Amplitude Modulation

A single-sideband (SSB) signal contains only one sideband (upper or lower) of a DSB-SC signal, so it needs only bandwidth $W$, half that of DSB.

Derivation of SSB in terms of Hilbert transform

Tools: the Hilbert transform $\hat m(t)$ of $m(t)$ has $\hat M(f)=-j\,\text{sgn}(f)M(f)$. The pre-envelope (analytic signal) is

$$m_+(t)=m(t)+j\hat m(t)\ \leftrightarrow\ M(f)+\text{sgn}(f)M(f)=2M(f)u(f)$$

So $m_+(t)$ contains only the positive-frequency part of $M(f)$, doubled. Similarly $m_-(t)=m(t)-j\hat m(t)\leftrightarrow 2M(f)u(-f)$ contains only negative frequencies.

Upper sideband: the USB spectrum is $M(f)$ shifted right by $f_c$ for its positive half and left by $f_c$ for its negative half:

$$S_{USB}(f)=\frac{A_c}{4}\left[2M(f-f_c)u(f-f_c)+2M(f+f_c)u(-f-f_c)\right]$$

Taking the inverse transform term by term:

$$\begin{aligned} s_{USB}(t)&=\frac{A_c}{4}\left[m_+(t)e^{j2\pi f_ct}+m_-(t)e^{-j2\pi f_ct}\right]\\ &=\frac{A_c}{4}\left[(m+j\hat m)e^{j\omega_ct}+(m-j\hat m)e^{-j\omega_ct}\right]\\ &=\frac{A_c}{4}\left[2m(t)\cos\omega_ct-2\hat m(t)\sin\omega_ct\right]\\ &=\frac{A_c}{2}\left[m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct\right] \end{aligned}$$

Lower sideband: using $m_-$ at $+f_c$ and $m_+$ at $-f_c$, in the same way:

$$s_{LSB}(t)=\frac{A_c}{2}\left[m(t)\cos\omega_ct+\hat m(t)\sin\omega_ct\right]$$

Combined: $s_{SSB}(t)=\frac{A_c}{2}\left[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct\right]$ (– for USB, + for LSB).

Check with a tone: $m=A_m\cos\omega_mt$, $\hat m=A_m\sin\omega_mt$. Then $s_{USB}=\frac{A_cA_m}{2}[\cos\omega_mt\cos\omega_ct-\sin\omega_mt\sin\omega_ct]=\frac{A_cA_m}{2}\cos(\omega_c+\omega_m)t$, a single line at $f_c+f_m$, as expected.

Generation: phase-shift (phase discrimination) method

This method implements the above equation directly.

           +->[Mod 1]<-- cos wct ---+
           |   m cos wct            |
 m(t)------+                   Carrier osc
           |                        |
           +->[-90 deg]->[Mod 2]<-[-90 deg]
              m^(t)       m^ sin wct
                              |
     [Mod1] --(+)-- [Mod2]: sum (-/+) -> SSB

1. Balanced modulator 1 multiplies $m(t)$ by $\cos\omega_ct$, giving DSB-SC $m\cos\omega_ct$.
2. A wideband $-90°$ phase shifter (Hilbert transformer) produces $\hat m(t)$; a second $-90°$ shifter gives $\sin\omega_ct$ from the carrier. Balanced modulator 2 gives $\hat m\sin\omega_ct$.
3. Subtracting the outputs gives USB; adding gives LSB, because one sideband adds in phase and the other cancels.

It needs no sharp sideband filter, so it works at any carrier frequency and with message spectra extending to low frequencies; its difficulty is building an exact $90°$ shifter over the whole audio band.

Hilbert transform

The Hilbert transform $\hat x(t)$ of a signal $x(t)$ is obtained by shifting the phase of every positive-frequency component by $-90°$ and every negative-frequency component by $+90°$, without changing amplitudes.

Time domain: it is the convolution of $x(t)$ with $\frac{1}{\pi t}$:

$$\hat x(t)=x(t)*\frac{1}{\pi t}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau$$

Frequency response: since $\frac{1}{\pi t}\leftrightarrow-j\,\text{sgn}(f)$,

$$H(f)=-j\,\text{sgn}(f)=\begin{cases}-j=1\angle-90°, & f>0\\ +j=1\angle+90°, & f<0\end{cases},\qquad \hat X(f)=-j\,\text{sgn}(f)X(f)$$

  |H(f)|              angle H(f)
 1 +-------+------    +90 ----+
   |       |                  |
 --+-------+------> f  -------+-------> f
           0                  |
                              +---- -90

Inverse: $x(t)=-\hat{\hat x}(t)$, i.e. $x(t)=-\frac{1}{\pi}\int\frac{\hat x(\tau)}{t-\tau}d\tau$.

Properties of the Hilbert transform

1. $x(t)$ and $\hat x(t)$ have the same amplitude spectrum, hence the same ESD/PSD, energy and power.
2. Applying it twice gives the negative: $\hat{\hat x}(t)=-x(t)$.
3. $x(t)$ and $\hat x(t)$ are orthogonal: $\int_{-\infty}^{\infty}x(t)\hat x(t)dt=0$.
4. $\widehat{\cos\omega t}=\sin\omega t$ and $\widehat{\sin\omega t}=-\cos\omega t$.
5. For a low-pass $m(t)$ and carrier $f_c$ greater than its bandwidth: $\widehat{m(t)\cos\omega_ct}=m(t)\sin\omega_ct$.
6. It is a linear operation, and an even signal gives an odd Hilbert transform (and vice versa).

Use: SSB generation $s(t)=m\cos\omega_ct\mp\hat m\sin\omega_ct$, pre-envelope and band-pass signal representation.

Distortionless transmission (line)

A transmission system or line is distortionless if the output is an exact replica of the input except for a constant scale factor and a constant delay:

$$y(t)=K\,x(t-t_d)\ \Rightarrow\ H(f)=K\,e^{-j2\pi ft_d}$$

Frequency response requirements:

• Amplitude: $\lvert H(f)\rvert=K$ constant for all frequencies in the signal band; otherwise amplitude distortion.
• Phase: $\theta(f)=-2\pi ft_d$, a straight line through the origin with slope $-2\pi t_d$; i.e. constant group delay $\tau_g=-\frac{1}{2\pi}\frac{d\theta}{df}=t_d$. Otherwise phase (delay) distortion.

  |H(f)|                theta(f)
 K +--------------        |\
   |                 -----+-\------> f
   +-------------> f      |  \ slope -2*pi*td

For a physical transmission line with parameters $R, L, G, C$ per unit length, this is achieved when the Heaviside condition $\frac{R}{L}=\frac{G}{C}$ holds. Then the attenuation constant $\alpha=\sqrt{RG}$ and the velocity $v=\frac{1}{\sqrt{LC}}$ are both independent of frequency, so all frequency components are attenuated and delayed equally. Telephone lines achieve this approximately by loading coils that increase $L$.

A superheterodyne receiver converts every received station frequency to a fixed intermediate frequency (IF) (455 kHz for AM broadcast, 10.7 MHz for FM) before the main amplification and detection. It is used in almost all radio and TV receivers.

Ant->[RF amp]->(Mixer)->[IF amp]->[Detector]->[AF amp]->Spkr
       ^          ^        455k        |
       |        [LO]                  AGC
       +--gang tuning--+   <-----------+

Working:

1. RF stage: tunes to the wanted station $f_s$, amplifies it and rejects the image frequency.
2. Mixer and local oscillator: the LO is ganged with RF tuning so that $f_{LO}=f_s+f_{IF}$. The mixer gives $f_{LO}-f_s=f_{IF}$ for every station.
3. IF amplifier: fixed-tuned, high-gain stages with steep skirts provide most of the gain and selectivity.
4. Detector: an envelope detector recovers audio; it also gives a DC level for AGC, which controls RF/IF gain to keep output steady.
5. AF amplifier and loudspeaker reproduce the sound.

Advantages: uniform gain and selectivity over the whole band, good adjacent-channel rejection, high sensitivity, stable fixed-frequency amplifiers. Disadvantages: image frequency $f_{si}=f_s+2f_{IF}$ interference (e.g. station at 1000 kHz has image at 1910 kHz), extra mixer noise and more complex circuitry.

A phase-locked loop (PLL) is a negative-feedback system that makes the phase and frequency of a local oscillator follow those of an input signal.

Essential components of a PLL

 vin ->[Phase    ]-> ve ->[Loop  ]-> vc --+
       [detector ]        [filter]        |
           ^                              |
           |                              v
           +-------------[  VCO  ]<-------+
                  vo (output)

1. Phase detector (multiplier): compares the input $v_{in}$ with the VCO output $v_o$ and produces an error voltage $v_e$ proportional to their phase difference, e.g. $v_e\propto\sin(\theta_i-\theta_o)$.
2. Loop filter (low-pass): removes the high-frequency (sum) terms of the phase detector output and sets the loop dynamics: lock range, capture range and noise bandwidth.
3. Voltage-controlled oscillator (VCO): its frequency changes in proportion to the control voltage: $f_o=f_{free}+K_vv_c$. It is pulled until its frequency equals the input frequency.
4. (Often) a loop amplifier to increase loop gain.

When locked, the VCO has the same frequency as the input, with a small fixed phase difference (90° for a multiplier phase detector at zero error).

PLL used to demodulate AM

Coherent (synchronous) AM detection needs a local carrier exactly matching the received carrier. The PLL extracts it.

 AM in --+-->[ PLL ]-->[ 90 deg ]--+  c(t)=cos(wc t)
         |   (locks to carrier)    |
         |                         v
         +----------------------->(X)-->[ LPF ]--> m(t)
                              product detector

Operation:

1. The received AM signal is $s(t)=A_c[1+k_am(t)]\cos\omega_ct$. It has a strong carrier component at $f_c$.
2. The PLL locks to this carrier. Because the multiplier phase detector settles at $90°$ phase difference, the VCO output is $\sin\omega_ct$ (quadrature).
3. A $90°$ phase shifter converts the VCO output to $\cos\omega_ct$, a clean carrier in phase with the received carrier.
4. A product detector multiplies the AM signal by this carrier:

$$\begin{aligned} v(t)&=A_c[1+k_am(t)]\cos\omega_ct\cdot\cos\omega_ct\\ &=\frac{A_c}{2}[1+k_am(t)]+\frac{A_c}{2}[1+k_am(t)]\cos2\omega_ct \end{aligned}$$

5. The low-pass filter removes the $2\omega_c$ term; a DC-blocking capacitor removes $A_c/2$, leaving $\frac{A_ck_a}{2}m(t)$, the message.

Advantages over envelope detection: works for modulation index above 100% and for weak, noisy signals, gives less distortion, and the PLL's narrow loop bandwidth rejects noise. The PLL automatically tracks carrier drift. For DSB-SC (no carrier), a Costas loop or squaring loop is used instead.

In coherent (synchronous) demodulation, the receiver multiplies the received signal by a locally generated carrier and low-pass filters. A PLL (or Costas loop) is used to generate this carrier. If the local oscillator is not exactly matched, i.e. it has a frequency error $\Delta\omega$ and phase error $\phi$, the output is degraded.

Let the local carrier be $c(t)=2\cos[(\omega_c+\Delta\omega)t+\phi]$, and define $\theta(t)=\Delta\omega t+\phi$.

 s(t) -->(X)-->[ LPF ]--> e0(t)
          ^
          | 2cos[(wc+dw)t + phi]
     [ PLL / local osc ]

Effect on DSB-SC

Received $s(t)=m(t)\cos\omega_ct$:

$$\begin{aligned} s(t)c(t)&=2m(t)\cos\omega_ct\cos(\omega_ct+\theta)\\ &=m(t)\cos\theta+m(t)\cos(2\omega_ct+\theta)\\ e_0(t)&=m(t)\cos(\Delta\omega t+\phi)\quad(\text{after LPF}) \end{aligned}$$

• Phase error only ($\Delta\omega=0$): $e_0=m(t)\cos\phi$. The output is undistorted but attenuated by $\cos\phi$. If $\phi$ varies randomly, the output fades; at $\phi=\pm90°$ the output is zero (quadrature null effect).
• Frequency error only ($\phi=0$): $e_0=m(t)\cos(\Delta\omega t)$. The message is multiplied by a slowly varying sinusoid, so the output beats (rises and falls periodically at rate $\Delta f$), causing serious distortion even for small $\Delta f$.

So DSB-SC demodulation needs both frequency and phase to be correct; a Costas loop or squaring PLL is used.

Effect on SSB

Received USB: $s(t)=m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct$. Multiplying and filtering:

$$\begin{aligned} e_0(t)&=m(t)\cos\theta+\hat m(t)\sin\theta\\ &=m(t)\cos(\Delta\omega t+\phi)+\hat m(t)\sin(\Delta\omega t+\phi) \end{aligned}$$

(For LSB the sign of the $\hat m$ term changes.)

• Phase error only: $e_0=m(t)\cos\phi+\hat m(t)\sin\phi$. Every frequency component of the message gets a constant phase shift $\phi$ (for a tone $A_m\cos\omega_mt$, $e_0=A_m\cos(\omega_mt-\phi)$). Amplitude is not reduced, but there is phase distortion. The human ear is insensitive to phase, so speech is acceptable, but data and video are affected.
• Frequency error only: for a tone, $e_0=A_m\cos(\omega_m-\Delta\omega)t$. Every component is shifted in frequency by $\Delta f$, so harmonic relationships are destroyed. Speech sounds unnatural ("Donald Duck" effect); a shift of a few Hz is tolerable for speech, but music becomes badly distorted.

Comparison

Remedy: a PLL locks the local oscillator to a transmitted pilot carrier (SSB, DSB) or recovers the carrier from the signal itself (Costas loop for DSB-SC), reducing both errors close to zero.

A phase-locked loop (PLL) is a closed-loop feedback circuit that locks the frequency and phase of a voltage-controlled oscillator to those of an input signal.

 vin ->[Phase det]->[Loop filter]->[Amp]--+--> vc (FM out)
           ^                              |
           +------------[ VCO ]<----------+
                       vo (locked output)

Components and working:

1. Phase detector compares the input and the VCO output and gives an error voltage proportional to their phase difference.
2. Loop filter (LPF) removes high-frequency terms and sets loop response.
3. VCO changes its frequency according to the control voltage, $f_o=f_{free}+K_vv_c$, and is pulled toward the input frequency.

When the loop is locked, the VCO frequency equals the input frequency with a constant small phase difference. If the input frequency changes, the error voltage changes and the VCO follows it.

Key terms:

• Free-running frequency: VCO frequency with no input.
• Capture range: range of input frequencies over which the loop can acquire lock.
• Lock (tracking) range: range over which it stays locked once locked; it is wider than the capture range.

Applications: FM demodulation (the control voltage is the message), carrier recovery for coherent AM/DSB/SSB detection (Costas loop), frequency synthesizers, frequency multiplication/division, clock recovery in digital receivers, and FSK demodulation. IC example: NE565.

The envelope detector is a simple non-coherent demodulator whose output follows the envelope of the AM wave. Since standard AM has envelope $A_c[1+k_am(t)]$, which is a shifted copy of $m(t)$, it recovers the message without any local carrier.

Circuit

        D
 o---->|---+------+--------o
           |      |
 AM in     C      R      vo(t) ~ envelope
           |      |
 o---------+------+--------o

Working

1. Positive half cycle: when the input exceeds the capacitor voltage, the diode conducts and $C$ charges quickly to the peak of the carrier through the small source resistance ($R_sC\ll1/f_c$).
2. Between peaks: when the input falls below the capacitor voltage, the diode becomes reverse-biased and $C$ discharges slowly through $R$.
3. On the next peak the diode conducts again and recharges $C$. Thus the capacitor voltage follows the peaks of the carrier, i.e. the envelope, with a small ripple at $f_c$.
4. A DC-blocking capacitor removes the DC ($A_c$) and a small RC low-pass filter removes ripple, giving $m(t)$.

Time constant condition:

$$\frac{1}{f_c}\ll RC\ll\frac{1}{W}$$

• If $RC$ is too small, the output follows the carrier cycles (large ripple).
• If $RC$ is too large, the capacitor cannot follow a falling envelope (diagonal clipping); for a tone with index $\mu$ we need $RC\le\frac{\sqrt{1-\mu^2}}{\mu\,\omega_m}$.

 AM wave       envelope      detector output
  /\  /\/\       ___           ___
 /  \/    \/\   /   \  __     /   \  __ (small ripple)

Requirements: $\mu\le1$ (no over-modulation) and $f_c\gg W$.

Can an envelope detector demodulate DSB-SC?

No. A DSB-SC wave is $s(t)=A_cm(t)\cos\omega_ct$. Its envelope is $A_c\lvert m(t)\rvert$, not $m(t)$. Whenever $m(t)$ becomes negative, the carrier phase reverses by $180°$ but the envelope stays positive, so the detector output is the rectified message $\lvert m(t)\rvert$, which is badly distorted. For example, a tone $\cos\omega_mt$ comes out as $\lvert\cos\omega_mt\rvert$, which contains $2f_m$ and higher harmonics. DSB-SC therefore needs coherent (synchronous) detection, using a carrier recovered by a Costas loop or squaring loop. (Envelope detection would only work if a large carrier were re-inserted, which converts it back to standard AM.)

The phase-shift (phase discrimination) method generates SSB directly from its mathematical form, without a sideband filter:

$$s_{SSB}(t)=\frac{A_c}{2}\left[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct\right]$$

where $\hat m(t)$ is the Hilbert transform of $m(t)$ (each component shifted by $-90°$). Minus gives USB, plus gives LSB.

Block diagram

             +-->[Bal. mod 1]-- m cos wct --+
             |        ^                     |
 m(t) -------+        | cos wct             v
             |   [Carrier osc]            ( + / - )--> SSB
             |        |                     ^
             |   [-90 deg]                  |
             |        | sin wct             |
             +->[-90]->[Bal. mod 2]-- m^ sin wct
                 m^(t)

Working

1. Balanced modulator 1 multiplies $m(t)$ by $\cos\omega_ct$, giving DSB-SC with both sidebands.
2. A wideband $-90°$ phase shifter (Hilbert transformer) produces $\hat m(t)$, and a carrier phase shifter gives $\sin\omega_ct$. Balanced modulator 2 gives $\hat m(t)\sin\omega_ct$.
3. The summer/subtractor combines them. One sideband appears in phase in both paths and adds; the other is in antiphase and cancels.

Proof with a tone $m=A_m\cos\omega_mt$, $\hat m=A_m\sin\omega_mt$:

$$\begin{aligned} m\cos\omega_ct&=\tfrac{A_m}{2}[\cos(\omega_c-\omega_m)t+\cos(\omega_c+\omega_m)t]\\ \hat m\sin\omega_ct&=\tfrac{A_m}{2}[\cos(\omega_c-\omega_m)t-\cos(\omega_c+\omega_m)t] \end{aligned}$$

Subtracting gives $A_m\cos(\omega_c+\omega_m)t$ (USB only); adding gives $A_m\cos(\omega_c-\omega_m)t$ (LSB only).

Pros

1. No sharp sideband filter is needed, so SSB can be generated directly at any carrier frequency (no multiple up-conversion).
2. Works for messages with energy near zero frequency (no need for a gap between sidebands, unlike the filter method).
3. Switching between USB and LSB is easy: just change adder to subtractor.
4. Lighter, smaller circuits for low carrier frequencies.

Cons

1. A wideband $90°$ phase shifter that gives exactly $90°$ at all audio frequencies (e.g. 300–3400 Hz) with equal gain is very difficult to build.
2. Any phase or amplitude imbalance leaves a residual unwanted sideband (poor sideband suppression, typically 30–40 dB vs 50–60 dB for filters).
3. Two balanced modulators must be closely matched, and the circuit needs careful adjustment and is sensitive to drift.

An envelope detector demodulates a standard AM wave $s(t)=A_c[1+\mu\cos\omega_mt]\cos\omega_ct$ by producing an output that follows its envelope $A_c[1+\mu\cos\omega_mt]$. It is a diode followed by a parallel RC load.

Circuit and operation

        D
 o---->|---+------+------o
    Rs     |      |
 AM in     C      RL    vo(t)
           |      |
 o---------+------+------o

1. Charging: on each positive peak of the carrier, the diode is forward-biased and $C$ charges rapidly to the peak value through the source and diode resistance $R_s$. Charging time constant $R_sC$ must be much smaller than the carrier period: $R_sC\ll1/f_c$.
2. Discharging: between peaks, the input drops below the capacitor voltage, the diode is cut off, and $C$ discharges slowly through $R_L$.
3. The capacitor voltage thus traces the envelope with a small sawtooth ripple at $f_c$. A coupling capacitor removes the DC part, giving the audio $m(t)$.

Necessary conditions on time constant

(a) Remove carrier ripple – discharge must be slow compared with the carrier period:

$$R_LC\gg\frac{1}{f_c}$$

(b) Follow the envelope – discharge must be fast compared with the message variations:

$$R_LC\ll\frac{1}{W}$$

Combined: $\frac{1}{f_c}\ll R_LC\ll\frac{1}{W}$, which is possible because $f_c\gg W$.

(c) Avoid diagonal clipping – for a tone, the envelope's fastest fall rate must not exceed the capacitor's discharge rate. This gives

$$R_LC\le\frac{\sqrt{1-\mu^2}}{\mu\,\omega_m}$$

For example, with $\mu=0.5$ and $f_m=5$ kHz, $R_LC\le\frac{0.866}{0.5\times2\pi\times5000}\approx55\ \mu\text{s}$.

Also, the modulation index must satisfy $\mu\le1$; otherwise the envelope crosses zero (over-modulation) and the output is distorted.

Waveforms

 (a) AM input      (b) RC too small   (c) RC correct
   |\ /\  /\ /|     follows carrier    smooth envelope
  /| V  \/  V |\          /\/\/\/\/\           .--.     .--.
 / |          | \                             /    \   /
                                                    '-'
 (d) RC too large: diagonal clipping
     '--.____         output cannot follow
             \___     the falling envelope

• (b) If $R_LC$ is too small, the capacitor discharges almost completely between cycles and the output contains large carrier ripple.
• (c) With the correct $R_LC$, the output follows the envelope with tiny ripple.
• (d) If $R_LC$ is too large, the capacitor voltage decays more slowly than the envelope falls, so the output misses the troughs (diagonal clipping).

Advantages: very simple, cheap, no local carrier needed; used in all AM broadcast receivers.

Given: message $m(t)=15\sin2\pi(1500t)$, so $A_m=15$ V, $f_m=1.5$ kHz. Carrier $c(t)=60\sin2\pi(100{,}000t)$, so $A_c=60$ V, $f_c=100$ kHz.

i) Modulated wave

For standard AM, the carrier amplitude varies as $A_c+m(t)$:

$$\begin{aligned} s(t)&=[A_c+m(t)]\sin2\pi f_ct\\ &=[60+15\sin2\pi(1500t)]\sin2\pi(100{,}000t)\\ &=60[1+0.25\sin2\pi(1500t)]\sin2\pi(100{,}000t) \end{aligned}$$

Expanding with $\sin A\sin B=\frac{1}{2}[\cos(A-B)-\cos(A+B)]$:

$$s(t)=60\sin2\pi(10^5t)+7.5\cos2\pi(98{,}500t)-7.5\cos2\pi(101{,}500t)$$

The envelope varies between $60+15=75$ V and $60-15=45$ V.

ii) Modulation index and percent modulation

$$\begin{aligned} \mu&=\frac{A_m}{A_c}=\frac{15}{60}=0.25\\ \text{Percent modulation}&=0.25\times100=25\% \end{aligned}$$

Check: $\mu=\frac{E_{max}-E_{min}}{E_{max}+E_{min}}=\frac{75-45}{75+45}=0.25$.

iii) Frequencies in the spectrum

Side-frequency amplitude $=\mu A_c/2=0.25\times60/2=7.5$ V. Bandwidth $=2f_m=3$ kHz.

          60 V
           |
   7.5 V   |   7.5 V
     |     |     |
 ----+-----+-----+-----> f (kHz)
   98.5   100  101.5

(Extra: on a 1 $\Omega$ load, $P_c=60^2/2=1800$ W, each sideband $28.125$ W, total $1856.25$ W.)

iv) Sketch of audio and carrier waves

Audio period $T_m=1/1500=0.667$ ms; carrier period $T_c=1/100{,}000=10\ \mu$s, so about 66.7 carrier cycles fit in one audio cycle.

 Audio: 15 sin(2*pi*1500 t), peak 15 V
  15 |    .--.
     |  /      \
   0 +-'--------'\--------/----> t
     |            \      /
 -15 |             '--'
     |<---- 0.667 ms ---->|

 Carrier: 60 sin(2*pi*1e5 t), peak 60 V
  60 | /\  /\  /\  /\  /\  /\
   0 +/--\/--\/--\/--\/--\/--\--> t
 -60 |
     |<>| 10 us

 AM wave: envelope between 45 V and 75 V
  75 |   /\/\/\
  45 |/\/      \/\/\      /\/
   0 +-------------------------> t
 -45 |\/\      /\/\/\    /\/\
 -75 |   \/\/\/      \/\/

A square-law modulator generates standard AM (DSB-FC) using a non-linear device (diode or transistor in its non-linear region) whose output contains a squared term, followed by a band-pass filter.

Block diagram

 m(t) -->(+)-v1->[Non-linear]-v2->[ BPF ]--> s(t)
          ^       [ device   ]     fc, 2W     AM
          |
 Ac cos wct

Analysis

The sum of message and carrier is applied to the device:

$$v_1(t)=A_c\cos\omega_ct+m(t)$$

The device characteristic (up to second order) is

$$v_2(t)=a_1v_1(t)+a_2v_1^2(t)$$

Substituting:

$$\begin{aligned} v_2(t)&=a_1A_c\cos\omega_ct+a_1m(t)+a_2\left[A_c^2\cos^2\omega_ct+2A_cm(t)\cos\omega_ct+m^2(t)\right]\\ &=\underbrace{a_1A_c\left[1+\frac{2a_2}{a_1}m(t)\right]\cos\omega_ct}_{\text{wanted AM}}+\underbrace{a_1m(t)+a_2m^2(t)+\frac{a_2A_c^2}{2}+\frac{a_2A_c^2}{2}\cos2\omega_ct}_{\text{unwanted}} \end{aligned}$$

A band-pass filter centred at $f_c$ with bandwidth $2W$ keeps only the first term:

$$s(t)=a_1A_c\left[1+k_am(t)\right]\cos\omega_ct,\qquad k_a=\frac{2a_2}{a_1}$$

This is exactly a DSB-AM (full carrier) signal with amplitude sensitivity $k_a$. Hence a square-law device plus a BPF works as an AM modulator.

Spectrum at the input of the BPF

For a message of bandwidth $W$, the components of $v_2(t)$ are:

 |V2(f)|      BPF passband
 |              +-------+
 |^             |   ^   |
 ||\_           |  /|\  |              ^
 ||  \__        | / | \ |              |
 +|-----\-------+/--+--\+--------------+--> f
  0  W  2W   fc-W  fc  fc+W           2fc
   m(t)  m^2(t)        AM             2fc

To separate the AM band from $m^2(t)$ without overlap, we need $f_c-W>2W$, i.e. $f_c>3W$.

Basic characteristics of AM

1. Carrier amplitude varies linearly with the message; envelope is $A_c[1+k_am(t)]$.
2. Spectrum has a carrier plus two sidebands (USB and LSB), each a copy of the message spectrum.
3. Bandwidth $B=2W$.
4. Modulation index $\mu=k_a\lvert m(t)\rvert_{max}\le1$ to avoid over-modulation.
5. Power: $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$; maximum efficiency only 33.3% at $\mu=1$ (most power is in the carrier).
6. Can be demodulated by a simple envelope detector, making receivers cheap.

Modulation

Modulation is the process in which some characteristic (amplitude, frequency or phase) of a high-frequency carrier wave is varied in accordance with the instantaneous value of the message (modulating) signal. In amplitude modulation (AM) the carrier amplitude is varied.

Need for modulation

1. Practical antenna height: antenna length $\approx\lambda/4$; for 10 kHz audio it would be 7.5 km, but for a 1 MHz carrier only 75 m.
2. Multiplexing: different stations use different carriers, so many signals share the same channel without interference.
3. Efficient radiation and long range: high-frequency signals radiate and propagate much better.
4. Narrowbanding: reduces the ratio of highest to lowest frequency, so one antenna and amplifier design works.
5. Noise reduction: wideband schemes (FM) improve SNR.
6. Channel matching: shifts the signal into the frequency band suited to the medium.

Time-domain expression of single-tone AM

Let the message be $m(t)=A_m\cos\omega_mt$ and carrier $c(t)=A_c\cos\omega_ct$, with $f_c\gg f_m$. In standard AM the envelope is $A_c+m(t)$:

$$\begin{aligned} s(t)&=[A_c+A_m\cos\omega_mt]\cos\omega_ct\\ &=A_c[1+\mu\cos\omega_mt]\cos\omega_ct,\qquad \mu=\frac{A_m}{A_c} \end{aligned}$$

where $\mu$ is the modulation index ($0\le\mu\le1$). Using $\cos A\cos B=\frac{1}{2}[\cos(A+B)+\cos(A-B)]$:

$$s(t)=\underbrace{A_c\cos\omega_ct}_{\text{carrier}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t}_{\text{USB}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t}_{\text{LSB}}$$

       Ac(1+mu)
   .-.   envelope    .-.
  /   \             /   \
 /     \    .-.    /     \   <- carrier inside envelope
        \  /   \  /
         '-     '-  Ac(1-mu)

Frequency-domain expression

Using $\cos2\pi f_0t\leftrightarrow\frac{1}{2}[\delta(f-f_0)+\delta(f+f_0)]$:

$$\begin{aligned} S(f)&=\frac{A_c}{2}\left[\delta(f-f_c)+\delta(f+f_c)\right]\\ &+\frac{\mu A_c}{4}\left[\delta(f-f_c-f_m)+\delta(f+f_c+f_m)\right]\\ &+\frac{\mu A_c}{4}\left[\delta(f-f_c+f_m)+\delta(f+f_c-f_m)\right] \end{aligned}$$

                 S(f)
       Ac/2               Ac/2
 muAc/4 |  muAc/4  muAc/4 |  muAc/4
   |    |    |       |    |    |
 --+----+----+---+---+----+----+--> f
  -fc-fm -fc -fc+fm 0 fc-fm fc fc+fm

Key results

• Bandwidth: $B=(f_c+f_m)-(f_c-f_m)=2f_m$.
• Power: $P_c=\frac{A_c^2}{2}$, each sideband $\frac{\mu^2A_c^2}{8}$, so total $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$.
• Efficiency: $\eta=\frac{\mu^2}{2+\mu^2}$; maximum 33.3% at $\mu=1$.

Example: $A_c=10$ V, $\mu=0.5$, $f_c=1$ MHz, $f_m=1$ kHz gives lines of 10 V at 1 MHz and 2.5 V at 0.999 MHz and 1.001 MHz; bandwidth 2 kHz.

A superheterodyne receiver converts the incoming RF signal of any station to a fixed intermediate frequency (IF), 455 kHz for AM broadcast, where most of the gain and selectivity are obtained.

Block diagram

 Antenna
   |
 [RF amplifier]--->( Mixer )--->[ IF amplifier ]
   ^  fs              ^             455 kHz
   |                  | fLO             |
   +--- ganged ---[Local osc]      [ Detector ]
        tuning                          |
                    AGC <---------------+
                                        |
                  Speaker <--[ AF amplifier ]

Working principle

1. Antenna and RF amplifier: the tuned RF stage selects the desired station frequency $f_s$ (535–1605 kHz), amplifies it, improves SNR and rejects the image frequency.
2. Local oscillator and mixer (frequency changer): the LO is ganged with the RF tuning so that it always runs IF above the signal:

$$f_{LO}=f_s+f_{IF}$$

The mixer produces $f_{LO}\pm f_s$; the difference $f_{LO}-f_s=455$ kHz is selected. The modulation (sidebands) is kept unchanged, only shifted to IF. 3. IF amplifier: several fixed-tuned stages at 455 kHz with bandwidth about 10 kHz. Since frequency is fixed, they give high, uniform gain and sharp selectivity (adjacent-channel rejection) for all stations. 4. Detector: a diode envelope detector recovers the audio from the IF signal. 5. AGC (automatic gain control): a DC voltage proportional to the carrier level from the detector controls the gain of RF and IF stages, keeping output nearly constant for strong and weak stations and preventing overload. 6. AF amplifier and loudspeaker: amplify the audio to the required power and convert it to sound.

Example: to receive 1000 kHz, $f_{LO}=1455$ kHz. The image is $f_s+2f_{IF}=1910$ kHz, which must be rejected by the RF stage.

Advantages: constant selectivity and sensitivity over the band, high gain without instability, good adjacent-channel rejection. Disadvantages: image-frequency problem, mixer noise, tracking alignment needed.

Why standard AM is used in AM broadcasting

1. Simple, cheap receivers: standard AM (DSB-FC) can be demodulated by a simple envelope detector; no carrier recovery or synchronization is needed.
2. One transmitter, millions of receivers: it is economical to put extra power (the carrier) in one costly transmitter so that every listener can use a cheap receiver.
3. The transmitted carrier also helps AGC and tuning.
4. Historical compatibility with the huge base of existing receivers.

The cost is poor power efficiency (at most 33.3%) and twice the bandwidth of SSB, which are acceptable for broadcasting.

The Hilbert transform of a signal $x(t)$, written $\hat x(t)$, is the signal obtained by shifting the phase of all its positive-frequency components by $-90°$ and all negative-frequency components by $+90°$, while keeping amplitudes unchanged.

Time domain:

$$\hat x(t)=x(t)*\frac{1}{\pi t}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau$$

Frequency domain: the Hilbert transformer is a filter with

$$H(f)=-j\,\text{sgn}(f),\qquad \hat X(f)=-j\,\text{sgn}(f)\,X(f)$$

so $\lvert H(f)\rvert=1$ and $\angle H(f)=-90°$ for $f>0$, $+90°$ for $f<0$. It is an ideal wideband $90°$ phase shifter.

Examples: $\widehat{\cos\omega_0t}=\sin\omega_0t$, $\widehat{\sin\omega_0t}=-\cos\omega_0t$.

Properties:

1. $x(t)$ and $\hat x(t)$ have the same amplitude spectrum, energy and power.
2. $\hat{\hat x}(t)=-x(t)$.
3. $x(t)$ and $\hat x(t)$ are orthogonal: $\int x(t)\hat x(t)dt=0$.
4. For low-pass $m(t)$ and $f_c$ above its bandwidth, $\widehat{m(t)\cos\omega_ct}=m(t)\sin\omega_ct$.

Applications:

• SSB generation: $s(t)=m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct$ (phase-shift method).
• Pre-envelope (analytic signal): $x_+(t)=x(t)+j\hat x(t)$, whose spectrum is zero for $f<0$.
• Band-pass signal representation in terms of in-phase and quadrature components, and complex envelope analysis.

Double-tone AM is amplitude modulation in which the message consists of two sinusoids:

$$m(t)=A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t,\qquad f_1<f_2\ll f_c$$

Expression

With carrier $A_c\cos\omega_ct$, the standard AM wave is

$$\begin{aligned} s(t)&=[A_c+A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t]\cos\omega_ct\\ &=A_c[1+\mu_1\cos\omega_1t+\mu_2\cos\omega_2t]\cos\omega_ct \end{aligned}$$

where $\mu_1=A_{m1}/A_c$ and $\mu_2=A_{m2}/A_c$ are the individual modulation indices. Expanding:

$$\begin{aligned} s(t)&=A_c\cos\omega_ct\\ &+\frac{\mu_1A_c}{2}\left[\cos(\omega_c+\omega_1)t+\cos(\omega_c-\omega_1)t\right]\\ &+\frac{\mu_2A_c}{2}\left[\cos(\omega_c+\omega_2)t+\cos(\omega_c-\omega_2)t\right] \end{aligned}$$

So the spectrum has five lines: the carrier at $f_c$, and side frequencies at $f_c\pm f_1$ and $f_c\pm f_2$.

                 Ac
      mu2Ac/2    |    mu2Ac/2
        |  mu1Ac/2 mu1Ac/2 |
        |   |    |    |    |
 -------+---+----+----+----+----> f
     fc-f2 fc-f1 fc fc+f1 fc+f2

Bandwidth

The bandwidth is set by the highest modulating frequency:

$$B=(f_c+f_2)-(f_c-f_2)=2f_2=2f_{max}$$

Modulation indices

• Individual indices: $\mu_1=\frac{A_{m1}}{A_c}$, $\mu_2=\frac{A_{m2}}{A_c}$.
• Net (total, effective) index: obtained from equal total sideband power:

$$P_t=P_c\left(1+\frac{\mu_1^2}{2}+\frac{\mu_2^2}{2}\right)=P_c\left(1+\frac{\mu_t^2}{2}\right)\ \Rightarrow\ \mu_t=\sqrt{\mu_1^2+\mu_2^2}$$

For no over-modulation, $\mu_t\le1$ (strictly, $\mu_1+\mu_2\le1$ ensures the envelope never goes negative).

Example: $\mu_1=0.3$, $\mu_2=0.4$ gives $\mu_t=0.5$.

Given: $s(t)=50\cos(2\pi10^6t+60°)-15\cos(2\pi10^6t-60°)+20\sin(4\pi10^4t-190°)+\sin(2\pi10^4t+190°)$

To draw line spectra, write every term as a cosine with positive amplitude, and combine terms of the same frequency.

Step 1: 1 MHz terms (phasor addition)

$$\begin{aligned} 50\angle60°&=25+j43.30\\ -15\angle-60°&=-7.5+j12.99\\ \text{Sum}&=17.5+j56.29=58.95\angle72.73° \end{aligned}$$

So $50\cos(\cdot+60°)-15\cos(\cdot-60°)=58.95\cos(2\pi10^6t+72.73°)$.

Step 2: 20 kHz term ($4\pi10^4t=2\pi\cdot2\times10^4t$)

Using $\sin x=\cos(x-90°)$:

$$20\sin(\omega t-190°)=20\cos(\omega t-280°)=20\cos(2\pi\,20000\,t+80°)$$

Step 3: 10 kHz term

$$\sin(\omega t+190°)=\cos(\omega t+100°)=\cos(2\pi\,10000\,t+100°)$$

Result

$$s(t)=\cos(2\pi10^4t+100°)+20\cos(2\pi\,2{\times}10^4t+80°)+58.95\cos(2\pi10^6t+72.73°)$$

For the two-sided spectrum, each line splits into two of half amplitude at $\pm f$ (0.5, 10, 29.47) with phase $+\theta$ at $+f$ and $-\theta$ at $-f$.

 One-sided amplitude spectrum
 58.95 |                         |
       |                         |
    20 |     |                   |
     1 |  |  |                   |
       +--+--+------//-----------+--> f
         10  20 kHz            1 MHz

 One-sided phase spectrum (deg)
   100 |  o
    80 |     o
 72.73 |                         o
       +--+--+------//-----------+--> f
         10  20 kHz            1 MHz

Double tone AM

In double tone (two-tone) AM the carrier is modulated by a message that contains two sinusoids of different frequencies.

Let the message and carrier be:

$$m(t) = A_{m1}\cos\omega_{m1}t + A_{m2}\cos\omega_{m2}t, \qquad c(t) = A_c\cos\omega_c t$$

The conventional (DSB-FC) AM wave is

$$\begin{aligned} s(t) &= [A_c + m(t)]\cos\omega_c t \\ &= A_c\left[1 + \frac{A_{m1}}{A_c}\cos\omega_{m1}t + \frac{A_{m2}}{A_c}\cos\omega_{m2}t\right]\cos\omega_c t \\ &= A_c\left[1 + \mu_1\cos\omega_{m1}t + \mu_2\cos\omega_{m2}t\right]\cos\omega_c t \end{aligned}$$

where $\mu_1 = A_{m1}/A_c$ and $\mu_2 = A_{m2}/A_c$ are the individual modulation indices. Expanding with $\cos A\cos B = \tfrac12[\cos(A+B)+\cos(A-B)]$:

$$\begin{aligned} s(t) = {} & A_c\cos\omega_c t \\ & + \frac{\mu_1A_c}{2}\cos(\omega_c+\omega_{m1})t + \frac{\mu_1A_c}{2}\cos(\omega_c-\omega_{m1})t \\ & + \frac{\mu_2A_c}{2}\cos(\omega_c+\omega_{m2})t + \frac{\mu_2A_c}{2}\cos(\omega_c-\omega_{m2})t \end{aligned}$$

So the spectrum has the carrier plus two upper and two lower side frequencies.

 amplitude
    |            Ac
    |            |
    |  u2Ac/2  u1Ac/2  u1Ac/2  u2Ac/2
    |   |    |   |   |    |
 ---+---+----+---+---+----+------> f
      fc-fm2 fc-fm1 fc fc+fm1 fc+fm2   (fm1 < fm2)

Power (load $R$):

$$P_t = \frac{A_c^2}{2R}\left[1 + \frac{\mu_1^2}{2} + \frac{\mu_2^2}{2}\right] = P_c\left(1 + \frac{\mu_t^2}{2}\right)$$

with the net (total) modulation index

$$\mu_t = \sqrt{\mu_1^2 + \mu_2^2}$$

For no over-modulation, $\mu_t \le 1$. Bandwidth $= 2f_{m,\max}$ (twice the higher message frequency).

DSB vs SSB