+R$ is sent in the normal 0–15 kHz band.

Stereo encoder (transmitter)

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

1. Matrix: an adder and a subtractor form $L+R$ and $L-R$.
2. LPF and pre-emphasis: each signal is limited to 15 kHz and pre-emphasized.
3. Balanced modulator: $L-R$ DSB-SC modulates a 38 kHz sub-carrier. Since the carrier is suppressed, its power goes into useful sidebands (23–53 kHz).
4. Pilot: a 19 kHz crystal oscillator gives the 38 kHz sub-carrier through a frequency doubler. The 19 kHz pilot itself is also added at low level (about 10% of deviation). It lies in an empty gap (15–23 kHz), is easy to filter, and tells the receiver the exact frequency and phase of the 38 kHz sub-carrier.
5. Adder: forms the composite signal that frequency-modulates the RF carrier:

$$m(t) = [L(t)+R(t)] + [L(t)-R(t)]\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

Spectrum of the composite baseband

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

The composite bandwidth is 53 kHz (with optional SCA services up to about 75 kHz).

Stereo decoder (receiver)

FM    +->[LPF 0-15k]------------L+R--+->(+)--> 2L
demod |                              |   ^
out --+->[BPF 23-53k]-->( X )->[LPF]-+   |
      |                   ^      L-R  |  |
      |                   | 38k       +--|-->(-)--> 2R
      +->[BPF 19k]-->[ x2 ]              |
         (pilot)                  L-R ---+

1. The FM discriminator output is the composite signal $m(t)$.
2. LPF (0–15 kHz) gives $L+R$. A mono receiver stops here.
3. Narrow BPF at 19 kHz extracts the pilot; a frequency doubler (or PLL) regenerates the 38 kHz sub-carrier, locked in phase with the transmitter.
4. BPF (23–53 kHz) selects the $L-R$ DSB-SC signal. It is multiplied by the 38 kHz carrier and low-pass filtered (coherent detection), giving $L-R$.
5. De-matrixing:

$$(L+R) + (L-R) = 2L, \qquad (L+R) - (L-R) = 2R$$

6. Each channel is de-emphasized and amplified to its own speaker. A stereo indicator lamp lights when the pilot is detected.

Advantages: full mono compatibility, small extra bandwidth, simple pilot-based synchronization. Drawback: the stereo SNR is somewhat lower than mono, since the $L-R$ channel sits at higher baseband frequencies where FM noise is larger.

Given: $m(t) = 5\cos(18849.55t)$, so $A_m = 5$ V and

$$f_m = \frac{18849.55}{2\pi} = 3000\ \text{Hz} = 3\ \text{kHz}$$

$k_f = 9$ kHz/V, $A_c = 25$ V, $f_c = 88.7$ MHz, $R = 10\ \Omega$.

Peak frequency deviation:

$$\Delta f = k_fA_m = 9 \times 5 = 45\ \text{kHz}$$

a) Modulation index

$$\beta = \frac{\Delta f}{f_m} = \frac{45}{3} = 15$$

(wide-band FM, since $\beta \gg 1$)

b) Carrier frequency swing

The carrier swings from $f_c - \Delta f$ to $f_c + \Delta f$, i.e. from 88.655 MHz to 88.745 MHz:

$$\text{Carrier swing} = 2\Delta f = 2 \times 45 = 90\ \text{kHz}$$

c) Bandwidth by Carson's rule

$$B_T = 2(\Delta f + f_m) = 2(45 + 3) = 96\ \text{kHz}$$

d) Total power in 10 Ω

The FM wave has constant amplitude, so its power does not depend on modulation:

$$P = \frac{A_c^2}{2R} = \frac{25^2}{2 \times 10} = \frac{625}{20} = 31.25\ \text{W}$$

Answer: $\beta = 15$; carrier swing = 90 kHz; $B_T = 96$ kHz; $P = 31.25$ W.

Role of the amplitude limiter

In a limiter–discriminator FM detector, the amplitude limiter removes all amplitude variations of the received FM signal before it reaches the discriminator.

• Noise, interference and fading change the amplitude of the FM wave, but the information is only in its frequency.
• A discriminator's output depends on both frequency and amplitude, so amplitude changes would appear as noise and distortion.
• The limiter (a hard clipper followed by a band-pass filter at the IF) produces a constant-amplitude signal with the same zero crossings, i.e. the same instantaneous frequency.

 FM + AM noise  -> [Hard limiter] -> [BPF] -> constant-
 /\/\/\  /\/\        square-ish     at fc    amplitude FM

So the discriminator sees only frequency variations, which gives FM its noise immunity.

FM demodulation using PLL

A phase-locked loop is a feedback system in which a voltage-controlled oscillator (VCO) is forced to follow the phase, and hence the frequency, of the input. The control voltage needed to do this is the demodulated message.

 s(t) -->[ Phase    ]-->[ Loop  ]--+--> v(t) = output
         [ detector ]   [ filter]  |    (message)
              ^                    |
              |                    |
              +------[  VCO  ]<----+

Blocks

• Phase detector (multiplier): compares input and VCO phases and gives an error voltage.
• Loop filter: low-pass; removes the $2f_c$ term and sets loop dynamics.
• VCO: its frequency is $f_c + k_vv(t)$.

Analysis

Input FM wave and VCO output:

$$s(t) = A_c\sin[\omega_ct + \phi_1(t)], \quad \phi_1(t) = 2\pi k_f\int_0^t m(\tau)d\tau$$ $$r(t) = A_v\cos[\omega_ct + \phi_2(t)], \quad \phi_2(t) = 2\pi k_v\int_0^t v(\tau)d\tau$$

The multiplier output, after removing the $2\omega_c$ term, is

$$e(t) = K_m\sin[\phi_1(t) - \phi_2(t)] = K_m\sin\phi_e(t)$$

When the loop is locked, $\phi_e$ is small, so $\sin\phi_e \approx \phi_e$ (linear model). With high loop gain the loop drives $\phi_e \to 0$, so

$$\phi_2(t) \approx \phi_1(t) \;\Rightarrow\; 2\pi k_v\int v(t)dt \approx 2\pi k_f\int m(t)dt$$

Differentiating both sides:

$$v(t) \approx \frac{k_f}{k_v}m(t)$$

The loop filter output is therefore proportional to the message. The PLL demodulator has good threshold performance, needs no tuned circuits, and is available as a single IC (e.g. 565).

A phase-locked loop (PLL) is a negative-feedback system whose voltage-controlled oscillator (VCO) tracks the phase of the input signal. When the input is an FM wave, the voltage that steers the VCO is proportional to the message, so the PLL acts as an FM demodulator.

Block diagram

  FM in   +-----------+   e(t)  +--------+  v(t)
 s(t) --->|  Phase    |-------->|  Loop  |---+---> output
          | detector  |         | filter |   |    ~ m(t)
          +-----------+         | H(f)   |   |
               ^                +--------+   |
               |   r(t)   +-------+          |
               +----------|  VCO  |<---------+
                          +-------+

Blocks

1. Phase detector: a multiplier comparing input and VCO output; its output depends on the phase difference.
2. Loop filter: a low-pass filter that removes the double-frequency term and controls loop bandwidth and stability.
3. VCO: free-running at $f_c$ when $v = 0$; its frequency changes by $k_v$ Hz per volt.

Analysis

Input FM signal:

$$s(t) = A_c\sin[\omega_ct + \phi_1(t)], \qquad \phi_1(t) = 2\pi k_f\int_0^t m(\tau)\,d\tau$$

VCO output (90° offset so the error is zero at lock):

$$r(t) = A_v\cos[\omega_ct + \phi_2(t)], \qquad \phi_2(t) = 2\pi k_v\int_0^t v(\tau)\,d\tau$$

Multiplier output:

$$s(t)r(t) = \frac{A_cA_v}{2}\left\{\sin[2\omega_ct + \phi_1 + \phi_2] + \sin[\phi_1 - \phi_2]\right\}$$

The loop filter removes the $2\omega_c$ term. The error signal is

$$e(t) = K_m\sin\phi_e(t), \qquad \phi_e = \phi_1 - \phi_2$$

where $K_m = A_cA_v/2$ (times the multiplier gain).

Linear model (loop in lock)

In lock $\phi_e$ is small, so $\sin\phi_e \approx \phi_e$:

 phi1 -->(+)--> K_m --> H(f) --+--> v(t)
          ^ -                  |
          |                    |
         phi2 <--[2 pi kv / s]-+

$$v(t) = K_m\,[\phi_e(t) * h(t)], \qquad \phi_2(t) = 2\pi k_v\int v\,dt$$

In the frequency domain,

$$\Phi_e(f) = \frac{\Phi_1(f)}{1 + L(f)}, \qquad L(f) = \frac{K_mk_vH(f)}{jf}$$

When the loop gain $|L(f)| \gg 1$ over the message band, $\Phi_e \approx 0$, so $\phi_2(t) \approx \phi_1(t)$:

$$2\pi k_v\int v(t)\,dt = 2\pi k_f\int m(t)\,dt \;\Rightarrow\; v(t) = \frac{k_f}{k_v}\,m(t)$$

So the loop filter output is a scaled copy of the message.

Physical explanation

• If the input frequency rises, the phase error grows; the error voltage increases and pushes the VCO frequency up until it matches.
• Thus the control voltage always follows the instantaneous input frequency, $f_c + k_fm(t)$, i.e. it follows $m(t)$.

Lock and capture range

• Lock range: frequency range over which a locked loop stays locked.
• Capture range: range over which an unlocked loop can acquire lock (smaller than lock range).
• The peak deviation $\Delta f$ must lie within the lock range.

Advantages

• No tuned LC circuits; easy IC realization (e.g. NE565).
• Better (lower) FM threshold than a limiter–discriminator.
• Good linearity and tracking of slow carrier drift.

Why pre-emphasis and de-emphasis are used

In FM, noise at the discriminator output is not flat: its power spectral density rises as $f^2$ (parabolic noise). Audio signals, on the other hand, have most energy at low frequencies and little at high frequencies. So the high audio frequencies have a very poor SNR.

• Pre-emphasis (at the transmitter, before the modulator): a high-pass RC network, $H_{pe}(f) = 1 + jf/f_1$, boosts high audio frequencies so that they stand well above the noise.
• De-emphasis (at the receiver, after the discriminator): a low-pass RC network, $H_{de}(f) = 1/(1 + jf/f_1)$, brings the boosted frequencies back to their original level. It attenuates the high-frequency noise by the same amount.
• Since $H_{pe}H_{de} = 1$, the message is undistorted, but the noise is reduced. The improvement is about 10–13 dB.
• Standard time constant: 75 µs ($f_1 = 2.12$ kHz) in USA, 50 µs in Europe and most other countries.

 Pre-emph          De-emph
 o-+-[R]-+-o       o-[R]-+-o
   +-||--+ |             |
     C     > r          === C
 o---------+-o     o-----+-o

Stereo encoder

A stereo encoder forms one composite baseband signal from the left and right audio channels for FM stereo broadcasting.

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

1. Matrix network: adder and subtractor make $L+R$ (for mono compatibility) and $L-R$.
2. LPF (15 kHz) and pre-emphasis on both.
3. Balanced modulator: $L-R$ DSB-SC modulates a 38 kHz sub-carrier (sidebands 23–53 kHz).
4. 19 kHz pilot oscillator and doubler: gives the 38 kHz sub-carrier; the 19 kHz pilot is also added at low level so receivers can regenerate the 38 kHz carrier.
5. Adder: composite signal

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

This 53 kHz composite signal frequency-modulates the main carrier.

Armstrong's method is an indirect method of generating wide-band FM: first a narrow-band FM (NBFM) wave is made with a crystal-controlled phase modulator, then frequency multipliers and a mixer raise the deviation and set the carrier frequency.

 m(t)->[Integrator]->[Balanced]--( Σ )-> NBFM
                     [modulator] ^ -
                         ^       | +
          +--[-90 deg]---+       |
 [Crystal osc fc1]---------------+
          (NB phase modulator)

 NBFM ->[ x n1 ]->[ Mixer ]->[ x n2 ]->[PA]-> WBFM
                      ^
                 [Osc fc2]

Steps

1. NBFM generation: the message is integrated and applied to a balanced modulator with a −90° shifted carrier. The output is added to the carrier, giving

$$s_{NB}(t) \approx A_c\cos\omega_{c1}t - \beta A_c\sin\omega_mt\,\sin\omega_{c1}t, \quad \beta < 0.3$$

which is NBFM. A crystal oscillator keeps $f_{c1}$ very stable. 2. Frequency multiplication (×n₁): a non-linear stage multiplies both carrier and deviation: $n_1f_{c1}$, $n_1\Delta f_1$. 3. Mixer: shifts the carrier down to a convenient value ($n_1f_{c1} - f_{c2}$) without changing the deviation. 4. Second multiplier (×n₂): gives the final carrier $f_c = n_2(n_1f_{c1} - f_{c2})$ and deviation $\Delta f = n_1n_2\Delta f_1$ (e.g. 75 kHz).

Merit: excellent carrier stability (crystal oscillator). Demerit: many multiplier stages; noise and distortion may increase.

Pre-emphasis and de-emphasis are needed in FM to improve the signal-to-noise ratio at high audio frequencies.

1. Noise in FM rises with frequency. The FM discriminator output noise PSD is $S_{N}(f) \propto f^2$ (triangular noise voltage, parabolic noise power). So high-frequency audio components get much more noise.
2. Audio signals are weak at high frequencies. Most speech and music energy is below 1–2 kHz.
3. Together, these make the SNR of the high audio frequencies very poor.

Solution

• Pre-emphasis at the transmitter: a high-pass RC network boosts high audio frequencies before modulation, $H_{pe}(f) = 1 + jf/f_1$.
• De-emphasis at the receiver: a low-pass RC network after demodulation, $H_{de}(f) = 1/(1 + jf/f_1)$, restores the original response and attenuates the high-frequency noise at the same time.
• Since $H_{pe}H_{de} = 1$, the signal is unchanged but the output noise falls, giving about 10–13 dB SNR improvement. Standard time constant: 75 µs (USA) or 50 µs (Europe), $f_1 = 1/(2\pi RC) \approx 2.1$ kHz for 75 µs.

 Gain   pre-emph  _/
  |           __/
  |----------/--------- 0 dB
  |          \__
  |             \_ de-emph
  +---------+---------> f
           f1

Relation between PSD and autocorrelation

By the Wiener–Khinchin theorem, the power spectral density (PSD) $S_X(f)$ of a wide-sense stationary process and its autocorrelation function $R_X(\tau)$ form a Fourier transform pair:

$$S_X(f) = \int_{-\infty}^{\infty}R_X(\tau)\,e^{-j2\pi f\tau}d\tau, \qquad R_X(\tau) = \int_{-\infty}^{\infty}S_X(f)\,e^{j2\pi f\tau}df$$

Consequences:

• Total average power: $P = R_X(0) = \int S_X(f)\,df$.
• $S_X(f)$ is real, even and non-negative because $R_X(\tau)$ is real and even.
• Example: white noise $R(\tau) = \frac{N_0}{2}\delta(\tau) \Leftrightarrow S(f) = \frac{N_0}{2}$.

Stereo FM encoder

Stereo FM sends left (L) and right (R) channels on one carrier while staying compatible with mono receivers.

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

1. Matrix forms $L+R$ and $L-R$; both are band-limited to 15 kHz and pre-emphasized.
2. $L-R$ DSB-SC modulates a 38 kHz sub-carrier in a balanced modulator.
3. The 38 kHz sub-carrier comes from a 19 kHz oscillator and a doubler; a small 19 kHz pilot is also added.
4. Composite signal:

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

Spectrum

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

Stereo FM decoder

FM    +->[LPF 0-15k]------------L+R--+->(+)--> 2L
demod |                              |   ^
out --+->[BPF 23-53k]-->( X )->[LPF]-+   |
      |                   ^      L-R  |  |
      |                   | 38k       +--|-->(-)--> 2R
      +->[BPF 19k]-->[ x2 ]              |
         (pilot)                  L-R ---+

1. FM discriminator gives the composite signal.
2. LPF (0–15 kHz) gives $L+R$ (all a mono set needs).
3. A narrow BPF picks the 19 kHz pilot; a doubler makes a phase-locked 38 kHz carrier.
4. BPF (23–53 kHz) selects the $L-R$ DSB-SC signal, which is coherently detected with the 38 kHz carrier and low-pass filtered.
5. Matrix: $(L+R)+(L-R) = 2L$ and $(L+R)-(L-R) = 2R$, then de-emphasis and audio amplifiers.

Compare with the standard single-tone angle-modulated wave $v(t) = A_c\sin(\omega_ct + \beta\sin\omega_mt)$:

$A_c = 12$ V, $\omega_c = 10^6$ rad/s, $\omega_m = 10^4$ rad/s, $\beta = 5$.

a) Carrier frequency

$$f_c = \frac{\omega_c}{2\pi} = \frac{10^6}{2\pi} = 159.15\ \text{kHz}$$

b) Modulating frequency

$$f_m = \frac{\omega_m}{2\pi} = \frac{10^4}{2\pi} = 1591.55\ \text{Hz} \approx 1.59\ \text{kHz}$$

c) Modulation index

The peak phase deviation is the coefficient of $\sin 10^4t$:

$$\beta = 5\ \text{rad}$$

(Treated as FM, the peak deviation is $\Delta f = \beta f_m = 5 \times 1591.55 = 7957.7$ Hz.)

d) Power in 100 Ω

The amplitude of an angle-modulated wave is constant, so

$$P = \frac{A_c^2}{2R} = \frac{12^2}{2 \times 100} = \frac{144}{200} = 0.72\ \text{W}$$

Answer: $f_c = 159.15$ kHz, $f_m = 1.59$ kHz, $\beta = 5$, $P = 0.72$ W.

Given: $A_m = 10$ V, $f_m = 4000$ Hz, $A_c = 80$ V, $f_c = 10^7$ Hz = 10 MHz, $k_f = 400$ Hz/V.

a) Peak frequency deviation

$$\Delta f = k_fA_m = 400 \times 10 = 4000\ \text{Hz} = 4\ \text{kHz}$$

b) Modulation index

$$\beta = \frac{\Delta f}{f_m} = \frac{4000}{4000} = 1$$

c) Bandwidth containing over 98% of power

Carson's rule is based on keeping about 98% of the power:

$$B_T = 2(\Delta f + f_m) = 2(4 + 4) = 16\ \text{kHz}$$

Check with Bessel functions for $\beta = 1$: $J_0 = 0.7652$, $J_1 = 0.4401$, $J_2 = 0.1149$, $J_3 = 0.0196$.

So sidebands up to $n = 2$ are needed for more than 98% power:

$$B = 2nf_m = 2 \times 2 \times 4 = 16\ \text{kHz}$$

which agrees with Carson's rule.

d) Total power in unit impedance

FM has constant amplitude, so

$$P = \frac{A_c^2}{2R} = \frac{80^2}{2 \times 1} = 3200\ \text{W}$$

Answer: $\Delta f = 4$ kHz; $\beta = 1$; $B = 16$ kHz; $P = 3200$ W.

NBFM versus WBFM

Stereo FM broadcasting

Stereo FM transmits left (L) and right (R) audio on one carrier, while mono receivers can still receive the program.

Encoder (transmitter)

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

1. Matrix makes $L+R$ and $L-R$, each limited to 15 kHz and pre-emphasized.
2. $L-R$ DSB-SC modulates a 38 kHz sub-carrier (balanced modulator).
3. 38 kHz is obtained by doubling a 19 kHz oscillator; the 19 kHz pilot is added at low level for receiver synchronization.
4. The composite signal

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

frequency-modulates the RF carrier (peak deviation 75 kHz).

Spectrum

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

Decoder (receiver)

FM    +->[LPF 0-15k]------------L+R--+->(+)--> 2L
demod |                              |   ^
out --+->[BPF 23-53k]-->( X )->[LPF]-+   |
      |                   ^      L-R  |  |
      |                   | 38k       +--|-->(-)--> 2R
      +->[BPF 19k]-->[ x2 ]              |
         (pilot)                  L-R ---+

• LPF gives $L+R$ (mono output).
• Pilot is filtered and doubled to 38 kHz; the 23–53 kHz band is coherently detected to give $L-R$.
• Adding and subtracting give $2L$ and $2R$.

In the Armstrong method, frequency multipliers multiply both carrier frequency and deviation; the mixer shifts the carrier frequency but leaves the deviation unchanged.

Diagram

 NBFM        x65          Mixer       x50
 200 kHz -->[n1]-->13 MHz-->[X]-->2.2M-->[n2]--> WBFM
 df=25 Hz       df=1625 Hz   ^   df=1625    110 MHz
                             |              df=81.25k
                      [Osc 10.8 MHz]

Step 1: after first multiplier (n₁ = 65)

$$\begin{aligned} f_1 &= 65 \times 200\ \text{kHz} = 13\ \text{MHz} \\ \Delta f_1 &= 65 \times 25 = 1625\ \text{Hz} \end{aligned}$$

Step 2: mixer with 10.8 MHz

The mixer gives $13 \pm 10.8$ MHz, i.e. 2.2 MHz (difference) or 23.8 MHz (sum). Deviation stays 1625 Hz.

Step 3: second multiplier (n₂ = 50)

$$\begin{aligned} \Delta f &= 50 \times 1625 = 81250\ \text{Hz} = 81.25\ \text{kHz} \\ f_c &= 50 \times 2.2 = 110\ \text{MHz} \quad (\text{difference}) \\ f_c &= 50 \times 23.8 = 1190\ \text{MHz} \quad (\text{sum}) \end{aligned}$$

Valid carrier frequency

The sum term gives 1190 MHz (UHF), which is far outside the VHF FM broadcast range and is rejected by filtering after the mixer. The difference term is the usable one, giving

Answer: maximum frequency deviation Δf = 81.25 kHz; valid WBFM carrier f_c = 110 MHz (from the 2.2 MHz difference output).

Note: 110 MHz lies just above the 88–108 MHz broadcast band and 81.25 kHz is slightly above the 75 kHz standard, so in practice the oscillator frequency or multiplier would be adjusted slightly; with the given data, 110 MHz is the only practical (VHF) choice.

Narrow band and wide band FM

• Narrow band FM (NBFM): FM with a small modulation index, $\beta = \Delta f/f_m \ll 1$ (practically $\beta < 0.3$). Its bandwidth is about $2f_m$, like AM, with only the carrier and one pair of sidebands:

$$s(t) \approx A_c\cos\omega_ct - \beta A_c\sin\omega_mt\,\sin\omega_ct$$

• Wide band FM (WBFM): FM with $\beta \gg 1$. It has many significant sidebands and bandwidth $B_T \approx 2(\Delta f + f_m)$ (Carson's rule). Example: FM broadcast with $\Delta f = 75$ kHz, $W = 15$ kHz, $\beta = 5$.

Generation of WBFM by Armstrong (indirect) method

Direct FM using an LC oscillator is not stable enough. Armstrong's method first makes a stable NBFM wave with a crystal oscillator, then uses frequency multiplication to obtain the required large deviation.

 m(t)->[Integrator]->[Balanced]--( Σ )-> NBFM
                     [modulator] ^ -
                         ^       | +
          +--[-90 deg]---+       |
 [Crystal osc fc1]---------------+
          (NB phase modulator)

 NBFM ->[ x n1 ]->[ Mixer ]->[ x n2 ]->[PA]-> WBFM
                      ^
                 [Osc fc2]

Step 1: NBFM by phase modulation of the integrated message. The message is integrated (so that phase modulation of $\int m\,dt$ gives FM) and fed to a balanced modulator with a carrier shifted by −90°. Adding the carrier gives

$$s_1(t) = A_c\cos\omega_{c1}t - A_c\beta_1\sin\omega_mt\,\sin\omega_{c1}t \approx A_c\cos(\omega_{c1}t + \beta_1\sin\omega_mt)$$

with $\beta_1$ kept small (< 0.5) to limit distortion. Typical values: $f_{c1} = 100$–200 kHz, $\Delta f_1 \approx 25$ Hz.

Step 2: frequency multiplier (×n₁). A non-linear device followed by a BPF tuned to the n-th harmonic. The output is $\cos(n_1\omega_{c1}t + n_1\beta_1\sin\omega_mt)$: both carrier and deviation are multiplied by $n_1$.

Step 3: mixer (frequency converter). Mixing with a crystal oscillator $f_{c2}$ shifts the carrier to $n_1f_{c1} - f_{c2}$ while the deviation stays $n_1\Delta f_1$. This lets the final carrier be set independently of the deviation.

Step 4: frequency multiplier (×n₂) to reach the final values:

$$f_c = n_2(n_1f_{c1} - f_{c2}), \qquad \Delta f = n_1n_2\Delta f_1$$

Step 5: power amplifier and antenna.

Example: $\Delta f_1 = 25$ Hz needs $n_1n_2 = 3000$ for $\Delta f = 75$ kHz.

Advantages: high carrier stability from crystal oscillators; good linearity. Disadvantages: many multiplier stages; noise is multiplied too; more complex.

Stereo FM carries left (L) and right (R) audio on one FM carrier as a composite baseband signal, keeping compatibility with mono receivers.

Encoder

 L --+-->(+)--L+R-->[LPF]----------->( Σ )-->
     |    ^                            ^ ^  to FM
 R --|----+                            | |  mod.
     +-->(-)--L-R-->[LPF]-->[Bal.mod]--+ |
          ^ (R)                 ^ 38k    |
                  [19k osc]->[x2]        |
                      |                  |
                      +--19 kHz pilot----+

• Matrix (adder/subtractor): forms $L+R$ and $L-R$.
• LPF (15 kHz) and pre-emphasis: band-limit each audio signal and boost high frequencies.
• 19 kHz pilot oscillator: crystal-controlled reference.
• Frequency doubler (×2): gives the 38 kHz sub-carrier, phase-locked to the pilot.
• Balanced modulator: DSB-SC modulates $L-R$ on 38 kHz, producing 23–53 kHz.
• Adder: sums $L+R$, the DSB-SC $L-R$ and a low-level 19 kHz pilot into the composite signal, which drives the FM modulator.

$$m(t) = (L+R) + (L-R)\cos(2\pi\,38\text{k}\,t) + A_p\cos(2\pi\,19\text{k}\,t)$$

 |  L+R      pilot    L-R DSB-SC
 |______       |    ____    ____
 |      |      |   |    |  |    |
 +------+------+---+----+--+----+---> f (kHz)
 0     15     19  23   38      53

Decoder

FM    +->[LPF 0-15k]------------L+R--+->(+)--> 2L
demod |                              |   ^
out --+->[BPF 23-53k]-->( X )->[LPF]-+   |
      |                   ^      L-R  |  |
      |                   | 38k       +--|-->(-)--> 2R
      +->[BPF 19k]-->[ x2 ]              |
         (pilot)                  L-R ---+

• FM discriminator: recovers the composite signal.
• LPF (0–15 kHz): extracts $L+R$ (mono signal).
• Narrow BPF (19 kHz): extracts the pilot.
• Doubler (×2): regenerates a 38 kHz carrier in phase with the transmitter's.
• BPF (23–53 kHz): selects the $L-R$ DSB-SC signal.
• Product detector and LPF: coherent detection with 38 kHz gives $L-R$.
• Matrix: $(L+R)+(L-R) = 2L$, $(L+R)-(L-R) = 2R$.
• De-emphasis and AF amplifiers: restore flat response and drive the left and right speakers.

Properties of Bessel functions $J_n(\beta)$

1. $J_{-n}(\beta) = (-1)^nJ_n(\beta)$: negative-order coefficients equal positive ones, with sign change for odd $n$.
2. $\sum_{n=-\infty}^{\infty}J_n^2(\beta) = 1$: total FM power is constant.
3. For small $\beta$: $J_0(\beta) \approx 1$, $J_1(\beta) \approx \beta/2$, $J_n(\beta) \approx 0$ for $n \ge 2$ (NBFM).
4. For a given $\beta$, $J_n(\beta)$ becomes very small when $n > \beta + 1$, so only a finite number of sidebands are significant.
5. $J_n(\beta)$ is oscillatory; $J_0(\beta) = 0$ at β = 2.405, 5.52, 8.65, ... (carrier disappears).

FM wave as a sum of cosines at $f_c + nf_m$

Single-tone FM:

$$s(t) = A_c\cos(2\pi f_ct + \beta\sin2\pi f_mt), \qquad \beta = \frac{\Delta f}{f_m}$$

Write it using the complex envelope:

$$s(t) = \text{Re}\left[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}\right]$$

$\tilde s(t) = A_ce^{j\beta\sin\omega_mt}$ is periodic in $T_m = 1/f_m$, so

$$\tilde s(t) = \sum_{n=-\infty}^{\infty}c_ne^{jn\omega_mt}, \qquad c_n = f_m\int_{-1/2f_m}^{1/2f_m}A_ce^{j\beta\sin\omega_mt}e^{-jn\omega_mt}dt$$

Put $x = \omega_mt$:

$$c_n = \frac{A_c}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx = A_cJ_n(\beta)$$

Hence

$$s(t) = \text{Re}\left[\sum_{n=-\infty}^{\infty}A_cJ_n(\beta)e^{j2\pi(f_c + nf_m)t}\right] = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c + nf_m)t]$$

This proves that the FM signal is a sum of infinitely many cosine components at $f_c + nf_m$, $n = 0, \pm1, \pm2, \dots$, with amplitudes $A_cJ_n(\beta)$.

Spectrum:

$$S(f) = \frac{A_c}{2}\sum_{n}J_n(\beta)[\delta(f - f_c - nf_m) + \delta(f + f_c + nf_m)]$$

         |     |     |     |     |
     |   |     |     |     |     |   |
 ----+---+-----+-----+-----+-----+---+---> f
   fc-3fm  fc-fm    fc   fc+fm  fc+2fm
 (lines spaced fm, heights A_c|J_n(beta)|)

Although infinite in theory, sidebands with $|n| > \beta + 1$ are negligible, giving Carson's bandwidth $2(\beta + 1)f_m$.

Angle modulation

Angle modulation is modulation in which the angle (phase) of the carrier is varied according to the message while its amplitude stays constant:

$$s(t) = A_c\cos\theta_i(t)$$

It has two forms: phase modulation (PM), $\theta_i = \omega_ct + k_pm(t)$, and frequency modulation (FM), $\theta_i = \omega_ct + 2\pi k_f\int m(\tau)d\tau$.

Time-domain expression of single-tone FM

Let $m(t) = A_m\cos2\pi f_mt$. Instantaneous frequency:

$$f_i(t) = f_c + k_fA_m\cos2\pi f_mt = f_c + \Delta f\cos2\pi f_mt$$

Instantaneous phase:

$$\theta_i(t) = 2\pi\int_0^tf_i(\tau)d\tau = 2\pi f_ct + \frac{\Delta f}{f_m}\sin2\pi f_mt$$

So

$$s(t) = A_c\cos(2\pi f_ct + \beta\sin2\pi f_mt), \qquad \beta = \frac{\Delta f}{f_m} = \frac{k_fA_m}{f_m}$$

Frequency-domain expression

Write $s(t) = \text{Re}[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}]$. The periodic term has Fourier series

$$e^{j\beta\sin\omega_mt} = \sum_{n=-\infty}^{\infty}J_n(\beta)e^{jn\omega_mt}, \quad J_n(\beta) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx$$

Therefore

$$s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c + nf_m)t]$$

Taking the Fourier transform:

$$S(f) = \frac{A_c}{2}\sum_{n=-\infty}^{\infty}J_n(\beta)\left[\delta(f - f_c - nf_m) + \delta(f + f_c + nf_m)\right]$$

The spectrum has a carrier of amplitude $A_cJ_0(\beta)$ and sideband pairs at $f_c \pm nf_m$ with amplitudes $A_c|J_n(\beta)|$. Total power is $A_c^2/2$, and practical bandwidth is $2(\beta + 1)f_m$.

Given: $f_c = 102.4$ MHz, $f_m = 5$ kHz, $\Delta f = 75$ kHz.

(a) Carrier swing

The carrier frequency swings between $f_c - \Delta f = 102.325$ MHz and $f_c + \Delta f = 102.475$ MHz:

$$\text{Carrier swing} = 2\Delta f = 2 \times 75 = 150\ \text{kHz}$$

(b) Bandwidth

By Carson's rule:

$$B_T = 2(\Delta f + f_m) = 2(75 + 5) = 160\ \text{kHz}$$

(Bessel-table check: for $\beta = 15$, significant sidebands extend to about $n = \beta + 1 = 16$, so $B = 2 \times 16 \times 5 = 160$ kHz.)

(c) Modulation index

$$\beta = \frac{\Delta f}{f_m} = \frac{75}{5} = 15$$

Answer: carrier swing = 150 kHz; bandwidth = 160 kHz; $\beta = 15$.

Single-tone FM in terms of Bessel functions

For $m(t) = A_m\cos\omega_mt$, the instantaneous frequency is $f_i = f_c + k_fA_m\cos\omega_mt$, so the phase is $\omega_ct + \beta\sin\omega_mt$ with $\beta = k_fA_m/f_m = \Delta f/f_m$:

$$s(t) = A_c\cos(\omega_ct + \beta\sin\omega_mt) = \text{Re}\left[A_ce^{j\beta\sin\omega_mt}e^{j\omega_ct}\right]$$

$e^{j\beta\sin\omega_mt}$ is periodic with period $1/f_m$; its Fourier series is

$$e^{j\beta\sin\omega_mt} = \sum_{n=-\infty}^{\infty}c_ne^{jn\omega_mt}, \qquad c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(\beta\sin x - nx)}dx = J_n(\beta)$$

(substituting $x = \omega_mt$). Therefore

$$s(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[(\omega_c + n\omega_m)t]$$

i.e. a carrier $A_cJ_0(\beta)$ at $f_c$ and sidebands $A_cJ_n(\beta)$ at $f_c + nf_m$.

Average power with components from −n to +n

Each component is a cosine of amplitude $A_cJ_k(\beta)$; the components are at different frequencies, so their powers add (1 Ω load):

$$P_n = \sum_{k=-n}^{n}\frac{A_c^2J_k^2(\beta)}{2} = \frac{A_c^2}{2}\left[J_0^2(\beta) + 2\sum_{k=1}^{n}J_k^2(\beta)\right]$$

using $J_{-k}^2 = J_k^2$. When all components are included, $\sum_{k=-\infty}^{\infty}J_k^2(\beta) = 1$, so

$$P = \frac{A_c^2}{2}$$

the same as the unmodulated carrier. The fraction of power within $\pm n$ sidebands is $J_0^2 + 2\sum_{1}^{n}J_k^2$. Example: for $\beta = 1$, $n = 2$ gives $0.7652^2 + 2(0.4401^2 + 0.1149^2) = 0.9992$, i.e. 99.9% of the power.